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FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM MOLECULAR BEAM MICROWAVE SPECTROSCOPY

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300 N ZEEBROAD, ANN ARBOR, Ml 48106
18 BEDFORD ROW, LONDON WC1 R 4EJ, ENGLAND
8108446
BALLE, TERRILL JOSEPH
FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM MOLECULAR
BEAM MICROWAVE SPECTROSCOPY
University of Illinois at Urbana-Champaign
University
Microfilms
I n t G r n f l t i O n S U 300N ZeebRoad,AnnArbor,MI48106
PH.D.
1980
FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM
MOLECULAR BEAM MICROWAVE SPECTROSCOPY
BY
TERRILL JOSEPH BALLE
B.S., University of Oregon, 1975
THESIS
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Chemistry
in the Graduate College of the
University of Illinois at Urbana-Champaign, 1980
Urbana, Illinois
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
THE GRADUATE COLLEGE
May, 1980
W E HEREBY RECOMMEND T H A T T H E THESIS BY
TERRILL JOSEPH BALLE
F.NTTTT.FT)
FABRY-PEROT CAVITY, PULSED FOURIER TRANSFORM MOLECULAR
BEAM MICROWAVE SPECTROSCOPY
BE ACCEPTED IN PARTIAL F U L F I L L M E N T OF T H E REOUIREMENTS FOR
T H E DEGREE O F
DOCTOR OF PHILOSOPHY
\/Djflcctor of Thesis Research
HOPCI of Department
Committee on Final Examumjionf
N&£
t Required for doctor's degree but not for master's
iii
Acknowledgments
I first want to thank Dr. W. H. Flygare, for without him none of
this would have happened.
I thank him for his faith and support over the
time it took to bring this project into fruition.
I wish to thank several members of the research group after I first
joined who taught me how to get things done in spite of everything.
I
thank the research group after the success of the spectrometer for doing
all the things I couldn't do and for carrying on the work.
I thank Dr.
J. D. McDonald for his invaluable advice concerning the pulsed valve and
many areas of spectroscopy.
The help of the Materials Research Laboratory is gratefully acknowledged.
They have contributed to the success of this spectrometer through
support of me, through new equipment money and most importantly through
their excellent machine shop.
I want to thank Wayne Craig, the machine
shop supervisor, for all his advice and help.
Special thanks goes to
Bud Dittman and Tom Koerner for all the machining and welding on the
spectrometers.
I wish to thank the Chemistry Machine Shop under Elmer Lash and the
Student Machine Shop under Bill Hempler and Ron Harrison for teaching me
everything I know about machining.
Another key part of this and every project in physical chemistry is
the contribution of Chuck Hawley and the Chemistry Electronics Shop.
The
chemistry department is truly fortunate in having two superb electrical
engineers, Chuck Hawley and Carl Reiner, who designed all the special
iv
electronics that were needed.
Special thanks also goes to Al Saldeen for
all his help.
I thank Art Gaylord of the Chemistry computer services for doing the
initial interfacing of the experiment to the departmental VAX computer
and all other help.
The help of Evelyn earlier and Kim Burr is appreciated for typing
this and our papers.
The support of the University of Illinois through teaching and research
assistantships and to Allied Chemical for a fellowship is gratefully acknowledged.
Finally, I wish to thank my first wife, Andrea, for all that she taught
me and for leaving me.
she does.
And to my second wife, Mary Beth, for doing all that
V
TABLE OF CONTENTS
Page
INTRODUCTION TO CAVITY SPECTROMETERS
1
FABRY-PEROT THEORY
3
A.
B.
C.
D.
Introduction
3
Quasi-optical Fabry-Perot Theory
4
1. Electromagnetic Fields in a Fabry-Perot Cavity. . . 4
2. Resonant Frequencies
. 8
3. Q Factor
9
Circuit Theory of a Microwave Cavity
io
Maxwell's Equations in a Microwave Cavity
18
PULSED MOLECULAR BEAM
21
A.
B.
C.
D.
E.
21
22
25
26
28
Introduction
Molecular Density Distribution
Molecular Flow Velocities
Numbers of Molecules Pulsed
Cooling of Rotational Energy
MOLECULAR POLARIZATION
31
A.
B.
C.
D.
E.
31
32
33
36
40
40
42
51
55
Introduction
Bloch Equations
Transient Absorption
Transient Emission
Signal Characteristics
1. Spontaneous Emission
2. Doppler Dephasing and Transit Time
3. Signal Detection
4. Resolution and Sensitivity
THE SPECTROMETER
58
A.
B.
C.
D.
E.
F.
58
58
67
70
71
75
Introduction
Block Diagram
Design Considerations
Molecular Spectra
Potential of the Method
Discussion
VI
Page
VI.
VII.
VIII.
A JOSEPHSON JUNCTION MIXER
78
A.
Introduction
78
B.
Josephson Junction Detector-Mixer
78
LIST OF REFERENCES
89
VITA
95
1
I.
INTRODUCTION TO CAVITY SPECTROMETERS
There have been many cavity spectrometers built since the development
of radar and radar techniques in World War I I .
Bleaney and Penrose m e a -
sured the inversion spectrum o f ammonia in a closed cavity spectrometer
in 1946.
The simplest w a y o f detecting a rotational transition is to
note a change in the reflected o r transmitted microwave power.
Due to
the smallness o f this change m o s t people choose to modulate the molecular
absorption with either the Stark o r Zeeman effects and detect with a phase
sensitive detector. ' ' ' ' '
A superheterodyne receiver can also b e
Q
easily used to detect absorption lines.
Scanning cavity spectrometers
6 9
have also been built. '
Closed microwave cavities are standard equip-
ment in all electron spin resonance spectrometers,
utilizing the effec-
tive concentration of the field for small samples.
Dielectric measurements
have many times employed microwave cavities to obtain the real and imagin* of- the
«. dielectric
* i J.
4- 4- 11,12,13,14,15
ary part
constant.
JT
Microwave
cavities are
16 17
also the heart of molecular beam masers '
and beam maser spectrometers . '
Open resonators of the Fabry-Perot type have been used increasingly
because of their high Q, the easy access to their interior field regions
6 1 ft
and the ability to work from the microwave to optical frequencies.
20,21,22,23,24,25
Lee and White.
A double resonance Fabry-Perot cell has been built by
26
Pulse techniques in microwave cavities were first discussed b y Dicke
and Romer.
27 28
'
Utilizing these pulse techniques in the microwave region
2
one can create a coherent macroscopic polarization in the gas which subsequently emits radiation.
This radiation is given off at a frequency
that is proportional to the difference in energy between two quantized
states of the molecules m
the gas.
This radiation is damped out by
relaxation process in the gas such as collisions, Doppler dephasing and
transit time effects.
Because of this damping, all excitation and
measurements have to be done in times short compared to the fastest relaxation process.
Dicke and Romer
27
were the first to measure the emission
from a low pressure gas sample of ammonia in a closed cavity.
29
from OCS in a waveguide cell was observed in 1967.
Emission
Laine observed
emission from ammonia in a closed cavity by using fast passage techniques.
A pulsed Fabry-Perot microwave spectrometer was built
32
relaxation times.
31
and used to measure
The transient absorption and emission theory was
derived in terms of the density matrix of the system by McGurk, et. al.
•20
This theory
nA
'
30
33
OC
'
is fundamental to the work presented in this thesis.
Pulsed time domain spectroscopy in a waveguide cell of rotational levels has
been developed as a routine spectroscopic tool.
36 37
'
A further advancement
combines the techniques of pulsed Fourier transform spectroscopy with a
Fabry-Perot cavity and a pulsed nozzle expansion.
38,39
3
II.
A.
FABRY-PEROT THEORY
Introduction
The theory of the Fabry-Perot resonator has been developed on two
fronts.
Historically the Fabry-Perot interfrometer with plane reflecting
mirrors has been used in the optical and infrared regions.
m
Connes
40 41
'
1956 and 1958 suggested the use of spherical mirrors in the optical
42
region.
In 1961, Boyd and Gordon
43
also Fox and Li,
developed the theory
for plane and spherical mirror Fabry-Perot cavities in the microwave region.
This approach to a microwave resonator, from the high frequency limit treats
the resonator as two mirrors with a wave bounding back and forth between
them.
Optics is used m
the diffraction limit.
The other approach to the
theory of a Fabry-Perot resonator is from circuit theory, or from the low
frequency limit.
The resonator is considered to be a tuned series RLC or
shunt GLC circuit, driven by an alternating voltage or current source.
Closed resonators have primarily been described from this point of view.
Both approaches will be used here.
The quasi-optical theory will be used to obtain the field modes and
resonant frequencies.
This approach to find the field distribution and
eigenvalues does not include the coupling to the resonator and hence does
not account for power flow into and out of the resonator.
This dependence
needs to be known to be able to calculate magnitudes of polarizating input
power and signal power coupled out into the detector.
Circuit theory
combined with Maxwell's equations enable one to treat these questions.
