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Electro-optic probing of gallium arsenide microwave circuits

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O rd er N u m b e r 9 4 2 0 0 1 4
E le c tr o -o p tic p r o b in g o f G a A s m icro w a v e c ir c u its
M echtel, Deborah Marie, Ph.D .
The Johns Hopkins University, 1994
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 4X106
Electro-optic Probing of GaAs
Microwave Circuits
by
D eb o rah M arie M echtel
A d issertatio n su b m itte d to T he Johns H o p k in s U n iv ersity
in conform ity w ith th e re q u ire m e n ts for the d e g re e of
D octor of P h ilo so p h y
B altim ore, M ary lan d
1993
Abstract
A c o n tin u o u s w a v e la se r b a s e d electro -o p tic p ro b in g in s tru m e n t
a n d a p u lse d laser b ased e lectro -o p tic p ro b in g in s tru m e n t w e re d e sig n e d
a n d c o n stru cte d for electro-o p tic p ro b in g of circuits fab ricated o n g alliu m
a rse n id e (GaAs).
T he c o n tin u o u s w av e (CW ) electro-optic p ro b in g in s tru m e n t h a d a
u n iq u e a d v a n ta g e th a t p e rm itte d th e a v erag e laser lig h t in te n s ity to be
m o n ito re d w ith o u t a loss in in s tru m e n t sen sitiv ity . M e a su re m e n ts of the
electric field profiles of circuits fab ricated o n G aA s rev ealed th e in fluence
o f m u ltip le b eam re flectio n s (MBR) o n elec tro -o p tic p ro b in g re su lts.
M u ltip le b e a m re fle c tio n s w e r e a n a ly z e d n u m e ric a lly , a n d a n e w
c a lib ra tio n tech n iq u e to re m o v e th e effects of MBR fro m e le c tro -o p tic
p ro b in g te st re s u lts w a s p r o p o s e d .
T h e C W e le c tro -o p tic p ro b in g
in s tr u m e n t re a d ily le n d s itse lf to a c a lib ra tio n te c h n iq u e b a s e d o n
m o n ito rin g the average light in te n sity of the p ro b e beam a n d can en h an ce
th e sensitivity of electro-optic p ro b in g .
The p u lse d electro-o p tic p ro b in g in s tru m e n t u sed a laser d io d e as
th e p ro b e beam . The p ro p o s e d c a lib ra tio n tech n iq u e for re m o v in g the
effects of MBR w o u ld also a p p ly to th e p u ls e d e le c tro -o p tic p ro b in g
in s tru m e n t.
A c o m p a riso n o f C W e le c tro -o p tic p ro b in g w ith p u ls e d
electro-optic p ro b in g m e a su re m e n ts d e m o n stra te d the u sefu ln ess of a low
freq u en cy C W in s tru m e n t for p re d ic tin g p ro b lem s th a t m ay o ccu r in h ig h
freq u en cy testing.
G aA s circuits tested w ith the C W electro-optic p ro b in g in stru m e n t
e x h ib ite d a n o n lin e a r d e p e n d e n c e of th e o u tp u t lig h t in te n s ity of the
in s tr u m e n t o n th e la se r lig h t in te n s ity s u p p lie d to th e e le c tro -o p tic
p ro b in g in stru m e n t. A th e rm a l m o d e l b a se d o n the ch an g es in th e in d ex
of refractio n of G aA s as the te m p e ra tu re v aries is p ro p o se d to ex p lain the
non lin earity .
To Gary,
my husband and constant support
iv
Acknowledgements
I th a n k Dr. H a rry K. C h arles, Jr. a n d Professor C. R o g er W estg ate
for their s u p p o rt an d g u id a n c e th ro u g h o u t the course of this w o rk . M any
p e o p le at the Johns H o p k in s U n iv e rs ity A p p lie d P h y sic s L ab h e lp e d
c o m p le te this w o rk by their tech n ical assistan ce in fab ricatin g p a rts of the
electro -o p tic pro b in g in s tru m e n t, technical advice an d e q u ip m e n t loans. I
esp ecially w ish to th a n k A rt H o g re fe for a circuit d e sig n , R ay Sova an d
M ik e T h o m a s fo r te sts r u n o n th e ir s p e c tra l p h o to m e te r, D e n n is
W ick e n d e n for testing G aA s sa m p le s o n his spectral p h o to m e te r, a n d the
m em b ers of TSE.
I also w ith to th a n k Jay W iesen feld a n d Dr. C. A. B u rru s of AT & T
Bell Labs for su p p ly in g laser d io d e s a n d m echanical d ra w in g s for a laser
fix tu re. I am grateful for th e v isits a n d d iscu ssio n s w ith Jay W iesen feld
th a t m a d e the p u lsed sy stem p ossible.
I thank m y fam ily for th eir s u p p o rt a n d the Rev. G e ra rd K noche for
th e s u p p o rt he re n d e red d u rin g the last p a rt of this w ork.
Finally, I th an k m y h u s b a n d , G ary , for his c o n sta n t s u p p o rt an d
en co u rag em en t.
v
Table of Contents
1
In tro d u ctio n
1
2
Electro-optic P ro b in g T heory
6
3
Electro-optic P ro b in g In stru m e n ts an d T est Sam ples
16
3.1
C o n tin u o u s w a v e electro-optic p ro b in g in s tru m e n t
17
3.2
P ulsed electro-optic p ro b in g in stru m e n t
25
3.3
G aA s circuits a n d test p ack ag e
39
4
The Role of M u ltip le Beam R eflections in Electro-optic P ro b in g
44
4.1
M ultiple b e am reflection theory
46
4.2
M ultiple b eam reflection m o d els
49
4.3
C alibration for m u ltip le b eam reflection effects
64
4.4
O u tp u t lig h t in te n sity a n d p h ase d elay
70
5
In ten sity D e p en d e n t Effects
81
6
Electro-optic P ro b in g Test R esults
96
6.1
CW electro-optic p ro b in g test resu lts
96
6.2
P ulsed electro-optic p ro b in g test resu lts
106
7
C onclusion
111
A p p e n d ix A
116
A p p e n d ix B
128
A p p e n d ix C
131
A p p e n d ix D
141
B ibliography
145
Vita
151
vi
List of Illustrations
2-1
Schem atic re p re se n ta tio n of electro-optic probing.
2-2
T he o rie n ta tio n of p o la riz a tio n d irectio n s fo r elec tro -o p tic
probing.
2-3
E lectro-optic effect re s p o n s e cu rv e w ith the ra tio of o u tp u t
light in tensity to in p u t lig h t in ten sity versus th e electric field
g en erated voltage sig n al at the sam p led point.
3-1
Schem atic block d ia g ra m of a co n tin u o u s w av e electro -o p tic
pro b in g system .
3-2
Schem atic re p re se n ta tio n of a typical G aA s test sa m p le a n d
the p ro b in g laser b e a m n e a r a tran sm issio n line, (a) th ree
dim ensional view (b) cross-sectional view .
3-3
Pre-am plifier circuit.
3-4
Schem atic block d ia g ra m of a co n tin u o s w av e electro -o p tic
probing system w ith q u a rte r a n d half w av e plates.
3-5
P h o to g rap h of AT&T InG aA s laser diode.
3-6
D igital sam p lin g oscillo sco p e m e asu re m e n t of laser in ten sity
28
w ith th e la se r DC b ia s e d slig h tly a b o v e th re s h o ld a n d
su p p lie d w ith less th a n -10 d b m of Rf p o w er.
29
3-7
A p h o to g ra p h of InG aA s p h o to d io d e p ack ag ed in m o u n t.
30
3-8
C lose-up view of the p h o to d io d e .
31
3-9
T op d o w n view of the p h o to d io d e .
32
3-10
D igital sam p lin g oscillo sco p e m e asu re m e n t of laser in ten sity
w ith th e la se r DC b ia s e d slig h tly a b o v e th re s h o ld a n d
su p p lie d w ith 8.13 d b m of Rf pow er.
3-11
33
D igital sam p lin g oscilloscope m e asu re m e n t of laser in ten sity
w ith th e la se r DC b ia s e d slig h tly a b o v e th re s h o ld a n d
su p p lie d w ith 24.22 d b m of Rf pow er.
3-12
34
D igital sam p lin g oscilloscope m e asu re m e n t of laser in ten sity
w ith th e la se r DC b ia s e d slig h tly a b o v e th re s h o ld a n d
su p p lie d w ith 31.40 d b m of Rf pow er.
viii
35
3-13
S chem atic b lo ck d ia g ra m of a p u lse d e lec tro -o p tic p ro b in g
system .
36
3-14
P h o to g rap h of th e p u ls e d electro-optic p ro b in g in stru m e n t.
37
3-15
P h o to g ra p h of th e p u ls e d laser an d o p tic tra in of th e p u lsed
electro-optic p ro b in g in stru m e n t.
3-16
38
P h o to g rap h of a test sam p le m o u n te d in the test fixture. The
s u p p o rt s u b s tra te is a lu m in a w ith a c o p la n a r w a v e g u id e
m e ta lizatio n p a tte rn .
41
3-17
P h o to g rap h of a G aA s tw o tran sm issio n line test sam p le.
42
3-18
P h o to g r a p h of a G aA s s tu b c a p a c ito r a n d in te rd ig ita l
capacitor test sam ple.
43
4-1
M u ltiple beam reflections in fro n t side p ro b in g .
45
4-2
P olar g ra p h of the n u m e ric a l a n aly sis of th e v a ria tio n of
o u tp u t in te n sity as a fu n ctio n of an aly zer p o sitio n th ro u g h a
ro ta tio n of 360 d e g re es for the non- MBR m odel.
4-3
52
P olar g ra p h of the n u m e ric a l a n aly sis of v a ria tio n o u tp u t
in te n s ity as a fu n c tio n of a n a ly z e r p o s itio n th ro u g h an
an aly zer ro ta tio n of 360 d eg rees for the MBR m o d el.
ix
53
4-4
L in ea r g ra p h of th e n u m e ric a l a n a ly sis of to ta l o u tp u t
in te n s ity as a fu n c tio n o f a n a ly z e r p o s itio n th r o u g h a n
analyzer rotation o f 360 d eg rees for the non- MBR m o d el.
4-5
54
L in ear g ra p h of th e n u m e ric a l a n a ly sis of to ta l o u tp u t
in te n s ity as a fu n c tio n o f a n a ly z e r p o s itio n th ro u g h an
an aly zer rotation of 360 d eg rees for th e MBR m o d el.
4-6
L ight in tensity as a fu n c tio n of su b strate thickness.
4-7
L in ea r g ra p h of th e n u m e ric a l a n a ly sis of to ta l o u tp u t
55
56
in te n s ity as a fu n c tio n of a n a ly z e r p o sitio n th ro u g h an
a n aly z e r ro ta tio n of 360 d e g re es for the MBR m o d e l w ith a
449pm thick su b stra te .
4-8
56
G ra p h of the a m p litu d e of the reflection coefficient R for the
infinite b e am m o d e l as th e loss coefficient B is v a rie d from
0.2 to 1 in in c re m e n ts of 0.2 a n d for the tw o b e a m m o d e ls
w h e re the solid lin e A re p re se n ts the m odel fro m Eq. (4-14)
a n d the solid line B re p re se n ts the m odel from Eq. (4-15).
4-9
G ra p h of the a rg u m e n t of the reflection coefficient R for the
infinite beam m o d e l as the loss coefficient B is v a rie d from
0.2 to 1 in in c re m e n ts of 0.2 a n d for the tw o b e am m o d e ls
x
58
w h e re A r e p re s e n ts th e m o d e l fro m Eq. (4-14) a n d B
rep resen ts the m o d e l from Eq. (4-15).
4-10
59
G ra p h of the m a g n itu d e of the reflection coefficient R for the
infinite b e am m o d e l as th e loss coefficient B is v a rie d from
0.2 to 1 in in c re m e n ts of 0.2 a n d for th e tw o b eam m o d els
w h e re solid line A re p re se n ts the m o d el from Eq. (4-14) a n d
solid line B re p re se n ts th e m o d e l from Eq. (4-15).
4-11
60
P o lar g ra p h of the n u m e ric a l a n a ly sis of th e v a ria tio n of
o u tp u t in te n sity as a fu n ctio n of a n aly zer p o sitio n th ro u g h a
ro tatio n of 360 d eg rees for the infinite b eam m o d el w h e n B =
1.
4-12
61
P o lar g ra p h of the n u m e ric a l a n a ly sis of th e v a ria tio n of
o u tp u t in ten sity as a fu n ctio n of a n aly zer p o sitio n th ro u g h a
ro tatio n of 360 d eg rees for the infinite b eam m o d el w h e n B =
0.8.
4-13
62
P o la r g ra p h of the n u m e ric a l a n aly sis of the v a ria tio n of
o u tp u t in ten sity as a fu n ctio n of an aly zer p o sitio n th ro u g h a
ro tatio n of 360 d eg rees for the infinite b eam m o d el w h e n B =
0.6.
63
xi
4-14
R eflected light in te n sity / In p u t lig h t in ten sity v e rsu s G aA s
sam p le thickness as B ch an g es from 0.2 to 1 in in crem en ts of
0.2
4-15
66
A m p litu d e of th e in te n s ity c h an g e for c h a n g e s in v o lta g e
from 0 to 1 V olts w h e n B = 0.6 an d the sa m p le th ick n ess is
450 |im .
4-16
67
A m p litu d e of th e in te n s ity ch an g e for ch an g e s in v o lta g e
from 0 to 1 V olts w h e n B = 0.6, the sam p le th ick n ess is 450
pm a n d the an aly z e r is ro ta te d 20 degrees
4-17
68
L in ea r g ra p h o f th e n u m e ric a l a n a ly sis of to ta l o u tp u t
in te n s ity as a fu n c tio n o f a n a ly z e r p o s itio n th ro u g h an
an aly z e r ro ta tio n of 360 d e g re es for the MBR m o d e l w ith a
half w av e p late at 45 d e g re es
4-18
69
I o u t/I in vs. ph ase delay for analyzer position s 0 to 45
degrees in 5 d e g re e in crem en ts.
4-19
4-20
71
Io u t/I in vs. phase delay for analyzer positions 45 to 90
degrees in 5 d e g re e in crem en ts.
71
Iout/Iin vs. phase delay for analyzer position at 45 degrees
74
xii
4-21
I o u t/ I in vs. p h a s e d e la y fo r a n a ly z e r p o s itio n e d a t 35
d e g re e s w ith s u b s tra te th ic k n e ss v a ria tio n s th a t p ro d u c e
p h ase delays of 0 to 45 d e g re es in in crem en ts of 15 d eg rees.
4-22
74
lout /1 in vs. phase delay for analyzer positioned at 45
degrees with a half wave plate rotated from 0 to 22.5
degrees.
4-23
75
Iout/Iin vs. phase delay for the infinite beam model with the
analyzer positioned at 45 degrees and B varied from 0.2 to 1
in 0.2 increments.
4-24
76
Iout/Iin vs. phase delay for the non MBR model with the
analyzer positioned at 45 degrees and with a quarter wave
plate at 45 degrees.
4-25
77
Iout/Iin vs. phase delay for the MBR model with the
analyzer positioned at 45 degrees and with a quarter wave
plate at 45 degrees.
4-26
78
lou t/I in vs. phase delay for the MBR model with the
analyzer rotated from 0 to 45 degrees in 5 degree increments
and with a quarter wave plate at 45 degrees.
xiii
79
4-27
I o u t / I i n vs. p h a s e d e la y fo r th e MBR m o d e l w ith th e
a n a ly z e r r o ta te d fro m 45 to 90 d e g re e s in 5 d e g re e
increm ents a n d w ith a q u a rte r w av e p late at 45 d egrees.
5-1
R eflected lig h t in te n s ity m e a s u re m e n ts v e rs u s laser in p u t
intensity.
5-2
84
C o n tra st b e tw e e n laser in te n sity m e a su re d fro m G aA s back
an d fro n t su rface m etalizatio n s.
5-3
85
F u rth er reflected lig h t in te n s ity m e a su re m e n ts v e rsu s laser
in p u t intensity.
5-4
80
86
R eflected lig h t in te n s ity m e a s u re m e n t v e rs u s laser in p u t
in tensity for a G aA s sam p le su p p lie d by C ry stal Specialties.
88
5-5
Index of refraction in G aA s as a function of te m p e ratu re.
90
5-6
R eflected lig h t in te n s ity m e a s u re m e n ts v e rs u s laser in p u t
in te n s ity w ith h e a te d to 70 d e g re e s a n d 110 d e g re e s
F ahrenheit.
93
V a ria tio n o v e r tim e o f to ta l re fle c te d lig h t in te n s ity
m e asu re d w ith th e o p tical p o w e r m eter.
xiv
94
V aria tio n o v e r tim e of re fle c ted lig h t in te n s ity m e a s u re d
w ith the lock-in am plifier.
Electro-optic field in te n sity v e rsu s p o sitio n in the v ic in ity of
a 50 Q transm ission line.
E le c tro -o p tic
f ie ld
in t e n s i ty
v e rs u s
p o s itio n
fo r
a
tran sm issio n line.
E lectro-optic field in te n s ity p ro files v ersu s p o s itio n in th e
vicinity of a stu b cap acito r. C u rv e A is the p ro file fro m the
stu b p o in t at a 45 d e g re e angle. C u rv e B is fro m th e cen tral
p o rtio n of the stu b p e rp e n d ic u la r to the stu b en d . C u rv e C is
a p ro b e trace p a ra llel to the stu b end.
Electro-optic field in te n sity profiles versus po sitio n for a stu b
capacitor.
E lectro-optic field in te n s ity p ro files v e rsu s p o s itio n for an
in te rd ig ital capacito r. C u rv e A is the resp o n se w ith th e left
e n d excited w h ile C u rv e B has the rig h t side excited. In each
case, the o p p o site e le c tro d e is term in ated in a 50 Q load.
Left side a c tiv a te d e le c tro -o p tic field in te n sity s u b tra c te d
from rig h t side a c tiv a te d electro -o p tic field in te n s ity v e rsu s
position for the g a p s n u m b e re d as in Fig. (6-5).
6-7
P u lse d elec tro -o p tic p ro b in g scan across tw o tra n sm issio n
lines.
6-8
106
Electric field in te n sity p ro file v e rsu s p o sitio n w ith th e signal
on the 50 Q tra n sm issio n line.
6-9
107
Electric field in te n sity p ro file v e rsu s p o sitio n w ith the signal
on the 75 Q. tra n sm issio n line.
6-10
108
Electric field in te n sity p ro files v ersu s p o sitio n w ith a signal
o n th e 50 г2 tra n s m is s io n lin e fo r th e p u ls e d p ro b in g
in stru m e n t a n d th e C W p ro b in g in stru m en t.
110
C -l
Laser dio d e sp e c tru m for 15 m A cu rren t.
132
C-2
Laser d io d e sp e c tru m for 25 m A cu rren t.
133
C-3
Laser d io d e sp e c tru m for 30 m A cu rren t.
134
C-4
Laser dio d e sp e c tru m for 35 m A cu rren t.
135
C-5
Laser d io d e sp e c tru m for 40 m A cu rren t.
136
C-6
Laser dio d e sp e c tru m for 45 m A cu rren t.
137
C-7
Laser d io d e sp e c tru m for 50 m A cu rren t.
138
xvi
C-8
L aser d io d e sp e c tru m for 60 m A cu rren t.
139
C-9
Laser d io d e sp ec tru m for 65 m A current.
140
D -l
A b so rb an ce v e rsu s w a v e le n g th for S u m ito m o G aA s w a fer
sam ple.
D-2
142
A bsorbance v e rsu s w a v e le n g th for C rystal Specialties w afer
sam ple.
D-3
143
A b so rb an c e v e rs u s w a v e le n g th for C o m in co G aA s w a fer
sam ple.
144
xvii
1
CHAPTER 1
INTRODUCTION
T rad itio n al m e th o d s for m e a s u rin g circuit an d d ev ice p e rfo rm an c e
a t h ig h s p e e d s a n d fre q u e n c ie s (e.g., e lec trica l p ro b e s, o scillo sc o p e s,
n e tw o rk a n a ly z e rs, etc.) h a v e lim ita tio n s b e c a u se th ey can in tro d u c e
p a r a s itic
e le m e n ts
th a t
change
c irc u it
o p e r a tio n
and
m e a su re
c h a ra c te ris tic s o r o b ta in d a ta o n ly a t th e in p u t a n d / o r o u tp u t p o rts
(term inals) of m icrow ave circuits.
A p ro m isin g n o n -in v a siv e p ro b in g p ro c e d u re called e lec tro -o p tic
p ro b in g , h a s b een re p o rte d [l,2 ].
T his p ro b in g m e th o d is b a se d on th e
lin e a r electro-optic effect: the c h a n g e in the index of refractio n of a crystal
in the p re sen c e of an electric field. G alliu m arsen id e (G aA s) a n d in d iu m
p h o s p h id e (InP) are e x am p les of c o m m o n sem ico n d u c to r m a te ria ls th at
ex h ib it this effect. E lectro-optic p ro b in g u ses the in fo rm atio n p ro d u c e d by
a su b -b a n d g a p laser w h e n the lig h t is p h a se m o d u la te d b y the ch an g e in
in d e x of re fra c tio n of a s e m ic o n d u c to r s u b s tra te in th e p re s e n c e of
m ic ro w a v e o r o th e r electric field s. A n electro -o p tic p ro b in g in s tru m e n t
p e rm its a p o in t-b y -p o in t e v a lu a tio n of th e elec tric fie ld in te rn a l to
m ic ro w a v e c irc u its fa b ric a te d o n G aA s ? in s te a d of lim itin g th e
2
in fo rm atio n to th a t g a th e re d at the in p u t o r o u tp u t p o rts o f a circuit. It is
th is ab ility to p ro b e th e in te rn a l field s of a circu it d ire c tly a n d w ith o u t
p a ra s itic a lly lo a d in g th e c irc u it th a t m a k es a n e le c tro -o p tic p ro b in g
in s tru m e n t an a d v an ta g e o u s a d d itio n to the test e q u ip m e n t for m icro w av e
circuit stru ctu res.
Electro-optic p ro b in g w as first re p o rte d by V aldm anis et al. w h e re a
la se r lig h t b e am w as m o d u la te d in an e x tern al electro -o p tic c ry stal th a t
w as c o u p le d electrically to th e test circuit[3]. K olner et al. a d v a n c e d the
tech n iq u e by u sin g the test circu it itself as th e electro-optic crystal to m ak e
the first in te rn a l m e a su re m e n ts of a G aA s circu it u sin g a N d .Y ag p u ls e d
laser to p ro b e the m icrow av e signal p re se n t in a G aA s FET trav elin g -w av e
a m p lifie r[4]. In te rn a l electro-optic p ro b in g w ith a N d:Y ag laser h a s been
d e m o n s tr a te d fo r th e m e a s u re m e n t of S- p a ra m e te rs o f c o p la n a r
w a v e g u id e a t 100 G H z [5] a n d for th e tw o -d im en sio n a l field m a p p in g of
co p lan a r w aveguide[6]. Z h u et al. in tro d u c e d a laser p ro b in g sy stem w ith
a c o n tin u o u s ra th e r th a n p u ls e d la se r p ro b e a n d m e a su re d the electric
field p rofile of co p lan ar w a v eg u id e at K H z frequencies[7]. The a d v a n ta g e
of th e c o n tin u o u s w ave electro-optic p ro b in g in stru m e n t w as its sim plicity
a n d low cost w h ile still p e rm ittin g electric field profiles of test circuits.
W iesen feld et al. su g g e ste d u sin g a gain sw itch e d laser d io d e to rep lace
the N d:Y ag laser in the p u lsed laser in stru m e n ts to d e m o n stra te a sm aller,
m o re c o n v e n ie n t in s tru m e n t th a t w o u ld h a v e the a d v a n ta g e o f b ein g
co n tin u o u sly tu n ab le o v er a w id e freq u en cy range[8]. The tra d e off for the
ease of u sin g the laser d io d e w as a lim itatio n in test frequencies to 20 G H z.
As
la s e r
d io d e s
b e c o m e c a p a b le
of p r o d u c in g
s h o r te r
p u ls e s ,
m e a su re m e n ts at h ig h e r freq u en cies w ill be possible. E lectro-optic p u lse d
3
p ro b in g w ith a laser d io d e h as b e en u se d in m a n y a p p licatio n s in c lu d in g
m e a s u rin g w a v e fo rm s fro m a G a A s fre q u e n c y d iv id e r c irc u it at 2.4
G H z[8], fro m an In G a A s /In P M ISFET in v erter[9 ], a n d from a p a c k a g e d
G aA s d ecisio n circuit[10]. For th e s tu d y of d ev ices at fre q u e n c ie s th a t
th eo retically are in the TFfz ran g e, e x tern al electro-optic p ro b in g h a s been
d e v e lo p e d w h e re a n electro-optic cry stal is p la ce d at the tip of a p ro b e th a t
is p la c e d n e a r th e su rface of th e te st sa m p le so that the frin g in g electric
fields m ay be stu d ie d [11].
T he o rig in a l m o tiv a tio n o f o u r w o rk w as to d e v e lo p b o th a
c o n tin u o u s w a v e e le c tro -o p tic p r o b in g in s tr u m e n t a n d a p u ls e d
sem ic o n d u c to r d io d e electro -o p tic p ro b in g in s tru m e n t to in v e stig a te the
m a p p in g of th e electric field in te n s ity vs. sp a tia l p o sitio n of m o n o lith ic
m ic ro w a v e in te g ra te d circ u its (M M IC ).
A s testin g o f G aA s s a m p le s
p ro g re s se d , w e id e n tifie d n ew p ro b le m s th a t n e e d e d to be u n d e rs to o d
before electro-optical p ro b in g c o u ld b eco m e practical. These p ro b le m s are
o n ly b e g in n in g to be a d d re sse d [12, for exam ple].
C o n c u rre n t w ith o u r w o rk , th e s tu d y of e le c tro -o p tic p ro b in g
tech n iq u es h as p ro g resse d w ith efforts to refin e the electro -o p tic p ro b in g
tech n iq u e m ore fully by focu sin g o n sev eral o u tsta n d in g p ro b lem s. O n e is
m u ltip le re fle c tio n s p re se n t in th e test s u b s tra te for c o n tin u o u s w a v e
te stin g or for long p u lse p u ls e d testin g . U n d e r these tw o circu m stan ces,
th e test sam p le creates an etalo n for th e lig h t b e am from the p ro b in g laser.
T h e p resen ce of m u ltip le reflectio n s can d ra m atically change te st re su lts if
they are n o t taken into account [13]. S om e tech n iq u es for d e -e m b e d d in g
th e test re su lts fro m m u ltip le b e a m s p re s e n t in the G aA s s u b stra te h av e
b een proposed[12]. W ith the a d v e n t of faster p h o to d io d e s a n d p ro sp e cts
4
for d e -e m b e d d in g the test re su lts fro m so m e m u ltip le b e am re fle c tio n
effects, u s in g c o n tin u o u s w av e laser p ro b in g for M M IC c h a ra c te riz a tio n
has been proposed[14].
W e d ire c te d o u r w o rk to r e s o lv in g so m e of th e p ro b le m s
a sso c ia te d w ith electro-optic p ro b in g in s tru m e n ts , a n d m a p th e electric
field p ro file s of se v e ra l p a ss iv e G a A s c irc u its.
A n o u tlin e of th is
d is s e rta tio n is as follow s. T he b asic th e o ry of electro -o p tic p ro b in g is
o u tlin e d in C h a p te r 2. C h ap ter 3 d escrib es th e co n tin u o u s w a v e e lec tro н
op tic p ro b in g in s tru m e n t a n d th e p u ls e d e lectro -o p tic in s tru m e n ts w e
built. T he c o n tin u o u s w av e (CW ) e lec tro -o p tic p ro b in g in s tru m e n t w as
used
to s tu d y e le c tro -o p tic p ro b in g at lo w fre q u e n c ie s to b e tte r
u n d e rs ta n d
how
to u s e
th is
te c h n iq u e
to m a k e
a c c u r a te
te s t
m e a su re m e n ts. T he CW in stru m e n t w as th en u sed to establish a b aselin e
for w o rk at G H z frequencies. T he p u ls e d electro-optic p ro b in g in s tru m e n t
w as b u ilt for hig h frequency (G H z) testing. A d escrip tio n of the G aA s test
circu its is in c lu d e d in C h a p te r 3.
C h a p te r 4 d iscu sses m u ltip le b e am
reflections (MBR) in the test sam p les a n d som e of the p ro b lem s c re a te d by
their p re se n c e in electro-o p tic p ro b in g in s tru m e n ts . A n e w a n a ly sis of
m u ltip le b e am reflections p resen ted in this c h ap te r show s a c o m p ariso n of
som e of th e a p p ro x im a tio n s u se d to ch arac te riz e the action of MBR a n d a
c a lib ra tio n te c h n iq u e to a cc o u n t fo r MBR in in s tru m e n t test re s u lts is
in tro d u c e d .