4
B.
Quasi-optical Fabry-Perot Theory
1.
Electromagnetic Fields in a Fabry-Perot Cavity
Inherent in the assumptions of the quasi-optical treatment are that
the dimensions of the mirrors are large compared to the wavelength, that
the fields are T.E.M. waves and are plane polarized.
43
distribution
To obtain the field
in the cavity, an initial wave, u , is considered leaving
one mirror, where u is
P
u
= ik J" uaexp(-ikR) (4iTR)~ (1 + cos8)ds-
This wave is the Fresnel field due to an illuminating aperature A, u
is
the aperture field, k is the propagation constant, R is the distance from
the aperture to an observation point and 9 is the angle which R makes with
respect to a unit normal to the aperture.
The field at a mirror after p
transits,up+1. is obtained by
up by
and ua by
up . up is
J replacing
tr
jr up +,.
J
1
the field across the other mirror which produced the field u
.
A steady
state is reached where the field distribution, u, between the two mirrors
is identical apart from a complex constant, y,
which depends on coordi-
nates .
up = (1/Y) P u and up+i = (1/Y) P + 1 u.
When this is substituted in the above equations, we get
u = Ylk A
u
exp(-ikR) (47rR)~ (1 + cos8)ds.
This equation can then be solved iteratively.
Another approach
44 45 46 47
' ' '
uses the solution to the wave equation following the concept of wave
48,49
beams.
5
The field distribution inside the resonator is given by these
methods for a TEM
mode as,
mnq
w
E(x,y,z,t) = E (t)H (»^)H (/fr—
exp(-r /w 2 )
o
m
w n
w w
cos (kz + (kr2/2R) - $ - irq/2) f
where
2
r
W
o
2
= x
=
W (z)
(
+ y
2
X
1/2
2 ? [ £ ( 2 R - *)]
>
= w
o
1/2
X z 2 1/2
[i + ^ - ^ - r j
7TW
£
o
-1
2
$ = tan (Xz/7Tw )
o
E (t) = E cos wt.
o
o
Referring to Figure 1, x,y,z are the rectangular coordinates from the
center of the cavity. H
are Hermite polynomials of order m,n.
m,n
*
radius of curvature of both mirrors is the same and equal to R.
fundamental or TEM
The
For the
mode the Hermite polynomials are unity and one has
basically a standing wave with a gaussian fall off going away from the
axis (x=y=0) of the cavity.
off as a function of z,
its value on axis.
so that when r = w the field has dropped to 1/e of
The value of w at z = 0 is w
beam waist diameter.
and 2w is called the
o
o
The beam waist is the minimum diameter that the
2
beam diameter attains.
wave front.
The function w(z) describes the gaussian fall
The factor kr /2R accounts for curvature of the
The phase front has to be curved because of the curved mirrors.
The only place the phase front is planar is at z = 0.
The term $ is a phase
shift difference between a gaussian beam and a plane wave.
The mode number
q is the number of nodes in the standing wave field, so that q + 1 is the
number of half wavelengths between the mirrors.
6
Figure 1.
The geometry and coordinate system of the Fabry-Perot cavity.
R is the radius of curvature of both mirrors.
between mirrors is &.
The distance
The center of the cavity is at x=y=z=0.
The beam waste parameter w
is measured at z=0, drawn in a
different position for clarity.
8
The factor 7lq/2 is included to change cos(kz) into sin(kz) depending on
whether the field is a maximum or zero at the center of the coordinate
system.
The beam waist dimension, 2w , is then the shortest distance
o
the molecules have to travel to enter and escape the field (assuming for
convenience that the molecules only interact with the field in this region)
This beam waist is a maximum when & = R, the confocal arrangement, and
falls to zero when A = 0 or Ji = 2R.
So that to maximize the time of
flight, one would want to work near the confocal geometry.
Using the
radius of curvature of one of our mirrors, 84 cm, and a mirror separation
of 70 cm, the beam waist at 10 GHz is 12.6 cm.
2.
Resonant Frequencies
The resonant frequencies V of the resonator for the TEM
mode
mnq
are
V = V [(q + 1) + (1/TT) (m + n + 1) cos" (1 - l/R)}
where V
= c/2Jl.
Using a mirror separation of 70 cm, V
that the dominant modes, TEM
,
is 214 MHz, so
, are separated by that amount.
The higher
order modes m or n ^ 0 can be easily seen,as the m,n even modes are excited by our coupling iris.
All modes are well separated in frequency
as the width of a mode 6v = V/Q is (for Q = 10 , V = 10 GHz) 1 MHz.
All
spectroscopic work is therefore done single moded, either in the dominant or
in a higher order mode. Experimental observations confirm the field distri45 47 50 51
butions and resonant frequencies given above. ' ' '
9
3.
Q Factor
The Q of a resonant system is a very useful and important property.
One of the ways to define Q is
Q = CO • W/P,
where to is the angular frequency of the radiation, W is the total energy
stored in the cavity and P is the power dissipation.
From the quasi-optical
point of view of a resonator, the power in the wave bounding back and forth
between two mirrors can be dissipated in two ways.
The wave can diffract
out the sides of the mirrors or can be dissipated in ohmic losses in the
metallic mirrors.
The diffraction losses can be made arbitrarily small by
just making the mirror diameter large.
42 43 44
has been done by many authors. ' '
The calculation of these losses
The power lost due to the finite
conductivity of the metal mirrors can be calculated, although it is difficult
to account for surface roughness or contamination.
For a good conductor, such as the aluminum we use in our mirrors, a
skin depth, 6, can be defined for O »
coe, where a is the conductivity
in l/(ohm m) and £ is the permittivity of the material.
52
<5 = (2/cou0) 1/2 ,
where y is the permeability of the material.
At this distance in the con-
ductor the amplitude of an electromagnetic wave has fallen to 1/e of its
value at the surface.
(a = 3.54 x 10
7
-5
For aluminum the skin depth is 8.5 x 10
cm
mhos/m) at 10 GHz.
The value of Q assuming this to be
the only source of power dissipation can be calculated to be,
52
10
5
For £ = 70 cm and 6 from before, 2 * 4.1 x 10 . Q'sas high as
5
2 x 10 have been measured in our cavity.
Cavities utilizing supercon-
11
53
ducting walls have reported Q's as high as 10
in the microwave region.
C.
Circuit Theory of a Microwave Cavity
A microwave cavity connected to an oscillator and a detector through
waveguide or coaxial cable can be represented by an equivalent circuit
54
as in Figure 2a.
The oscillator produces a voltage v and has an internal resistance R . The total power, P , available from this oscillator
0
o
2
is v /4R .
The iris couplings to the cavity and out to the load, R , are
represented by transformers of turns ratio n
and n„ respectively.
cavity itself is the series combination of R, L, and C.
The
Power dissipation
in the cavity is accounted for by R and energy stored in the magnetic and
electric fields is accounted for by L and C, respectively.
To simplify
the analysis, one can transform the input and output circuits to the middle
loop as m
Fibure 2b.
The generator and the detector load are assumed
matched to the input and output waveguides, hence R
= R .
U
The cavity
L
resonator is now a simple RLC tuned circuit driven by a generator.
The
impedance for the series circuit is,
2
2
Z = nn R + R + n„ R + i (toL - 1/coC) .
1
O
i.
L
The resonant frequency, CO , of the circuit is defined as that frequency
at which the reactance vanishes, therefore
to L = 1/to C o r co = 1//LC.
c
c
c
At this frequency the cavity appears as a pure resistive load to the input
circuit and the magnitude of the stored energy in the electric and magnetic
fields are equal.
R
•vww-
L
C
•UJUULLr
0
l:n.
l:n,
(2a)
£
n, R 0
JWWV-
R
•AA/WV
L
"UULUlr
C
n|R
(2b)
Figure 2.
Equivalent circuit of microwave cavity
12
As defined before the Q is,
to (Ene-gy Stored)
(Power Dissipated)
Four Q's will now be defined according to this definition.
The energy
2
2
stored in the inductor (1/2) LI , and in the capacitor (1/2) Cv , are
equal in magnitude and equal to the total energy stored W. The first Q,
the unloaded Q, Q , accounts for power dissipation in R which is (1/2)
2
RI so that,
to (1/2) L I 2
to L
Q = _S
_
£_ .
=
R
°
(1/2)RI2
Similarly, the input and output coupling Q's, Q .and Q
to L
and
2„i
Q*c2
cl = ~ ^2„—
n R
1 o
are given by
to L
° 2„
n„ RT
2 L
The sum of all dissipative elements defines the loaded Q, Q ,
L
1/QL - d/COcL) (R + n i 2 R Q + n
2
\)
= 1/QQ + V Q e l + l/Qo2-
The ratio of the unloaded Q to the two coupling Q's define two coupling
coefficients $ 1 and ft
a
2
n, R
1 O
Q
*0
2
n„ R
2 L
,a
xQ
O
The impedance of the equivalent circuit can now be written m
these parameters as
CO
c
CO
terms of
13
Using the impedance, the power flow through the circuit can be
calculated.
For instance, to calculate the power that is coupled into
the cavity from the generator, we find the power dissipated in R, P_.
R
P R = I 2 R = (n;Lv/|z|)2R
2 2„
n. v R
2
2w
2
2
R [a+31+e2)
2
td,_ o
2
+
Q o (---^) 2 ]
c
2
Dividing this by the power available from the oscillator P = v /4R gives,
ffl
P //P
R
o ~
2w
2
< » w
w
c 2 *
•*,<=--•#>
c
When the cavity is being driven on resonance (co = co ) this expression reduces to,
P„/P
R
=
43,
1
4Q
_ L
° <1+W*
2
2cl2o
The amount of power dissipated in the load, P , from the generator can be
derived in the same way and is,
4Q 2
43,3,
X
PT/P =
*
L
° (l + 3 1+ 3 2 ) 2
L
QclQc2
Further insight into the meaning of the coupling coefficients can be
gained by considering a reflection cavity.
output coupling iris.
This cavity has only one input-
Again using the circuit theory, the amount of power
coupled into the cavity on resonance is
43
P
R
4Q 2
/P =
= __h°
<1+e>2
QCQ0
14
where 3 is the one coupling coefficient and 1/Q_ = 1/Q„ + 1/Q • If 3 = 1
L
O
c
then PD/P^ = 1 and all the power that is coupled into the cavity is
i\
o
dissipated there. This is the matched load condition.
The transmis-
sion line is terminated m an impedance which is the complex conjugate
of its own impedance. No power is reflected and the VSWR = 1. Also
for this case, since 3 = Q /Q # Q = Q and Q_ = (1/2) Q . When 3 = 1
o
c
O
C
It
the cavity is said to be critically coupled.
undercoupled, so that Q
> Q
o
For 3 < 1 the cavity is
and more power is dissipated in the cavity
than in the coupling. When 3 > 1 the cavity is called overcoupled, so
that Q > Q and more power is now dissipated in the coupling than in the
o
c
cavity.
For a transmission cavity some power will always be coupled out
through the coupling iris that is not feeding power into the cavity.
If we call P , the power reflected from the input to the cavity
then,
P
0
=
P
F
+
P
+
R
P
L •
or
l
= P„/P
+ PT7P
F' o
Using the values for P„/P
R
cavity,
o
R
+ PT/P
o
and P„/P
L
L
•
o
obtained before for the transmission
o
we h a v e ,
43,
P /P
F
= 1 -
°
48,3 9
l
(l+e^Bj)
d-31+32)2
ci+e^Bg) 2
X
2
(I+SJ+BJJ)
2
15
With 3 1 = 3 2 = li one ninth of the input power is reflected and four
ninths is each dissipated in the load and in the cavity.
case Q Q = Q Q 1 = Q Q 2 and Q L = (1/3) Q Q .
flected from the input coupling i n s ,
Also for this
In order to have no power re1 - 3 + 3 = 0 .
Another way to think about cavity coupling is to consider the electric
waves or voltages traveling in the system.
Voltage reflection coefficients
can be obtained by taking the square root of the power reflection coefficients.
When one pulse excites a transmission cavity, there will be
an incident electric wave on the coupling iris.
a wave will be reflected and a wave transmitted.
radiate into the cavity.
Upon hitting the input iris
The transmitted wave will
When this wave encounters the opposite mirror,
part of it will be coupled out and part will be reflected with a 180°
phase shift.
Part of this reflected wave will be coupled out of the
original mirror and part re-reflected, building up the energy in the cavity.
The parts of the original wave that get coupled out the input iris after
being reflected back will be 180° out of phase with the input wave that
;just gets reflected from the iris without ever entering the cavity.
These
two waves will destructively interfere, making it appear that nothing is
coming out of the cavity input.
As the energy stored in the cavity builds
up, the cancelling wave builds up to a value depending upon the coupling
conditions and the Q's.
The totally transmitted wave also builds up with
this same cavity time constant X.
This time constant will be calculated
next.
Two other common ways of thinking about the Q factor of a cavity
are related to the bandwidth and decay time constant of a cavity.
To
16
derive the bandwidth of a cavity, the full expression for the power
dissipated in the load P is from before,
L
P
^ 2
2
2
P
l/ o ~
(i + 3 1 + 3 2 )
2
u
w
c 2
2
+
Q0 (--f)2
c
This will be rewritten as ,
P.
L "= P o
2
p
2^_VC_2
^o VV
V
c
where a = 4 3 , 3 , , 3 = 1 + 3, + 3 , , to = 2TTV and to = 2TW .
close t o the c a v i t y frequency,
2
2
V
V
V -V
(V+V )(V-V )
2V Av
c
c_ _
c
c ,,
c
v ~ v ~ v v ~
VV
~ VV
c
c
c
c
For frequencies
2Av
V
so t h a t , (Av = V-V )
c
p
L
- p
a
°02
2 2^2
*o
v
or
P.L = P
° 3 2 0y 2 + Av 2
o
This is the power transmitted to the load as a function of frequency.
2
When V = V ,P = P a/3 which was given previously. The frequencies above
and below resonance where the power has fallen to half its peak power on
resonance are given in the limit 4Q
2
2
» 8 by,
17
The bandwidth, 6v, at half transmitted power is then 6v = V /Q .
C
Ju
A measurement of the frequencies where the power has fallen to half the
maximum value is an easy way to find the loaded Q.
The decay time con-
stant, X, of a cavity can be obtained from the definition of the Q.
The
energy stored in the cavity will decay at a rate proportional to the amount
already present,
- d£ " W/T '
or
W = W e
o
-t/x
,
'
but this rate o^ energy loss must equal the power dissipated, or
as.
JI
P
" dt ~
" QL '
so that
W/x
= COQW/QL ,
or
X = QL/tOc .
In summary then,
Q L - Vc/<SV
°-L=T(V
and
6v = 1/2UX .
= 1 0 4 at V
For a cavity with Q
Jj
Usee.
= 1 0 GHz, Sv = 1 MHz and x = 0.16
c
18
D.
Maxwell's Equations in a Microwave Cavity
To complete our description of a Fabry-Perot cavity, we will use
Maxwell's equations to relate the electric field in the cavity to the Q
and hence to the power flow. Maxwell's equations for a lossy and polarizable dielectric are (in cgs units)
-XT
V x E =
c dt
->-
4ir -»•
1 3D
V x H = — J + ^^-^f
C
C dt
D= V • B = 0
-*•
->
D = E + 4irP
B = VH
-> .
J
OE ,
Taking the curl of the first equation and substituting to get the wave
equation for the electric field and the polarization, we get
V 2 E -i2H. a E _
c
(JL) i =
c
iIHp.
c
One way to solve this equation would be to expand the normal mode functions that we derived earlier from the quasi-optical approach.
Instead
of doing this, for the sake of clarity, let us expand in terms of a plane
standing wave,
E(r,t) = I An(t) Un(r) ,
19
where we take U (r) to be simply sin k Z.
n
n
Then k
n
has to satisfy
the boundary and mode number requirements of the cavity so that k
n
where n is some integer and i is the distance between the mirrors.
2
k
2
= -5—
*
Also
2
= to /c .
If this is done, we can get a simple equation for the time
dependence of the electric field driven by the polarization, (setting
u = 1)
An + 4rra An + ton2 An = -47T T
PUmdv.
«
This equation can be further simplified in two ways.
The finite conductivity
losses can be represented by the finite loaded Q of the cavity, so that 4TTC
will be replaced by CO /Q_. The spatial part of the polarization can also
n L
be expanded in terms of the same set of modes as the electric field,
P(?,t) = I G n (t) U n (?),
n
then
4/r JT P u dv =
m
The function G
4TT G
n
•
is in general complex to allow for components in phase and
out of phase with A(t).
In all of this we have assumed the spatial modes
to be orthonormal,
T Un UmdV = 6
n m
nm
«
Also since all modes are well separated in frequency we can restrict all
discussion to only one mode.
to
A(t) + ^ A(t)
Q
L
So that now,
+ CO 2A(t) = -4TT G(t).
(II-D
The polarization will be calculated in a following section and this equation will then be used to find the electric field.
20
The energy stored in the electric field can be calculated simply
from;
2
2
W = J* (1/8TT) | E ) dV = (1/8TT) E V .
(H-2)
This integral, over the volume of the cavity, has been done for the gaussian
standing wave given in Chapter II, section B,
W = (1/8TT) E
55
so that,
2
x 7T&(w / 2 ) 2
o
o
where H is the distance between the mirrors, w
is the maximum value of the electric field.