C h a p te r 5 sh o w s d a ta re la te d to a n o th er p ro b le m ? n o n н
lin ear in te n sity d e p e n d e n t b e h a v io r d isc o v e re d for the first tim e d u rin g
o u r e le c tro -o p tic p ro b in g for electric field p ro files of p a ssiv e G aA s test
circuits. C h a p te r 6 show s CW electro-optic in stru m e n t an d p u ls e d electro н
optic in s tru m e n t m a p p in g s of electric field in te n sity profiles for a v a rie ty
5
of G aA s te st sam ples. C h a p te r 7 n o te s th e co n clu sio n s d ra w n fro m this
w o rk a n d gives reco m m en d atio n s for fu rth e r stu d y .
6
CHAPTER 2
ELECTRO-OPTIC PROBING THEORY
E lectro-optic p ro b in g is b a se d o n th e lin e a r electro-optic effect [1].
In this c h a p te r, th e sim p lifie d d ia g ra m s h o w n in Fig. (2-1) is u s e d to
illu stra te th e basic p rin c ip le s of elec tro -o p tic p ro b in g of an electric field
a lo n g a m ic ro s trip tra n s m is s io n lin e .
T h e first p a rt of th e c h a p te r
d escrib es the o rie n ta tio n of the sy ste m w ith resp ect to the p rin c ip le axis
of G aA s.
N e x t, the lin e a r e le c tro -o p tic effect in G aA s c au se d b y th e
p re s e n c e of a n electric field in th e c ry s ta l is d e sc rib e d .
F in a lly , th e
a p p lic a tio n of th e lin e a r e le c tro -o p tic effect to the sim p lifie d e le c tro н
o p tic p ro b in g in s tru m e n t illu s tra te d in Fig. (2-1) is show n.
T he m ic ro strip in Fig. (2-1) is a s s u m e d to be fab ricated o n sem iin s u la tin g G aA s w ith th e (100) a n d (001) c ry s ta llo g ra p h ic d ire c tio n s
o rie n te d a lo n g the x-axis a n d z-axis, re sp ectiv ely . G aA s a n d InP b elo n g
to a class of cry stals th a t lack in v e rsio n s y m m e try [15] an d th u s ex h ib it
th e lin e a r e lec tro -o p tic effect.
T h is o rie n ta tio n is n o t o n ly c o n v e n ie n t
for e le c tro -o p tic p ro b in g , b u t is th e o rie n ta tio n c o m m o n ly u s e d for
w a fe rs m a n u f a c tu r e d fo r u s e as M M IC s (M o n o lith ic M ic ro w a v e
Fig. 2-1 Schematic representation of electro-optic p ro b in g
7
4_)
j
(U
bo g
o
8
In te g ra te d C ircuits). T he lig h t so u rce a n d p o la riz e r in Fig. (2-1) p ro d u c e
lig h t lin e a rly p o la riz e d in th e y -d ire c tio n th a t is u s e d to p ro b e th e
v o lta g e a lo n g th e tra n s m is sio n line.
T h is o rie n ta tio n o f p ro b e b e a m
p o la riz a tio n a n d sa m p le c ry s ta llo g ra p h ic d ire c tio n s y ield s th e la rg e st
a n d th e re fo re o p tim a l te st re sp o n se .
T h e d ia m e te r of th e lig h t b e am
in c id e n t o n the m ic ro s trip s a m p le d e fin e s th e area p ro b e d , a n d h en ce
th e s m a lle s t s p o t size is d iffra c tio n lim ite d .
W ith in th e G aA s, th e
lin early p o la riz e d lig h t m a y b e m o d e le d as tw o o rth o g o n a l p o la riz a tio n
c o m p o n e n ts e ac h of w h ic h e x p e rie n c e s a d iffe re n t in d e x of re fra c tio n
d u e to th e e ffe ct o f th e m ic ro w a v e e le c tric fie ld o n th e g a lliu m
a rse n id e 's in d ices of refractio n .
T he effect of th e electric field o n th e in d ices of re fra c tio n of an
e le c tro o p tic se m ic o n d u c to r m a te ria l is e a sily illu s tra te d by the in d e x
ellip so id .
G iv e n th e d ire c tio n of th e w a v e n o rm a l of th e p ro p a g a tin g
lig h t in c id e n t o n th e c ry s ta l, th e in d e x e llip s o id h as th e fo llo w in g
p ro p e r ty .
T h e in te rs e c tio n o f th e in d e x e llip s o id a n d
a p la n e
p e rp e n d ic u la r to the w a v e n o rm a l is an ellip se. T he tw o p o la riz a tio n
d ir e c tio n s
a s s o c ia te d
w ith
th e
w ave
n o rm a l c o in c id e
w ith
th e
d ire c tio n s of th e m ajo r a n d m in o r axes of th e ellip se. T he m a g n itu d e
of the ra d iu s vectors alo n g th e ellip se axes g iv e th e indices of refractio n
for th e tw o p o la riz a tio n d ire c tio n s [16]. A g en eral an aly sis of the in d ex
ellipsoid m a y be fo u n d in references [15, 17].
T he g en eral e q u a tio n for an e llip so id w ith c o n sta n ts ciy is
U\ yX
2
+
2
(122)?
A33^
2
2 ^ 2 3 2^j C-v + 2гi[2-rv = 1.
(2 - 1)
9
For G aAs in th e ab sence of a n electric field, Eq. (2-1) becom es
r2
.2 ++
v2
?
?
2 ++
no
no
~2
(2 - 2)
?
2
no
w h e re a\ \ = c*22 ~ a33 = ~ r a n d a \2 = л23 = a 3i = b ? For G aA s, n0 = 3 .6 for
ligh t w ith a w a v e le n g th of 1300 nm .
G a A s is a cubic crystal th u s its
in d e x of re fra c tio n is iso tro p ic , a n d its in d e x e llip s o id is a sp h e re .
G aA s's in d e x of refractio n n0 c han ges in a n electric field d e fo rm in g the
in d ex ellip so id fro m its sp herical shape.
This d e fo rm a tio n is the basis
of electro-optic p ro b ing .
T h e d e f o r m a t i o n of th e s p h e r ic a l i n d e x e llip s o id for G a A s
be ca u se of an electric field c h ang es the coefficients in Eq. (2-1).
The
c h a n g e in e a c h c o efficient ajj is re la te d to e a c h c o m p o n e n t of the
electric field E b y th e e lectro -o p tic m a trix for G a A s w h e re rk( is an
electro-optic coefficient[16]. For GaAs, r4l = 1.43x10
i 'y
m / V w h e n the
p ro b in g light w a v e le n g th is 1300 n m [16].
"0
0
0
0
0
0
Aa^
0
0
0
E.x
A a 23
r4\
0
0
Aл3i
Ey
E?
0
'41
()
_Aan _
0
0
r4 ]_
'A a n ~
Aл22
(2-3)
The ellipsoid p r o d u c e d b y the d e fo rm a tio n of th e s p h e re is described by
10
X
2
?2 +
2
2
v
+
z
~~2 + 2t'4\Ex y z + 2 /' 41 Eyzx + 2 r 4 iE zx y = 1.
(2-4)
For the e le c tro -o p tic p r o b in g s a m p le g e o m e t r y in Fig. (2-1), w h e n a
m ic ro w a v e field in the x d irectio n г r is p r e s e n t, Ey = EZ = 0 since the
lig h t b e a m o n ly in te ra c ts w ith the x c o m p o n e n t of the electric field,
hence Eq. (2-4) beco m es
2
2
?-j- H y H
2
(2-5)
y + 2/?4 \ E yyz ? 1.
A 45░ro ta tio n a b o u t the x axis tra n s fo rm s Eq. (2-5) into a c o n v e n ie n t
fo rm for d e t e r m i n i n g th e in d e x of re f r a c tio n for e a c h of the tw o
p o la riz a tio n c o m p o n e n ts :
.12
12
= 1.
( 2 -6 )
r4 \ E x
In the G a A s s a m p le , the tw o p o l a r i z a t i o n c o m p o n e n t s a re
located alo ng the y ? a n d z ? axes as d e fin e d in Eq. (2-6) a n d illustrated by
Fig. (2-2).
T h e in d ic e s of re fra c tio n for th e p o la riz a tio n c o m p o n e n ts
are:
n3
' V = ' ^ + Y ' 4 l Ex
n3
n o - f ?-4\Ex .
(2-7)
(2-8)
11
Fig. 2-2 The o rie n ta tio n of p o la riz a tio n d ire c tio n s for electro-optic
probing
A s a re s u lt of th e p o la r iz a t io n c o m p o n e n t s e x p e r ie n c in g d if f e r e n t
in d ic e s of re fra c tio n , a p h a s e d e la y T, th a t is p r o p o r t i o n a l to the
stre n g th of the electric field at the p ro b e d p o in t, is in tr o d u c e d b e tw e e n
the p o la riz a tio n c o m p o n e n ts of the p ro b e b e a m lig h t p a ss in g th ro u g h
the G aA s.
T h e r e la tio n s h ip b e tw e e n th e p h a s e d e la y a n d the s ig n al
v o ltage is
(2-9)
w h e re 0 is the p h a s e a sso ciated w ith e ach p o la riz a tio n c o m p o n e n t, A
is the w a v e le n g th of the in c id e n t light, a n d V is the signal voltage.
T he h a lf-w a v e v o lta g e Vn is the v o lta g e level in the test s a m p le
that c o rre sp o n d s to a p h a s e d elay of n. E q u a tio n (2-9) then becom es
( 2- 10)
since
( 2- 11)
For G aA s Vn ~ 5 k V .
For the tw o po larizer, q u a rte r-w a v e p la te test sy ste m in Fig. (2-1)
the ratio of o u tp u t to in p u t light intensity (IG / Ii) is [15]
13
w h e re T is the p h a s e d e la y in tro d u c e d by b o th the q u a rte r-w a v e plate( ti/2) a n d the G aA s crystal ( r m ) a n d V in c lu d e s the v o ltag e in the G aA s
crystal (Vm) a n d th e e q u iv a le n t v o lta g e of the q u a rte r-w a v e plate ( 1 / 2
Vto T hu s,
r =- + r m
(2-13)
and
? = sin2f ? + ?
/;
U
(2-14)
2
U sin g trig o n o m e tric identities,
i - = i ( I + s i n r m ).
(2-15)
From Eq. (2-10),
(2-16)
K
T h u s, th e r a tio of o u t p u t lig h t in te n s ity to i n p u t lig h t in te n s ity is
related to the v o lta g e signal, V m ' at th e G a A s m ic ro strip sam ple p o in t
by
/,
2
( nV ^
1+ sin H Km
\
V
yK J
(2-17)
14
U sin g the sm all a ng le ap p ro x im a tio n , Eq. (2-17) m a y be re w ritte n as
1+
(2-18)
v VK
F ig u re (2-3) s h o w s th e g ra p h ic a l r e p r e s e n ta tio n b e tw e e n in p u t a n d
o u t p u t in te n s ity a n d the a p p lie d v o lta g e as g iv e n b y Eq. (2-12) a n d
illustrates that the electro-o ptic p ro b in g s y s te m o p e ra te s w ith g reatest
sensitivity n e a r o n e -h a lf of the half w a v e v o lta g e VjtT h e a p p l i c a t i o n o f th e se re s u lts to a p ra c tic a l e le c tro -o p tic
p ro b in g in s tru m e n t w ill be d iscu ssed in C h a p te r 3.
O
50
AS
I
rH
fS
( !I/ I) Ajisuajui jqSij jnduj / Ajisuajui jqSii }ndjno
o
Fig. 2-3 Electro-optic effect response curve with the ratio of output light intensity to input light intensity versus
the electric field generated voltage signal at the sampled point
15
16
CHAPTER 3
ELECTRO-OPTIC PROBING INSTRUMENTS
AND TEST SAMPLES
W e d e s ig n e d a n d a ssem b led b o th a c o n tin u o u s w a v e laser electro-optic
p ro b in g
in stru m e n t
in stru m e n t.
and
a
p u lsed
laser
e le c tr o - o p tic
p ro b in g
S e c tio n 3.1 d e s c r ib e s th e d e s ig n a n d s e t- u p of th e
c o n t in u o u s w a v e (CW) e le c tro -o p tic p r o b in g in s tru m e n t.
The CW
e lec tro -o p tic p r o b in g in s tr u m e n t w a s u s e d to s tu d y the te c h n iq u e of
electro -o ptic p r o b in g at low fre q u e n c ie s a n d to establish a b a se lin e in
p r e p a r a t io n for te stin g G aA s circ u its w ith a p u ls e d sy stem at h ig h e r
freq u en cies.
T h e fre q u e n c y lim it o f the C W in s tru m e n t rests on the
s p e e d of th e p h o to d e te c to r u s e d to s e n s e lig ht intensity in the e lec tro н
op tic p ro b in g in stru m e n t. The u p p e r lim it for p h o to d io d e s is p re s e n tly
c laim ed to b e a r o u n d 60 G H z [14]. H o w e v e r issues such as sen sitivity
h a v e to be c o n sid e re d for practical a p p lic atio n s. The a d v a n ta g e of C W
e le c tro -o p tic p r o b in g is its re la tiv e e a s e to set up .
A p u ls e d la se r
p e r m its th e u s e of a slo w er p h o t o d i o d e a n d th e re fo re in c re ase s the
b a n d w i d t h a n d sensitivity of th e in s tr u m e n t.
Test results u s in g h ig h
17
p o w e r c o m p re s s e d laser pulses h av e b e e n p re s e n te d at 100 G H z [5].
A
p u ls e d laser sy ste m b ased o n a laser d io d e is lim ited b y the w id th of the
laser p ulses. This limits laser d io d e b a s e d sy ste m s to below 40 G H z for
m o d e lo ck ed o p e ra tio n a n d b e lo w 20 G H z for gain sw itch e d o p e ra tio n
[18].
W e c h o se to gain sw itch the la se r d io d e to e sta b lish a s im p le r
sy stem th a t w o u l d still be capable of te stin g th e vast m ajority of G a A s
circuits.
A n a d v a n ta g e of an e lectro -o p tic p ro b in g sy ste m b a s e d o n a
gain s w itc h e d laser is that the freq u e n cy re s p o n s e of m ic ro w a v e circuits
can be easily s tu d ie d since the laser d io d e b a s e d system can be tu n e d to
s w e e p t h r o u g h sev eral d ifferen t fre q u e n c ie s.
The c h a ra c te riz a tio n of
the p u ls e d laser d io d e used for p u ls e d electro-optic pro b in g is d iscu ssed
in section 3.2. T h e p u ls e d electro-optic p r o b in g in s tru m e n t is d e sc rib e d
alon g w ith an e x p la n a tio n of the d e te c tio n system .
G aA s
In section 3.3, the
c irc u its w e d e s ig n e d a n d te s te d a r e d e sc rib e d .
d is c u s s e s
th e c u s to m
d e s ig n e d
a lu m in a
c irc u it b o a r d
Section 3.3
used
for
m o u n tin g th e G a A s circuits a n d the m ic r o w a v e p a ck a g e w e d e s ig n e d
a n d fa b ric a te d for testing the circuits w ith the electro-o ptic p r o b in g
in s t r u m e n t .
3.1 C o n tin u o u s w ave electro-optic p ro b in g
instrum ent
F ig u re (3-1) sh o w s a sch e m a tic d i a g r a m of the CW in s tru m e n t.
Since e le c tro -o p tic p ro b in g is c o n d u c te d in th e in fr a r e d to o p e r a te
18
b elow the a b so rp tio n e d g e (b an dg ap ) of the test sam ple's sem ic o n d u c to r
s u b s tr a te (e.g., G aA s, InP, etc.), th e h e liu m - n e o n laser a n d th e b e a m
s p litte r a re n e c e s s a ry for a lig n m e n t a n d v is u a l v e rific a tio n of th e
optical p a th .
D u rin g s am p le m e a s u r e m e n ts , the h e liu m -n e o n la se r is
tu rn e d off a n d the 1.3 |im s e m ic o n d u c to r laser (M itsubishi, M o d e l N o .
ML 7701A laser d io d e ) in a M elles G rio t (d iod e) laser system , (M o d e l
06DLD003), p ro v id e s the s u b -b a n d g a p laser p ro b in g source. In Fig. (3-1)
light fro m the 1.3 p m laser p asses th r o u g h the p o la riz in g b e a m s p litte r
p r o d u c in g p - p o la r iz e d light.
T h e o n e - e ig h th w a v e p la te e llip tic a lly
p o la riz es the light, a n d the lens focuses the elliptically p o la riz e d lig h t
o n to
th e
te s t s a m p le .
T h is n e w
a r r a n g e m e n t of a p o l a r i z i n g
b e a m s p litte r w ith a o n e -e ig h th w a v e p la te m in im ize s the loss of lig h t
in the sy ste m as c o m p a re d to a q u a rte r w a v e plate, an alyzer, p o la riz e r
c o m b in a tio n b a s e d o n Fig. (2-1) [19], a n d in c re a s e s i n s t r u m e n t
sensitivity.
O n c e the light enters th e s a m p le , the elliptically p o la riz e d
lig ht e x p e rie n c e s a fu r th e r p h a s e d e la y b e tw e e n its tw o p o la riz a tio n
co m p o n en ts. T h e s tre n g th of the s ig n al v o lta g e d e te rm in e s the size of
the d elay at the p o in t of p ro b e im p in g e m e n t as d escrib ed in C h a p te r 2.
The laser lig ht p asses th ro u g h the s a m p le , reflects off the back su rfa ce
m etal (or g r o u n d p la n e for m o s t m ic r o w a v e circuits), p asses th r o u g h
the s am p le a g a in a n d o u t of the se m ic o n d u c to r. Fig ure (3-2) illu stra te s
this d o u b le p a ss th r o u g h the s a m p le w h e r e the cross-sectional v ie w
s h o w s th e la se r b e a m e n te r in g th e test s a m p le a lo n g o n e of th e
tran sm issio n lines. T he spot size of the laser b e am w a s m e a s u re d to b e
less th a n 6 q m b y s c a n n in g th e la se r b e a m acro ss a p in hole.
An
19
Electrophysics in fra re d vidicon cam era w a s u s e d to track the location of
the p ro b e b e a m o n the surface of the G aA s circuit sam ple.
After ex itin g the sam ple, the light is re c o llim a ted by the focusing
lens a n d s u b s e q u e n t p a s s a g e th ro u g h th e e ig h th w a v e p late f u r th e r
increases the p h a s e delay.
T he p o la riz in g b e a m s p litte r s e p a ra te s the
re tu rn in g lig h t into its p-p o larized a n d s -p o la riz e d com ponents. T he sp o la riz e d lig h t p a sse s th ro u g h the fo c u s in g lens to the p h o to d e te c to r
(InGaAs P IN d io d e RCA# C30618E), p re a m p lifie r a n d lock-in am plifier
s y ste m (S ta n f o rd R e s e a rc h S ystem s, M o d e l N o. SR530).
The lig h t
in tensity at th e p h o to d e te c to r is d ire c tly r e la te d to the streng th of the
electric field at the p o in t p ro b e d in the test sam ple.
The pre -a m p lifier
a n d lock-in a m p lifie r lim it noise a n d s u p p l y an o u t p u t voltage th a t is
p r o p o r tio n a l to lig h t in te n s ity c h a n g e s th a t o ccu r at the re fe re n c e
voltage freq u en cy . W e d e sig n e d a n d fa b ric a ted the pre-am plifier circuit
s h o w n in Fig. (3-3)" . The circuit is d e s ig n e d so th at the DC p art of the
signal can b e d r a in e d a w a y from the in p u t signal. This design e le m e n t
e n a b le s u s
to m o n i t o r
the a v e r a g e l i g h t in te n s ity b y u s in g a n
oscilloscope to p r o b e the voltage level a t p o in t A as n o te d in Fig. (3-3).
This c ap a b ility gives u s a u n iq u e a d v a n t a g e w h e n issues such as the
m u ltip le b e a m reflections d iscu ssed in C h a p t e r 4 are ad d re ssed .
h av e m e a s u r e d 0.1 v o lt signals.
We
T h e s e n s itiv ity of laser d io d e b a s e d
p ro b in g i n s tr u m e n ts is claim ed to be in th e ra n g e of m V / V (H z) [8],
h o w e v e r p r o b le m s th a t a re d iscu ssed in C h a p t e r s 4 a n d 5 m a k e this
characteristic difficult to assess.
* We thank Art H ogrefe for the circuit design.
O th e r a rra n g e m e n ts of op tical e le m e n ts m a y be u s e d in electro н
optic p ro b in g . [2] T h e a rra n g e m e n t in Fig. (3-4) u ses q u a rte r a n d half
w a v e p lates w h e r e the fast axis of the q u a rte r w a v e p late is ro ta te d 22.5
d e g re e s fro m th e z axis a n d the fast ax is of the h a l f w a v e p la te is
r o ta te d 33.75 d e g re e s from the z' axis. In this set u p , w ith the q u a rte r
a n d half w a v e p la te s at th e se a n g u la r o r ie n ta tio n s , the sm all a n g le
a p p r o x im a tio n g iv e s the r e la tio n s h ip b e t w e e n the r a tio of o u t p u t
in te n s ity to i n p u t in te n s ity a n d v o lta g e s h o w n in Eq. (2-18) a n d
re p e a te d h e re as Eq. (3-1). W e h a v e n o t s t u d i e d the cap ability of this
a r r a n g e m e n t of o p tic a l e le m e n ts for p r a c tic a l p r o b le m s s u c h as
m u ltip le b e a m reflections.
a.
z л
o 60
6 >
of a continuous wave electro-optic probing system
00
Fig. 3-1 Schematic block diagram
21
<u
CD
.л Q
22
Laser beam
Au
GaAs
(a)
Incident
beam -
LASER
PROBE
Reflected
^ beam
50W transm ission line
75W transm ission line
Front surface
GaAs substrate
Back surface m etallization
(b)
Fig 3-2 Schem atic re p re se n ta tio n of a typical G aA s test s a m p le
a n d th e p ro b in g laser b e am n e a r a tra n sm issio n line, (a) three
d im e n s io n a l view (b) cross-sectional view .
c\ c
Fig. 3-3 Pre-amplifier circuit
23
Pm 0)
of a continuous wave electro-optic probing system
(Q CL,
Fig. 3-4 Schematic block diagram
Photodetector
with quarter and half wave plates
24
25
3.2 Pulsed electro-optic probing instrum ent
T h e la se r u s e d fo r p u l s e d p r o b i n g is a n A T & T [8] 1.3
m ic ro m e te r In G a A sP in jectio n laser d io d e m o u n te d o n th e e d g e of a
c o p p e r sh im .
W e f a b r ic a te d a 50 Q te st fix tu re for th e s h im a n d
m o u n te d it in a slot m a c h in e d into a c o p p e r cylinder. A therm o electric
cooler from M elcor is u s e d to k e e p the te m p e ra tu re of th e laser cooled
to w ith in 15.0 + 0.1 d e g re e s C. A th e rm isto r w as m o u n t e d o n the e n d
of the cy lind er to m o n ito r th e te m p e ra tu re of the laser d io d e . T he laser
d io d e is s h o w n m o u n te d in Fig. (3-5)**.
T he laser is gain s w itc h e d b y u s in g a h ig h fr e q u e n c y m ic ro w a v e
signal [8] s u p p lie d b y a n H P 8341B m ic ro w a v e s y n th e s iz e r.
T he gain
s w itc h e d laser p r o d u c e s p u ls e s w ith a re p e titio n r a t e e q u a l to the
m ic ro w a v e frequency s u p p lie d by the H P 8341B.
Since the signal from
th e m ic r o w a v e s y n th e s iz e r is o n ly 10 d b m , an A m p lif ie r R ese arch
m ic ro w a v e am p lifie r M o d e l N u m b e r 4W1000 is u s e d to in c re a s e the
signal p ow er. To gain s w itc h the laser, the laser d io d e is b ia se d o ne or
tw o tim e s a b o v e the th r e s h o ld c u r r e n t of 25 m A [8] u s in g an ILX
** W e thank J.M. W iesenfeld and C. A. Burrus o f AT & T Bell Laboratories for su p p lyin g
the laser d io d e and m echanical d ra w in g s for the test fixture. We are grateful to J. M.
W iesenfeld for con versations that supported the im plem entation o f the laser d io d e in
the pulsed system .
26
L ig h tw av e C u rre n t S o urce m o d e l LDX-3412. The RF signal a n d the bias
signal are c o m b ined w ith a h ig h frequ en cy bias tee.
T h e sm all s ig n a l r e s p o n s e of the la se r d i o d e lig h t in te n s ity
o u t p u t w h e n th e la s e r is m o d u l a t e d a t m ic r o w a v e f r e q u e n c ie s is
s h o w n in Fig. (3-6).
T h e m e a s u r e m e n t w a s m a d e w ith a n E pitaxx
In G a A s p h o to d io d e th a t w e c u s to m m o u n te d in th e fix tu re s h o w n in
Fig. (3-7). The fixture is p a r t of an H P 83040 series m o d u la r m icrocircuit
package.
The p h o to d io d e w a s m o u n te d w ith c o n d u c tiv e e p o x y to the
g o ld g ro u n d p a d of the a lu m in a s u b stra te a n d a via w a s b o re d th ro u g h
the su b stra te to e n a b le laser b e am access to the 25 p m active area of the
p h o to d io d e . T h e p h o to d io d e w a s w ire b o n d e d to the 50Q tran sm issio n
line o n th e H P fix tu re.
T w o d iffe re n t v ie w s of th e p h o t o d i o d e are
s h o w n in Figs. (3-8) a n d (3-9). T he fixture w a s c o n n ec te d directly to the
in p u t of a n H P 54020T d ig ital s a m p lin g oscilloscope. As the RF p o w e r
to the laser d io d e is in c re a s e d , th e la rg e s ig n al r e s p o n s e b e c o m e s a
series of sh ort p u lses w ith a p u ls e w id th less th a n 100 pico secon d at a 1
G H z re p e titio n rate. F ig u re s (3-10), (3-11), a n d (3-12) illu s tra te these
results.
For the p u ls e d sy ste m , o u r detec tio n is a c c o m p lis h e d u s in g
a
tech n iq u e called h a rm o n ic - m ixin g electro-optic p ro b in g (H M EO P) [20].
This is a freq u e n cy d o m a in te c h n iq u e th a t h a s b e e n u s e d by sev eral
g ro u p s [2, 20, 21].
m icro w av e
s ig n a l
T h e in te ra c tio n of the laser p r o b in g b e a m a n d the
in
th e
te s t
sam p le
can
be
v ie w e d
as
th e
m u ltip lic a tio n of th e i n p u t lig h t in te n s ity a n d th e m ic ro w a v e signal
v o ltage [22], th e re fo re cre atin g a m ixin g action. A s a resu lt, if a sm all
freq u e n cy chan g e Af o n the o rd e r of a few Khz is in tr o d u c e d so th a t the
27
laser p u ls e frequ en cy if f l a n d the m ic ro w a v e signal fr e q u e n c y is m fl +
Af w h e r e m is s o m e in te g e r , th e s p e c tr u m of the lig h t e x itin g the
o ptical train a n d e n te rin g th e d e tec tio n system m a y be d e sc rib e d as m fl
w ith sid e b a n d s at m f l + Af w h e r e m = 0,1,2,... [2] If th e p h o to d e te c to r is
fo llo w e d b y a n a r r o w b a n d filter sim ilar to
th a t f o u n d in a s p e c tru m
a n a ly z e r, th e n the a m p l i t u d e a n d p h a s e of th e s ig n a l at Af m a y be
id e n tifie d .
It is p r o p o r t i o n a l to the a m p l i t u d e a n d p h a s e of the
m ic ro w a v e signal in th e te st s a m p le [2].
W e u s e d th e p re a m p lifie r
d e sc rib e d in section 3.1 to increase the a m p litu d e of th e signal a n d limit
noise.
This also p e r m its u s to m o n ito r the a v erag e lig h t in te n s ity as
n o te d in section 3.1
T h e s c h e m a tic of the test s e t-u p u s e d for p u ls e d
p r o b in g is s h o w n in Fig. (3-13).
T he p u ls e d e le c tro -o p tic p ro b in g
in s tr u m e n t is s h o w n in Fig. (3-14) a n d a closer view o f the o ptical train
is s h o w n in Fig. (3-15).
29
in
-p
?
in
?p
a
a
a
a
r\j
??wot#
o c > c
>
г
EO
O h O
m o n o
cm i n a i m
to . i n a
.CD . .
in ? ? ?
n ii ii ii
> i-
?p
qj x o a
in a -p p
C
-4
?
r-t i~
?
t i?
i
<+- Ql 01 Q)
?
???
w
-p
r?
i
a id
> c
E
a
xo
(0
a w
a iiD
in .
z
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a
Q
o
in ?
in
ii
CD
ii
CM
L
Ql
JC
L a
O 0
E -P
> 1 /1
>
TJ
X
ID
ID
-P
?P > ?
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O"O> z
>\ E
e in
a o
a
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a
o
o
?
cd
.