(II-3)
is the beam waist and E_
o
o
From before the energy stored
is related to Q by,
to w
and the power dissipated on resonance in the cavity is given by,
48,
p /p
R
4Q
l
-
=
°
M1+82)
2
2
L
QclQ0 "
The maximum electric field can thus be calculated to be,
*o=
(327r
VL2/Va)c2cl)1/2-
For our cavity, H = 70 cm and as previously calculated w
V
c
= 10 GHz, so that V = 2.2 x 10 3 cm 3 .
Then, for Q
L
= 6.3 cm for
= 10 , Q
ci
= 3 x 10
and an input power of 10 m watts, the peak electric field is 4.7 volts/cm.
With a medium power TWT, input powers of 40 watts are possible, which leads
to a peak electric field of almost 300 volts/cm.
21
III.
A.
PULSED MOLECULAR BEAM
Introduction
There have been many papers on the gas dynamics of molecular beams.
The term molecular beam, as used here, will mean the expanding gas from
the nozzle, even though there is really no collimated beam of particles.
56
57 58
Sources for much of the theory and data are in books,
reviews '
or in the Rarefied Gas Dynamics series.
For our microwave spectrometer,
the important quantities are the spatial density of molecules in the
expanding beam, the molecular velocities, the number of molecules in a
pulse and finally their temperature.
The molecular or dimer spatial
distribution and velocity are involved with the Doppler effects which
will be discussed later.
The number of molecules and their temperature
are used to compute the number density difference between two rotational
levels at thermal equilibrium.
This number density difference is used
in the next chapter to compute the power emitted by the molecules.
All
experiments are done with a flat thin plate orifice bolted on the bottom
of a solenoid pulsed valve.
As used here, the gas dynamic theory is
identical for the pulsed or continuous expansions.
Steady state flow
conditions are established quickly compared to the pulse valve open
time.
61
We do not use any skimmer, the so-called free jet.
all work is done with seeded beams.
between 95% and 97% of a rare gas.
In addition,
Most expansion mixtures contain
Therefore the gas dynamics will be
dominated by the expansion properties of rare gases.
22
B.
Molecular Density Distribution
A simplified density distribution is used for distances far from
,_.
60,61
n
the nozzle,
p(r,e> = P n (D 2 /r 2 ) cos m B ,
(III-I)
where D is the nozzle diameter, r and 9 are the radial distance and
polar angle from the nozzle orifice, as in Figure 3.
The density of
molecules at the nozzle is given by p .
For an effusive source it can be shown by the kinetic theory of
gases that the power of cos 9 is m = 1.
occurs m
Effusive flow through a nozzle
the realm where each molecule's motion is independent of all
other molecules.
The condition for effusive flow through a circular
nozzle of diameter D is,
X'/D »
1 ,
where X' is the mean free path, and X'/D is called the Knudsen number.
The
mean free path is given by,
X' =
(/2~TT p n
d2)"1
,
where d is the molecular diameter.
For an effusive source p = p , the
n
o
—8
source or stagnation molecular number density. Using d = 4 x 10
cm and
-5
1 atmosphere, the mean free path is 4.3 x 10
mm. The smallest nozzle
diameters we have used are ~.l mm.
For practically all our experiments we use source pressures greater
than one atmosphere, therefore we are far from the effusive limit.
NOZZLE
A
•". v,
Figure 3.
Geometry of the nozzle relative to the Fabry-Perot cavity. The
radial distance and velocity from the nozzle are r and v respectively. The polar angle from the nozzle is 0. The projection
of v on the z axis is v = v sin 8.
o
z
o
24
For a supersonic jet expansion, the basic isentropic relations
for a perfect gas are
p/Po = (1 + 3ti. M 2 ) " 1 / ^ - ! )
(III-2)
= (T/T ) 1 / 2 = (1 + ^ ) "
(III-3)
a/a
O
O
1 / 2
2
v = Ma.
(III-4)
In these equations, the o subscript denotes source conditions, M is the
Mach number, y is the ratio of specific heats (y = 5/3 for a rare gas),
a is the local speed of sound and v is the flow velocity. The definition
of a is
a
i
= /vkT /m.
i
Returning to equation (III-2), we can now compute p by noting that
the Mach number is one at the nozzle plane,
v
n
P
o
v
2 '
,
(III-5)
and for a rare gas p = p (0.65). With 1 atmosphere and 300°K as our
n
o
19
3
19
source conditions rp = 2.69 x 10 molecules/cm , so that rp = 1.7 x 10
o
n
3
molecules/cm . Using a nozzle with a diameter of 1 mm, the number densaty
on the nozzle axis and at the cavity center is p(r = 17 cm, 6 = 0 ) =
14
3
6 x 10 molecules/cm . This number density would correspond to a static
pressure of 17 microns.
We will discuss later, in the section on Doppler broadening and in
two upcoming publications, the ability to use the observed time domain
rotational line shape to measure the power of the cosine distribution in
equation (III-l).
25
Another question is whether or not the dimers follow this same distribution. Mass separation effects have been observed in beam expansions.
Condensation reactions that produce not only dimers and trimers but
clusters containing thousands of molecules have been observed and
studied.
'
An experiment could be done to measure the power of the
cosine for a monomer and a dimer in the same gas mixture to see if the
dimers were distributed differently.
C. Molecular Flow Velocities
In the reservoir the static gas behind the nozzle has, of course,
zero flow velocity. A static gas has a Maxwellian distribution around
zero velocity and an average speed, v , of,
3i
~kJF
a
VTT
m
During the expansion the random translational kinetic energy and
any internally stored energy (rotations and vibrations of polyatomics)
is converted into mass flow through binary collisions. The effective
rotational and vibrational temperature drops, the velocity distribution
narrows and moves out along the flow velocity axis.
Since local thermo-
dynamic equilibrium was assumed to exist at all times m the derivation
of equations (111-2,3) that describe this process, we can calculate the
flow velocity as a function of beam temperature. From equation (III-3)
and (III-4) we have
v =
!L ^o a- f )
Y-l
m
o
1/2
(III-6)
26
If the expansion were to convert all internal energy to directed
mass flow the temperature would be zero and equation (III-6) simplifies to
2Y kT
.
v m = (—T — - ) /Z ,
T
y-1 m
(III-7)
the terminal velocity. Note that if the temperatures were zero, the sonic
velocity would be zero and since the flow velocity is still finite the Mach
number is infinity. Also, the terminal velocity is only a factor of 1.4
larger, for a rare gas, than the average velocity in the source. For Kr
4
gas, the terminal velocity is 3.8 x 10 cm/sec.
D. Numbers of Molecules Pulsed
The number of molecules that are released, N, in a single pulse of
our nozzle is,
N = Cp v A t ,
n n n v
where v is the flow velocity at the nozzle, A is the nozzle area, t
n
*
n
v
eg
is the valve pulse time and C is a discharge coefficient
Y = 5/3).
(C = 0.55 for
At the nozzle, the Mach number is unity so that from equations
(III-3)
vn = an = ao ,Y+1x-1/2
(Jr
2r-)
This and Equation (III-5) gives ,
N = p a A t (0.31) .
*o o n v
(III-8)
For a pulse time of 3 msec, a circular nozzle of diameter 1 mm, 1 atmosphere
18
and 300°K source conditions, there will be 4 x 10 particles released.
4
Since the particles are traveling around v = 3.8 x 10 cm/sec, there are
27
about one tenth of this total inside the beam waist at any one time.
Also, because we use a seeded beam only 2%-5% of this number are dipolar
molecules. If one is interested in the molecular dimers, the number of
these dimers formed is then again some fraction of the number of molecules.
Just what fraction of the molecules are dimers has been the subject
e
i, 4.T.
*
• .- 64,65,66,67,68 _. .
..
. ,
of much theory and experiment.
It is generally accepted
that dimer formation proceeds by a three-body collision,
k
l
M + M + M^ D + M,
k„
where D is the dimer. The third body is needed to carry off excess
kinetic energy, allowing the complex to fall into the shallow potential
well.
The forward rate constant is proportional to the termolecular
collision rate and the back reaction is proportional to the bimolecular
collision rate. The termolecular collision frequency is proportional
2
to p D, where p is the pressure or number density at the nozzle and D,
as before, is the nozzle diameter. Although this simple model would
predict a plot of dimer concentration versus pressure and diameter to
fit as pD where q is 0.5, measurements are close. Reports of measureg*7
£C
CD
ments on dimer concentrations find q = 0 . 5 5 ,
q=0.63,
q = 0.5
and
69
q = 0.55.
Maximum mole fractions of dimers have been reported to be
65 67
as high as 0.1 for certain expansion conditions.
'
Other factors af-
fecting dimer concentration are the source temperature, the geometry of
the nozzle
'
and clearly the species of the molecule involved. Just
as every molecule has a different boiling temperature depending on its
polarizability, multipole moments, mass, etc., so every molecule has
different expansion dynamics.
28
E.
Cooling of Rotational Energy
As mentioned before, energy in the internal degrees of freedom is
converted into increased mass flow by binary collisions.
collision frequency is proportional to pD.
The binary
Because energy exchange be-
tween translation and rotation is very efficient, rotational temperatures
should be equivalent to translational temperatures.
An expression giving
the terminal translational temperature of a beam can be derived from
equation (III-3).
The terminal Mach number can be given by
71 72 75
' '
MT=£(X'/D)(1-^,
where e is a collisional effectiveness constant.
given
as
to be M
For Argon this was
= 133 (pD) ' so that equation (III-3) can be written
71
T = T [1 + 5896 (pD) 0 - 8 ]" 1 ,
where p is in atmospheres and D m
cm.
Using this formula and for our
conditions, 1 atmosphere, 1 mm and 300"K,. the terminal translational
temperature calculates to be 0.3°K.
tures for various expansions
Measurements of rotational tempera-
71 73 74 75
' ' '
are higher but still in the
1°-4°K range.
A study of the vibrational and rotational relaxation of I„ in
75
seeded beams shows a way to control the rotational temperature.
Rota-
tional temperatures from 3°K, I- seeded in Ar, to 66°K, I_ seeded in
n-butane, were measured.
Because the processes of dimer formation and
rotational cooling cannot be separated, one will always have to trade
off dimer concentration and beam temperature.
A higher beam temperature
29
for dimer studies is needed to populate vibrational levels of the dimer.
If a higher than ground vibrational level is populated enough to observe
and assign rotational transitions in that state, this would enable us to
get a far better idea of what the true dimer potential energy well is
like.
No one has seen any of these vibrational satellites for rare gas
small molecule van der Waals complexes although they have been seen in
hydrogen bonded type complexes such as HCN-HF.
76
Presumably the reason
no one has seen these vibrational satellites is that the beam is too cold;
the HCN-HF complex was observed in a cooled static gas cell.
Vibrational
cooling proceeds with a slower rate and hence final vibrational temperatures are always higher than rotational temperatures.
75
One can bracket the rotational temperature in the cavity experiment
by measuring several rotational transitions.
A molecule can be picked
that has several transitions at nearly the same frequency but one that
has the energy levels of these transitions spanning a large energy range.
Since the molecules populate the rotational levels by the Boltzmann factor,
one can observe various transitions whose energy levels are at successively
higher energy levels until one can no longer observe the transition.
states of a symmetric top molecule could be conventionally used.
m
K
Even
a beam expansion, molecules are observed to follow the Boltzmann dis-
tribution.
For later reference, the number density difference between two rotational levels will be calculated.
j
th
The number density of molecules in the
..
state is,
N
where p
=
PJ
211 exp(-EVkT)
is the total molecular number density, g
is the degeneracy of
30
the level, q the partition function and E_ the energy of the level.
The
u
number density difference of molecules available for the transition
J •*• J + 1 is then
AN
= N_ - N -
O
J
J*rl
= (p /q)(2J+1)(e" E J /kT -e" E J+l /kT)
(III-9)
O
As an example, consider the pseudo-diatomic van der waals molecule
KrHCl with a rotational constant B a 1200 MHz and energy levels E_ =
u
hB J(J+1).
is
The rotational partition function for a diatomic molecule
77
q = | (1 + 1/3 <f> + ^ ( | ) 2 + . - . . ) ,
where 9 = hB/k, the equivalent rotational temperature.
valid only for T > 6, for our case 9 = 0.06°K.
This formula is
To show the effect on the
number density difference that temperature makes, we will calculate
AN /p
for several values of J and two temperatures.
0.5°K
300 °K
ANQ(0,l)/po
2.3 x 10~ 2
7.4 x 10~ 8
ANo(3,4)/pQ
1.2 x 1 0 _ 1
2.1 X 10""5
ANo(9,10)/Po
5.9 x 1 0 - 5
1.4 x 10~ 5
31
IV. MOLECULAR POLARIZATION
A.
Introduction
The last of the three main elements of the spectrometer will be discussed m
this chapter.
The groundwork and introductions to the theory as
OO "3/ 3C
we use it have been done.
'
'
This theory was developed for pulse
Fourier Transform Spectroscopy in a waveguide cell with a static gas.
The
extension of the theory to a Fabry-Perot cavity with a molecular beam is the
subject of a forthcoming publication.
Although this chapter will ignore
most of the subtle changes that occur, it will give most major effects and
point out the kinds of differences one would expect.
are presented and discussed first.
P
The Bloch equations
is re-expressed as the solution to a
driven and damped harmonic oscillator equation.
Since P
is mainly respon-
sible for the detected signal, its time behavior is followed from pulse
preparation to emission.
The solutions to the Bloch equations used in this
chapter are for traveling waves and have been given before.
'
'
Although
a standing wave can be written as the sum of two traveling waves oppositely
directed, the polarization for the standing wave in the cavity cannot be
analytically re-expressed as the sum of two polarizations from two traveling
waves.
Even though the polarization cannot be exactly obtained from the
traveling wave solution, it is a good approximation and will be used'here.
Other approximations used m
this chapter are that the molecules are consider-
ed to be stationary during their excitation and emission, and that the fields
are of uniform strength inside the beam waist diameter.
32
A.
Bloch Equation's
A derivation of the molecular polarization that is produced by pulse
excitation of microwave rotational transitions was first done by Dicke. 27
Derivations similar to those done in nuclear magnetic resonance
have been developed in this group.
' '
78
theory
These same types of equations
are used in the infrared and optical regions of the spectrum as well.
79 80
'
The three coupled differential equations that describe the coupling
of the molecular dipoles to the radiation field are
P
P
+ Ato P. + ~
1
P. - ACOP
i
r
2
+K
E
33
= 0
(IV-1)
2
fejd
+I±-o
o \ 4/
T2
(1V-2)
,
. /AN-AN \
S AN - E P. + T
° = 0
m
4
o i
4 \ T, /
(IV-3)
These equations are written in a reference frame rotating at the applied
frequence so that there is no time dependence of the electric field.
K is 2y.
E /h,
r
ij o
u
r
i;j
is the dipole
moment matrix element and Ato is
c
to - CO. to is the rotational transition angular frequency.
AN is the
population density difference between two rotational states and AN is
that difference at thermal equilibrium.
T1 and T_ are phenomenological relaxation times.
T accounts for
the relaxation of the population difference to equilibrium.
the loss of the coherent polarization.
T„ accounts for
There are many mechanisms for these
relaxations, the most common one being collisions.
Because kT at room temper-
ature is so much larger than hv at microwave frequencies, most collisions
have enough energy to change the rotational state of the molecules involved.
33
This change of rotational state thus contributes to the T
relaxation of
the population difference and also to the T 2 relaxation of the coherent
polarization, as the relative phase of the rotational oscillations is randomized.
For our nozzle expansion to first order the collisions are drastically
reduced, so that other types of relaxation processes are important.
These
processes are Doppler dephasing, spontaneous emission and transit time
effects, which will be discussed more fully later.
P
ization.
and P
P
are the real and imaginary parts of the macroscopic polar-
is the in phase (in phase with the driving electric field)
part and allows for dispersive effects such as wavelength changes in a
dielectric.
Just letting the cavity fill with air up to atmospheric pres-
sure, from 0.01 microns, casues the cavity fill with air up to atmospheric
pressure, from 0.01 microns, causes the cavity resonant frequency to change
6 MHz.
This means that the change m
the cavity resonance frequency due
to a gas pulse at about 10-20 microns average pressure would be around
2 x 10
times the 6 MHz or around 100 Hz.
This change is well below our
normal line position uncertainty ( 1 - 5 kHz) and is ignored.
P
is the out
of phase component and accounts for power flow from the field to the molecules and back.
It can be simply related to the absorption coefficient of
a gas, Y, by
T
B.
p.
_
4irto i
" " c E
o
Transient absorption
The three coupled differential equations are usually solved by Laplace
transforms for P., P and AN.
l
r
33
For the case that T. = T 0 , a simple second
1
2
^
order differential equation can be written for P . This is accomplished by
34
differentiating again with respect to time equation (IV-2) and substituting
into this the three original equations, to give
P
i + ( 2/T 2) P i
+
(AW
+ 1/T
2
+
o ' Pi
=
4T"" 2 '
(IV
~ 4)
Because this equation is also written in the rotating frame, the right
hand side has no time dependence.
In this frame, when the electric field
is turned on it looks like a step function. Initially, at time zero, P
and P
are both zero and AN is at its equilibrium value, so that,
P.(0) = 0
P (0) = -K2£
o (^
•
from eq. (IV-2). When these initial conditions are used to solve the
driven and damped harmonic oscillator equation for P, we get
l
ilK2AN E ( l / T J [ e
P (t) =
o o
1
1
-t/T
2
(costO't-tO'T n sintO't)-l]
1
4[(l/T2)2+tO'2]
where co' 2 = Ato2 + K2E
2
.
o
In the pulse Fourier transform experiment one naturally wants to
maximize P
and due to the exponential decay this only happens at times
very short compared to T„. In the limit that T. = T 2 = °° equation (IV-4)
reduces to,
P. + (Ato + K E
i
o
)P.=0
i
,
and using the same initial conditions the solutipn is ,
hK2E A N
P
i(t)
=
-
40)' ° s i n
M>t
'
2
2
2 2
where CO' = Aco + K E . in addition, the experiment is operated so that
o
Kff » Ato, which further simplifies the result to be
35
P. (t) =
i
KE
ftKAN
r-£ sin KE t.