.to
aoo
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ID
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n n ii ii
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a
a
a
tuн
rn L
cd
0 ai
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01 L L
. EDO
-C ? E +J
- > cn
u p
Fig. 3-6 Digital sampling oscilloscope m easu rem en t of laser intensity with the laser DC
biased slightly above threshold and supplied with less than -10 dbm of Rf p o w e r.
U)
c
Fig. 3-7 A photograph
of InGaAs photodiode packaged
m m ount
30
31
Fig. 3-9 Top down view
of the p h o to d io d e
32
33
(n
c
a
a
o
a
o c> c
EQ
Q Q Q
tn oa a
tMinain
to .ina
.00
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.
<4-'?
f? >
г
?^
1/3
c
ai
0 (I)
Vh
oi
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o
o
o
C
a╗
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m
a>
ao
s
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E +J
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otj
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I/)
0
60
+j
r-1-л 0 0)
0TJ> C
C
'a,
гto
C/3
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to
a -*
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S
(I)
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C
r-H
1
Q
a
o
Q
ID
CD
0) L
D 01
J0JC +J
01 L L
ui?>w
bb
E-Lh
with 8.13 dbm
AUmJ
W? +J 4->
and supplied
01 X O D
laser DC biased slightly above threshold
a╗
A
?*->
of Rf p o w e r.
C\J
0
O U)
> c
U1
L CL
>cn
E P
p >
OTI> C
0
*?i
Fig. 3-11 Digital sampling oscilloscope m easurem en t of laser intensity with the
laser DC biased slightly above threshold and supplied with 24.22 dbm of RF
p ow er.
34
(J)
c> c
LO??
╗??
?
in
c
o
a
a
o
t\j
0) x o o
U) o +j +j
<+-? ?
i
> c
(f)
c
a
a
a
in
CD
l a.
>i/i
E -P
w
c
a
o
a
a
to
Fig. 3-12 Digital sampling oscilloscope m e asu re m e n t of laser intensity with the laser
DC biased slightly above threshold and supplied with 31.40 dbm of Rf p o w e r.
35
U)
O C > CL
Mi crowave S y n t h e s i z e r
Mcrowave Synthesizer
P o w e r a m p l if i er
Photodetector
and
P r e a mp l i fie r
Sp e c tru m Analyzer
DC C u r r e n t
source
Bia s T
F o c u s i n g Lens
1 . 3 m m L a s e r Di ode
( Pr obi ng I a s e r )
Pola ri zi ng Be a m s p l i t t e r
1/8W aveplate
F o c u s i ng Lens
Fig.3-13 S chem atic block d ia g ra m of a p u ls e d electro-optic pro b in g
sy ste m
Sam
' fi nL
rt IJ
O
eeee,
electro-optic probing in stru m e n t.
' l c I
of the pulsed
MeJOa
Fig. 3-14 Photograph
37
39
3.3 GaAs circuits and test package
A g alliu m a rs e n id e test sam ple u s e d in o u r stu d ie s is sh o w n in Fig. (316). P a ssiv e m ic ro w a v e circuits w e r e c o n s tr u c te d o n 450-500 p m thick
G a A s s e m i- in s u la tin g s u b s tr a te s th a t w e r e h ig h ly p o lis h e d o n b o th
sides to e n h a n c e th e p e n e tr a tio n of the p r o b e b e a m into the s a m p le .
T h e s u b s t r a t e s w e r e liq u id e n c a p s u l a t e d C z o c h ra ls k i ty p e g a lliu m
a rs e n id e w ith the fo llo w in g specifications: s e m i-in su la tin g G a A s w ith
re s is tiv ity :> 1.0 x 10? U *cm ; e tc h p it d e n s ity < 5.0 x 10^ p e r c n rd ;
o rie n ta tio n (100) + 0.5░; thickness 450-500 pm ; thickness v a ria tio n < 4
pm ; s u rfa c e fin ish -b o th sides p o lis h e d ; d ia m e te r n o m in a l 50 m m w ith
m a x im u m b o w < 4 pm .
The w a fe r s w e r e p u r c h a s e d from S u m ito m o
Electric In d u s trie s Ltd. (Tokyo, Japan). W e d e sig n e d a n d fab ricated an
a lu m in a s u b s tr a te w ith a c o p la n a r w a v e g u id e m eta liza tio n p a tte r n to
m o u n t the G a A s test structures.
T h e G a A s circuits w ere b o n d e d w ith
c o n d u c tiv e e p o x y to the a lu m in a s u b s tr a te a n d rib b o n b o n d e d to the
c o p lan a r
w av eg u id e.
We p la ce d
th e
alu m in a
m ic ro w a v e test fix tu re w e d e s ig n e d a n d fab ric a ted .
s u b s tr a te s
in
a
The m ic r o w a v e
fix tu re s h o w n in Fig. (3-16) is 50 m m x 35 m m a n d the G aA s c h ip is
a p p ro x im a te ly 10 m m x 5mm.
A s c a n n in g e le c tro n p h o to m i c r o g r a p h of a G aA s s a m p l e is
s h o w n in Fig. (3-17). The s a m p le c o n ta in s tw o transm ission lines 50 Q
(350 p m w id e ) a n d 75 Q (112 p m w id e ) w ith a thickness of 2 p m at a
40
c e n te r f r e q u e n c y of 10 G H z.
T h e r i b b o n b o n d (250 [im w id e )
in te rc o n n e c t to the c o p la n a r w a v e g u i d e is sh o w n .
The m e ta lliz atio n
system for the G a A s sa m p le s consists of an e v a p o r a te d c h ro m iu m g o ld
p la tin g b a s e (50 n m of C r a n d 150 n m of A u ) fo llo w e d by 2 g m of
p a tte rn p la te d gold.
T h e thick p la te d g o ld is u s e d as an etch m a s k to
re m o v e the C r / A u p la tin g base. O th e r s a m p le s h a v e u s e d a s p u tte r e d
tita n iu m g o ld p la tin g b a se (front s id e o n ly ).
Fig. (3-18) illu strates an
ex am p le of the interd ig ital a n d stu b c ap acito rs w e tested.
Fig. 3-16 Photograph of a test sample mounted in the test fixture. The s u p p o r t
substrate is alumina with a coplaner wave guide metallization pattern.
44
CHAPTER 4
THE ROLE OF MULTIPLE BEAM
REFLECTIONS IN ELECTRO-OPTIC
PROBING
In the sim p lified d e s c rip tio n of e lectro -o ptic p ro b in g g iv e n in C h a p te r
2, the laser b e a m p a s s e s th r o u g h th e G a A s a n d e x p e rie n c e s a p h a s e
d elay Y b e tw e e n th e p o la riz a tio n c o m p o n en ts.
B ecause the app lic atio n
of this tech n iq u e to m ic ro s trip test s am p les p re c lu d e s the possibility of
p a ssin g a p ro b e b e a m d ire c tly th ro u g h the G aA s since the m etal g r o u n d
p la n e of th e m ic r o s tr ip acts as a m ir r o r ra th e r th a n a tr a n s p a r e n t
su rfa ce , fro n t s id e p r o b i n g is u s e d as d e s c rib e d in C h a p te r 3 a n d
illu s tra te d in Fig. (3-2).
T h e p h a s e d e la y a sc rib e d to the fro n t s id e
p ro b in g g e o m e try in C h a p te r 3 differs fro m th e s im p le d e sc rip tio n in
C h a p te r 2 by a c o n sta n t th a t takes into acco u nt the d o u b le pass th ro u g h
the GaAs. T he d e sc rip tio n of electro-optic p ro b in g in C h a p te r 2 neglects
the e x istence of a p a r tia l reflection of th e p ro b e b e a m off the fro n t
su rface of the G aA s a n d m u ltip le b e am reflections in te rn a l to the G aA s
sa m p le as s h o w n in Fig. (4-1). Li et al. [23] first s h o w e d the d ra m a tic
in fluen ce of m u ltip le b e a m reflections (MBR) o n e lectro -o p tic p ro b in g
45
results.
MBR is th e m o s t im p o r ta n t c a lib ra tio n issu e for electro-optic
p ro b in g w ith c o n tin u o u s w a v e lasers or p u ls e d lasers w ith long p u lses
[ 12].
M icrostrip
transm ission
lines
i----
------------------------------------------------Au
G round plane
Fig. 4-1 M u ltip le b e a m reflections in fro n t sid e p ro bin g
In this c h a p te r , sec tio n 4.1 e x p la in s m u ltip l e b e a m reflection
theory .
In s e c tio n 4.2, tw o s im p le tw o -b e a m re fle c tio n m o d e ls a re
d iscussed .
W e u s e the tw o -b e a m m o d e l w e d e v e lo p e d in co njunction
w ith a n u m e r ic a l a n a ly s is b a se d o n Jon es c a lc u lu s to illu stra te th e
influence of MBR o n e lectro-o p tic test resu lts.
N ex t, an infinite b e a m
m o d el that w e im p le m e n te d a n d n u m e ric a lly a n a ly z e d is c o m p ared to
the tw o -b e a m m o d e ls a n d is u s e d to s t u d y MBR.
In section 4.3, w e
p r o p o s e a c a lib ra tio n te c h n iq u e b a s e d o n th e in fin ite b e am m o d e l
nu m e ric al an aly sis to correct for the in flu en c e of MBR on electro-optic
test results. Section 4.4 is a s tu d y of the in p u t a n d o u tp u t light intensity
v e rsu s p h a se d e la y c u rv e first p re se n te d in C h a p te r 2 a n d illustrated in
46
Fig. (2-3).
T h e in flu en c e of MBR a n d c h a n g e s in the e le m e n ts of the
electro-optic p ro b in g in s tru m e n t optical tra in are co nsidered.
4.1 Multiple beam reflection theory
C o n s id e r the o rie n ta tio n of the p r o b e b e a m as d e sc rib e d in Figs.
(2-1) a n d (2-2) w h e r e the b e am is lin e a rly p o la riz e d in the y d ire c tio n
a n d p ro p a g a te s in the x direction a n d the y' a n d z' directions are ro ta te d
45 d e g re es fro m the y a n d z directions in th e yz plane. The electric field
vector of the b e a m m a y be d escribed by
(4-1)
w h e re co is th e a n g u la r freq u e n cy , k is the w a v e n u m b e r a n d A is a
com plex v e cto r g iv en by
A = Az'e iS:' z' +Av'C,<5v?y
(4-2)
w h e re z' a n d y ' are u n it vectors in the z' a n d y' directions respectively ,
8 z 'a n d 8y' a re p h a s e angles a n d A:>, A y = 1 / a /2 since the laser b e a m is
linearly p o la r iz e d a n d a s s u m e d to h a v e a u n ity m a g n itu d e in the y
d ire c tio n .
U s in g the sim p lifie d d e sc rip tio n (non-M BR) of the p h a s e d e la y
in the G a A s fro m C h a p te r 2, the laser b e a m e n ters a n d p asses th ro u g h
47
th e G a A s so th at th e p h a se d e la y T in tr o d u c e d b e tw e e n the z' a n d y'
p o la riz a tio n c o m p o n e n ts to be in s e rte d in Eq. (4-2) are
?г
_I
riV
w h e re V = ? fro m Eq. (2-10). C o n s id e r first ju st the z' c o m p o n e n t of
V j:
th e c o m p le x v e c to r A.
T h e im p lic it a s s u m p t i o n in the n o n -M B R
m o d e l is that if the electric field v ecto r of the in p u t b e a m is d e sc rib e d as
E (t tu ,
= Re[Az. e>((ot-tx+Sz' )]
(4. 4)
z in
w h e r e 5Z' = 0, th en the o u tp u t b e a m reflected from the g r o u n d p la n e of
the G aA s m a y be described by
г < л > -. out = * г ? ? ' in
(4 ?5 )
w h e r e R is the c o m plex reflection coefficient d e fin e d as
(4-6)
R = \R\ei(^
a n d \R\ a n d з a re th e m a g n itu d e a n d p h a s e respectively.
In the non-
M BR m o d e l, R = \ \ \ e i(t> w h e re p is th e c h a n g e in p h a s e c a u s e d b y
p a s s a g e th r o u g h th e G aA s so th a t
re w ritte n as
b =r/2.
H e n c e Eq. (4-5) m a y be
48
i
r
__
J(<u/-fcv+?)
-J2 f
out
for th e n o n -M B R m o d e l.
E ( X t) >
(4-7)
2
If M BR is in c lu d e d in the m o d e l, th e n
c h an g e s since the reflection coefficient of the m o del changes.
2 out
F o r th e M BR m o d e l, the re fle c tio n co efficient fo r f r o n t sid e
p ro b in g is
R
~
+ /'21f12r21г' ,2<& - ,'212h 2 / 21t?,3<^ +
r \ 2 ~
?
...+ (-
w h ere
1) /21
r i2 and ti2
t \ 2 r2 \ e
Q,
a
+???
a re th e F r e s n e l re fle c tio n
and
tra n sm issio n
coefficients respectively for the air / G a A s interface, a n d r 2 l a n d t21 are
the F re sn e l
coefficient for the G a A s / air interface.
By s u m m i n g the
series Eq. (4-8) m a y be re w ritte n as
R = l^ ~ T1 + ~r 2 T\e vf
(4 ' 9 )
w here
, 4xn h F
?=?
+I
..
(4-10)
is the p h a s e d e lay for o n e pass th r o u g h the GaAs, h is th e th ick ness of
the G aA s sam p le, a n d X is the w a v e le n g th of the light.
If th e G a A s is a s s u m e d to be lossy d u e to defects or s c a tte rin g ,
e q u a tio n (4-8) becom es [23]
49
' . M - \ ) ar2in-'tl2t2lBnein*+...
w h e r e B is the loss coefficient a n d (4-10) m ay be re w ritte n as
R = yn - W n B e ' *
12 l + r ^ B e ' *
(4 _1 2 )
4.2 Multiple beam reflection models
W ith an in d e x of re fra c tio n of G a A s of 3.6, the v a lu e s of the
F resn el reflection a n d tra n sm issio n coefficients are [24]
n 2 = --565
t'21 =.565
h2 =.4347
(4-13)
>21=1-56
The reflection coefficient h a s b e en a p p ro x im a te d b a s e d o n these
v a lu e s for the Fresnel reflection a n d tra n sm issio n coefficients as a twob e a m m o d e l, w h e re the o n ly reflection s c o n sid e re d a re the reflection
off th e front s u rfa c e of the G a A s a n d o n e reflection fr o m th e back
su rfa c e of sam ple, as
50
(4-14)
th u s
(4-15)
\R\ = (1 + Co.y0)1/2
w h e n the G aA s loss coefficient is B=.8 [23].
T h e justification for this
a p p ro x im a tio n is th a t th e se tw o reflections a re m u c h la rg e r th a n any
s u b s e q u e n t re fle c tio n s [23].
F ro m Eq. (4-15), th e tw o -b e a m m o d e l
d e s c rib e d b y Eq. (4-14) vio lates th e la w of c o n s e rv a tio n of e n e rg y for
so m e values of (|). To a v o id this, w e use the a p p ro x im a tio n
(4-16)
w here
(4-17)
T he a p p ro x im a tio n u s e d in e q u a tio n (4-14) h as b e e n u sed to s h o w the
effect of MBR on e le c tro -o p tic p ro b in g te st r e s u lts by d e r iv in g the
c h a n g e in o u tp u t in te n s ity of the electro-optic p r o b in g in s tru m e n t as a
fu n c tio n of the a n g u la r p o s itio n of the a n a ly z e r in the in s tru m e n t [23].
F ro n t side p ro b in g w a s u s e d w ith a s y ste m g e o m e try sim ila r to that
s h o w n in Fig. (2-1). W e use, as a n a ltern ativ e, n u m e ric a l re su lts from
a n an aly sis b a s e d o n Jones C a lc u lu s [16], a te c h n iq u e d is c u s s e d in
51
A p p e n d ix
A.
U sin g
th is
approach,
w ith
th e
in p u t
in t e n s i t y
n o rm a liz e d , a lo n g w ith e q u a tio n (4-16) s h o w s s o m e of the c h ang es th a t
occur in elec tro -o p tic p ro b in g o u t p u t in te n s ity w h e n MBR is in c lu d e d
in th e e le c tro - o p tic p r o b i n g m o d e l.
Fig. (4-2) s h o w s th e o u t p u t
intensity v a ria tio n s d u e to the p re sen c e of a 1 v o lt sign al in a 450 p m
thick G aA s s a m p l e as a fu n c tio n of a n a ly z e r a n g le for the non-M B R
m o d e l.
W h e n the M BR m o d e l is u sed , th e g r a p h in Fig. (4-2) c h a n g e s
significantly. The c h a n g e in in ten sity as a fu n c tio n of a n aly zer ang le is
s h o w n in Fig (4-3). Both the a n g u la r location of the m a x im u m a n d the
m a g n itu d e o f th e c h a n g e in o u t p u t in te n s ity fro m the 1 v o lt sig nal
h av e shifted.
In Fig (4-4) a n d Fig (4-5) total o u t p u t in te n sity is s h o w n
for the non-M BR a n d MBR m o d e ls respectively. Figures (4-4) a n d (4-5)
sh o w clearly th a t the a v e ra g e o u t p u t light in te n s ity changes as well as
the a m p litu d e of th e in te n sity c h an g e a n d the a n g u la r locations of the
m a x im u m a n d m i n i m u m in tensities.
The a p p ro x im a te m o d e l of MBR is d e p e n d e n t o n the p h a se d elay in the
G aA s w h e n n o m ic ro w a v e signal is p re s e n t as s h o w n by Eq. (4-10) as
well as the p h a s e d e la y in tro d u c e d by the m ic ro w a v e signal V.
F rom
Eq. (4-10), w e k n o w th a t the o u t p u t in te n s ity is d e p e n d e n t o n the
thickness of the G aA s substrate.
Fig. (4-6) illustrates the chang e in light
intensity fro m v a ria tio n s in su b stra te thickness fro m 448pm to 451pm .
G aA s thickness v a ria tio n s also c han ge th e lo c atio n of the m a x im u m in
Fig. (4-5). A c h a n g e in G aA s thickness to 449pm c h an g e s the results to
those s h o w n in Fig. (4-7).
52
n o ob
- o .0001
Fig. 4-2 P o la r g r a p h of th e n u m e r ic a l a n a ly sis of the
v a r ia tio n of o u t p u t in te n s ity as a fu n c tio n of a n a ly z e r
p o s itio n th r o u g h a ro ta tio n of 360 d e g re e s for th e no n M BR m o d e l.
53
0001
0 . 00001
- Of . 0 0 0 0 1
Fig. 4-3 Polar g ra p h of the n u m e ric a l analysis of v a ria tio n
o u t p u t in te n s ity as a f u n c t i o n of a n a ly z e r p o s i t i o n
th ro u g h an a n aly z e r r o ta tio n of 360 d eg rees for the M BR
m o d e l.
0 . boo05
50
2 00
00
2 50 \
300
3
O
Degrees
Fig. 4-4 L in ear g r a p h of the n u m e ric a l a n a ly s is of total
o u t p u t in te n s ity
as
a f u n c tio n
of a n a l y z e r
p o sitio n
th ro u g h an a n a ly z e r r o ta tio n of 360 d e g re e s for the n onMBR m odel.
55
11
4-J
? ╗?H
C/5
C
0)
-▒_t
?╗
?H
4-*
rC
bD
a,
3
O
3 b()
bo
11
4-╗
Degrees
Fig. 4-5 L in e a r g r a p h of the n u m e ric a l a n a ly s is of total
o u tp u t in te n sity
as a f u n c tio n
of a n a l y z e r p o s itio n
th ro u g h a n a n a ly z e r ro ta tio n of 360 d e g re e s for the MBR
m o d e l.
Substrate height ( m icrom eters)
Fig. 4-6 L igh t in ten sity as a function o f s u b stra te thickness
?&
??
*
G
01
4-╗
C
329
? fH
X
bJD
? fH
?' I
p
3
CL,
3
O
Degrees
Fig. 4-7 L in e a r g r a p h of the n u m e ric a l a n a ly s is of total
o u tp u t in te n sity
as a f u n c tio n of a n a l y z e r
p o s itio n
th ro u g h a n a n a ly z e r ro ta tio n of 360 d e g r e e s for the MBR
m odel w ith a 449 p m thick substrate.
57
The tw o -b e a m m o d e l d e m o n s tra te s th e influ en ce of MBR o n the
o u t p u t lig h t in te n s ity of a n e le c tro -o p tic p r o b in g in s tru m e n t.
MBR
c h a n g e s th e a v e r a g e o u t p u t lig h t in te n s ity , th e a m p l i t u d e of the
variatio ns in o u t p u t in te n s ity c au se d b y a m ic ro w a v e signal in the test
sam p le, a n d ro ta te s th e m a x im u m of th e o u t p u t in te n sity from the 45
d e g re e a n a ly z e r p o s itio n .
W e e x p a n d e d th e tw o -b e a m m o d e l to an
in fin ite b e a m m o d e l for a m o re c o m p le te a n a ly s is of MBR. T he
a p p r o x im a tio n u s e d for th e tw o -b e a m m o d e l reflection coefficient R
changes b o th th e m a g n itu d e a n d a r g u m e n t of R fro m the infinite b e am
m o d e l d e fin e d b y e q u a tio n (4-12). F ig u res (4-8), (4-9) a n d (4-10) s h o w a
c o m p a riso n of the m a g n itu d e a n d a r g u m e n t of R as well as the sq u a re
of the m a g n itu d e I R^ I . It is s h o w n in Fig. (4-8) that the m a g n itu d e of
R varies w id e ly as B is c h a n g e d a n d it can differ significantly fro m the
m o d e ls d e s c rib e d b y Eq. (4-14) a n d (4-16) th a t rely o n the a s s u m p tio n
that B = 0.8.
Fig. (4-10) s h o w s clearly th e vio la tio n of c o n se rv a tio n of
en erg y by the m o d e l d e sc rib e d by Eq. (4-14). T h e a p p ro x im a te m o dels
a re b a s e d o n th e in fin ite b e a m m o d e l w i t h B = 0.8.
H ow ever,
variation s in B also c h a n g e the angle of the m a x im u m intensity as well
as the a m p litu d e of the intensity c h a n g e as s h o w n in Figs. (4-11), (4-12),
a n d (4-13).
58
B = 1.0
B = 0.2
o
o
o
bO
2o0
2 b0
3 00
3bO
D egrees
Fig. 4-8 G ra p h of the a m p litu d e of the reflection coefficient
R for th e infin ite b e am m o d e l as the loss coefficient B is
v a rie d fro m 0.2 to 1 in in c re m en ts of 0.2 a n d for the twob e am m o d e ls w h e re the solid line A re p re s e n ts the m o d e l
fro m Eq. (4-15) a n d the solid line B re p re s e n ts the m o d e l
from Eq. (4-17).
59
A,B
o
л*?
*
c
a>
bO
00
?b0
300
3 b0
B
&
B = 1.0
<
B = 0.2
D egrees
Fig. 4-9 G r a p h of the a rg u m e n t of the reflection coefficient
R for the infin ite b e am m o d e l as th e loss coefficient B is
v a rie d from 0.2 to 1 in in c re m e n ts of 0.2 a n d for the tw ob e a m m o d e ls w h e re A r e p r e s e n ts th e m o d e l from Eq. (414) a n d B rep resents the m o d e l fro m Eq. (4-16).
60
г = 1.0
B = 0 .2
3 b()
Degrees
Fig. 4-10 G r a p h of th e m a g n i t u d e of th e r e fle c tio n
c o effic ien t R fo r the in fin ite b e a m m o d e l as th e lo ss
coefficient B is v a rie d fro m 0.2 to 1 in in c re m e n ts of 0.2
and
for
th e
tw o -b eam
m o d e ls
w here
s o lid
lin e
A
r e p r e s e n ts the m o d e l fr o m Eq. (4-14) a n d so lid lin e B
re p re se n ts the m o del fro m Eq. (4-16).
61
- o . oooi,-o. 0004-
Fig. 4-11 P o la r g r a p h of th e n u m e ric a l a n a ly s is of the
v a ria tio n of o u t p u t in te n s ity as a fu n c tio n of a n a ly z e r
p o sitio n th ro u g h a r o ta tio n of 360 d egrees for th e in fin ite
b e a m model w h e n B = 1.
62
o
Fig. 4-12 P o la r g r a p h of the n u m e ric a l a n a ly s is of the
v a ria tio n of o u t p u t in te n s ity as a fu n c tio n of a n a ly z e r
p o sitio n th ro u g h a ro ta tio n of 360 d e g re es for the infinite
b e a m m o d e l w h e n B = 0.8.
Fig. 4-13 P o la r g r a p h of the n u m e r ic a l a n a ly s is of the
v a ria tio n of o u t p u t in te n s ity as a f u n c tio n of a n a ly z e r
p o sitio n th r o u g h a ro ta tio n of 360 d e g re e s for the infinite
b eam m o d e l w h e n B = 0.6.
64
4.3 C alibration for m ultiple beam reflection
effects
U sin g th e infin ite b e am m odel, w e h a v e d e v e lo p e d a c alib ratio n
tech niqu e u s in g n u m e ric a l analysis.
T o c a lib ra te test results w ith th is
m odel, B, the loss coefficient, m u s t be d e te r m in e d . This d e te r m in a tio n
m a y com e fro m th eo retical results or b y te stin g [12]. U sing o u r in fin ite
b e am m o d e l a n d n u m e ric a l analysis, w e s u g g e s t a w a y of u s in g test
results to d e te r m in e B. Figure 4-14 s h o w s a g r a p h of average reflected
light in te n s ity re fe re n c e d to the in p u t lig h t in te n s ity for d ifferen t loss
coefficients as th e th ickn ess of the G aA s is v aried .
By d e te rm in in g th e
in p u t lig h t in te n s ity a n d m o n ito rin g th e re fle c ted light intensity as th e
laser p ro b e b e a m is s c a n n e d across a G a A s s a m p le , the v a ria tio n in
p e ak reflected in te n sity indicates the v a lu e of B.
A fter B is d e te r m in e d , the relativ e h e ig h t of a test sam ple can b e
d e te r m in e d b y m o n ito r in g the a v e ra g e lig h t in te n s ity received at th e
p h o to d e te c to r.
W ith the p re a m p lifie r d e s c rib e d in C h a p te r 3, th is is
easily a c c o m p lis h e d as d e sc rib e d in se c tio n 3.1 w ith o u t c h an g in g th e
optical a lig n m e n t of the in s tru m e n t or d ire c tin g light out of the s y s te m
to an optical p o w e r m eter. There is no loss of light intensity in o rd e r to
m o n ito r th e a v e r a g e lig h t in te n s ity a n d th u s th e re is n o lo ss in
in s tr u m e n t s e n s itiv ity .
65
O n c e B is e s ta b lis h e d a n d the re la tiv e h e ig h t of th e s a m p le is
k n o w n , a set of c a lib ra tio n c u rv e s th a t are re fe re n ce d to th e s a m p le
th ick ness can b e calculated.
Fig. (4-15) s h o w s su ch a c u rv e for B = 0.6
a n d a s u b stra te thickness of 450 pm . For a p a rtic u la r h eig ht, the c h an ge
in o u t p u t in te n s ity a m p litu d e c a n b e d ire c tly re la te d to th e v o lta g e
level a t the p r o b e d point.
F rom the results s h o w n in section 4.2, a n d n o te d by Li et a l.[18],
ro ta tin g the a n a ly z e r in the s y s te m fr o m 45 d e g re es can in c re a s e the
o u t p u t in te n s ity .
W e can ta k e a d v a n t a g e of MBR to e n h a n c e the
s e n s itiv ity of th e test r e s u lts b y i n c l u d i n g c a lib ra tio n c u r v e s for
a n a ly z e r p ositio n s oth e r th a n 45 d e g re es. A n e x a m p le of this is s h o w n
in Fig. (4-16) w h e r e the a n a ly z e r fro m the e x am p le in Fig. (4-15) has
been ro ta te d b y 20 d eg rees th e re b y d o u b lin g the m a g n itu d e of the test
re su lts.
T est re su lts m a y also be e n h a n c e d b y testin g w ith a tu n a b le
laser th a t is tu n e d to the w a v e l e n g t h th a t g iv e s m a x i m u m o u t p u t
intensity [13], b u t this m e th o d re q u ire s a tu n ab le laser a n d a p p e a r s to be
less c o n v e n ie n t th en sim p ly ro ta tin g the analyzer.
A n o th e r po ssib ility
for o b ta in in g the p e ak in te n sity v a lu e s w h e n MBR creates a shift in the
a n g u la r location of the m a x im u m , is to in tro d u c e a h alf w a v e p la te in
the o p tic a l train before the G aA s sam p le.
T he half w a v e p la te rotates
th e p o h u iz a t io n of the in p u t sig n al to the G a A s s a m p le a n d shifts the
locatio ns of the in te n sity p eak s.
W ith a half w a v e p late a t 45 d e g re es,
the o u t p u t in Fig (4-5) shifts to th a t s h o w n in Fig. (4-17). R esults w ith a
tu n a b le la se r u s e d for h ig h fr e q u e n c y C W p ro b in g s u g g e s t th a t this
c alib ratio n te c h n iq u e c o u ld be u s e d
w ith a C W e lectro -o ptic p ro b in g
in s tr u m e n t for testing in the GFiz range.
66
AA9 . ?
AA9 . A
AA9 . b
449 . 8
Sub strate height (m icrom eters)
Fig. 4-14 R eflected lig h t in te n sity / I n p u t lig h t in te n sity
v e rsu s G aA s s a m p le thickness as B c h an g e s fro m 0.2 to 1
in inc re m en ts of 0.2
67
&
?D5
G
OJ
0
0
O,
G
0 03
' B = 0.6
H e ig h t = 450
oo
0 0 02
00
n3
?rH
>
0 00 1
-6
l/i
G
QJ
-U-*
G
M
M-h
o
0)
T3
G
-i- i
V oltage
"a,
6
<
Fig. 4-15 A m p litu d e of the in tensity c h a n g e for changes in
v o lta g e fro m 0 to 1 Volts w h e n B = 0.6 a n d the s am p le
thickness is 450 gm.
68
?H
Vi
c
a)
3
p-
c
0 . 0 00 0 6
o
<ti
'C
od
0 . 0000b
i?t
- f-H
>
B = 0.6
(J. 000 04
H e ig h t = 450
0 . 0 00 03
-r**4
Vi
G
0)
0 . 00002
0 . 00001
o
d)
0
0-4
a
<
0
.
2
0 .4
o .b
0 .8
Voltage
Fig. 4-16 A m p litu d e of the in te n sity ch ange for c h ang es in
v o lta g e fro m 0 to 1 V olts w h e n B = 0.6, th e s a m p l e
thickness is 450 p m a n d the a n aly z e r is ro tated 20 degrees.
Degrees
Fig. 4-17 L in ear g r a p h of the n u m e ric a l a n a ly s is of total
o u tp u t in te n sity
as a f u n c tio n of a n a l y z e r
p o s itio n
th ro u g h an a n a ly z e r ro ta tio n of 360 d e g re e s for the MBR
m odel w ith a half w a v e p late at 45 degrees
70
4.4 O utput light intensity and phase delay
It has been s h o w n [23] that ro ta tin g the a n a ly z e r enables the p e a k
in te n sity to b e o b ta in e d .
If th e a n a ly z e r is ro ta te d in o rd e r to o b ta in
p e ak in ten sity v alu es, the l o u t / f i n c urve s h o w n in Fig. (2-3) c h ang es to
the c u rv e s s h o w n in Figs. (4-18) a n d (4-19) w h e r e th e c u rv e s are
d e p e n d e n t o n the a n a ly z e r a n g le as the a n a ly z e r is p o sitio n e d from 0 to
45 degrees a n d 45 to 90 d e g re es respectively in 5 d e g re e increm ents.
If the a n a ly z e r w a s n o t ro ta te d a n d re m a in e d at 45 d egrees, a n d
MBR is co n sid e re d , th e re la tio n sh ip th a t w a s d e sc rib e d in C h a p te r 2 for
l o u t / I i n vs. p h a s e d e la y in the G aA s is th a t s h o w n in Fig. (4-20). The
analytic exp re ssio n for this c u rv e is
(4-18)
as s h o w n in A p p e n d i x B.
R ecall th a t th e no n -M B R c u rv e fro m
C h ap ter 2 is describ ed by
?
in
foul ! I in ~ s *n
2b
?
2
(4-19)
If the analy zer is n o t ro ta te d from 45 d e g re es, lo u t d e p e n d s only on the
p h a se d elay T a n d c h an g e s b y a co n stan t v a lu e irreg a rd le ss of su b stra te
thickness.
71
m o .?
C
a>
o
d
Oh
0 o.i
0
? P? H
m
d
QJ
0 . I)
d
4O-H
h
d
o
P h ase d e lay (radians)
Fig. 4-18 I o u t / I i n vs. p h a s e d e la y for a n aly z e r p osition s 0
to 45 deg rees in 5 d e g re e in crem ents.
j ?*
? hH
in
C
0╗
o
0
2
.
d
O
h
O
0
m
d
0)
0.0
d
O
H-H
0
h
d
O
Phase d e lay (radians)
Fig. 4-19 I o u t / I i n vs. p h a s e d elay for a n aly z e r p osition s 45
to 90 deg rees in 5 d e g re e increm ents.
72
F ig u re (4-21) is a n e x a m p le w ith the a n a ly z e r at 35 d e g re e s for several
c h a n g e s in s u b stra te thickness. T h e se c u rv e s are sensitive to v a ria tio n s
in th e s u b s tr a te thickness.
T h e fact th a t the MBR m o d e l w ith the
a n a ly z e r a t 45 degrees is in s e n s itiv e to c h an g e s in s u b s tr a te thickness
s u g g e s ts th a t if s o m e th in g o th e r th a n a q u a rte r w a v e p la te is u s e d to
bias th e sy stem , variations in s u b s tr a te thickness could be ig n o re d .
T h e in se rtio n of a half w a v e p la te into the o p tic a l tra in , th a t
w o u l d b e ro ta te d to get a p e a k in te n s ity re s p o n se as d e sc rib e d in section
4.3, also m o d ifies the non-M B R c u rv e th a t describes I o u t / I i n vs. p h a se
d e la y as s h o w n in Fig (4-22) w h e r e the a n g le of the h a lf w a v e p late
v aries fro m 0 to 22.5 d eg rees a n d M BR effects are ignored.
T he c u rves also c h a n g e for th e infin ite (MBR) b e a m m o d e l as B
is v a rie d . Fig. (4-23) sh o w s the v a ria tio n of IQu t / l i n vs. p h a s e d elay in
the G a A s as B varies from 0.2 to 1 in in c re m en ts of 0.2.
T h e lo u t vs. p h a se d e la y c u rv e s can be v ie w ed fro m a d ifferen t
p e rs p e c tiv e if the q u a rte r w a v e p la te is in c lu d e d in the analysis. Recall
th a t th e o rig in a l I o u t/ I in v e rs u s p h a s e d e la y curve in C h a p te r 2 is not
c a lc u la te d w ith a qu a rte r w a v e p la te in the system . The q u a r te r w a v e
p la te is i n t e n d e d to bias th e s y s te m so th a t linear c h a n g e s in the
m ic r o w a v e s ig n al p re s e n t in th e G a A s p r o d u c e a lin e a r c h a n g e in
o u t p u t in tensity .
If the a n a ly z e r is at 45 d e g re es a n d a q u a r te r w a v e
p la te is b ia sin g the system , th e n Figs (4-24) a n d (4-25) s h o w the nonMBR a n d MBR results resp ec tiv e ly as the p h a se delay v aries fro m 0 for
b o th p o sitiv e a n d negative p h a s e delays.
If th e a p p ro x im a te m o d e l for M BR is u s e d a n d th e a n a ly z e r
a n g le is v a rie d , the c u rv e s c h a n g e a lo n g w ith the a n a ly z e r ang le.
73
Figures (4-26) a n d (4-27) s h o w the results as the a n aly zer a n g le is v aried
from 0 to 45 d eg rees a n d 45 to 90 d eg rees respectively in in c re m en ts of 5
degrees.
The an alysis of the lig h t in ten sity curves s h o w that electro-optic
p r o b i n g in s tr u m e n ts a re s e n s itiv e to v a ria tio n s in h e ig h t w h e n the
a n a ly z e r is r o ta te d fr o m
th e 45 d e g r e e p o s itio n .
A lso , c h a n g e s
i n t r o d u c e d b y in s e r tin g a q u a r t e r w a v e p la te o r h a lf w a v e p la te
in flu en c e the light in te n s ity c u rv e s as the half w a v e p la te or a n a ly z e r
are rotated.
74
I/)
0 .2
c
a╗
0
3
Pu
G
0 . 1
0
in
G
0╗
0 .0
G
4a,
-1
G
o
Ph ase d elay (radians)
Fig. 4-20 Io u t/Iin vs. phase delay for analyzer position at 45 degrees
0
.
2
0.1
I) . o
3
O
Phase delay (radians)
Fig. 4-21 I o u t / I i n vs. p h a se d elay for a n a ly z e r p osition ed at
35 d e g r e e s
w ith
su b strate
th ic k n e s s
v a ria tio n s
th a t
p ro d u c e p h a s e delays of 0 to 45 d eg rees in in c re m e n ts of 15
degrees.
75
0.0
*rH
CA
г
<?
u'
?+
-S 0
-*?
1
2
CL.
гI 0
?
?
5.0
10.0
?15.0
20.0
?22.5
- rH
C
<D 0
0
3
O
P h ase d e lay (radians)
Fig. 4-22 I o u t / I i n vs. p h ase d e la y for a n aly zer positioned at
45 d e g re e s w ith a half w a v e p la te ro ta te d from 0 to 22.5
degrees.
0.b
1
1.b
2
2 .b
3
Ph ase delay (radians)
Fig. 4-23 I o u t / I i n vs. p h a s e d e la y for th e in fin ite b e a m
m o d e l w ith the a n a ly z e r p o s itio n e d at 45 d e g re e s a n d B
v a rie d from 0.2 to 1 in 0.2 increm ents.
77
>>
1/3
г
<u
P h ase delay (radians)
Fig. 4-24 I o u t / I i n vs. p h a s e delay for th e n o n-M B R m od el
w ith the a n a ly z e r p o s itio n e d at 45 d e g r e e s a n d w ith a
q uarter w a v e p la te a t 45 degrees.
78
e
3
O
Phase d e lay (radians)
Fig. 4-25 Io u t/Iin vs. phase d elay for the MBR m odel w ith
the analyzer positioned at 45 d eg rees and w ith a quarter
w ave plate at 45 degrees.
79
\
>N
4-╗
3
a,
c
Phase d elay (radians)
Fig. 4-26 I o u t / I i n vs. p h a s e d e lay for the MBR m o d e l w ith
th e a n a ly z e r r o ta te d fro m 0 to 45 d e g re e s in 5 d e g re e
increm en ts a n d w ith a q u a rte r w a v e plate at 45 d eg rees
80
Phase delay (radians)
Fig. 4-27 I o u t / I i n vs. p h a s e d elay for the M BR m o d e l w ith
the a n a ly z e r r o t a t e d fro m 45 to 90 d e g re e s in 5 d e g re e
increm ents a n d w ith a q u a rte r w ave plate a t 45 degrees
81
CHAPTER 5
INTENSITY DEPENDENT EFFECTS
G aAs
sam p les
te s te d w i t h
th e
CW
e le c tr o - o p tic
p ro b in g
in s tru m e n t e x h ib ite d results that s h o w e d a d e p e n d e n c e of the in te n s ity
of the reflected ligh t from the sam p les o n the laser intensity s u p p lie d to
the electro-optic p ro b in g in s tru m e n t.
T h e d e p e n d e n c e of the re fle c ted
light in te n s ity o n in c re a s in g i n p u t lig h t in te n s ity d e v ia te d fro m th e
e x p e c te d lin e a r re la tio n s h ip .
T h is e ffe ct is n o t d is c u s s e d in th e
literature re la te d to electro-optic p ro b in g . In this chapter, e x p e rim e n ta l
d a ta e x h ib itin g a m o r e c o m p lic a te d d e p e n d e n c e of o u t p u t li g h t
in te n sity o n i n p u t lig h t in te n s ity a re s h o w n .
R esu lts are s h o w n for
G aA s te st s a m p l e s fro m tw o d if f e r e n t s u p p lie r s . W e s u g g e s t tw o
possible s o u rc e s of the "nonlinearity".
T h e first is flu c tu a tio n s in the
sp ec tru m
in
o f th e la s e r d io d e
in s tru m e n t.
used
th e e le c tr o - o p tic p r o b i n g
T h e se c o n d is local h e a tin g in th e G a A s c a u s e d by the
pro b e laser beam .
Figure (5-1) s h o w s the results fro m p ro b in g a G aAs s a m p le ta k en
from a w a fe r s u p p lie d by S u m ito m o Electric In d u s trie s Ltd. T h e total
light in te n sity w a s m o n ito r e d at the d e te c to r location s h o w n in Fig. (31) w ith a N e w p o r t optical p o w e r m eter. T h e o u t p u t of the electro-optic
82
in s tr u m e n t w a s m e a s u re d w ith a lock-in am plifier as s h o w n in Fig. (31) w ith o u t using the p re a m p lifie r. T he laser light in te n s ity s u p p lie d by
th e laser d io d e is m o n ito r e d b y the level of c u rre n t u s e d to d r iv e the
laser diod e. The d e v ia tio n fro m lin e a r b e h av io r of the G aA s s am p les is
fu r th e r illu stra te d by the c o n tra s t b e tw e e n the b e h a v io r w h e n th e laser
beam
e n te r s th e G a A s a n d
is re fle c te d off of th e b a c k s u rfa c e
m e ta liz a tio n a n d the linear b e h a v io r w h e n the b e a m is reflected off the
fr o n t su rfa ce m e ta l tra n s m is s io n lines o n the s a m p le in Fig. (5-2).
To
v e rify th a t the d e v iatio n fro m linear b e h av io r w a s p re s e n t th r o u g h o u t
th e w a fe r , a n o th e r G a A s c irc u it s a m p le fro m the s a m e S u m ito m o
w a f e r w a s tested a n d the o b s e r v e d "nonlinear" b e h a v io r is s h o w n in
Fig. (5-3).
T he G aA s s e m ic o n d u c to r s a m p le m a y be m o d e le d as a n etalon.
H je lm e a n d M ic k e ls o n [12] h a v e d e s c rib e d th e s a m p l e s u b s tr a te
a n a ly tic a lly to s h o w its c h a ra c te ris tic s as a n a s y m m e tr ic F a b ry -P e ro t
e talo n
th a t s h o w s
reso n an t
b e h a v i o r in o u t p u t
d e p e n d i n g o n s u b s tr a te th ic k n e s s .
l i g h t in t e n s i t y
O u r m o d e l of m u l t i p l e b e a m
reflections in a G aA s s u b s tr a te p re s e n te d in C h a p te r 4 s h o w s th a t the
o u t p u t light intensity is s en sitiv e to su b stra te thickness as illu stra te d by
Fig. (4-6). T h e re s o n a n t b e h a v io r of the light in te n sity reflected b y the
G a A s e ta lo n is d e p e n d e n t n o t o n ly on the thickness of the s u b stra te ,
b u t also o n the in d ex of re fra c tio n of the s u b strate m a te ria l [25] o r the
w a v e le n g th of the light e n te r in g the s u b stra te [14].
T h e g ra p h in Fig. (5-1) of total laser ligh t in te n s ity m e a s u r e d by
th e o p tic a l p o w e r m e te r a t th e o u t p u t of the e le c tro -o p tic p ro b in g
i n s t r u m e n t s h o w s a d e v ia t io n fro m lin e a rity in the re fle c te d lig ht
in te n s ity w ith r e s p e c t to in c re a s in g la s e r in te n s ity .
T h e lo c k -in
am p lifie r m e a s u r e m e n ts th a t are in te n d e d to m e a s u r e the v o lta g e at
the p ro b e d p o in t in the G a A s s u b stra te s h o w re s o n a n t b e h a v io r in Fig.
(5-1) th a t c o in c id e s w ith r a p i d in c re a s e s in th e s lo p e of the total
reflected op tical in te n sity w ith respect to th e i n p u t laser light intensity.
Fig ure (5-2) verifies that the b e h a v io r of the reflected light is c au se d by
reflection fro m th e G a A s s u b s tr a te a n d n o t re fle c tio n fro m a m irro r
like surface s u c h as gold. T h e cause of these re su lts is either a c h an g e
in the o ptical p r o b e laser w a v e le n g th w ith in c re a s in g in te n s ity or a
p hy sical c h arac te ristic of the G aA s or the p r o b e b e a m that m a k e s the
in d ex of refraction d e p e n d e n t o n light intensity.
As s h o w n in C h a p te r 4, a n d as e x p la in e d analytically b y H jelm e
a n d M ickelson [12] the effect of m u ltip le b e a m in terferen ce m a k e s the
re su lts of e lec tro -o p tic p r o b in g d e p e n d e n t o n the p h a s e d e la y in the
GaAs. The p h a s e d e la y is d e p e n d e n t n o t o n ly o n the
thickness of the
s u b stra te as m e n tio n e d earlier b u t also o n the w a v e le n g th of the p ro b e
beam . U sing Eq. (4-10) that is re p e ate d h e re as Eq. (5-1) to d escribe the
p h a s e d e lay in G a A s w ith a thickness h of 500 p m a n d a n in d e x of
refraction n of 3.6, s h o w s that variation s o n th e o r d e r of a fraction of a
m ic ro m e te r c a n c h a n g e the o u t p u t fro m its m a x i m u m v a lu e to its
m i n i m u m v a lu e .
84
5
c
o>
г
01
<
0/>
3
0>
6
i01
QJ
г
u
QJ
г
o
D?гu
-8 .0 0 e - 1
4
lp
-6 .0 0 e - 1
3
-4 .0 0 e - 1
2
lock-in
- 2 .00e-1
CX
o
amplifier measurement (uV)
1 .0 0 e+ 0
optical meter
lock-in
1
2 .7 1 e -2 0
20
30
40
50
60
70
Current supplied to laser (mA)
Fig. 5-1 R eflected lig h t in te n s ity m e a s u r e m e n t s v e rs u s
laser in p u t in ten sity
85
5
-h- T ran sm ission line
G aA s
c
a╗
g
a╗
0in
aJ
a╗
g
Vh
<U
-w
a>
g
>1
<u
г
0
O
1
4
3
2
h
1I
fS
U
4-i
? pH
O
O
h
1
20
30
40
50
60
C u rre n t s u p p lie d to laser (mA)
Fig. 5-2 C o n tr a s t b e tw e e n laser in te n s ity m e a s u r e d from
G aAs back a n d fro n t surface m etalizations.
70
86
0.18
(u V )
Lock -in
0.14-
-
0.8
-
0.6
0.1 2 0 . 10 -
0.08-
-0 .4
Lock-in
(X
m easu rem en ts
0.16-
O p tical m eter
am plifier
-a -
0.06a.
0.04
0.2
20
30
40
50
60
70
C urrent s u p p lie d to laser (m A )
Fig. (5-3) F u r th e r reflected lig h t in te n s ity m e a s u r e m e n ts
v e rsu s laser in p u t in ten sity
87
This b e h a v io r h a s b e e n s h o w n by tu n in g th e p r o b in g laser to
d iffe re n t w a v e le n g th s [14].
W e tested the la se r p r o b e b e a m w ith a
B om en Inc. DA3.02 v a c u u m s p e c tro p h o to m e te r s y s te m to o b ta in the
spectral characteristics of the laser as laser in te n s ity w a s increased. The
re su lts are s h o w n in A p p e n d ix C.
T hey in d ic a te th a t th e re m a y b e a
slig h t shift as laser in te n s ity in creases w h ic h c o u ld b e in flu en c in g the
o u t p u t of the electro-optic p ro b in g in stru m e n t.
A n o th e r e x p la n a tio n for the d e v ia tio n fr o m lin e a r b e h a v io r in
reflected lig h t in te n s ity is a n in te ra ctio n b e tw e e n the laser b e a m a n d
th e G a A s s u b s tr a te th a t m a k e s the in d e x of re fra c tio n of th e G a A s
d e p e n d e n t o n ligh t in ten sity .
W e in v e stig ate d tw o possibilities ? the
p re s e n c e of d e fe cts in th e G a A s cry stal a n d th e r m a l effects.
Som e
a lte rn a tiv e s for th e b e h a v io r w e re e x p lo re d in c lu d in g : c h ec k in g the
G a A s s am p les for c o n ta m in a n ts that w o u ld r e s p o n d to la se r lig h t at
1300 nm , testing s a m p le s s u p p lie d by o th e r m a n u fa c tu r e r s , testing for
c h a n g e s in the laser s p e c tr u m w ith in c re a s e d lig h t in te n s ity o u tp u t,
a n d s tu d y in g th erm al effects w h e n the te m p e ra tu re of the G aA s sam p le
is changed.
W e te sted for d e fe cts in th e G aA s s u p p l i e d by S u m ito m o by
c h e c k in g for d e v i a t i o n s
in lin e a r b e h a v i o r
from
G aAs
s a m p le s
o b ta in e d from C rystal Specialties. Sam ples from C ry stal Specialties also
ex h ib ite d this b e h a v io r as s h o w n in Fig. (5-4). If th e re is a defect, it is
n o t u n iq u e to S u m ito m o w afers.
Tests w e re also c o n d u c te d w ith a s p e c tro m e te r for w afers from
th re e s u p p lie rs ? S u m ito m o , C ry s ta l S pe c ialties a n d C o m in c o .
A
s w e e p of the s p e c tro m e te r from 890 n m to 2000 n m s h o w s the expected
88
la rg e r e s p o n s e n e a r th e w a v e le n g th (890 n m ) w ith e n e r g y n e a r the
b a n d g a p of G aA s. T h e results at lo n g e r w a v e le n g th s s h o w the ab se n c e
of c o n ta m in a n ts a t a level th a t w o u l d c re a te a n o n lin e a r r e s p o n s e
b e lo w the b a n d g a p of G aA s w h e n the s a m p le s are p ro b e d by the 1300
n m laser be am . These tests results are in A p p e n d ix D.
3
>
g
c01
sa>
V-i
3c/i
a>
2
J-H
a>
Cl
1
g
uo
0
10
20
30
40
C u rren t s u p p lie d to la ser (m A )
Fig. 5-4 Reflected light in ten sity m e a s u r e m e n t v ersus laser
i n p u t in te n s ity for a G aA s s a m p l e s u p p lie d by C ry stal
Specialties.
T herm al
e ffe c ts
c an in f l u e n c e
th e in d e x
of r e f r a c tio n
of
s e m ic o n d u c to r m aterials [26, 27], Since th e p ro b e b e am is focused to a
89
s p o t size of less th a n 10 m ic ro m e te rs, the in te n s ity a t that p o in t can be
in th e ra n g e of h u n d r e d s of w a tts p e r m illim eter s q u a re d . T he c han ge
in in d e x of refraction of G a A s w ith te m p e ra tu re can b e e stim ate d from
th e re la tio n sh ip that relates the dielectric c o n s ta n t er to the b a n d g a p Eg
of the sem ico n d u c to r [28]
w h e r e a is a c o n sta n t.
T h e in d e x of re fra c tio n , n, is re la te d to the
dielectric c o n sta n t by
S u b stitu tin g Eq. (5-2) into Eq. (5-3) gives
(5-4)
T he ch an ge in b a n d g a p of a G aA s s e m ic o n d u c to r w ith increasing
te m p e ra tu re can be a p p ro x im a te d as a linear re la tio n s h ip [29] as sh o w n
in Eq. (5-5)
Eg = E g (0 ) - BT
(5-5)
w h e r e Eg(0) is the b a n d g a p at 0 K, T is the te m p e ra tu re in Kelvin, a n d B
= 3.3 x 10?4 eV /K . [29] U sing Eq. (5-5), Eq. (5-4) m a y be re w ritten as
90
n =
1+
a
-
(Eg ( 0 ) - B T ) 2
(5-6)
A g r a p h of this e q u a tio n is s h o w n in Fig. (5-5).
n
T e m p e ra tu re (K)
Fig. 5-5 In d e x of refraction in G aA s as a function of te m p e ra tu re
F ro m Eq. (5-1), for a 500 p m G a A s sub strate, changes in the in d ex
of refraction b y a few ten th o u s a n d th s can c h a n g e the p h a se d e la y b y n
ra d ia n s.
F r o m Fig. 5-5 it is clear th a t a n increase in te m p e ra tu re by a
few d e g re e s can d ra m a tic a lly c h a n g e the p h a s e d elay in G aA s.
For
exam p le, a 1 d e g re e change in the te m p e r a tu r e changes the p h a s e d elay
by n ra d ia n s T he inten sity of the laser can easily raise the te m p e ra tu re
at the s a m p le p oin t.
W e also s t u d ie d the effect of te m p e r a t u r e changes in th e b u lk
G aA s, b y u s in g a c artrid g e h e a te r to h e a t the G aA s s am p le test fixture.
T he la se r in te n s ity s u p p lie d to th e i n s t r u m e n t w a s in c r e a s e d a n d
91
m e a s u r e m e n ts w e re ta k e n w ith th e lock-in a m plifier. T h e test results
for te m p e ra tu re s at 21 a n d 43 d e g re e s C are s h o w n in F igu re (5-6). The
o u t p u t s h o w s a ra n g e of v a lu es for each d a ta p o in t tak en at a p articu lar
te m p e ra tu re .
As d a ta w as ta k e n for e ach c u rre n t p o in t at a p a rtic u la r
t e m p e r a t u r e , th e o u t p u t l ig h t in t e n s i t y r e c o r d e d b y th e lo c k -in
am p lifie r d id not give o ne s ta tio n a ry value, b u t flu c tu a te d b e tw e e n two
v a lu e s.
T h erefore, tw o lines a re g r a p h e d for each te m p e r a tu r e w h e re
the h ig h e st a n d lo w e st v a lu es are s h o w n for each s a m p le p o in t taken at
a p a rtic u la r te m p e ra tu re .
F ig ure (5-6) s h o w s th a t h e a tin g of the b u lk G aA s in fluen ces the
o u t p u t of the electro -o ptic s a m p lin g in s tr u m e n t as w o u ld b e e x p ected
fro m Eq. (5-6). These results s h o w th a t te m p e ra tu re c h an g e s in the bulk
G a A s s a m p le influence test re su lts as w ell as localized h e a tin g from the
la se r p ro b e beam . T h e test re su lts a t 43 d e g re e s C s h o w a w id e r range
b e tw e e n the h ig h e st a n d lo w e s t lock-in am p lifie r re su lts th a n the test
re s u lts at 21 d e g re es C. W e also o b s e rv e d th a t the v a ria tio n from the
h ig h e s t v a lu e to the lo w est v a lu e d took a b o u t 30 seco nd s at 21 degrees
C, b u t at 43 d e g re es C, lock-in am p lifie r o u t p u t re a d in g s c h a n g e d from
h ig h to low in a fraction of a second.
The c h ang e in electro-optic p ro b in g test resu lts as the b u lk G aAs
is h e a te d , in a d d itio n to local h e a tin g from the laser b e am , led us to test
for h e a tin g of the b u lk G aA s by the laser b e a m over long tim e periods.
W e te sted the refle c ted lig h t in te n s ity w ith the laser s ta tio n a r y at a
p a r tic u la r s a m p lin g p o in t a n d th u s s u p p lie d local h e a tin g w ith the
b u lk G aA s at ro o m te m p e ra tu re . T h e o u tp u t of the in s tr u m e n t v aried
s in u s o id a lly o ver time.
Fig. (5-7) a n d Fig. (5-8) s h o w the v a ria tio n in
92
o u t p u t w h e n th e la s e r is s ta tio n a r y a n d to ta l o p tic a l in te n s ity is
re c o rd e d w ith the N e w p o r t optical p o w e r m e te r a n d the e lectro-op tic
in s tr u m e n t o u t p u t is r e c o rd e d w ith the lock-in a m p lifier.
T h e se test
results s h o w a long te rm th erm al effect that o ccu rs o v e r a few h o u rs of
h e a tin g a n d m u s t b e c o n s id e re d alo n g w ith the th e rm a l c h an g e s th a t
in fluen ce the re s u lts of tests c o n d u c te d o v e r a s h o rt p e rio d of time.
T he lo n g te rm th e rm a l effects co u ld b e c a u s e d by a c h a n g e in p a th
length related to the th e rm a l coefficient of e x p a n s io n for GaAs.
The re la tio n s h ip b e tw e e n the reflected lig h t in te n sity a n d i n p u t
intensity s h o w s b e h a v io r that d eviates fro m a lin ear d e p e n d e n c e . This
"n o n lin earity "
in flu en c es
th e
o u tp u t
of
elec tro -o p tic
p ro b in g
in s tru m e n ts by c re a tin g re s o n a n t p e a k s in the o u t p u t v o lta g e results.
A c o m p ariso n w ith o th e r s u p p lie rs d id n o t reveal a n y defects that w e re
u n iq u e to th e S u m ito m o w afers.
T h e sh ift in laser w a v e le n g th can
change electro-optic p ro b in g results, b u t local h e atin g by the p ro b e laser
b e a m c re a te s t e m p e r a t u r e c h a n g e s th a t can c h a n g e th e in d e x of
refraction of G aA s d ra m a tic ally .
H e a tin g of the b u lk G aA s o v e r lo n g
p e rio d s of tim e also c h ang es test results. F u rth e r testing w ith the G aA s
in a h e a t s u n k fix tu re ta k en o v er lim ite d tim e p e rio d s is re q u ire d to
learn m o r e a b o u t th e in te n s ity d e p e n d e n t b e h a v io r of the G a A s test
sam ples.