4
(IV-5)
o
is called the Rabi frequency.
For microwave transitions, where
the dipole matrix element can easily be 1 Debye and the electric field
easily 5 volts/cm the Rabi frequency is,
Kff = 2\X E fa = 31.6 MHz .
O
"1J o "
Since the bandwidth of our cavity is around 1 MHz, if the oscillator is
exactly on the cavity resonance the molecular transition has to be less
than 500 kHz away, which makes Aco =
2TTAV
around 3 MHz.
A similar derivation for AN can be done using the same assumptions,
AN(t) = AN cos Kff t
o
o
.
(IV-6)
At time equal to zero, the polarization is zero and the population difference is at its equilibrium point.
When the electric field is turned on the
polarization starts to build and AN falls, at Kff t = TT/2 the polarization
is maximized and the population difference is zero.
This 7T/2 pulse is the
normal operating condition for pulsed Fourier transform NMR, microwave and
optical spectroscopy.
In a waveguide microwave experiment where the elec-
tric field is fairly uniform this maximum polarization condition is easily
seen, for constant input power the coherent emission signal magnitude follows the sinusoidal behavior in equation (IV-5) as one varies the pulse
length.
In the cavity experiment where there is a standing wave field the
electric field goes from zero to its maximum in 1/4 of a wavelength.
Mole-
cules that are at a anti-node see the large electric field and are rotated
t
through the angle 0 = KZ? t or more correctly 0 = J KB dt. Molecules at a
o
36
node see zero electric field.
So that for a static gas one has a distri-
bution of polarized molecules from zero to a maximum.
The signal from this
distribution of polarized molecules in the cavity increases as one increases
the pulse length, goes to a maximum but generally levels off as the pulse
length is increased further.
Increasing the pulse length past the IT/2 point, according to our
simple equations, should result in a decrease in the magnitude of the polarization.
At KE" t = TT the polarization should be zero and the population difo
ference should be the negative of its original value.
Since the population
is now inverted one can give an average time it takes a molecule to absorb
a photon.
TT/KB
If every molecule were in the lower level at time zero, at time
every molecule is in the upper level.
All of the previous discussion was in the rotating frame.
In the lab
frame the electric field does not appear as D.C. so that the driving term
m
equation (IV-4) will be oscillating at to. The polarization will be
oscillating at to and modulated by Kff .
D.
Transient emission
The effect of the microwave pulse is to create a macroscopic polarization in the gas sample.
The molecules in each successive field anti-node
will be polarized X/2 wavelength apart and 180° out of phase.
Classically
such a phased array of dipoles will coherently radiate in a direction along
the line of the anti-nodes.
Each molecule is phased in such a way that all
emissions add up (in the absence of movement) together.
This coherent
emission from the molecules after the microwave pulse has passed occurs at
the rotational transition frequency.
Also the power due to the coherent
emission is proportional to the number density difference squared.
37
Returning to equation (IV-4) to obtain the time evolution of P.
after the pulse we set E
= 0 and get
P\ + (2/T2) P ± + (Ato2 + 1/T22) P x = 0
.
(IV-7)
Equations (IV-5) and (IV-6) are used as the solutions for the polarization
and population difference that describe the preparation of the system
prior to pulse cut-off.
The pulse length, X , is chosen to be of a length
ir
such that KE" X = Tr/2. With this idealized situation the values of P ,
op
i'
Pr'. A N and Pi at t = xp will be,
hKAN
o
4
P (X )
1
p
P v(x ); * 0
r p
AN(x ) = 0
P
vv
=
h.KAN
o
4T„
The last equation follows from equation (IV-2).
The solution to the damped
harmonic oscillator equation (IV-7) for times t > X
using these initial
IT
c o n d i t i o n s i s then,
-t/T
P (t) = e
hKAN
——•
cos AtOt .
(IV-3)
In the lab frame Ato is replaced by to , the rotational transition frequency.
This polarization is what we use to drive the electric field in the cavity.
The spatial part of this polarization is given by the field modes for a
Fabry-Perot cavity.
This part is not needed to obtain the signal that is
coupled out and detected.
The magnitude of the time dependent part of the electric field is
obtained by solving equation (II-l),
38
to
A(t) + -~
W
L
A(t) + W
2
A(t) = - 4TT G(t) ,
C
where P.(t), in equation (IV-8), will be used for G(t). P (t) as explained
before, will be ignored.
In taking the second derivative with respect
2
to time, of P., only the term with to will be retained as to » 1/T„
10
4
(10 versus 10 ). Substituting this into equation (II-l) gives
to
A ( t ) + ~ A ( t ) + CO
Q_
c
-t/T
A ( t ) = 47T tO
o
e
2
= to P ,. c o s to t
O lM
o
hKAN
r-24
COS to t
o
.
The transient solution to this differential equation will have the term,
exp-(to t/Q ) , in it, the buildup or decay of the fields in the cavity.
Because this time Q_/to ~ 0.2 ysec is short compared to the exponential
fall off of the driving fields T„ ~ 100 ysec, the cavity fields will be
considered to be steady state. In the steady state solution
to A(t) we
will take cos to t as the only time dependent driving term because
to » 1/T„.
With these assumptions, the well known solution with initial
conditions A(0) = A(0) = 0 is,
to
A(t) = -
where
2 P„
'M
2 _ .-XM_
/co to\ 2 1/2
/ 2
2,2_,
(to -to ) +
c
o
39
If the molecular emission signal is exactly on the cavity resonance,
to = to , t h e n ,
c
o
'
6 -
TT/2 ,
and
A ( t ) = p . M Q„ s m to t
iM
L
o
The electric field is
E(r,t) = U(r) P.„ QT sin CO t
lM L
o
The amount of power delivered to the cavity, P , by this process can
R
be obtained by integrating over the volume, that the molecules occupy, V ,
the time averaged product of the electric field and P (t),
P
R
=
J" < E *Pi> dV
V
s
W
o
P = ——
R
2
2
P V
I> iM s
20
Since this is the only source of power we can calculate the energy stored
in the cavity from the definition of the Q,
W = Q^/tO = 1/2 Q 2 P 2 V o .
L R c
L lM s
From the definition of the coupling Q we can find the amount of power
coupled out of the cavity, P_, and available to our detector by substituting
L
in the value of W,
40
L
~ 2 c2
"
to Q 2
PT (t) = — — —
L
2 Qc2
2
2c2
L
E.
s
, ~2t/T
(TlhKAN ) e
o
B
P (t)
iM
2
= T 2L i + g ^
V
s
,„_.. 4 „ . 2 "
e
(27r
A
H* V
(IV-9)
2 t / T
2
v
s
Signal Characteristics
1.
Spontaneous Emission
One feature of the output power calculated in the last section is
seemingly very unphysical, that being the dependence of the output power
on Q.
From conservation of energy the integral of the power over the time
of the decay should be independent of Q.
If we could collect all the energy
emitted by the molecules and if there were only this one power dissipation
process, then this integral should equal Nhto where N is the number of molecules emitting a photon of energy h.to.
If there were only this one relax-
ation process its rate would have to be proportional to Q so that the Q's
would cancel upon taking the integral.
The decay process has to depend on
the molecules being in the high Q cavity and on the coherence between the
molecules.
It has been shown that the spontaneous emission rate is enhanced
in a cavity.
ESR theory.
84
This phenomena has been used in NMR theory
78 81 82 83
' ' '
and in
The Einstein A coefficient for spontaneous emission can be
85
written as
A = B W(to) (exp (hCoAT) - 1) ,
where B is the Einstein B coefficient for stimulated emission and W(to) is
41
the electromagnetic energy density per frequency interval.
The B coeffi-
cient is obtained from quantum mechanics and is ,
B = iry2 ./3h2 •
ij
The energy density per unit frequency for free space is
3
W(C0) = "l^y
(exp (hto/kT) - l ) " 1 .
TT C
When these two results are substituted in for A one obtains the usual
expression for the spontaneous emission rate.
But the molecules are not in
free space, they are in a cavity with highly reflecting mirrors.
The energy
density per unit frequency in the cavity is enhanced by Q. W(to) for the
7 8
«. is
• given
•
u
cavity
by
4Q
Lh
W(co) = — —
-1
(exp (hto/kT) - 1)
The other factor that leads to an enhanced spontaneous emission rate is the
coherence established between molecular oscillations by the pulse.
This
coherence causes the emission rate to be faster by the factor N, the number
81 86
of oscillators. '
This enhancement in spontaneous emission rate by N
occurs for molecules in free space as well.