93
3
" * ^ 2 1 degree G ran ge
2
^
43 d e g r e e C ran ge
1
Lock-in
am plifier
m ea su rem en ts
(m V )
4
0
10
20
30
40
C urrent su p p lie d to la ser (m A )
Fig. 5-6 R eflected lig h t in te n s ity m e a s u r e m e n ts v e r s u s
laser in p u t intensity at 21 d e g re e s a n d 43 degrees C.
94
0.43
г
6
c
01
6
0)
>-(
0.42"
лQJ
0.41"
|
0.40-
<D
г
o
O.
0.39-
0.38
0
10
20
30
40
Tim e (1 u n it = 5 m inutes)
Fig. 5-7 V a r ia tio n o v e r tim e of to ta l re fle c te d lig h t
in ten sity m e a s u r e d w ith the optical p o w e r m e te r
50
95
m e a s u r e m e n ts
(uV )
1 .4 ?
1.2- Q
i
??
i .o -
?
G EE Q
?
?
Q
?
Lock-in
am plifier
El
0.8 -
?
?
?
?
m
?
?
Q
0.6 -
Q
?
?
I
Q
EE1
? ??
0.4 ?
0
?i
ED
g
BBjB--------1-------1--------- 1-------- 1-------- 1? ? fpB------- 1-------
10
20
30
40
T i m e (1 u n i t = 5 m in u te s )
Fig. 5-8 V a ria tio n o v e r tim e of refle c ted lig h t in te n s ity
m e a s u re d w ith the lock-in am p lifier
50
96
CHAPTER 6
ELECTRO-OPTIC PROBING TEST RESULTS
In this c h a p te r e lectro-op tic p r o b i n g test re su lts are p re s e n te d .
The first p a r t of th e c h a p te r re p o rts o n te st re su lts fro m c o n tin u o u s
w a v e e lec tro -o p tic p r o b in g a n d in c lu d e s a d is c u s s io n of the effect of
m u ltip le
beam
reflectio n s
on
th e
te st
re su lts
fo r
m ic ro strip
tran sm issio n lines. The effect of m u ltip le reflections o n test re su lts for
so m e a d d itio n a l p a s s iv e test s tr u c tu r e s is s h o w n for th e first tim e.
O ther effects s u c h as the m etalizatio n th ic k n e ss of circuit test s tru c tu re s
that in flu en c e th e o u tc o m e of test re s u lts are also in tr o d u c e d for the
first time.
T h e s e c o n d p a r t of the c h a p te r d escrib es the e lec tro -o p tic
p ro b in g r e s u lts a c h ie v e d w ith the p u l s e d la se r sy ste m .
M u ltip le
reflection effects a re r e p o r te d o n a n d a c o m p a r is o n to the C W test
results is in c lu d e d .
6.1 CW electro-optic probing test results
The C W e lec tro -o p tic in s tr u m e n t d e s c rib e d in sectio n 3.1 w a s
u se d to m a p th e electric field in te n s ity vs. sp atial p o sitio n of s e v e ra l
97
G aA s s a m p le s s o m e of w h ic h w e r e s h o w n in Figs. (3-17) a n d (3-18).
Testing w a s c o n d u c te d in the K H z ra n g e w ith the sam p les m o u n t e d in
the test fixture s h o w n in Fig. (3-16)
In Fig ure (6-1) a field profile a r o u n d a typical tran sm issio n line is
illu strated .
T he d a ta profile is s h o w n a r o u n d the 50 Q tr a n s m is s io n
line of Figs. (3-16) a n d (3-17).
T h e field u n d e r the line c a n n o t b e
m e a s u r e d since th e tra n s m is sio n lin e m e ta l blocks the p a s s a g e of the
light probe.
T he d a ta s h o w s d ifferent p e a k m a g n itu d e s that are th e re s u lt of
m u ltip le b e a m in te rfe re n c e c a u s e d b y a c h a n g e in the h e ig h t of the
G aA s su b strate .
As d iscu ssed in C h a p t e r s 4 a n d 5, the o u t p u t of the
e lec tro -o p tic p r o b in g in s tr u m e n t is s e n s itiv e to s u b s tr a te th ic k n e ss.
C h an g e s in the s u b s tr a te thickness of a b o u t 0.2 m icro ns can c h a n g e the
o u t p u t of th e i n s t r u m e n t fro m a m a x i m u m to a m in im u m .
The
e lectric fie ld p ro f ile s h o w n in Fig. (6-1) s h o w s a c h a n g e in p e a k
m a g n itu d e s th a t is c o m m e n s u ra te w ith th e thickness v a ria tio n s of the
S u m i to m o G a A s
te st w a fers.
A n o t h e r fie ld p ro f ile of a G a A s
tran sm issio n line is s h o w n in Fig. (6-2), w h e r e the ra p id d e crea se s a n d
in c re ase s in the electric field p ro file m a y be d u e to m u ltip le b e a m
reflections. The d a ta in Fig. (6-2) w a s tak en b y a d justing the in te n sity of
the laser b e a m slig h tly at each d a ta p o in t to e n s u re that d a ta w a s n o t
taken at a re s o n a n t p e a k as d iscu ssed in C h a p te r 5.
Figure (6-3) de tails the field profiles a t the tip of a s tu b capacitor.
T he field s tr e n g th in the vicin ity of th e c o rn e r p o in ts (c u rv e A) are
stro n g e r for a n e q u iv a le n t d ista n ce fro m the s tu b m e ta liz a tio n th a n a
sim ilar profile in the center of the s tu b (cu rv e B). The d ifferin g p e a k
98
v a lu es in c u rv e B a re a n o th e r e x a m p le of th e in fluen ce of m u l t i p l e
b e a m re fle c tio n s in th e G aA s s u b s tr a te .
A trace p a s t th e s t u b e n d
p a ra llel to th e tr a n s m is s io n line (c u rv e C) a g a in illu stra te s th a t the
field intensity a t th e c o rn e r p oin ts is g re a te r o n a relative basis t h a n at
the stu b center as w o u ld be expected.
Fig. (6-4) s h o w s the profiles ta k e n for h o riz o n ta l scans a r o u n d a
s tu b capacitor.
C u r v e A illu stra te s th e re la tiv e ly s tro n g e r fie ld th a t
exists closer to the tra n sm issio n line.
Fig. (6-5) is the electro-optical r e s p o n s e c u rv e for an in te rd ig ita l
capacitor.
T h e la se r p r o b in g re s u lts in d ic a te th a t the re la tiv e field
intensity is m a x im u m in the g ap s b e tw e e n the m etalization.
D a ta w a s
taken w ith the left s id e of the cap a c ito r a c tiv a te d w h ile the r ig h t side
w as te rm in a te d in a 50 Q load, a n d th e n w ith the set-up re v e rse d .
In
so m e g ap s the a c tiv a tio n of the left s id e h a d m o re influence w h ile in
o th er gaps the rig h t sid e h a d m ore influence. F u rth er s tu d y of the g a p s
w as c o n d u c te d b y s c a n n in g each g a p vertically a n d by taking tw o d a ta
p o in ts ? o n e w ith th e left side a c tiv a te d a n d o n e w ith the rig h t s id e
a ctiv ated w ith th e o p p o s ite elec tro d e te r m in a te d in a 50 Q lo a d ? at
each p h y sica l lo c a tio n scan ned.
T h e re s u lts s h o w n in Fig. (6-6) a re
p re sen te d w ith the left side activated d a ta s u b tra c te d from the rig h t side
activated data.
It is clear th a t the s tro n g e st in flu e n c e n e a r the b e g in n in g a n d
e n d of each scan is d e te r m in e d b y the p h y s ic a l p ro x im ity of th e s id e
activated.
H o w e v e r, the influence as the sca n p ro c e ed s into the c e n te r
of each s w e e p is d e te r m in e d o n an a lte r n a tin g b asis b e tw e e n th e left
a n d right activ ation test results. In o u r o p in io n , this b e h av io r is d u e to
99
th e in flu en c e of th e m e ta liz a tio n th ic k n e ss o r th e e d g e s h a p e of the
m e ta l s tru c tu re s .
T h e s e n s itiv ity to th e e d g e m e ta liz a tio n p a t t e r n
s h o w s a n e w fa c to r th a t m u s t b e c o n s id e re d w h e n p ro b in g n e a r the
e d g e of the circuit s tru c tu re s.
The g e o m e tric re la tio n sh ip b e tw e e n the
"foot print" of th e la se r p r o b e b e a m a n d c irc u it s tru c tu re s h as b e e n
co n sid e re d [12], b u t the influence of the m e ta l d e p o sitio n for circuits o n
G a A s has n o t b e e n a d d re s s e d .
b e y o n d the sco p e of o u r studies.
F u r th e r s tu d y of the m e ta liza tio n w a s
1.0
Relative intensity
0.8 '
0. 6 '
0.4'
0. 2 '
0.0
1
2
3
4
5
Position (m m )
Fig. 6-1 Electro-optic field in te n sity v e rs u s po sitio n in
vicinity of a 50 Q tra n s m is sio n line.
1.0
Relative
in te n sity
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
Position (m m )
Fig. 6-2 Electro-optic field intensity v e rs u s p o s itio n for a
tra n s m is sio n line.
1.5
o
o
o
rH
O
001
00i
I
O / O
V
O/
o / o
Aiisueju! aAjieieu
o
in
in
<N
in
o
o
in
o
o
o
o
00
o
Aijsuaiuj aA|iB|ay
й
Fig. 6-3 Electro-optic field intensity profiles versus position in the vicinity of a stub capacitor. Curve A
is the profile from the stub point at a 45 degree angle. Curve B is from the central portion of the stub
perpendicular to the stub end. Curve C is a probe trace parallel to the stub end.
102
vo
o
in
o
^3
o
cn
o
CO
й
vO
й
o
fN
o
E
E
лN
O
░o
o
Aiisudiu; 0A!;e|ay
Aijsuaiu; aAjieiay
Fig. 6-4 Electro-optic field intensity profiles versus position for a stub capacitor
103
104
502434
1 2
3
4
5
6
7
Relative intensity
A-A
left
right
0.00
0.25
0.50
0.75
1.00
1.25
mm
Fig. 6-5 E le c tro -o p tic fie ld
in te n sity
p o s itio n for a n in te rd ig ita l c a p a c ito r.
p ro f ile s
versus
C u r v e A is th e
r e s p o n s e w ith the left e n d ex cited w h ile C u rv e B h as th e
rig h t s id e excited.
In each case, the o p p o s ite elec tro d e is
te rm in a te d in a 50 Q load.
- Left intensity
(mV)
105
g a p 1 rt - 1ft
g a p 2 rt - 1ft
g a p 9 rt - 1ft
g a p 5 rt - 1ft
g a p 8 rt - 1ft
Right intensity
g a p 7 rt - 1ft
g a p 6 rt - 1ft
g a p 4 rt - 1ft
g a p 3 rt - 1ft
Fig. 6-6 Left sid e a c tiv a te d e le c tro -o p tic field in te n s ity
s u b tr a c te d fro m rig h t s id e a c tiv a te d e le c tro -o p tic field
in te n sity v e rsu s p o sitio n for the g a p s n u m b e re d as in Fig.
(6-5).
106
6.2 Pulsed laser electro-optic probing test results
Electric field intensity profiles of the s am p le sh o w n in Fig. (3-16)
w e re also ta k e n w ith the p u ls e d laser electro-optic p ro b in g in s tru m e n t.
A 1.00021 G H z s ig n al w as s u p p lie d to th e s a m p le b y an H P 8341B
m ic r o w a v e s y n th e s iz e r .
T h e la s e r w a s b ia s e d w i t h a 36.1 m A
(thresh old is 25 m A ) c u rre n t fro m a DC c u rre n t so urce a n d 30.69 d b m
of RF p o w e r a t 1 G h z w a s s u p p lie d by a m ic ro w a v e p o w e r am plifier.
T he d e te c tio n fr e q u e n c y w a s 2.1 K H z.
F ig u re (6-7) s h o w s the p a th
sca n n e d b y th e p u ls e d electro-optic p r o b in g in stru m e n t.
Electric field
intensity p ro file s are s h o w n in Figs. (6-8) a n d (6-9) for a sig n al o n the
w id e r 50 Q
t r a n s m is s io n lin e a n d
t h e n o n the n a r r o w e r 75 Q
tra n sm issio n line re sp ectiv ely for the 12 m m scan sh o w n in Fig. (6-7).
In b o th electric field profiles, n o s ig n al is d e te c te d as the la se r p ro b e
scans across th e m e ta l tran sm issio n line.
r50L
75 il
12 m m
Fig. 6-7 P u ls e d e le c tro -o p tic p r o b i n g scan a cro ss tw o
tra n s m is s io n
lines
107
1.0
Relative
In te n sity
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
Position (mm)
Fig. 6-8 Electric field in tensity p rofile v e rsu s p osition w ith
the signal o n the 50 Q tra n sm issio n line.
12
108
1.0
0. 8 "
0. 6 "
a>
>
0 .4 -
0.2
-
0.0
0
2
4
6
8
10
12
Position (m m )
Fig. 6-9 Electric field in te n sity profile v ersu s position w ith
the signal o n the 75 г2 tra n s m is sio n line.
109
The electric field in te n sity profile in Fig. 6-8 s h o w s different p e ak
m a g n itu d e s on e ither s id e of the 50 Q tra n s m is sio n line from the effect
of m u ltip le b e a m re fle c tio n s as first s h o w n w ith th e C W in s tr u m e n t
test re su lts for this s a m p l e in Fig. (6-1).
T h e elec tric field in te n s ity
p ro file for the 75 Q. tr a n s m is s io n line a lso s h o w s th e in flu e n c e of
m u ltip le beam reflections o n test results.
T h e la rg e st p e a k m a g n itu d e
a lw a y s a p p e a rs o n the s a m e side of the tra n s m is sio n lines for b o th the
50 Q. a n d 75 ill tr a n s m is s io n w h ic h w o u l d b e e x p e c te d for a G a A s
s a m p l e w ith a c o n t i n u o u s c h a n g e in h e ig h t d u e to the th ic k n e s s
v a ria tio n of the G a A s w a fe r th at the s a m p le s u b s tr a te w a s diced from .
F ig u re 6-10 s h o w s a c o m p a r is o n of test re s u lts fro m the CW a n d the
p u ls e d electro-optic p r o b in g in s tru m e n t.
T h e a p p e a r a n c e of m u ltip le
reflection effects in p u ls e d electro-optic p ro b in g test re su lts u n d e rsco re s
th e u s e f u ln e s s of a lo w fr e q u e n c y C W i n s t r u m e n t for p r e d ic tin g
p ro b le m s that m a y o ccu r in h ig h frequency testing.
110
1.0
0.8 -
0.61
01
>
0.4-
0.2 -
P u lsed
гW
0.0
0
2
4
6
8
10
Position (m m )
Fig. 6-10 Electric field in te n s ity p ro files v e r s u s p o s itio n
w ith a signal o n the 50 Q tra n sm issio n line for the p u ls e d
p ro b in g in s tru m e n t a n d the C W p ro b in g in s tru m e n t.
12
Ill
CHAPTER 7
CONCLUSION
E x p e rim e n ta l m e a s u r e m e n t s of e lec tro -o p tic p r o b i n g re v e a le d
s u b s ta n tia l d e v iatio n fro m e x p e c te d resu lts b ased o n c u rre n t m odels. A
c o n tin u o u s w a v e e lec tro -o p tic p r o b in g in s tru m e n t w a s b u ilt to
m ap
electric field in te n sity p ro file s a n d to p ro v id e a b a s e lin e for s tu d y at
m i c r o w a v e f r e q u e n c ie s .
T e s tin g
of m ic r o w a v e
p a ssiv e
c irc u its
r e v e a le d m u ltip l e b e a m re fle c tio n effects a n d n o n - l i n e a r in te n s ity
d e p e n d e n t effects.
W e stu d ie d
th e s e p r o b le m s
to a d v a n c e the
te c h n iq u e of electro -op tic p r o b in g so th a t acc u ra te test re su lts can be
o b ta in e d .
A p u ls e d elec tro -o p tic p ro b in g in s tru m e n t w a s b uilt to test
s a m p le s a t G H z frequencies. Electric field in tensity profiles taken w ith
b o th in s tru m e n ts w e re p re s e n te d for several G aAs sam p les.
M u ltip le b e a m re fle c tio n s w e re m o d e le d n u m e ric a lly by u sin g
Jones calculus. Tw o d ifferen t m o d e ls for the reflection coefficient w ere
p r e s e n te d ? a tw o-beam m o d e l a n d a n infinite b e a m m o d e l. O u r twob e a m m o d e l d o es n o t v io la te c o n s e rv a tio n of e n e r g y in c o n tra st to
o th e r tw o -b e a m m o d e ls [23].
The effect of m u ltip le b e a m reflections
w a s s h o w n n u m erically b y c o m p a r in g the results of a non-M B R m o del
to th o se of o u r tw o -b e a m M BR m o d e l.
MBR c h a n g e s the total light
112
intensity as w ell as the a m p litu d e of the c h an g in g light in te n sity fro m a
sin u so id a l sig n al p re s e n t o n a test s a m p le .
M u ltip le b e a m reflections
m a k e the o u t p u t in te n s ity s en sitiv e to c h an g e s in s u b s tr a te th ic k n e ss
w h e n the electro-optic sy stem is b ia se d w ith a q u a rte r w a v e plate. The
in fin ite b e a m
m odel
ta k e s in to
a c c o u n t a n in f i n ite n u m b e r of
reflections a n d the influence of a v a ria b le loss coefficient. R esults w e re
p r e s e n te d fo r th e infin ite b e a m m o d e l w ith several loss coefficients.
T he total lig h t in te n sity as w ell as th e a m p litu d e of th e c h a n g in g light
in ten sity fro m a sin u so id a l signal p r e s e n t o n a test s a m p le w e re s h o w n
to also be in flu en ced by a c h an g in g loss coefficient.
W e s u g g e s t n e w c a lib ra tio n te c h n iq u e s b a s e d o n th e in fin ite
b e a m m o d e l th a t take in to a c c o u n t v a ria tio n s in loss co efficient a n d
s u b s t r a t e th ic k n e s s
to a c c o u n t fo r th e effects of m u l t i p l e b e a m
reflections in test results.
T he te c h n iq u e s co u ld b e e x te n d e d to h ig h
freq u e n cy C W p ro b in g for sy ste m s th a t in c o rp o ra te a fast p h o to d io d e
detector.
To e n h a n c e th e s e n s itiv ity of th e c a lib ra tio n te c h n iq u e , tw o
m e th o d s w e re su gg ested . O n e w a s to ro ta te the a n a ly z e r to g e t a p e ak
lig h t in te n s ity [23], a n d the o th e r w a s to ro ta te a h alf w a v e p la te
in s e rte d in to the o ptical train.
In o u r o p in io n , these tw o o p tio n s are
m u c h s im p le r than v a ry in g the w a v e le n g th of the p ro b e laser [14].
We also conducted a study of the changes in the Iout/Iin versus
phase delay curves when the analyzer is rotated from its 45 degree
position.
We show that relation creates a family of curves for Iout/Iin
versus phase delay that are dependent on analyzer position.
Numerical analysis shows that if the analyzer is not moved from its 45
113
degree position that the I o u t /I in
versus phase delay curve is
insensitive to changes in substrate thickness. An area of further study
would be to use the information from that curve to bias the system for
substrate thickness insensitive operation. The influence of the rotation
of a halfwave plate on the Iou t/Iin versus phase delay curve and a
changing loss coefficient were also studied.
A p o ssib le top ic for fu tu re s tu d y of th e influence of MBR w o u ld
b e to im p l e m e n t th e c a lib ra tio n m e t h o d s w e s u g g e s te d .
A n o th er
in te re stin g s tu d y w o u ld be to coat the s a m p le s w ith an anti-re fle c tio n
c o a tin g to lim it th e in flu e n c e of m u l t i p l e b e a m re fle c tio n s o n te st
results.
N o n lin e a r in te n s ity d e p e n d e n t effects w e re d isco v e re d w h e n the
in te n sity of the la se r b e a m i n tr o d u c e d in to the elec tro -o p tic p r o b in g
i n s t r u m e n t w a s v a r ie d .
This e ffect w a s d e m o n s t r a t e d for G a A s
s am p les fro m tw o d iffe re n t m a n u fa c tu r e r s .
T h e sa m p le s w e re te ste d
for c o n ta m in a n ts th a t w o u l d excite the G a A s a lth o u g h the p r o b e laser
b e a m w a v e le n g th of 1300 n m h a s a n e n e r g y b e lo w the b a n d g a p of
GaAs.
T h e tests w ith a s p e c tro m e te r w e re n e g a tiv e for this ty p e of
c o n ta m in a n t.
T h e laser w a s tested for c h a n g e s in its s p e c tr u m th a t
w o u ld o ccu r w ith in c re a s in g ligh t in te n s ity o u tp u t.
Tests s h o w e d a
slight shift in the s p e c tru m w ith in c re a s in g lig ht in tensity so the laser
itself m a y be the s o u rc e of the n o n lin e a rity [14]. F u rth e r s tu d y w o u ld
re q u ire c o n c u r re n t m e a s u r e m e n ts of the lig h t in te n sity reflected fro m
th e G a A s s a m p l e as th e s p e c t r u m
of th e la s e r w a s m o n i t o r e d .
H o w e v e r, in o u r s tu d ie s , the n o n lin e a r b e h a v io r is d o m i n a t e d b y
th erm al effects.
W e s h o w that c h a n g in g th e te m p e ra tu re of th e G a A s
114
b y o n ly a few d e g re es c h a n g e s the index of refractio n e n o u g h to h ave a
d ra m a tic effect o n th e o u t p u t of the electro-op tic p r o b in g in s tru m e n t.
This sug gests that th e m e a s u r e m e n t w o u ld b e n efit if the s a m p le w e re
m o u n t e d in a h e a t sin k .
L o n g te rm v a r ia tio n s o v e r 1 h o u r w e re
o b serv ed . The o rig in of this effect is not clear, b u t m a y b e related to a
c h a n g e in the p a th le n g th of th e sam p le d u e to th e th e rm a l coefficient
o f e x p a n s io n for G aA s.
C h o p p in g the p ro b e b e a m , u s in g sh o rt laser
p u ls e s , or u s in g a lo w in te n s ity laser w o u l d m itig a te s o m e of the
th e rm a l effects.
E lectro-optic p r o b i n g w a s u s e d to ta k e electric field in te n sity
p ro file s of sev e ra l s a m p l e s w i t h the CW s y s te m a n d m a k e relativ e
m e a s u r e m e n ts b e tw e e n c irc u it stru c tu re s.
T h e p re s e n c e of m u ltip le
b e a m in te rfe re n c e w a s n o t e d o n the e le c tric fie ld p ro files.
The
in flu en c e of m e ta liz a tio n o n th e electro-optic p r o b in g re su lts w a s also
p ro p o s e d .
A p u ls e d e le c tro -o p tic p ro b in g in s tr u m e n t w a s b u ilt for testing
c ircuits in the G H z ra n g e .
T h e p u ls e d laser w a s c h arac te riz ed w ith a
p h o t o d i o d e c u s to m m o u n t e d o n a h igh f r e q u e n c y test fix tu re a n d
te ste d w ith a d ig ita l s a m p l i n g oscilloscope.
Electric field in ten sity
profiles of tra n sm issio n lines d e m o n s tra te d test resu lts taken at 1 GHz.
A c o m p a r is o n of th e te sts fr o m the C W a n d p u l s e d in s tr u m e n ts
d e m o n s tra te s the u s e f u ln e s s of the C W in s tr u m e n t for e sta b lish in g a
b a se lin e for fu tu re s t u d y of b o th the low a n d h ig h freq u e n cy test setн
up s.
Electro-optic p r o b in g h a s been a m p ly d e m o n s t r a t e d as a useful
tool for c h arac te riz in g h ig h s p e e d circuits a n d devices.
Both external
115
a n d in te rn a l m e th o d s h a v e b e e n u s e d to m a k e a v a rie ty of tests. M u ch
of th e te s tin g h a s b e e n q u a l i t a t i v e as c a lib ra tio n is s u e s a re just
b e g in n in g to be a d d r e s s e d [12].
In te rn a l e le c tro -o p tic p r o b in g as a
re la tiv e ly lo w cost, s im p le m e t h o d of e v a lu a tin g th e e lectric fields
in t e r n a l to a circ u it is a p r o m i s i n g m e t h o d
th a t b e c o m e s m o r e
p o w e rfu l as it continues to b e m o re co m pletely characterized.
116
Appendix A
Jones calculus d e term in e s th e final po larization state of the p ro b e
b e a m a fter p a ss a g e th ro u g h a series of o p tic a l elem ents a n d h e n c e the
v ariatio n of the intensity of the p ro b e b e a m at the detector.
If w e a ssu m e th at the electro-optic p ro b e beam is d e sc rib e d by
Ac
i(o)t-tx)
(A -l)
w h e re [17]
A = Az<e'^z' z' +Av' e ^ y y'
(A-2)
a n d z' a n d y' are u n it vectors, th e n th e Jones vector r e p re s e n ta tio n
d escribing the polarization state of th e p la n e w a v e is
aV
5-'
iS y ?
(A-3)
A y<e
N o te th a t the use of (cot - kz) in d ic a te s th a t positive 5 re p re s e n ts a
p hase lead. The z' a n d y' axes are d e fin e d in C h ap ter 2.
117
For the optical tra in s h o w n in Fig. (2-1), a fter the initial passag e of
th e b e am th ro u g h th e p o la riz in g b e am s p litte r, the n o rm a liz e d Jones
vector is
Bi n
(A-4)
T h e p o larization s ta te of th e p ro b e b e a m for th e sim p le e x am p le of
p a ssin g th ro u g h a X /4 w a v e p late a n d the G a A s s a m p le a n d reflecting
back thro ug h the G aA s a n d a n analyzer is c alcu lated as
Bлut = p l - r ( 0 3 ) T ( - 0 2 ) - g - r ( 0 2)
(A-5)
w h e re the rotation m atrices r(0) are defined as
r(0) =
cos 8
sin 0
-sin 0
cos 6
(A-5)
the A,/4 w ave plate is re p re s e n te d by [17]
wl =
? 1 O'
0
i
(A-6)
the analyzer is re p re s e n te d by
1 O'
Pi
0
0
(A-6)
118
a n d th e G aA s test sam p le is d escrib ed by
id
0
0
.-id
(A-7)
w h e re d is the p h a se d e lay asso c iate d w ith each p o la riz a tio n c o m p o n e n t
of th e G aA s.
T h e v a lu e of g ( l , l ) a n d g(2,2) in the G a A s m a trix are
d e te rm in e d by the m o d el cho sen to re p re se n t the reflection coefficient for
GaAs.
A m athem atica n o teb o o k s h o w s an ex am p le of a n u m e ric al analysis
to d e te rm in e total o u tp u t light in te n sity b a se d o n Jones C alcu lu s for the
infinite b e am m odel.
лл>
119
M ATH EM A TICA NOTEBOOK
The input vector v and quarter waveplate are defined
v = {1,0}
t = -45 Degree //N
k = Pi / 4 //N
w4 = {{E x p [-1 k], 0}, {0, Exp[I k] } >
rtheta = {{Cos[t], Sin[t]}, {Sin[-t],
w4thp = w4 . rtheta
{1,
Cos[t]}}
0}
-0.785398
0.785398
{{0.707107
-
{{0.707107,
{{0.5
-
0.5
0.707107
I,
-0.707107},
I,
-0.5
0],
{0,
0.707107
{0.707107,
+ 0.5
I},
{0.5
+
0.707107
0.707107}}
+
0.5
I, 0.5
+
0.5
The phase delay u for a single pass through a 450 |im
thick GaAs sample with an index of refraction of
3.6.
Hgt = 450
Volt = f * amp * Sin[w]
Phase = Hgt * (3.3 * (10 A 7)) * (1 * (10 A -6))
Phmod = Volt / (1.78 * (10 A 3))
u = Phase + Phmod/2
450
a mp
f
Sin[w]
14850.
0.000561798
14850.
+
amp
f
Sin[w]
0.000280899
a mp
f
I}}
Sin[w]
The reflection coefficient mbi for the
infinite beam model
I}}
120
R1
R2
T1
T2
=
=
=
=
-.555
.555
.445
1.555
Cosp = Cos[u]
Sinp = Sinful
Ra = ((Tl*T2*B*(Cosp + I Sinp))/
(1 + (R2 * B*(Cosp + I Sinp))))
mbi[f_] = Rl - Ra
-0.555
0. 555
0. 445
1. 555
Cos[14850.
+ 0.000280899
a mp
f
Sinfw]]
Sin[14850.
+ 0.000280899
a mp
f
Sinfw ]]
(0.691975
I
(1
+
B
(Cos[14850.
S in [14850.
0.555
-0.555
B
+
I
S in [14850.
-
(0.691975
I
Sin[14850.
+
0.555
I
0.000280899
0.000280899
(Cos[14850.
+
B
+
a mp
(Cos[14850.
a mp
+
f
Sin[14850.