Combining all these factors gives for the A coefficient,
Try?
A = N
A
N
A =
4Q h
=J2
—
v
3h
15lK2AN Q T V
*
o L s
3V
where N was replaced by AN V . V
o s s
is the volume that the molecules (sample)
occupy, and V is the field volume of the cavity, see equations (II-2) and
(II-3).
This additional decay process has to be combined with equation
42
(IV-9) for the power coupled out of the cavity.
Lumping the molecular con-
stants together, the equation is now,
P
T
Q2
= *,n-FT-
(t)
L
-2t/T
e
e
,
(IV-10)
10 Q c 2
where
p . o = (1/2) (oovs (TftKAN/ .
There are two more important effects that limit the time behavior of the
emitted power.
2.
Doppler Dephasing and Transit Time
The Doppler effect in microwave spectroscopy causes a shift in the
apparent resonant frequency of a molecule due to its translational motion.
In a laboratory fixed frame, where the radiation is propagating along the
+z axis the emission frequency of a molecule moving with a component of
velocity along the +z axis will be higher than co by kv . Likewise a
molecule that is emitting radiation along the +z axis but is traveling
with a velocity component in the -z direction will have the emission
frequency shifted down by kv
to the fixed observer.
As shown in chapter III section B there is some distribution of molecules leaving the nozzle p(r,0).
This distribution will be simplified to
D(0) and will be normalized so as to be dimensionless,
D(0) = D cos 2n 9 .
o
Figure 3 shows v
and 0, where v
is the velocity of the molecules coming
out of the nozzle, assumed here to be a constant, and v
= v
z
sin0.
o
We now define a polarization for molecules at each angle 9 and both
propagation directions,
43
-t/T
P.(0,t) = e
i
hKAN
— j - 2 - cos (to ± kv sin9)t .
4
o
o
The polarization is obtained by integrating this from 0 to TT/2 ,
Tf/2
P (t) = 21* P. (0,t) D(0) d9 .
i
«
i
o
Substituting in and dropping the zero terms gives,
-t/T
P. (t) = e
faiKAN
1
TT/2
)D cos to t P
cos (kv t sin0) cos
O
O
O «J
o
o
0d0 .
This is a known integral and can be expressed as ,
-t/T
P (t) = e
where J
(hKAN )D
J (kv t)
cos CO t —
-—
°
(kvt)n
o
is a Bessel function of integer order n.
,
(IV-11)
The time domain signal
from the J=0 to J=l OCS transition is shown in figure 4.
There is only one
transition but the beating due to the Doppler effect is clearly seen.
Bessel functions are similar to sinusoidal functions in that both oscillate
about zero.
Bessel functions, apart from their first zero, have almost
equal differences between zero crossings to all orders.
Although J
starts
at one while all others start at zero, the major distinction between different orders is the location of the first non trivial zero.
This property can
be used to try to obtain the power of the cosine distribution from fitting
observer time domain signals to equation (IV-11).
Another result of this slow oscillation is that when the data is Fourier
transformed there will be two peaks in the frequency domain.
The peak sepa-
ration in the frequency domain is then a measure of the velocity v .
We
have recently measured the microwave spectrum of OCS and the dimer Ar'OCS
under identical gas dynamic expansion conditions.
We found the ratio of
the doublet frequency splittings to be equal to the inverse ratio of the-
44
Figure 4.
The time domain digitized emission signal from the J=0 to
J=l rotational transition in
O C S .
There are 256
points digitized at a rate of 1 ysec per point. The gas
mixture is 4% OCS in Kr.
^VWW^'^W./W r
0
32
64
96
128 160 ^92
DIGITAL POINTS
224 256
46
masses of the two species.
This is in complete agreement with equation
(III-6) given for the flow velocity of particles in a nozzle expansion.
The time domain and Fourier transform of the Ar'DBr, J=3, F=9/2 to J=4
F=9/2 transition is shown in figures 5 and 6 respectively.
It is easier to explain a doublet if the molecules were pulsed
down the cavity axis.
Let us assume for the present that there is no
angular distribution to these molecules and therefore v = v .
z
o
The
standing wave fields in the cavity can be considered a superposition of
two oppositely directed traveling waves.
The molecules in their own
moving reference frame are emitting at one frequency but this frequency is
Doppler shifted up by kv
to the wave traveling in the direction of the
molecules and shifted down to the other wave.
There are therefore two
emission frequencies present in the cavity which beat together and are
transformed to a doublet.
The peak separation in the frequency domain would
be just 2kv , which for Kr velocities and at 10 GHz gives the separation
to be 25 kHz.
Adding in an angular distribution to this beam would just
broaden out the two peaks.
The fact that pulsing the gas perpendicular to
the cavity axis also produces an effective beat (the Bessel function) is
remarkable.
The measured doublet separation for the perpendicular beam is
about 80% of 2kv .
This doubling is a great hindrance to the measurement
of transition frequencies when the different transitions are separated by
frequencies of the order of the Doppler splitting.
The third effect of equation (IV-11) on the signal is the decay of the
polarization produced by the term (kv t)
.
This decay is due to the
distribution of the emission frequencies destructively interfering and is
the dominant emission decay process.
87
of a forthcoming publication.
The lineshape problem is the subject
47
Figure 5.
The time domain digitized signal from the J=3, F=9/2 to J=4,
79
F=9/2 transition in ArD Br. There are 256 points at 0.5 ysec
per point. The gas mixture is 4% DBr in Ar.
96
128
160
DIGITAL POINTS
192
224
256
CO
The frequency domain Fourier transform of the data in figure 5.
This spectrum is the sum of the squares of the real and imaginary
parts of the transform.
Also 256 zeros were added to the time
domain data before transforming.
The frequency axis is in MHz
and the center of the dip gives the transition frequency.
two peaks are caused by the Doppler effect.
The
\,V^/W, ^d
8764.9508765.000 8765.050 8765.100 8765.15
FREQUENCY
o
51
The final limit to the duration of the emission signal is the finite
time the molecules spend in the field regions of the cavity. Although it
is not clear at which point to consider the molecules in or out of the
field we will utilize the beam waist 2w to compute the transit time X .
o
T
Use of the terminal velocity will give a minimum transit time,
x m = 2w /vm .
T
o T
For Krypton v
4
~ 3.8 x 10 cm/sec and for 2w = 12 cm this minimum transit
time is 316 ysec. For molecules that have strong signals such as monomer
OCS seeded in Kr as shown in figure 4, the maximum time the emission signal
lasts is around 250 ysec. To overcome the 300 ysec transit time limitation mirrors with larger beam waists could be used or the nozzle could be
directed down the cavity axis.
3.
Signal Detection
The detectors used in these experiments are double balanced mixers.
To describe what is detected by these diode mixers, consider the currentvoltage plot of one of these diodes. To explain the conversion of the
microwave frequency to an intermediate frequency (IF), we expand the current in a Taylor series around zero voltage v ,
/ N
/ i ^ /dA
i(v) = i(v ) + (—J
°
W v
0
^ •> /-> A 2 A
2 . i A3 A
3 .
v + 1/2 — ^
v + — —^-j
v + ... ,
6
\dv2/vo
\dv3/Vo
where di/dv has the dimensions of a conductance, g, and d i/dv
will
be written g . There are two inputs to the mixer, the signal coming from
the cavity, v = v.. cos (to. t + <J).) and the local oscillator (LO) signal
s i
l l
vT„ = v„ cos(co„t + d>-), so that v = v
+ v__.
The term linear in v will
52
give frequency components to the current that are just the input ones.
The quadratic term gives,
(v +vr/J
S
IiO
2
2
2
2
2
= vn cos (to.t+tj).) + v_ cos (C0ot+tj>_)
1
1
+ 2 v v
-
1
costco^+^J
/.
<£
£•
cos(to 2 t+^ 2 )
(v 2 / 2 ) (l+cos(2C0]t+2!J)1)) + (v 2 2 /2) (l+cos(2C02t+2(J>2))
+ v v [cos ((to -to2)t+cj) -t|> 2 )+cos((to +to2)t+(jJ1+(j)2)] .
There are current frequency components at 2to1» 2co2,to-^-to,,a n d
W
l+aV
Similarly the cubic term will give current frequency components at a^,
to„, 2C0n -to„, 2co. +to„, 2C0.-C0 , 2co +to , 3co and 3to .
2
1 2
1 2
2 1
*21
1
&
So that in general,
one can obtain all frequencies nto +mco where n and m can be any positive
or negative integer or zero.
The power in the higher harmonics will fall
off depending on the diode but one can easily produce and use the n, m = 30
to 40 harmonics.
The circuit of the mixer filters out all frequency components except
to -to , the intermediate or difference frequency.
The current output of the
mixer is then,
l = 1/2 g
v x v 2 cos (to -to )t.
Depending on the impedance in the output circuit a voltage will be produced
by this current that is amplifier, mixed down again, and digitized.
fore, it is the voltage or electric field that is detected.