+
f
amp
Sinfw]]
+
f
Sinfw]]
Sinfw]]))
a mp
f
+
/
Sinfw]]
f
Sinfw]]))
f
Sinfw]]))
+
+
+
0.000280899
(Cos [1 4 8 5 0 .
amp
0.000280899
0.000280899
B
0.000280899
(1
+
amp
0.000280899
0.000280899
amp
f
a mp
f
/
Sinfw]]
+
Sinfw]]))
Calculation of the components of the GaAs matrix
a = Absfmbifl]]
PowerfE,
I
* Exp[I Argfmbifl]]]
Arg[-0.555
(Cos[14850.
I
(1
I
I
B
+
(0.691975
B
+
I
B
Sinfw]]
amp
+
+
Sinfw]]))
+ 0.000280899
a mp
+ 0.000280899
(C o s [ 148 5 0 .
S in [14850.
amp
0.000280899
0.000280899
S i n [14850.
+ 0.555
B
+ 0.000280899
(Cos[14850.
S in [14850.
-
(Cos[14850.
(1
(0.691975
0.000280899
S i n [14850.
+ 0.555
Abs[-0.555
+
a mp
amp
+
+
Sinfw]]))
0.000280899
+ 0.000280899
/
Sinfw]]
Sinfw ]]))]]
Sinfw]]
amp
a mp
a mp
/
Sinfw]]
Sinfw]]))]
+
121
b = Abs [mbit-1]] * Exp[I Arg[mbi[-1]]]
Power[E,
I
Arg[-0.555
(Cos[14850.
I
(1
+
I
B
I
B
-
B
-
-
Sin[w]]
amp
a mp
-
amp
amp
+
a mp
{{Costs],
{{1 ,
0 },
{ {C os[s],
S in fs]},
{0 ,
{-Sin [s],
{Sin[-s], Costs]}}
Costs]}}
0 }}
S i n f s ] },
{0,
0}}
Output Jones vector and the calculation of
the output intensity at the detector for a
loss coefficient B = 0.6 and a 1 Volt signal
/
Sin[w]]
Sinfw ]]))]
Analyzer definition
poltheta = {{Cosfs], Sin[s]},
pola = {{1,0}, {0,0}}
wholepol = pola . poltheta
/
Sin[w]]
Sin[w ]]))
0.000280899
0.000280899
a mp
S infw ]]))]]
Sin[w]]
amp
+
Sin[w ]]))
0.000280899
0.000280899
(Cos [ 1 4 8 5 0 .
S i n [14850.
a mp
0.000280899
0.000280899
S in [14850.
B
0.000280899
(0.691975
+ 0.555
I
-
(Cos[14850.
S i n [14850.
-
(Cos[14850.
(1
(0.691975
0.000280899
S in [14850.
0.555
Abs[-0.555
-
+?
+
122
bap[cunp_, w , B_] = wholepol .
{{a, 0}, (0, b}} . w4thp . v
{(0.5
-
0.5
I)
Power[E,
Arg[-0.555
-
(Cos[14850.
I
(1
+
I
(0.691975
+ 0.000280899
Sin[14850.
0.555
+
(Cos[14850.
B
Abs[-0.555
I
(1
+
+
(0.691975
B
-
+
0.555
(0.5
+
B
+ 0.5
I
+
Power[E,
(0.691975
-
B
-
I
S i n [14850.
-
(0.691975
(Cos[14850.
I
(1
+
0}
B
amp
-
a mp
amp
a mp
/
Sinfw]]
Sinfw ]]))]
+
C os[s]\
Sinfw]]
a mp
+
Sinfw]]))
/
+
0.000280899
a mp
0.000280899
-
+
Sinfw]]))
a mp
Sinfw ]]))]]
B
(Cos[14850.
Sin[14850.
Sinfw ]]))]]
0.000280899
Sinfw]]
0.000280899
S i n [14850.
0.555
I
,
-
a mp
Sinfw]]
0.000280899
-
/
B
(Cos[14850 .
0.000280899
Abs[-0.555
+
0.000280899
S i n f 14850.
+
Sinfw]]))
+
a mp
+ 0.000280899
-
0.555
amp
0.000280899
I)
(Cos[14850.
Sinfw]]
0.000280899
(Cos[14850.
Sinfl4850.
I Arg[-0.555
(1
+
Sinfw]]
a mp
+
0.000280899
S i n [14850.
I
amp
S i n [14850.
(Cos[14850.
amp
0.000280899
0.000280899
I
B
-
Sinfw]]
amp
Sinfw]]))
0.000280899
0.000280899
a mp
+
a mp
/
Sinfw]]
Sinfw ]]))]
+
S in [s]\
123
bone = b a p [1,90 Degree,.6]
{(0.5
-
0.5
I)
Power[E,
Arg[-0.555
-
(Cos[14850.
I
(1
+
I
(0.415185
+ 0.000280899
S i n [14850.
0.333
(Cos[14850.
0.000280899
I
Abs[-0.555
S i n [14850.
-
I
+
I
Sin[14850.
(0.5
Power[E,
I
+
0.5
I
S i n [14850.
I
I
+
-
-
S i n [s ],
0}
+
Sin[90
Degree]]
0.000280899
Degree]]))]
-
+
Sin[90
Degree]]))
/
-
Sin[90
-
Degree]]
0.000280899
S i n [14850.
Sin[90
0.000280899
(Cos [ 1 4 8 5 0 .
0.000280899
I
I
-
0.000280899
S i n [14850.
0.333
Degree]]))
+
Sin[90
D egree]]))]]
(0.415185
(Cos[14850.
(1
Degree]]
Sin[90
(Cos[14850.
S i n [14850.
-
+
-
0.000280899
Abs[-0.555
Sin[90
+ 0.000280899
(Cos[14850.
0.333
Degree]]
I)
0.000280899
+
D egree]]))]]
+
Sin[90
Arg[-0.555
(0.415185
(1
/
+
Sin[90
Sin[90
+ 0.000280899
(Cos[14850.
0.000280899
+
Degree]]
0.000280899
0.000280899
S in [14850.
+ 0.333
Cos[s]
+
Degree]]))
+
Sin[90
+
Degree]]
Sin[90
(0.415185
(Cos[14850.
(1
Sin[90
+ 0.000280899
Sin[90
+
Degree]]))
-
Sin[90
-
Degree]]
Degree]]
0.000280899
+
Sin[90
D egree]]))]
/
124
ex = Chop[bone[[1]]]
(0.5
-
0.5
I)
Power[E,
Arg[-0.555
-
(Cos[14850.
I
(1
I
(0.415185
+ 0.000280899
S i n [14850.
+ 0.333
+
0.000280899
(C os[14850.
0.000280899
I
Abs[-0.555
S in [14850.
-
(1
I
+
I
Sin[90
+
0.5
+
-
Sin[s]
Degree]]
+ 0.000280899
+
Degree]]))
Sin[90
Sin[90
-
0.000280899
0.000280899
(C os[14850.
S in [14850.
-
/
Degree]]
+
Degree]]))]
Degree]]
Degree]]))
+
/
-
Sin[90
-
Sin[90
Sin[90
Degree]]
0.000280899
+
Sin[90
Degree]]))]]
(0.415185
-
0.000280899
S in [ 14850.
+ 0.333
I
Degree]]))]]
-
(Cos[14850.
(Cos[14850.
I
+
Sin[90
0.000280899
0.000280899
(1
/
I)
S i n [14850.
I
Degree]]))
Sin[90
Sin[90
+ 0.000280899
Arg[-0.555
+ 0.333
Abs[-0.555
Degree]]
+ 0.000280899
(Cos[14850.
(0.415185
I
Sin[90
+
+
0.000280899
S i n [14850.
(0.5
Power[E,
(1
+
S i n [14850.
+ 0.333
Cos[s]
Degree]]
(0.415185
(Cos[14850.
I
Sin[90
-
0.000280899
(C os[14850.
S i n [14850.
Sin[90
-
-
Degree]]
Sin[90
0.000280899
0.000280899
+
Degree]]))
Sin[90
Sin[90
/
Degree]]
Degree]]))]
+
125
cex = Conjugate[ex]
Conjugate[(0.5
I
-
0.5
Arg[-0.555
I)
-
Power[E,
(0.415185
(Cos[14850.
I
(1
+
+ 0.000280899
S i n [14850.
0.333
+
0.000280899
(Cos[14850.
0.000280899
I
-
(1
+
0.333
(Cos[14850.
S i n [14850.
+
(0.5
Power[E,
I
+
(0.415185
+
0.5
(1
+
S i n [14850.
/
+
Sin[90
D egree]]))]
Degree]]
0.000280899
Degree]]))
/
-
Sin[90
-
+
Sin[90
Degree]]
0.000280899
+
Sin[90
D egree]]))]]
(0.415185
-
0.000280899
S i n [14850.
0.333
-
(Cos[14850.
S i n [14850.
Sin[90
0.000280899
0.000280899
I
+
Degree]]))
-
(Cos[14850.
(Cos[14850.
I
Degree]]
Sin[90
-
0.000280899
-
Degree]]
Sin[90
-
(Cos[14850.
0.333
Abs[-0.555
D egree]]))]]
I)
S i n [14850.
I
/
+
Sin[90
Sin[90
0.000280899
0.000280899
(1
Degree]]))
+
Sin[90
+
Arg[-0.555
I
0.000280899
+ 0.000280899
0.000280899
Cos[s]
Degree]]
+ 0.000280899
Sin[14850.
I
Sin[90
+
(0.415185
(Cos[14850.
I
+
Degree]]
+
Sin[90
S i n [14850.
Abs[-0.555
Sin[90
+
Degree]]))
/
-
Sin[90
-
Degree]]
Sin[90
Degree]]
0.000280899
+
Sin[90
D egree]]))]
S i n [ s ]]
i = ex * cex
Conjugate[(0.5
I
-
0.5
Arg[-0.555
I)
-
Power[E,
(0.415185
(Cos[14850.
I
(1
+
+ 0.000280899
S i n [14850.
0.333
+
(Cos[14850 .
0.000280899
I
S i n [14850.
Abs[-0.555
-
I
+
+
Cos[s]
+
r T?
S i n [14850.
(0.5
T
A m
+
0.5
f _ H
Sin[90
0.000280899
(Cos[14850.
0.000280899
I
Degree]]
0.000280899
+ 0.000280899
S i n [14850.
0.333
+
Degree]]))
/
+
Sin[90
+
Degree]]
Sin[90
+
Sin[90
D egree]]))]]
(0.415185
(Cos[14850.
(1
Sin[90
0.000280899
C C E[
Sin[90
_
Sin[90
+
Degree]]))
+
Degree]]
+ 0.000280899
I)
Degree]]
+
Sin[90
Degree]]))]
/
126
(0.415185
(C o s [ 1 4 8 5 0 .
0.000280899
I
(1
+
Sin[90
S i n [14850.
0.333
-
(Cos[14850.
S i n [14850.
Abs[-0.555
-
I
+
-
-
S in [14850.
I
-
Sin[90
Degree]]
0.000280899
+
Degree]]))
/
+
Sin[90
Degree]]))]
-
(Cos [1 4 8 5 0 .
0.000280899
I
(1
+
0.000280899
-
+
+ 0.333
+
(Cos[14850.
S in [14850.
(0.5
Power[E,
I
+
I
0.5
-
S i n [14850.
-
Degree]]
Sin[90
-
+
Sin[90
Degree]]))
/
Degree]]
0.000280899
0.000280899
S i n [14850.
-
(C os[14850.
Sin[14850.
Sin[90
0.000280899
0.000280899
I
Degree]]))]
+
Sin[90
Degree]]))]]
(0.415185
(Cos[14850.
+ 0.333
/
+
Sin[90
0.000280899
0.000280899
I
Degree]]
Sin[90
(Cos[14850.
-
+
Degree]]))
-
S i n [14850.
Abs[-0.555
Degree]]
I)
(C o s [ 1 4 8 5 0 .
0.333
I
D egree]]))]]
+
Sin[90
0.000280899
+
/
+
Sin[90
+ 0.000280899
Arg[-0.555
(0.415185
(1
Degree]]))
Sin[90
Sin[90
0.000280899
0.000280899
+
Degree]]
0.000280899
0.000280899
S i n [14850.
I
Sin[90
+
Sin[90
+
+
(0.415185
(Cos[14850.
I
Degree]]
+ 0.000280899
(Cos[14850.
S i n [14850.
Abs[-0.555
+
Sin[90
S i n [14850.
0.333
I
(1
Degree]]
-
Sin[90
Arg[-0.555
(0.415185
Cos[s]
D egree]]))]]
( ( 0 . 5 - 0 . 5 1 )
Power[E,
(1
/
+
Sin[90
Sin[90
0.000280899
(Cos [1 4 8 5 0 .
0.000280899
I
Degree]]
0.000280899
0.000280899
S i n [14850.
0.333
S in [s]]
Degree]]))
-
Sin[90
-
+
Sin[90
(0.415185
(Cos[14850 .
(1
Degree]]
0.000280899
0.000280899
I
-
+
Degree]]))
-
Sin[90
-
Degree]]
Sin[90
Degree]]
0.000280899
+
Sin[90
S in [s])
Plot of the output intensity as the analyzer
is rotated 271 radians
Degree]]))]
/
127
r[s_] := i
Plot[r[s], {s, 0, 2Pi}]
0 . 03518
0 . 03516
0 .03514
0 .03512
0 . 03508
0 . 03506
-Graphics-
128
A ppendix B
The I0 u t / lin v e rsu s p h a se d e la y c u rv e ch an g es fro m th a t d e sc rib e d by
Eq. (2-12) a n d illu stra te d in Fig. (2-3) w h e n th e tw o -b eam m o d e l is u se d
to d e sc rib e th e o u tp u t lig h t in te n s ity w h e n m u ltip le b e a m re fle c tio n s
are c o n sid e re d . T he reflectio n coefficient for the tw o -b eam m o d e l is
a n d R m a y be re w ritte n as
R = \R\ei0 = C o s ^ e 2 .
(B-2)
T hus, the com plex vector A fro m Eq. (4-2) m ay be w ritte n as
?<t>+
.<!>-
A = -j=Cos ? e 2 z' + -4 = C o s ^ - e
V2
2
V2
2
2 y'
(B-3)
w h e re from Eq. (4-10)
,
4^лh
*+=?
T
*2
(B' 4)
129
and
4?r/7h r
*-= ?
- Y
,t> C!\
U sin g Jo n es c a lc u lu s as d e s c rib e d in A p p e n d ix A, the Jones
v ecto r for Eq. (B-3) is
V2
C o s </>+ e '
0+
(B-6)
1 Cos ? e ' 2
V2
T he Jones v ecto r, Bo u t/ d e sc rib in g th e o u tp u t a fte r p a ssa g e th ro u g h the
a n aly zer m ay be calc u la te d fro m
Bout
1
1 O'
? 7 21
0 0
.72
1
"72 ?
1
72 .
0+
C o s^ e 'T
?
V2
(B-7)
-4y C o s ? e ' 2
V2
?
th erefo re the o u tp u t in te n sity lo u t m ay be calcu lated as [17]
I = |Bout(bD|2 +|Bout(2J)f
\
0+
^C os^e
2 -\C o s^ -i~ '~ 2
2
v
l
\
X
2
/
.0+
1
n
$+
1
?kCos?t-e 2
z - - k1Cn o s
.0- ^
1
- "1>
/
(B-8)
130
E q u atio n (B-8) m ay be re w ritte n as
i f <p ~
r
i ?
r
- i ?
M tM t)
e ^ +e
2
(B-9)
) + Cos2( ^
?C o s ( ? ^\Cos
2
\2 )
<PV2
Cos-
If p is d e fin e d as
Ann h
p
=
(B-10)
?
then Eq. (B-9) m ay be re w ritte n as
21
= Sin 2 ? Cos2 ?
4
4
= ? Si n2 ? .
4
2
Sin n Sin ? +
4
(B -ll)
131
Appendix C
S pectral characteristics for th e laser d io d e d e te rm in e d by a B om en
Inc. DA3.02 v a c u u m sp e c tro p h o to m e te r sy stem are sh o w n in Figs. (C -l) to
(C-9). Each g ra p h show s a re la tiv e m e a su re m e n t o n the v ertical axis th at
is u n iq u e to each graph.
MAX=
6. 68E+16.
Fig. C-l Laser diode spectrum
for 15 mA c u rre n t
132
o
cn
o
a
in
a
m
a
MAX=
6. 12E+16
Fig. C-2 Laser diode spectrum
for 25 mA input c u rre n t
133
m
a
d
d
o?
o'
d
MAX=
4. 32E+1B.
Fig. C-3 Laser diode spectrum
m
CD
O)
for 30 mA input c u rre n t
134
OD
MAX=
3. 41E+1B.
Fig. C-4 Laser diode spectrum
for 35 mA input c u rre n t
135
m
MAX=
3. 68E+1B.
Fig. C-5 Laser diode spectrum
for 40 mA input c u rre n t
136
cn
o
o
in
o
m
MAX=
2. D8E+1B.
Fig. C-6 Laser diode spectrum
for 45 mA input c u rre n t
137
MAX=
2. 69E+16.
cn
CM
__ 03
r^
CM
o
co
a
cn
a
a
a
in
a
m
IM
for 50 mA input c u rre n t
ro
Fig. C-7 Laser diode spectrum
138
CD
=)
m
MAX=
3. 52E+16.
Fig. C-8 Laser diode spectrum
for 60 mA input c u rre n t
139
a
o)
o
in
m
MAX=
2. 75E+16.
c\j
o
d
d
d
o
for 65 mA input c u rre n t
o
Fig. C-9 Laser diode spectrum
140
141
Appendix D
The ab so rb an ce of th re e G aA s w a fer sa m p le s o v e r a ran g e of lig h t
w av elen g th s is sh o w n in Figs. (D -l) to (D-3). T he ab so rb an ce Abs is
A bs = log my-
(D -l)
w h e re I is the light in te n sity w ith o u t a G aA s sam p le in se rte d into the lig h t
an d I0
^ 8 ^ in te n sity w ith the G aA s sam p le in serted .
Fig. D-l Absorbance versus wavelength
for Sumitomo
GaAs wafer sa m p le
142
Fig. D-2 Absorbance versus wavelength
for Crystal Specialties GaAs wafer sam p le
143
Fig. D-3 Absorbance versus wavelength
for Cominco GaAs wafer sa m p le
144
145
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[18]
J. A. W iesenfeld, A. J. Taylor, R. S. T ucker, G. Eisenstein, a n d C. A.
B u rru s, "E lectro-optic sa m p lin g u s in g in jectio n lasers," SPIE, vol.
795 C h a ra c te riz a tio n of V ery H ig h S p eed S em ico n d u cto r D evices
a n d In te g ra te d C ircuits, pp. 339-344,1986.
[19]
A. J.T aylor, J. M. W iesen feld , G. E ise n stein , R. S. T u c k e r, J. R.
T a lm a n , a n d U. K oren, "E lectro-optic S a m p lin g of Fast E lectrical
Signals U sing a n InG aA sP Injection Laser," Electronics L etters, vol.
22, p p. 61-62, 1986.
[20]
Z. H . Z h u , C. L. P an, Y. H. Lo, M. C. W u, an d S. W ang, "E lectroн
o p tic m e a s u r e m e n t of s ta n d i n g w a v e s in a G aA s c o p la n a r
w av eg u id e," A p p lie d Physics L etters, vol. 50, pp. 1228-1230,1987.
[21]
J. M. W iesenfeld, a n d M. S. H e u tm a k e r, "Frequency R esp o n se of an
In te rn a l A m p lifier in a H ig h -S p eed In te g ra te d C ircuit M e a su re d by
E lectro -o p tic Sam pling," E lectro n ics L etters, vol. 24, pp. 106-107,
1988.
149
[22]
J. M. W ie se n fe ld , a n d R. K. Jain, "D irect O p tic a l P ro b in g of
In te g ra te d C ircu its a n d H ig h -S p eed D evices," S em ico n d u cto rs a n d
Sem im etals, vol. 28,1990.
[23]
M.G. Li, E. A. C h a u c h a rd , a n d C. H. Lee, "In term ix in g O p tical an d
M icrow ave S ignals in G aA s M icro strip C irc u its for Phase-L ocking
A p p lic a tio n s," IEEE T ra n s a c tio n s o n M ic ro w a v e T h e o ry a n d
T echniques, vol. 38, p p . 1921-1924,1990.
[24]
Pochi Yeh, "O ptical W aves in L ay ered M edia," W iley, N e w York,
1988.
[25]
D. A. B. M iller, S. Des Sm ith, a n d C. T. Seaton, "O ptical bistability in
sem ico n d u cto rs," IEEE J. of Q u a n tu m E lectronics, vol. QE-17, pp.
312-317,1981.
[26]
H. M. G ibbs, G. K hitro v a, an d N. P e y g h a m b arian (Eds), "N onlinear
Photonics," S pringer-V erlag, Berlin H eid elb erg , 1990.
[27]
H. M. G ibbs, S. L. M cC all, T. N . C. V en k atesan , A. C. G o ssard , A.
Passner, a n d W. W ieg m an n , "O ptical b istab ility in sem iconductors,"
A p plied P hysics L etters, vol. 35, pp. 451-453,1979.
[28]
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C am bridge, 1965.
150
[29]
N . W . A sh c ro ft a n d N . D. M e rm in , "Solid State P h y sics," H o lt,
R inehart, an d W insto n , P h ilad e lp h ia , 1976.
151
Vita
D e b o rah M ech tel w a s b o rn S e p te m b e r 30, 1960 in B altim o re,
M ary lan d . She received h e r BSME w ith h ig h h o n o rs fro m the U n iv ersity
of V irginia in 1982. W hile at th e U n iv ersity of V irginia, sh e w as a w ard e d
In te rm e d ia te H o n o rs. S he w as a P hysicist's A ssista n t w ith th e N atio n al
O c ea n ic a n d A tm o s p h e ric A d m in is tra tio n E n g in e e rin g D e v e lo p m e n t
L a b o ra to ry d u rin g th e s u m m e r of 1980 w h e re sh e p a rtic ip a te d in a
p ro g ra m
fo r d e v e lo p in g
te c h n iq u e s
and
te s t in g
e q u ip m e n t fo r
d e te rm in in g ocean w a te r levels. D u rin g the 1981 s u m m e r term she w as
a w a rd e d an E n g in ee rin g In te rn sh ip w ith the H e w le tt P a c k ard C o m p a n y
in P alo A lto, C alifo rn ia w h e re she w as re s p o n sib le for th e m ech an ical
d e sig n an d set-u p of a sy ste m th a t she u se d for lig h t m e a su re m e n ts of
LEDs.
A fter g ra d u a tio n sh e joined W estin g h o u se E lectric C o rp o ra tio n in
B altim ore as an e n g in ee r in th e Electro-O ptics G ro u p . She d e sig n e d test
fix tu re s for o p tic a l c o m p o n e n ts , a n d w as re s p o n s ib le for th e o p to н
m ech an ical lay o u t, d e sig n , m a n u fa c tu re , a n d a sse m b ly of electro-optical
system s.
She left W e stin g h o u se to join the Ph.D . p ro g ra m of the Electrical
a n d C o m p u te r E n g in eerin g D e p artm en t at th e Johns H o p k in s U niversity.
She w as a w a rd e d h e r MSE in 1988. She receiv ed a n APL F ello w sh ip for
g ra d u a te study.
Ms. M echtel is a m e m b e r of T au Beta Pi N a tio n a l E n g in ee rin g
H o n o r Society an d Pi T au Sigm a N atio n al M echanical E n g in eerin g H o n o r
Society.
She has p re s e n te d h e r w o rk at th e 1991 ISH M In te rn a tio n a l
152
S y m p o siu m on M icro ele c tro n ic s, th e 1990 Jo h n s H o p k in s M ic ro w a v e
M aterial M ea su rem en ts S em inar, a n d th e 1988 C en ter for N o n d e stru c tiv e
E v alu atio n of M aterials S ym posium .
P u blications
D.M. M ech tel, H .K . C h arles, Jr., a n d C.R. W estg a te, 1991 P ro c e e d in g s
In te rn a tio n a l S y m p o siu m on M icro electro n ics, p p . 271-276, "E lectro-optic
pro bing: A n e w tool for the electronic p a ck a g in g engineer".
D .M .
M e c h te l,
A .G . A n d r e o u ,
D .N .
C h ris to d o u lid e s ,
J.
W ag n er,C .R .W estg ate,C .H .P alm er, T.O . P o eh ler, R eview of P ro g re ss in
Q u a n tita tiv e N o n d e s tru c tiv e E v a lu a tio n v 7B 1988 p 1133-1140 "Subb a n d g a p laser p ro b in g of G aA s devices a n d circuits".
c tiv e if the q u a rte r w a v e p la te is in c lu d e d in the analysis. Recall
th a t th e o rig in a l I o u t/ I in v e rs u s p h a s e d e la y curve in C h a p te r 2 is not
c a lc u la te d w ith a qu a rte r w a v e p la te in the system . The q u a r te r w a v e
p la te is i n t e n d e d to bias th e s y s te m so th a t linear c h a n g e s in the
m ic r o w a v e s ig n al p re s e n t in th e G a A s p r o d u c e a lin e a r c h a n g e in
o u t p u t in tensity .
If the a n a ly z e r is at 45 d e g re es a n d a q u a r te r w a v e
p la te is b ia sin g the system , th e n Figs (4-24) a n d (4-25) s h o w the nonMBR a n d MBR results resp ec tiv e ly as the p h a se delay v aries fro m 0 for
b o th p o sitiv e a n d negative p h a s e delays.
If th e a p p ro x im a te m o d e l for M BR is u s e d a n d th e a n a ly z e r
a n g le is v a rie d , the c u rv e s c h a n g e a lo n g w ith the a n a ly z e r ang le.
73
Figures (4-26) a n d (4-27) s h o w the results as the a n aly zer a n g le is v aried
from 0 to 45 d eg rees a n d 45 to 90 d eg rees respectively in in c re m en ts of 5
degrees.
The an alysis of the lig h t in ten sity curves s h o w that electro-optic
p r o b i n g in s tr u m e n ts a re s e n s itiv e to v a ria tio n s in h e ig h t w h e n the
a n a ly z e r is r o ta te d fr o m
th e 45 d e g r e e p o s itio n .
A lso , c h a n g e s
i n t r o d u c e d b y in s e r tin g a q u a r t e r w a v e p la te o r h a lf w a v e p la te
in flu en c e the light in te n s ity c u rv e s as the half w a v e p la te or a n a ly z e r
are rotated.
74
I/)
0 .2
c
a╗
0
3
Pu
G
0 . 1
0
in
G
0╗
0 .0
G
4a,
-1
G
o
Ph ase d elay (radians)
Fig. 4-20 Io u t/Iin vs. phase delay for analyzer position at 45 degrees
0
.
2
0.1
I) . o
3
O
Phase delay (radians)
Fig. 4-21 I o u t / I i n vs. p h a se d elay for a n a ly z e r p osition ed at
35 d e g r e e s
w ith
su b strate
th ic k n e s s
v a ria tio n s
th a t
p ro d u c e p h a s e delays of 0 to 45 d eg rees in in c re m e n ts of 15
degrees.
75
0.0
*rH
CA
г
<?
u'
?+
-S 0
-*?
1
2
CL.
гI 0
?
?
5.0
10.0
?15.0
20.0
?22.5
- rH
C
<D 0
0
3
O
P h ase d e lay (radians)
Fig. 4-22 I o u t / I i n vs. p h ase d e la y for a n aly zer positioned at
45 d e g re e s w ith a half w a v e p la te ro ta te d from 0 to 22.5
degrees.
0.b
1
1.b
2
2 .b
3
Ph ase delay (radians)
Fig. 4-23 I o u t / I i n vs. p h a s e d e la y for th e in fin ite b e a m
m o d e l w ith the a n a ly z e r p o s itio n e d at 45 d e g re e s a n d B
v a rie d from 0.2 to 1 in 0.2 increm ents.
77
>>
1/3
г
<u
P h ase delay (radians)
Fig. 4-24 I o u t / I i n vs. p h a s e delay for th e n o n-M B R m od el
w ith the a n a ly z e r p o s itio n e d at 45 d e g r e e s a n d w ith a
q uarter w a v e p la te a t 45 degrees.
78
e
3
O
Phase d e lay (radians)
Fig. 4-25 Io u t/Iin vs. phase d elay for the MBR m odel w ith
the analyzer positioned at 45 d eg rees and w ith a quarter
w ave plate at 45 degrees.
79
\
>N
4-╗
3
a,
c
Phase d elay (radians)
Fig. 4-26 I o u t / I i n vs. p h a s e d e lay for the MBR m o d e l w ith
th e a n a ly z e r r o ta te d fro m 0 to 45 d e g re e s in 5 d e g re e
increm en ts a n d w ith a q u a rte r w a v e plate at 45 d eg rees
80
Phase delay (radians)
Fig. 4-27 I o u t / I i n vs. p h a s e d elay for the M BR m o d e l w ith
the a n a ly z e r r o t a t e d fro m 45 to 90 d e g re e s in 5 d e g re e
increm ents a n d w ith a q u a rte r w ave plate a t 45 degrees
81
CHAPTER 5
INTENSITY DEPENDENT EFFECTS
G aAs
sam p les
te s te d w i t h
th e
CW
e le c tr o - o p tic
p ro b in g
in s tru m e n t e x h ib ite d results that s h o w e d a d e p e n d e n c e of the in te n s ity
of the reflected ligh t from the sam p les o n the laser intensity s u p p lie d to
the electro-optic p ro b in g in s tru m e n t.
T h e d e p e n d e n c e of the re fle c ted
light in te n s ity o n in c re a s in g i n p u t lig h t in te n s ity d e v ia te d fro m th e
e x p e c te d lin e a r re la tio n s h ip .
T h is e ffe ct is n o t d is c u s s e d in th e
literature re la te d to electro-optic p ro b in g . In this chapter, e x p e rim e n ta l
d a ta e x h ib itin g a m o r e c o m p lic a te d d e p e n d e n c e of o u t p u t li g h t
in te n sity o n i n p u t lig h t in te n s ity a re s h o w n .
R esu lts are s h o w n for
G aA s te st s a m p l e s fro m tw o d if f e r e n t s u p p lie r s . W e s u g g e s t tw o
possible s o u rc e s of the "nonlinearity".
T h e first is flu c tu a tio n s in the
sp ec tru m
in
o f th e la s e r d io d e
in s tru m e n t.
used
th e e le c tr o - o p tic p r o b i n g
T h e se c o n d is local h e a tin g in th e G a A s c a u s e d by the
pro b e laser beam .
Figure (5-1) s h o w s the results fro m p ro b in g a G aAs s a m p le ta k en
from a w a fe r s u p p lie d by S u m ito m o Electric In d u s trie s Ltd. T h e total
light in te n sity w a s m o n ito r e d at the d e te c to r location s h o w n in Fig. (31) w ith a N e w p o r t optical p o w e r m eter. T h e o u t p u t of the electro-optic
82
in s tr u m e n t w a s m e a s u re d w ith a lock-in am plifier as s h o w n in Fig. (31) w ith o u t using the p re a m p lifie r. T he laser light in te n s ity s u p p lie d by
th e laser d io d e is m o n ito r e d b y the level of c u rre n t u s e d to d r iv e the
laser diod e. The d e v ia tio n fro m lin e a r b e h av io r of the G aA s s am p les is
fu r th e r illu stra te d by the c o n tra s t b e tw e e n the b e h a v io r w h e n th e laser
beam
e n te r s th e G a A s a n d
is re fle c te d off of th e b a c k s u rfa c e
m e ta liz a tio n a n d the linear b e h a v io r w h e n the b e a m is reflected off the
fr o n t su rfa ce m e ta l tra n s m is s io n lines o n the s a m p le in Fig. (5-2).
To
v e rify th a t the d e v iatio n fro m linear b e h av io r w a s p re s e n t th r o u g h o u t
th e w a fe r , a n o th e r G a A s c irc u it s a m p le fro m the s a m e S u m ito m o
w a f e r w a s tested a n d the o b s e r v e d "nonlinear" b e h a v io r is s h o w n in
Fig. (5-3).
T he G aA s s e m ic o n d u c to r s a m p le m a y be m o d e le d as a n etalon.
H je lm e a n d M ic k e ls o n [12] h a v e d e s c rib e d th e s a m p l e s u b s tr a te
a n a ly tic a lly to s h o w its c h a ra c te ris tic s as a n a s y m m e tr ic F a b ry -P e ro t
e talo n
th a t s h o w s
reso n an t
b e h a v i o r in o u t p u t
d e p e n d i n g o n s u b s tr a te th ic k n e s s .
l i g h t in t e n s i t y
O u r m o d e l of m u l t i p l e b e a m
reflections in a G aA s s u b s tr a te p re s e n te d in C h a p te r 4 s h o w s th a t the
o u t p u t light intensity is s en sitiv e to su b stra te thickness as illu stra te d by
Fig. (4-6). T h e re s o n a n t b e h a v io r of the light in te n sity reflected b y the
G a A s e ta lo n is d e p e n d e n t n o t o n ly on the thickness of the s u b stra te ,
b u t also o n the in d ex of re fra c tio n of the s u b strate m a te ria l [25] o r the
w a v e le n g th of the light e n te r in g the s u b stra te [14].
T h e g ra p h in Fig. (5-1) of total laser ligh t in te n s ity m e a s u r e d by
th e o p tic a l p o w e r m e te r a t th e o u t p u t of the e le c tro -o p tic p ro b in g
i n s t r u m e n t s h o w s a d e v ia t io n fro m lin e a rity in the re fle c te d lig ht
in te n s ity w ith r e s p e c t to in c re a s in g la s e r in te n s ity .
T h e lo c k -in
am p lifie r m e a s u r e m e n ts th a t are in te n d e d to m e a s u r e the v o lta g e at
the p ro b e d p o in t in the G a A s s u b stra te s h o w re s o n a n t b e h a v io r in Fig.
(5-1) th a t c o in c id e s w ith r a p i d in c re a s e s in th e s lo p e of the total
reflected op tical in te n sity w ith respect to th e i n p u t laser light intensity.
Fig ure (5-2) verifies that the b e h a v io r of the reflected light is c au se d by
reflection fro m th e G a A s s u b s tr a te a n d n o t re fle c tio n fro m a m irro r
like surface s u c h as gold. T h e cause of these re su lts is either a c h an g e
in the o ptical p r o b e laser w a v e le n g th w ith in c re a s in g in te n s ity or a
p hy sical c h arac te ristic of the G aA s or the p r o b e b e a m that m a k e s the
in d ex of refraction d e p e n d e n t o n light intensity.
As s h o w n in C h a p te r 4, a n d as e x p la in e d analytically b y H jelm e
a n d M ickelson [12] the effect of m u ltip le b e a m in terferen ce m a k e s the
re su lts of e lec tro -o p tic p r o b in g d e p e n d e n t o n the p h a s e d e la y in the
GaAs. The p h a s e d e la y is d e p e n d e n t n o t o n ly o n the
thickness of the
s u b stra te as m e n tio n e d earlier b u t also o n the w a v e le n g th of the p ro b e
beam . U sing Eq. (4-10) that is re p e ate d h e re as Eq. (5-1) to d escribe the
p h a s e d e lay in G a A s w ith a thickness h of 500 p m a n d a n in d e x of
refraction n of 3.6, s h o w s that variation s o n th e o r d e r of a fraction of a
m ic ro m e te r c a n c h a n g e the o u t p u t fro m its m a x i m u m v a lu e to its
m i n i m u m v a lu e .
84
5
c
o>
г
01
<
0/>
3
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6
i01
QJ
г
u
QJ
г
o
D?гu
-8 .0 0 e - 1
4
lp
-6 .0 0 e - 1
3
-4 .0 0 e - 1
2
lock-in
- 2 .00e-1
CX
o
amplifier measurement (uV)
1 .0 0 e+ 0
optical meter
lock-in
1
2 .7 1 e -2 0
20
30
40
50
60
70
Current supplied to laser (mA)
Fig. 5-1 R eflected lig h t in te n s ity m e a s u r e m e n t s v e rs u s
laser in p u t in ten sity
85
5
-h- T ran sm ission line
G aA s
c
a╗
g
a╗
0in
aJ
a╗
g
Vh
<U
-w
a>
g
>1
<u
г
0
O
1
4
3
2
h
1I
fS
U
4-i
? pH
O
O
h
1
20
30
40
50
60
C u rre n t s u p p lie d to laser (mA)
Fig. 5-2 C o n tr a s t b e tw e e n laser in te n s ity m e a s u r e d from
G aAs back a n d fro n t surface m etalizations.
70
86
0.18
(u V )
Lock -in
0.14-
-
0.8
-
0.6
0.1 2 0 . 10 -
0.08-
-0 .4
Lock-in
(X
m easu rem en ts
0.16-
O p tical m eter
am plifier
-a -
0.06a.
0.04
0.2
20
30
40
50
60
70
C urrent s u p p lie d to laser (m A )
Fig. (5-3) F u r th e r reflected lig h t in te n s ity m e a s u r e m e n ts
v e rsu s laser in p u t in ten sity
87
This b e h a v io r h a s b e e n s h o w n by tu n in g th e p r o b in g laser to
d iffe re n t w a v e le n g th s [14].
W e tested the la se r p r o b e b e a m w ith a
B om en Inc. DA3.02 v a c u u m s p e c tro p h o to m e te r s y s te m to o b ta in the
spectral characteristics of the laser as laser in te n s ity w a s increased. The
re su lts are s h o w n in A p p e n d ix C.
T hey in d ic a te th a t th e re m a y b e a
slig h t shift as laser in te n s ity in creases w h ic h c o u ld b e in flu en c in g the
o u t p u t of the electro-optic p ro b in g in stru m e n t.
A n o th e r e x p la n a tio n for the d e v ia tio n fr o m lin e a r b e h a v io r in
reflected lig h t in te n s ity is a n in te ra ctio n b e tw e e n the laser b e a m a n d
th e G a A s s u b s tr a te th a t m a k e s the in d e x of re fra c tio n of th e G a A s
d e p e n d e n t o n ligh t in ten sity .
W e in v e stig ate d tw o possibilities ? the
p re s e n c e of d e fe cts in th e G a A s cry stal a n d th e r m a l effects.
Som e
a lte rn a tiv e s for th e b e h a v io r w e re e x p lo re d in c lu d in g : c h ec k in g the
G a A s s am p les for c o n ta m in a n ts that w o u ld r e s p o n d to la se r lig h t at
1300 nm , testing s a m p le s s u p p lie d by o th e r m a n u fa c tu r e r s , testing for
c h a n g e s in the laser s p e c tr u m w ith in c re a s e d lig h t in te n s ity o u tp u t,
a n d s tu d y in g th erm al effects w h e n the te m p e ra tu re of the G aA s sam p le
is changed.
W e te sted for d e fe cts in th e G aA s s u p p l i e d by S u m ito m o by
c h e c k in g for d e v i a t i o n s
in lin e a r b e h a v i o r
from
G aAs
s a m p le s
o b ta in e d from C rystal Specialties. Sam ples from C ry stal Specialties also
ex h ib ite d this b e h a v io r as s h o w n in Fig. (5-4). If th e re is a defect, it is
n o t u n iq u e to S u m ito m o w afers.
Tests w e re also c o n d u c te d w ith a s p e c tro m e te r for w afers from
th re e s u p p lie rs ? S u m ito m o , C ry s ta l S pe c ialties a n d C o m in c o .
A
s w e e p of the s p e c tro m e te r from 890 n m to 2000 n m s h o w s the expected
88
la rg e r e s p o n s e n e a r th e w a v e le n g th (890 n m ) w ith e n e r g y n e a r the
b a n d g a p of G aA s. T h e results at lo n g e r w a v e le n g th s s h o w the ab se n c e
of c o n ta m in a n ts a t a level th a t w o u l d c re a te a n o n lin e a r r e s p o n s e
b e lo w the b a n d g a p of G aA s w h e n the s a m p le s are p ro b e d by the 1300
n m laser be am . These tests results are in A p p e n d ix D.
3
>
g
c01
sa>
V-i
3c/i
a>
2
J-H
a>
Cl
1
g
uo
0
10
20
30
40
C u rren t s u p p lie d to la ser (m A )
Fig. 5-4 Reflected light in ten sity m e a s u r e m e n t v ersus laser
i n p u t in te n s ity for a G aA s s a m p l e s u p p lie d by C ry stal
Specialties.
T herm al
e ffe c ts
c an in f l u e n c e
th e in d e x
of r e f r a c tio n
of
s e m ic o n d u c to r m aterials [26, 27], Since th e p ro b e b e am is focused to a
89
s p o t size of less th a n 10 m ic ro m e te rs, the in te n s ity a t that p o in t can be
in th e ra n g e of h u n d r e d s of w a tts p e r m illim eter s q u a re d . T he c han ge
in in d e x of refraction of G a A s w ith te m p e ra tu re can b e e stim ate d from
th e re la tio n sh ip that relates the dielectric c o n s ta n t er to the b a n d g a p Eg
of the sem ico n d u c to r [28]
w h e r e a is a c o n sta n t.
T h e in d e x of re fra c tio n , n, is re la te d to the
dielectric c o n sta n t by
S u b stitu tin g Eq. (5-2) into Eq. (5-3) gives
(5-4)
T he ch an ge in b a n d g a p of a G aA s s e m ic o n d u c to r w ith increasing
te m p e ra tu re can be a p p ro x im a te d as a linear re la tio n s h ip [29] as sh o w n
in Eq. (5-5)
Eg = E g (0 ) - BT
(5-5)
w h e r e Eg(0) is the b a n d g a p at 0 K, T is the te m p e ra tu re in Kelvin, a n d B
= 3.3 x 10?4 eV /K . [29] U sing Eq. (5-5), Eq. (5-4) m a y be re w ritten as
90
n =
1+
a
-
(Eg ( 0 ) - B T ) 2
(5-6)
A g r a p h of this e q u a tio n is s h o w n in Fig. (5-5).
n
T e m p e ra tu re (K)
Fig. 5-5 In d e x of refraction in G aA s as a function of te m p e ra tu re
F ro m Eq. (5-1), for a 500 p m G a A s sub strate, changes in the in d ex
of refraction b y a few ten th o u s a n d th s can c h a n g e the p h a se d e la y b y n
ra d ia n s.
F r o m Fig. 5-5 it is clear th a t a n increase in te m p e ra tu re by a
few d e g re e s can d ra m a tic a lly c h a n g e the p h a s e d elay in G aA s.
For
exam p le, a 1 d e g re e change in the te m p e r a tu r e changes the p h a s e d elay
by n ra d ia n s T he inten sity of the laser can easily raise the te m p e ra tu re
at the s a m p le p oin t.
W e also s t u d ie d the effect of te m p e r a t u r e changes in th e b u lk
G aA s, b y u s in g a c artrid g e h e a te r to h e a t the G aA s s am p le test fixture.
T he la se r in te n s ity s u p p lie d to th e i n s t r u m e n t w a s in c r e a s e d a n d
91
m e a s u r e m e n ts w e re ta k e n w ith th e lock-in a m plifier. T h e test results
for te m p e ra tu re s at 21 a n d 43 d e g re e s C are s h o w n in F igu re (5-6). The
o u t p u t s h o w s a ra n g e of v a lu es for each d a ta p o in t tak en at a p articu lar
te m p e ra tu re .
As d a ta w as ta k e n for e ach c u rre n t p o in t at a p a rtic u la r
t e m p e r a t u r e , th e o u t p u t l ig h t in t e n s i t y r e c o r d e d b y th e lo c k -in
am p lifie r d id not give o ne s ta tio n a ry value, b u t flu c tu a te d b e tw e e n two
v a lu e s.
T h erefore, tw o lines a re g r a p h e d for each te m p e r a tu r e w h e re
the h ig h e st a n d lo w e st v a lu es are s h o w n for each s a m p le p o in t taken at
a p a rtic u la r te m p e ra tu re .
F ig ure (5-6) s h o w s th a t h e a tin g of the b u lk G aA s in fluen ces the
o u t p u t of the electro -o ptic s a m p lin g in s tr u m e n t as w o u ld b e e x p ected
fro m Eq. (5-6). These results s h o w th a t te m p e ra tu re c h an g e s in the bulk
G a A s s a m p le influence test re su lts as w ell as localized h e a tin g from the
la se r p ro b e beam . T h e test re su lts a t 43 d e g re e s C s h o w a w id e r range
b e tw e e n the h ig h e st a n d lo w e s t lock-in am p lifie r re su lts th a n the test
re s u lts at 21 d e g re es C. W e also o b s e rv e d th a t the v a ria tio n from the
h ig h e s t v a lu e to the lo w est v a lu e d took a b o u t 30 seco nd s at 21 degrees
C, b u t at 43 d e g re es C, lock-in am p lifie r o u t p u t re a d in g s c h a n g e d from
h ig h to low in a fraction of a second.
The c h ang e in electro-optic p ro b in g test resu lts as the b u lk G aAs
is h e a te d , in a d d itio n to local h e a tin g from the laser b e am , led us to test
for h e a tin g of the b u lk G aA s by the laser b e a m over long tim e periods.
W e te sted the refle c ted lig h t in te n s ity w ith the laser s ta tio n a r y at a
p a r tic u la r s a m p lin g p o in t a n d th u s s u p p lie d local h e a tin g w ith the
b u lk G aA s at ro o m te m p e ra tu re . T h e o u tp u t of the in s tr u m e n t v aried
s in u s o id a lly o ver time.
Fig. (5-7) a n d Fig. (5-8) s h o w the v a ria tio n in
92
o u t p u t w h e n th e la s e r is s ta tio n a r y a n d to ta l o p tic a l in te n s ity is
re c o rd e d w ith the N e w p o r t optical p o w e r m e te r a n d the e lectro-op tic
in s tr u m e n t o u t p u t is r e c o rd e d w ith the lock-in a m p lifier.
T h e se test
results s h o w a long te rm th erm al effect that o ccu rs o v e r a few h o u rs of
h e a tin g a n d m u s t b e c o n s id e re d alo n g w ith the th e rm a l c h an g e s th a t
in fluen ce the re s u lts of tests c o n d u c te d o v e r a s h o rt p e rio d of time.
T he lo n g te rm th e rm a l effects co u ld b e c a u s e d by a c h a n g e in p a th
length related to the th e rm a l coefficient of e x p a n s io n for GaAs.
The re la tio n s h ip b e tw e e n the reflected lig h t in te n sity a n d i n p u t
intensity s h o w s b e h a v io r that d eviates fro m a lin ear d e p e n d e n c e . This
"n o n lin earity "
in flu en c es
th e
o u tp u t
of
elec tro -o p tic
p ro b in g
in s tru m e n ts by c re a tin g re s o n a n t p e a k s in the o u t p u t v o lta g e results.
A c o m p ariso n w ith o th e r s u p p lie rs d id n o t reveal a n y defects that w e re
u n iq u e to th e S u m ito m o w afers.
T h e sh ift in laser w a v e le n g th can
change electro-optic p ro b in g results, b u t local h e atin g by the p ro b e laser
b e a m c re a te s t e m p e r a t u r e c h a n g e s th a t can c h a n g e th e in d e x of
refraction of G aA s d ra m a tic ally .
H e a tin g of the b u lk G aA s o v e r lo n g
p e rio d s of tim e also c h ang es test results. F u rth e r testing w ith the G aA s
in a h e a t s u n k fix tu re ta k en o v er lim ite d tim e p e rio d s is re q u ire d to
learn m o r e a b o u t th e in te n s ity d e p e n d e n t b e h a v io r of the G a A s test
sam ples.
93
3
" * ^ 2 1 degree G ran ge
2
^
43 d e g r e e C ran ge
1
Lock-in
am plifier
m ea su rem en ts
(m V )
4
0
10
20
30
40
C urrent su p p lie d to la ser (m A )
Fig. 5-6 R eflected lig h t in te n s ity m e a s u r e m e n ts v e r s u s
laser in p u t intensity at 21 d e g re e s a n d 43 degrees C.
94
0.43
г
6
c
01
6
0)
>-(
0.42"
лQJ
0.41"
|
0.40-
<D
г
o
O.
0.39-
0.38
0
10
20
30
40
Tim e (1 u n it = 5 m inutes)
Fig. 5-7 V a r ia tio n o v e r tim e of to ta l re fle c te d lig h t
in ten sity m e a s u r e d w ith the optical p o w e r m e te r
50
95
m e a s u r e m e n ts
(uV )
1 .4 ?
1.2- Q
i
??
i .o -
?
G EE Q
?
?
Q
?
Lock-in
am plifier
El
0.8 -
?
?
?
?
m
?
?
Q
0.6 -
Q
?
?
I
Q
EE1
? ??
0.4 ?
0
?i
ED
g
BBjB--------1-------1--------- 1-------- 1-------- 1? ? fpB------- 1-------
10
20
30
40
T i m e (1 u n i t = 5 m in u te s )
Fig. 5-8 V a ria tio n o v e r tim e of refle c ted lig h t in te n s ity
m e a s u re d w ith the lock-in am p lifier
50
96
CHAPTER 6
ELECTRO-OPTIC PROBING TEST RESULTS
In this c h a p te r e lectro-op tic p r o b i n g test re su lts are p re s e n te d .
The first p a r t of th e c h a p te r re p o rts o n te st re su lts fro m c o n tin u o u s
w a v e e lec tro -o p tic p r o b in g a n d in c lu d e s a d is c u s s io n of the effect of
m u ltip le
beam
reflectio n s
on
th e
te st
re su lts
fo r
m ic ro strip
tran sm issio n lines. The effect of m u ltip le reflections o n test re su lts for
so m e a d d itio n a l p a s s iv e test s tr u c tu r e s is s h o w n for th e first tim e.
O ther effects s u c h as the m etalizatio n th ic k n e ss of circuit test s tru c tu re s
that in flu en c e th e o u tc o m e of test re s u lts are also in tr o d u c e d for the
first time.
T h e s e c o n d p a r t of the c h a p te r d escrib es the e lec tro -o p tic
p ro b in g r e s u lts a c h ie v e d w ith the p u l s e d la se r sy ste m .
M u ltip le
reflection effects a re r e p o r te d o n a n d a c o m p a r is o n to the C W test
results is in c lu d e d .
6.1 CW electro-optic probing test results
The C W e lec tro -o p tic in s tr u m e n t d e s c rib e d in sectio n 3.1 w a s
u se d to m a p th e electric field in te n s ity vs. sp atial p o sitio n of s e v e ra l
97
G aA s s a m p le s s o m e of w h ic h w e r e s h o w n in Figs. (3-17) a n d (3-18).
Testing w a s c o n d u c te d in the K H z ra n g e w ith the sam p les m o u n t e d in
the test fixture s h o w n in Fig. (3-16)
In Fig ure (6-1) a field profile a r o u n d a typical tran sm issio n line is
illu strated .
T he d a ta profile is s h o w n a r o u n d the 50 Q tr a n s m is s io n
line of Figs. (3-16) a n d (3-17).
T h e field u n d e r the line c a n n o t b e
m e a s u r e d since th e tra n s m is sio n lin e m e ta l blocks the p a s s a g e of the
light probe.
T he d a ta s h o w s d ifferent p e a k m a g n itu d e s that are th e re s u lt of
m u ltip le b e a m in te rfe re n c e c a u s e d b y a c h a n g e in the h e ig h t of the
G aA s su b strate .
As d iscu ssed in C h a p t e r s 4 a n d 5, the o u t p u t of the
e lec tro -o p tic p r o b in g in s tr u m e n t is s e n s itiv e to s u b s tr a te th ic k n e ss.
C h an g e s in the s u b s tr a te thickness of a b o u t 0.2 m icro ns can c h a n g e the
o u t p u t of th e i n s t r u m e n t fro m a m a x i m u m to a m in im u m .
The
e lectric fie ld p ro f ile s h o w n in Fig. (6-1) s h o w s a c h a n g e in p e a k
m a g n itu d e s th a t is c o m m e n s u ra te w ith th e thickness v a ria tio n s of the
S u m i to m o G a A s
te st w a fers.
A n o t h e r fie ld p ro f ile of a G a A s
tran sm issio n line is s h o w n in Fig. (6-2), w h e r e the ra p id d e crea se s a n d
in c re ase s in the electric field p ro file m a y be d u e to m u ltip le b e a m
reflections. The d a ta in Fig. (6-2) w a s tak en b y a d justing the in te n sity of
the laser b e a m slig h tly at each d a ta p o in t to e n s u re that d a ta w a s n o t
taken at a re s o n a n t p e a k as d iscu ssed in C h a p te r 5.
Figure (6-3) de tails the field profiles a t the tip of a s tu b capacitor.
T he field s tr e n g th in the vicin ity of th e c o rn e r p o in ts (c u rv e A) are
stro n g e r for a n e q u iv a le n t d ista n ce fro m the s tu b m e ta liz a tio n th a n a
sim ilar profile in the center of the s tu b (cu rv e B). The d ifferin g p e a k
98
v a lu es in c u rv e B a re a n o th e r e x a m p le of th e in fluen ce of m u l t i p l e
b e a m re fle c tio n s in th e G aA s s u b s tr a te .
A trace p a s t th e s t u b e n d
p a ra llel to th e tr a n s m is s io n line (c u rv e C) a g a in illu stra te s th a t the
field intensity a t th e c o rn e r p oin ts is g re a te r o n a relative basis t h a n at
the stu b center as w o u ld be expected.
Fig. (6-4) s h o w s the profiles ta k e n for h o riz o n ta l scans a r o u n d a
s tu b capacitor.
C u r v e A illu stra te s th e re la tiv e ly s tro n g e r fie ld th a t
exists closer to the tra n sm issio n line.
Fig. (6-5) is the electro-optical r e s p o n s e c u rv e for an in te rd ig ita l
capacitor.
T h e la se r p r o b in g re s u lts in d ic a te th a t the re la tiv e field
intensity is m a x im u m in the g ap s b e tw e e n the m etalization.
D a ta w a s
taken w ith the left s id e of the cap a c ito r a c tiv a te d w h ile the r ig h t side
w as te rm in a te d in a 50 Q load, a n d th e n w ith the set-up re v e rse d .
In
so m e g ap s the a c tiv a tio n of the left s id e h a d m o re influence w h ile in
o th er gaps the rig h t sid e h a d m ore influence. F u rth er s tu d y of the g a p s
w as c o n d u c te d b y s c a n n in g each g a p vertically a n d by taking tw o d a ta
p o in ts ? o n e w ith th e left side a c tiv a te d a n d o n e w ith the rig h t s id e
a ctiv ated w ith th e o p p o s ite elec tro d e te r m in a te d in a 50 Q lo a d ? at
each p h y sica l lo c a tio n scan ned.
T h e re s u lts s h o w n in Fig. (6-6) a re
p re sen te d w ith the left side activated d a ta s u b tra c te d from the rig h t side
activated data.
It is clear th a t the s tro n g e st in flu e n c e n e a r the b e g in n in g a n d
e n d of each scan is d e te r m in e d b y the p h y s ic a l p ro x im ity of th e s id e
activated.
H o w e v e r, the influence as the sca n p ro c e ed s into the c e n te r
of each s w e e p is d e te r m in e d o n an a lte r n a tin g b asis b e tw e e n th e left
a n d right activ ation test results. In o u r o p in io n , this b e h av io r is d u e to
99
th e in flu en c e of th e m e ta liz a tio n th ic k n e ss o r th e e d g e s h a p e of the
m e ta l s tru c tu re s .
T h e s e n s itiv ity to th e e d g e m e ta liz a tio n p a t t e r n
s h o w s a n e w fa c to r th a t m u s t b e c o n s id e re d w h e n p ro b in g n e a r the
e d g e of the circuit s tru c tu re s.
The g e o m e tric re la tio n sh ip b e tw e e n the
"foot print" of th e la se r p r o b e b e a m a n d c irc u it s tru c tu re s h as b e e n
co n sid e re d [12], b u t the influence of the m e ta l d e p o sitio n for circuits o n
G a A s has n o t b e e n a d d re s s e d .
b e y o n d the sco p e of o u r studies.
F u r th e r s tu d y of the m e ta liza tio n w a s
1.0
Relative intensity
0.8 '
0. 6 '
0.4'
0. 2 '
0.0
1
2
3
4
5
Position (m m )
Fig. 6-1 Electro-optic field in te n sity v e rs u s po sitio n in
vicinity of a 50 Q tra n s m is sio n line.
1.0
Relative
in te n sity
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
Position (m m )
Fig. 6-2 Electro-optic field intensity v e rs u s p o s itio n for a
tra n s m is sio n line.
1.5
o
o
o
rH
O
001
00i
I
O / O
V
O/
o / o
Aiisueju! aAjieieu
o
in
in
<N
in
o
o
in
o
o
o
o
00
o
Aijsuaiuj aA|iB|ay
й
Fig. 6-3 Electro-optic field intensity profiles versus position in the vicinity of a stub capacitor. Curve A
is the profile from the stub point at a 45 degree angle. Curve B is from the central portion of the stub
perpendicular to the stub end. Curve C is a probe trace parallel to the stub end.
102
vo
o
in
o
^3
o
cn
o
CO
й
vO
й
o
fN
o
E
E
лN
O
░o
o
Aiisudiu; 0A!;e|ay
Aijsuaiu; aAjieiay
Fig. 6-4 Electro-optic field intensity profiles versus position for a stub capacitor
103
104
502434
1 2
3
4
5
6
7
Relative intensity
A-A
left
right
0.00
0.25
0.50
0.75
1.00
1.25
mm
Fig. 6-5 E le c tro -o p tic fie ld
in te n sity
p o s itio n for a n in te rd ig ita l c a p a c ito r.
p ro f ile s
versus
C u r v e A is th e
r e s p o n s e w ith the left e n d ex cited w h ile C u rv e B h as th e
rig h t s id e excited.
In each case, the o p p o s ite elec tro d e is
te rm in a te d in a 50 Q load.
- Left intensity
(mV)
105
g a p 1 rt - 1ft
g a p 2 rt - 1ft
g a p 9 rt - 1ft
g a p 5 rt - 1ft
g a p 8 rt - 1ft
Right intensity
g a p 7 rt - 1ft
g a p 6 rt - 1ft
g a p 4 rt - 1ft
g a p 3 rt - 1ft
Fig. 6-6 Left sid e a c tiv a te d e le c tro -o p tic field in te n s ity
s u b tr a c te d fro m rig h t s id e a c tiv a te d e le c tro -o p tic field
in te n sity v e rsu s p o sitio n for the g a p s n u m b e re d as in Fig.
(6-5).
106
6.2 Pulsed laser electro-optic probing test results
Electric field intensity profiles of the s am p le sh o w n in Fig. (3-16)
w e re also ta k e n w ith the p u ls e d laser electro-optic p ro b in g in s tru m e n t.
A 1.00021 G H z s ig n al w as s u p p lie d to th e s a m p le b y an H P 8341B
m ic r o w a v e s y n th e s iz e r .
T h e la s e r w a s b ia s e d w i t h a 36.1 m A
(thresh old is 25 m A ) c u rre n t fro m a DC c u rre n t so urce a n d 30.69 d b m
of RF p o w e r a t 1 G h z w a s s u p p lie d by a m ic ro w a v e p o w e r am plifier.
T he d e te c tio n fr e q u e n c y w a s 2.1 K H z.
F ig u re (6-7) s h o w s the p a th
sca n n e d b y th e p u ls e d electro-optic p r o b in g in stru m e n t.
Electric field
intensity p ro file s are s h o w n in Figs. (6-8) a n d (6-9) for a sig n al o n the
w id e r 50 Q
t r a n s m is s io n lin e a n d
t h e n o n the n a r r o w e r 75 Q
tra n sm issio n line re sp ectiv ely for the 12 m m scan sh o w n in Fig. (6-7).
In b o th electric field profiles, n o s ig n al is d e te c te d as the la se r p ro b e
scans across th e m e ta l tran sm issio n line.
r50L
75 il
12 m m
Fig. 6-7 P u ls e d e le c tro -o p tic p r o b i n g scan a cro ss tw o
tra n s m is s io n
lines
107
1.0
Relative
In te n sity
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
Position (mm)
Fig. 6-8 Electric field in tensity p rofile v e rsu s p osition w ith
the signal o n the 50 Q tra n sm issio n line.
12
108
1.0
0. 8 "
0. 6 "
a>
>
0 .4 -
0.2
-
0.0
0
2
4
6
8
10
12
Position (m m )
Fig. 6-9 Electric field in te n sity profile v ersu s position w ith
the signal o n the 75 г2 tra n s m is sio n line.
109
The electric field in te n sity profile in Fig. 6-8 s h o w s different p e ak
m a g n itu d e s on e ither s id e of the 50 Q tra n s m is sio n line from the effect
of m u ltip le b e a m re fle c tio n s as first s h o w n w ith th e C W in s tr u m e n t
test re su lts for this s a m p l e in Fig. (6-1).
T h e elec tric field in te n s ity
p ro file for the 75 Q. tr a n s m is s io n line a lso s h o w s th e in flu e n c e of
m u ltip le beam reflections o n test results.
T h e la rg e st p e a k m a g n itu d e
a lw a y s a p p e a rs o n the s a m e side of the tra n s m is sio n lines for b o th the
50 Q. a n d 75 ill tr a n s m is s io n w h ic h w o u l d b e e x p e c te d for a G a A s
s a m p l e w ith a c o n t i n u o u s c h a n g e in h e ig h t d u e to the th ic k n e s s
v a ria tio n of the G a A s w a fe r th at the s a m p le s u b s tr a te w a s diced from .
F ig u re 6-10 s h o w s a c o m p a r is o n of test re s u lts fro m the CW a n d the
p u ls e d electro-optic p r o b in g in s tru m e n t.
T h e a p p e a r a n c e of m u ltip le
reflection effects in p u ls e d electro-optic p ro b in g test re su lts u n d e rsco re s
th e u s e f u ln e s s of a lo w fr e q u e n c y C W i n s t r u m e n t for p r e d ic tin g
p ro b le m s that m a y o ccu r in h ig h frequency testing.
110
1.0
0.8 -
0.61
01
>
0.4-
0.2 -
P u lsed
гW
0.0
0
2
4
6
8
10
Position (m m )
Fig. 6-10 Electric field in te n s ity p ro files v e r s u s p o s itio n
w ith a signal o n the 50 Q tra n sm issio n line for the p u ls e d
p ro b in g in s tru m e n t a n d the C W p ro b in g in s tru m e n t.
12
Ill
CHAPTER 7
CONCLUSION
E x p e rim e n ta l m e a s u r e m e n t s of e lec tro -o p tic p r o b i n g re v e a le d
s u b s ta n tia l d e v iatio n fro m e x p e c te d resu lts b ased o n c u rre n t m odels. A
c o n tin u o u s w a v e e lec tro -o p tic p r o b in g in s tru m e n t w a s b u ilt to
m ap
electric field in te n sity p ro file s a n d to p ro v id e a b a s e lin e for s tu d y at
m i c r o w a v e f r e q u e n c ie s .
T e s tin g
of m ic r o w a v e
p a ssiv e
c irc u its
r e v e a le d m u ltip l e b e a m re fle c tio n effects a n d n o n - l i n e a r in te n s ity
d e p e n d e n t effects.
W e stu d ie d
th e s e p r o b le m s
to a d v a n c e the
te c h n iq u e of electro -op tic p r o b in g so th a t acc u ra te test re su lts can be
o b ta in e d .
A p u ls e d elec tro -o p tic p ro b in g in s tru m e n t w a s b uilt to test
s a m p le s a t G H z frequencies. Electric field in tensity profiles taken w ith
b o th in s tru m e n ts w e re p re s e n te d for several G aAs sam p les.
M u ltip le b e a m re fle c tio n s w e re m o d e le d n u m e ric a lly by u sin g
Jones calculus. Tw o d ifferen t m o d e ls for the reflection coefficient w ere
p r e s e n te d ? a tw o-beam m o d e l a n d a n infinite b e a m m o d e l. O u r twob e a m m o d e l d o es n o t v io la te c o n s e rv a tio n of e n e r g y in c o n tra st to
o th e r tw o -b e a m m o d e ls [23].
The effect of m u ltip le b e a m reflections
w a s s h o w n n u m erically b y c o m p a r in g the results of a non-M B R m o del
to th o se of o u r tw o -b e a m M BR m o d e l.
MBR c h a n g e s the total light
112
intensity as w ell as the a m p litu d e of the c h an g in g light in te n sity fro m a
sin u so id a l sig n al p re s e n t o n a test s a m p le .
M u ltip le b e a m reflections
m a k e the o u t p u t in te n s ity s en sitiv e to c h an g e s in s u b s tr a te th ic k n e ss
w h e n the electro-optic sy stem is b ia se d w ith a q u a rte r w a v e plate. The
in fin ite b e a m
m odel
ta k e s in to
a c c o u n t a n in f i n ite n u m b e r of
reflections a n d the influence of a v a ria b le loss coefficient. R esults w e re
p r e s e n te d fo r th e infin ite b e a m m o d e l w ith several loss coefficients.
T he total lig h t in te n sity as w ell as th e a m p litu d e of th e c h a n g in g light
in ten sity fro m a sin u so id a l signal p r e s e n t o n a test s a m p le w e re s h o w n
to also be in flu en ced by a c h an g in g loss coefficient.
W e s u g g e s t n e w c a lib ra tio n te c h n iq u e s b a s e d o n th e in fin ite
b e a m m o d e l th a t take in to a c c o u n t v a ria tio n s in loss co efficient a n d
s u b s t r a t e th ic k n e s s
to a c c o u n t fo r th e effects of m u l t i p l e b e a m
reflections in test results.
T he te c h n iq u e s co u ld b e e x te n d e d to h ig h
freq u e n cy C W p ro b in g for sy ste m s th a t in c o rp o ra te a fast p h o to d io d e
detector.
To e n h a n c e th e s e n s itiv ity of th e c a lib ra tio n te c h n iq u e , tw o
m e th o d s w e re su gg ested . O n e w a s to ro ta te the a n a ly z e r to g e t a p e ak
lig h t in te n s ity [23], a n d the o th e r w a s to ro ta te a h alf w a v e p la te
in s e rte d in to the o ptical train.
In o u r o p in io n , these tw o o p tio n s are
m u c h s im p le r than v a ry in g the w a v e le n g th of the p ro b e laser [14].
We also conducted a study of the changes in the Iout/Iin versus
phase delay curves when the analyzer is rotated from its 45 degree
position.
We show that relation creates a family of curves for Iout/Iin
versus phase delay that are dependent on analyzer position.
Numerical analysis shows that if the analyzer is not moved from its 45
113
degree position that the I o u t /I in
versus phase delay curve is
insensitive to changes in substrate thickness. An area of further study
would be to use the information from that curve to bias the system for
substrate thickness insensitive operation. The influence of the rotation
of a halfwave plate on the Iou t/Iin versus phase delay curve and a
changing loss coefficient were also studied.
A p o ssib le top ic for fu tu re s tu d y of th e influence of MBR w o u ld
b e to im p l e m e n t th e c a lib ra tio n m e t h o d s w e s u g g e s te d .
A n o th er
in te re stin g s tu d y w o u ld be to coat the s a m p le s w ith an anti-re fle c tio n
c o a tin g to lim it th e in flu e n c e of m u l t i p l e b e a m re fle c tio n s o n te st
results.
N o n lin e a r in te n s ity d e p e n d e n t effects w e re d isco v e re d w h e n the
in te n sity of the la se r b e a m i n tr o d u c e d in to the elec tro -o p tic p r o b in g
i n s t r u m e n t w a s v a r ie d .
This e ffect w a s d e m o n s t r a t e d for G a A s
s am p les fro m tw o d iffe re n t m a n u fa c tu r e r s .
T h e sa m p le s w e re te ste d
for c o n ta m in a n ts th a t w o u l d excite the G a A s a lth o u g h the p r o b e laser
b e a m w a v e le n g th of 1300 n m h a s a n e n e r g y b e lo w the b a n d g a p of
GaAs.
T h e tests w ith a s p e c tro m e te r w e re n e g a tiv e for this ty p e of
c o n ta m in a n t.
T h e laser w a s tested for c h a n g e s in its s p e c tr u m th a t
w o u ld o ccu r w ith in c re a s in g ligh t in te n s ity o u tp u t.
Tests s h o w e d a
slight shift in the s p e c tru m w ith in c re a s in g lig ht in tensity so the laser
itself m a y be the s o u rc e of the n o n lin e a rity [14]. F u rth e r s tu d y w o u ld
re q u ire c o n c u r re n t m e a s u r e m e n ts of the lig h t in te n sity reflected fro m
th e G a A s s a m p l e as th e s p e c t r u m
of th e la s e r w a s m o n i t o r e d .
H o w e v e r, in o u r s tu d ie s , the n o n lin e a r b e h a v io r is d o m i n a t e d b y
th erm al effects.
W e s h o w that c h a n g in g th e te m p e ra tu re of th e G a A s
114
b y o n ly a few d e g re es c h a n g e s the index of refractio n e n o u g h to h ave a
d ra m a tic effect o n th e o u t p u t of the electro-op tic p r o b in g in s tru m e n t.
This sug gests that th e m e a s u r e m e n t w o u ld b e n efit if the s a m p le w e re
m o u n t e d in a h e a t sin k .
L o n g te rm v a r ia tio n s o v e r 1 h o u r w e re
o b serv ed . The o rig in of this effect is not clear, b u t m a y b e related to a
c h a n g e in the p a th le n g th of th e sam p le d u e to th e th e rm a l coefficient
o f e x p a n s io n for G aA s.
C h o p p in g the p ro b e b e a m , u s in g sh o rt laser
p u ls e s , or u s in g a lo w in te n s ity laser w o u l d m itig a te s o m e of the
th e rm a l effects.
E lectro-optic p r o b i n g w a s u s e d to ta k e electric field in te n sity
p ro file s of sev e ra l s a m p l e s w i t h the CW s y s te m a n d m a k e relativ e
m e a s u r e m e n ts b e tw e e n c irc u it stru c tu re s.
T h e p re s e n c e of m u ltip le
b e a m in te rfe re n c e w a s n o t e d o n the e le c tric fie ld p ro files.
The
in flu en c e of m e ta liz a tio n o n th e electro-optic p r o b in g re su lts w a s also
p ro p o s e d .
A p u ls e d e le c tro -o p tic p ro b in g in s tr u m e n t w a s b u ilt for testing
c ircuits in the G H z ra n g e .
T h e p u ls e d laser w a s c h arac te riz ed w ith a
p h o t o d i o d e c u s to m m o u n t e d o n a h igh f r e q u e n c y test fix tu re a n d
te ste d w ith a d ig ita l s a m p l i n g oscilloscope.
Electric field in ten sity
profiles of tra n sm issio n lines d e m o n s tra te d test resu lts taken at 1 GHz.
A c o m p a r is o n of th e te sts fr o m the C W a n d p u l s e d in s tr u m e n ts
d e m o n s tra te s the u s e f u ln e s s of the C W in s tr u m e n t for e sta b lish in g a
b a se lin e for fu tu re s t u d y of b o th the low a n d h ig h freq u e n cy test setн
up s.
Electro-optic p r o b in g h a s been a m p ly d e m o n s t r a t e d as a useful
tool for c h arac te riz in g h ig h s p e e d circuits a n d devices.
Both external
115
a n d in te rn a l m e th o d s h a v e b e e n u s e d to m a k e a v a rie ty of tests. M u ch
of th e te s tin g h a s b e e n q u a l i t a t i v e as c a lib ra tio n is s u e s a re just
b e g in n in g to be a d d r e s s e d [12].
In te rn a l e le c tro -o p tic p r o b in g as a
re la tiv e ly lo w cost, s im p le m e t h o d of e v a lu a tin g th e e lectric fields
in t e r n a l to a circ u it is a p r o m i s i n g m e t h o d
th a t b e c o m e s m o r e
p o w e rfu l as it continues to b e m o re co m pletely characterized.
116
Appendix A
Jones calculus d e term in e s th e final po larization state of the p ro b e
b e a m a fter p a ss a g e th ro u g h a series of o p tic a l elem ents a n d h e n c e the
v ariatio n of the intensity of the p ro b e b e a m at the detector.
If w e a ssu m e th at the electro-optic p ro b e beam is d e sc rib e d by
Ac
i(o)t-tx)
(A -l)
w h e re [17]
A = Az<e'^z' z' +Av' e ^ y y'
(A-2)
a n d z' a n d y' are u n it vectors, th e n th e Jones vector r e p re s e n ta tio n
d escribing the polarization state of th e p la n e w a v e is
aV
5-'
iS y ?
(A-3)
A y<e
N o te th a t the use of (cot - kz) in d ic a te s th a t positive 5 re p re s e n ts a
p hase lead. The z' a n d y' axes are d e fin e d in C h ap ter 2.
117
For the optical tra in s h o w n in Fig. (2-1), a fter the initial passag e of
th e b e am th ro u g h th e p o la riz in g b e am s p litte r, the n o rm a liz e d Jones
vector is
Bi n
(A-4)
T h e p o larization s ta te of th e p ro b e b e a m for th e sim p le e x am p le of
p a ssin g th ro u g h a X /4 w a v e p late a n d the G a A s s a m p le a n d reflecting
back thro ug h the G aA s a n d a n analyzer is c alcu lated as
Bлut = p l - r ( 0 3 ) T ( - 0 2 ) - g - r ( 0 2)
(A-5)
w h e re the rotation m atrices r(0) are defined as
r(0) =
cos 8
sin 0
-sin 0
cos 6
(A-5)
the A,/4 w ave plate is re p re s e n te d by [17]
wl =
? 1 O'
0
i
(A-6)
the analyzer is re p re s e n te d by
1 O'
Pi
0
0
(A-6)
118
a n d th e G aA s test sam p le is d escrib ed by
id
0
0
.-id
(A-7)
w h e re d is the p h a se d e lay asso c iate d w ith each p o la riz a tio n c o m p o n e n t
of th e G aA s.
T h e v a lu e of g ( l , l ) a n d g(2,2) in the G a A s m a trix are
d e te rm in e d by the m o d el cho sen to re p re se n t the reflection coefficient for
GaAs.
A m athem atica n o teb o o k s h o w s an ex am p le of a n u m e ric al analysis
to d e te rm in e total o u tp u t light in te n sity b a se d o n Jones C alcu lu s for the
infinite b e am m odel.
лл>
119
M ATH EM A TICA NOTEBOOK
The input vector v and quarter waveplate are defined
v = {1,0}
t = -45 Degree //N
k = Pi / 4 //N
w4 = {{E x p [-1 k], 0}, {0, Exp[I k] } >
rtheta = {{Cos[t], Sin[t]}, {Sin[-t],
w4thp = w4 . rtheta
{1,
Cos[t]}}
0}
-0.785398
0.785398
{{0.707107
-
{{0.707107,
{{0.5
-
0.5
0.707107
I,
-0.707107},
I,
-0.5
0],
{0,
0.707107
{0.707107,
+ 0.5
I},
{0.5
+
0.707107
0.707107}}
+
0.5
I, 0.5
+
0.5
The phase delay u for a single pass through a 450 |im
thick GaAs sample with an index of refraction of
3.6.
Hgt = 450
Volt = f * amp * Sin[w]
Phase = Hgt * (3.3 * (10 A 7)) * (1 * (10 A -6))
Phmod = Volt / (1.78 * (10 A 3))
u = Phase + Phmod/2
450
a mp
f
Sin[w]
14850.
0.000561798
14850.
+
amp
f
Sin[w]
0.000280899
a mp
f
I}}
Sin[w]
The reflection coefficient mbi for the
infinite beam model
I}}
120
R1
R2
T1
T2
=
=
=
=
-.555
.555
.445
1.555
Cosp = Cos[u]
Sinp = Sinful
Ra = ((Tl*T2*B*(Cosp + I Sinp))/
(1 + (R2 * B*(Cosp + I Sinp))))
mbi[f_] = Rl - Ra
-0.555
0. 555
0. 445
1. 555
Cos[14850.
+ 0.000280899
a mp
f
Sinfw]]
Sin[14850.
+ 0.000280899
a mp
f
Sinfw ]]
(0.691975
I
(1
+
B
(Cos[14850.
S in [14850.
0.555
-0.555
B
+
I
S in [14850.
-
(0.691975
I
Sin[14850.
+
0.555
I
0.000280899
0.000280899
(Cos[14850.
+
B
+
a mp
(Cos[14850.
a mp
+
f
Sin[14850.
+
f
amp
Sinfw]]
+
f
Sinfw]]
Sinfw]]))
a mp
f
+
/
Sinfw]]
f
Sinfw]]))
f
Sinfw]]))
+
+
+
0.000280899
(Cos [1 4 8 5 0 .
amp
0.000280899
0.000280899
B
0.000280899
(1
+
amp
0.000280899
0.000280899
amp
f
a mp
f
/
Sinfw]]
+
Sinfw]]))
Calculation of the components of the GaAs matrix
a = Absfmbifl]]
PowerfE,
I
* Exp[I Argfmbifl]]]
Arg[-0.555
(Cos[14850.
I
(1
I
I
B
+
(0.691975
B
+
I
B
Sinfw]]
amp
+
+
Sinfw]]))
+ 0.000280899
a mp
+ 0.000280899
(C o s [ 148 5 0 .
S in [14850.
amp
0.000280899
0.000280899
S i n [14850.
+ 0.555
B
+ 0.000280899
(Cos[14850.
S in [14850.
-
(Cos[14850.
(1
(0.691975
0.000280899
S i n [14850.
+ 0.555
Abs[-0.555
+
a mp
amp
+
+
Sinfw]]))
0.000280899
+ 0.000280899
/
Sinfw]]
Sinfw ]]))]]
Sinfw]]
amp
a mp
a mp
/
Sinfw]]
Sinfw]]))]
+
121
b = Abs [mbit-1]] * Exp[I Arg[mbi[-1]]]
Power[E,
I
Arg[-0.555
(Cos[14850.
I
(1
+
I
B
I
B
-
B
-
-
Sin[w]]
amp
a mp
-
amp
amp
+
a mp
{{Costs],
{{1 ,
0 },
{ {C os[s],
S in fs]},
{0 ,
{-Sin [s],
{Sin[-s], Costs]}}
Costs]}}
0 }}
S i n f s ] },
{0,
0}}
Output Jones vector and the calculation of
the output intensity at the detector for a
loss coefficient B = 0.6 and a 1 Volt signal
/
Sin[w]]
Sinfw ]]))]
Analyzer definition
poltheta = {{Cosfs], Sin[s]},
pola = {{1,0}, {0,0}}
wholepol = pola . poltheta
/
Sin[w]]
Sin[w ]]))
0.000280899
0.000280899
a mp
S infw ]]))]]
Sin[w]]
amp
+
Sin[w ]]))
0.000280899
0.000280899
(Cos [ 1 4 8 5 0 .
S i n [14850.
a mp
0.000280899
0.000280899
S in [14850.
B
0.000280899
(0.691975
+ 0.555
I
-
(Cos[14850.
S i n [14850.
-
(Cos[14850.
(1
(0.691975
0.000280899
S in [14850.
0.555
Abs[-0.555
-
+?
+
122
bap[cunp_, w , B_] = wholepol .
{{a, 0}, (0, b}} . w4thp . v
{(0.5
-
0.5
I)
Power[E,
Arg[-0.555
-
(Cos[14850.
I
(1
+
I
(0.691975
+ 0.000280899
Sin[14850.
0.555
+
(Cos[14850.
B
Abs[-0.555
I
(1
+
+
(0.691975
B
-
+
0.555
(0.5
+
B
+ 0.5
I
+
Power[E,
(0.691975
-
B
-
I
S i n [14850.
-
(0.691975
(Cos[14850.
I
(1
+
0}
B
amp
-
a mp
amp
a mp
/
Sinfw]]
Sinfw ]]))]
+
C os[s]\
Sinfw]]
a mp
+
Sinfw]]))
/
+
0.000280899
a mp
0.000280899
-
+
Sinfw]]))
a mp
Sinfw ]]))]]
B
(Cos[14850.
Sin[14850.
Sinfw ]]))]]
0.000280899
Sinfw]]
0.000280899
S i n [14850.
0.555
I
,
-
a mp
Sinfw]]
0.000280899
-
/
B
(Cos[14850 .
0.000280899
Abs[-0.555
+
0.000280899
S i n f 14850.
+
Sinfw]]))
+
a mp
+ 0.000280899
-
0.555
amp
0.000280899
I)
(Cos[14850.
Sinfw]]
0.000280899
(Cos[14850.
Sinfl4850.
I Arg[-0.555
(1
+
Sinfw]]
a mp
+
0.000280899
S i n [14850.
I
amp
S i n [14850.
(Cos[14850.
amp
0.000280899
0.000280899
I
B
-
Sinfw]]
amp
Sinfw]]))
0.000280899
0.000280899
a mp
+
a mp
/
Sinfw]]
Sinfw ]]))]
+
S in [s]\
123
bone = b a p [1,90 Degree,.6]
{(0.5
-
0.5
I)
Power[E,
Arg[-0.555
-
(Cos[14850.
I
(1
+
I
(0.415185
+ 0.000280899
S i n [14850.
0.333
(Cos[14850.
0.000280899
I
Abs[-0.555
S i n [14850.
-
I
+
I
Sin[14850.
(0.5
Power[E,
I
+
0.5
I
S i n [14850.
I
I
+
-
-
S i n [s ],
0}
+
Sin[90
Degree]]
0.000280899
Degree]]))]
-
+
Sin[90
Degree]]))
/
-
Sin[90
-
Degree]]
0.000280899
S i n [14850.
Sin[90
0.000280899
(Cos [ 1 4 8 5 0 .
0.000280899
I
I
-
0.000280899
S i n [14850.
0.333
Degree]]))
+
Sin[90
D egree]]))]]
(0.415185
(Cos[14850.
(1
Degree]]
Sin[90
(Cos[14850.
S i n [14850.
-
+
-
0.000280899
Abs[-0.555
Sin[90
+ 0.000280899
(Cos[14850.
0.333
Degree]]
I)
0.000280899
+
D egree]]))]]
+
Sin[90
Arg[-0.555
(0.415185
(1
/
+
Sin[90
Sin[90
+ 0.000280899
(Cos[14850.
0.000280899
+
Degree]]
0.000280899
0.000280899
S in [14850.
+ 0.333
Cos[s]
+
Degree]]))
+
Sin[90
+
Degree]]
Sin[90
(0.415185
(Cos[14850.
(1
Sin[90
+ 0.000280899
Sin[90
+
Degree]]))
-
Sin[90
-
Degree]]
Degree]]
0.000280899
+
Sin[90
D egree]]))]
/
124
ex = Chop[bone[[1]]]
(0.5
-
0.5
I)
Power[E,
Arg[-0.555
-
(Cos[14850.
I
(1
I
(0.415185
+ 0.000280899
S i n [14850.
+ 0.333
+
0.000280899
(C os[14850.
0.000280899
I
Abs[-0.555
S in [14850.
-
(1
I
+
I
Sin[90
+
0.5
+
-
Sin[s]
Degree]]
+ 0.000280899
+
Degree]]))
Sin[90
Sin[90
-
0.000280899
0.000280899
(C os[14850.
S in [14850.
-
/
Degree]]
+
Degree]]))]
Degree]]
Degree]]))
+
/
-
Sin[90
-
Sin[90
Sin[90
Degree]]
0.000280899
+
Sin[90
Degree]]))]]
(0.415185
-
0.000280899
S in [ 14850.
+ 0.333
I
Degree]]))]]
-
(Cos[14850.
(Cos[14850.
I
+
Sin[90
0.000280899
0.000280899
(1
/
I)
S i n [14850.
I
Degree]]))
Sin[90
Sin[90
+ 0.000280899
Arg[-0.555
+ 0.333
Abs[-0.555
Degree]]
+ 0.000280899
(Cos[14850.
(0.415185
I
Sin[90
+
+
0.000280899
S i n [14850.
(0.5
Power[E,
(1
+
S i n [14850.
+ 0.333
Cos[s]
Degree]]
(0.415185
(Cos[14850.
I
Sin[90
-
0.000280899
(C os[14850.
S i n [14850.
Sin[90
-
-
Degree]]
Sin[90
0.000280899
0.000280899
+
Degree]]))
Sin[90
Sin[90
/
Degree]]
Degree]]))]
+
125
cex = Conjugate[ex]
Conjugate[(0.5
I
-
0.5
Arg[-0.555
I)
-
Power[E,
(0.415185
(Cos[14850.
I
(1
+
+ 0.000280899
S i n [14850.
0.333
+
0.000280899
(Cos[14850.
0.000280899
I
-
(1
+
0.333
(Cos[14850.
S i n [14850.
+
(0.5
Power[E,
I
+
(0.415185
+
0.5
(1
+
S i n [14850.
/
+
Sin[90
D egree]]))]
Degree]]
0.000280899
Degree]]))
/
-
Sin[90
-
+
Sin[90
Degree]]
0.000280899
+
Sin[90
D egree]]))]]
(0.415185
-
0.000280899
S i n [14850.
0.333
-
(Cos[14850.
S i n [14850.
Sin[90
0.000280899
0.000280899
I
+
Degree]]))
-
(Cos[14850.
(Cos[14850.
I
Degree]]
Sin[90
-
0.000280899
-
Degree]]
Sin[90
-
(Cos[14850.
0.333
Abs[-0.555
D egree]]))]]
I)
S i n [14850.
I
/
+
Sin[90
Sin[90
0.000280899
0.000280899
(1
Degree]]))
+
Sin[90
+
Arg[-0.555
I
0.000280899
+ 0.000280899
0.000280899
Cos[s]
Degree]]
+ 0.000280899
Sin[14850.
I
Sin[90
+
(0.415185
(Cos[14850.
I
+
Degree]]
+
Sin[90
S i n [14850.
Abs[-0.555
Sin[90
+
Degree]]))
/
-
Sin[90
-
Degree]]
Sin[90
Degree]]
0.000280899
+
Sin[90
D egree]]))]
S i n [ s ]]
i = ex * cex
Conjugate[(0.5
I
-
0.5
Arg[-0.555
I)
-
Power[E,
(0.415185
(Cos[14850.
I
(1
+
+ 0.000280899
S i n [14850.
0.333
+
(Cos[14850 .
0.000280899
I
S i n [14850.
Abs[-0.555
-
I
+
+
Cos[s]
+
r T?
S i n [14850.
(0.5
T
A m
+
0.5
f _ H
Sin[90
0.000280899
(Cos[14850.
0.000280899
I
Degree]]
0.000280899
+ 0.000280899
S i n [14850.
0.333
+
Degree]]
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