There-
The electric
field in the cavity output waveguide is related to the power in the dominant
mode by
-ST.
53
where P
is the power coupled out of the cavity, a and b are the inside
dimensions of the waveguide and Z is the waveguide impedance,
_
7
n
^r •
In this formula r| is the impedance of free space, 377 ohms, and X is the
c
cutoff wavelength for the guide (A = 2a, TE
mode). Combining these
c
xu
results, the digitized signal is proportional to the square root of P as
L
given in equation (IV-9).
In order to successfully digitize and Fourier transform the signal
emitted from the molecules, the most important parameter that has to be
optimized is the signal to noise ratio.
The power signal to noise ratio is
used because the noise figure is defined in terms of powers.
The voltage
signal to noise ratio is the square root of the power signal to noise ratio.
Since we do not record the signal directly but mix it down, additional noise
is introduced by the mixers and amplifiers.
The ratio of the signal power
to noise power, at the input of the mixer, to the signal power to noise
power at the output, is defined as the noise figure of the mixer, F,
S
S./N
- x x
~ S /N
o o
The noise figure of a mixer is a function of the internal parameters of the
mixer, the losses in the circuitry, the quality of the nonlinear device,
matching problems and local oscillator bias.
For the input signal power, S ,
we take the power coupled out of the davity as given in equation (IV-9).
The input noise power is given by kTB, where k is Boltzmann's constant and
B is the noise bandwidth.
then,
The signal to noise ratio out of the mixer is
54
S./N.
P
N
(IV"12>
V o " "V^ - kTiF •
The mixer is just the first stage of the detection circuit.
through n devices in cascade the total noise figure F
(F - 1 ) B
T?
F
- w
F
T" I
4.
+
2
^i^
1 l,n
where F
i
(F,-l)B,
3
3,n
2,n
+
1 2 l,n
88
(F - 1 ) B
n
n
+
^G^
is
For one
•" * £ — i 7 ~ '
l,n
/Tt7 n_x
(IV 13)
"
l,n
is the noise figure of the first device, B,
is the bandwidth of
l,n
the first through nth devices and G
is the gain of the first through
±,n
nth device.
The gain of several devices in cascade is just the product
of the gains for each device alone.
For a passive device like a mixer the
gain is equal to 1/L where L is the conversion loss, L = S /S .
i o
As can
be seen from equation (IV-13) if the g a m of the first element is high the
total noise figure will be essentially the noise figure of the first
element.
The double balanced mixer we use has a low noise IF preamplifier
built into it and has an overall noise figure of 6dB, or in other words a
signal to noise degradation of four.
This mixer also has a RF/IF g a m
of 20 dB so that the gain of the IF amplifier divided by the conversion
loss of the mixer is 100.
Considering this mixer-IF amplifier as the
first device, the second device is another amplifier with at most a lOdB
noise figure and a gain of 27 dB.
The bandwidth of the second through
last device is essentially the bandwidth of the entire cascade because
the narrowest bandwidth device is last.
Using just these numbers the first
term in equation (IV-13) is four, the second term 0.09 and the third term
even smaller.
55
4.
Resolution and Sensitivity
The resolution, R, in frequency interval per point of a Fourier transformed time domain signal is just the reciprocal of the total time of the
record, so that,
R = (time per point x total number of points)
One can add zeros onto the end of a time record and increase the resolution but of course no new infromation is gained.
In order to observe
smaller splittings the time that one observes the molecular emission has to
increase.
If the emission from the molecules were a simple exponentially
damped, exp(-t/T 2 ), sinusoidal then the half width at half maximum of the
transformed line would be l/(2Tf T_).
The basic limitation is the transit
time through the cell, although the Doppler dephasing is now the major
limitation.
If we could attain a 300 ysec T„ which would entail say a 500
ysec record length then the frequency per point would be 2 kHz.
We mostly
operate at 0.5 ysec per point and take 256 points, giving a 128 ysec record
length.
Two definitions of sensitivity will be used here, the first in terms
of a minimum number of molecules and the second in terms of the smallest
dipole moment matrix element detectable.
These calculations will assume
the power signal to noise ratio, single shot, to be one, at a time t = T„.
Equation (IV-9) with the simple exponential decay will be used for P_, a
similar calculation could easily be done using P_ as given in equation
L
(IV-11) with a similar result.
Use of the time domain signal to noise,
single shot is in keeping with the way signals were usually found.
The
signal to noise improvement by averaging many scans and by Fourier trans89
..
forming with filter functions has been given before.
Combining equations
56
(IV-12) and (IV-9) and solving for the product of the dipole moment matrix
element times the number density gives,
F
u AN
11
where Q /Q
=( 2 ™
° UA-VB^L
(1+0 +B >\l/2
r-*-\
(IV 14)
'
~
2
= 32/(l + 3, + 3-) and K = 2y. fa were also used.
A numerical value of the right hand side of equation (IV-14) can be
obtained by substituting in T = 300°K, B (the overall bandwidth) = 300 kHz,
4
F = 4, to = 2TTV = 2TT x 10 GHz and XQ_ = 10 . For V_, the sample
volume,
c
o
o
L
S
we use the expression given in equation (II-2) and (II-3), the field volume,
V
s
= V = TfJl (w / 2 ) 2 .
o
3
3
Using A = 70 cm and w = 6 cm the volume is 1.9 x 10 cm . One would like
to minimize the function (1 + 3, + 3-)/32, the smallest it gets is unity,
when 3 2 = °°- For this calculation we will take 3T = 3~ = 1 which makes the
function equal to three. Using the above values we get,
yH
ij
-13
2
AN = 2.16 x 10
statcoul/cm .
o
If we assume that the molecules are evenly divided through the volume V,
then the number of molecules, N, needed for a given rotational temperature
can be calculated from N = p V, and equation (III-9).
Using y.. = 1 Debye
and AN(0,1) at 0.5°K and 300°K from the calculations following equation
(III-9) we get for a J=0 to J=l rotational transition,
N = 1.7 x 10
molecules at 0.5°K
15
N = 5.4 x 10 molecules at 300°K .
57
Going back to the calculations following Equation (III-5), we gave the
14
3
number density at the cavity center to be 6 x 10 particles /cm . If we
13 -3
are using a seeded beam of 2% molecules we have p = 1.3 x 10 cm . Then
using AN(0,1) at 0.5°K and 300°K as before the minimum dipole moment matrix
element required is,
y.. = 7.8 x 10~ 7 Debye at 0.5°K
ID
y
=0.24 Debye at 300°K .
ID
In order to have a polarization bandwidth B = 2y. E fa equal to the
-7
cavity bandwidth, 1 MHz,with a 7.8 x 10 Debye matrix element one would
need an electric field of around 200 k volts/cm. We have previously calculated a peak electric field of 300 volts/cm with 40 watts input power.
With this electric field the smallest matrix element that gives a polariza-4
tion bandwidth the same as the cavities bandwidth would be 5 x 10 Debye.
These calculations show that if the rotational temperature is 0.5°K we
need only 1.7 x 10
molecules in the cell to observe a signal with the
5
given S/N ratio. We would need about 5 x 10 more molecules if the temperature were 300°K, to get the same S/N. The calculation of the smallest molecular dipole matrix element required to observe a transition shows our
ability to study small dipole moment species such as Ar'Kr.
58
V.
A.
THE SPECTROMETER
Introduction
This chapter discusses the synthesis of the cavity, the pulse
Fourier transform and the pulsed molecular nozzle into a working
spectrometer.
Underlying the feasibility of the experiment, x, x and
P
T„ have to be ordered as T_ > X > X. The polarization
has to remain
c
2
2
p
coherent for a time, T„, longer than the microwave pulse time, x ,
2
p
which has to be longer than the cavity decay time X.
The block diagram is discussed first, so that how the spectra are
measured should be clear. This microwave circuit is an analogy of the
90
classical Rayleigh refractometer.
Several of the features that govern
the design and construction are discussed next.
The section on molecular
spectra traces the major results that have been generated since the first
observation of a van der Waals molecule.
In the potential of the method,
some possible future experiments that seem feasible will be outlined.
Several areas are mentioned where application of this method could contribute to further chemical insight.
The last section discusses the
general advantages and disadvantages of the three techniques, whose combination has yielded this new spectrometer.
B.
Block Diagram
The block diagram of the spectrometer is shown in Figure 7.
The
master oscillator (MO) is frequency stabilized by the lock box (LB) to
The block diagram of the spectrometer.
(MO) master oscillator,
(1GHz) a one GHz signal, (VFO) variable frequency oscillator,
(COUNT) frequency counter, (LB) lock box, (LO) local oscillator,
(Ml) harmonic mixer 1, (ATT) attenuator, (AMP) amplifier, (ISO)
isolator, (M2) mixer 2, (PIN1) PIN diode 1, (C) circulator,
(SST) slide screw tuner, (DET) diode detector, (DRV) valve
driver, (PUL BOX) pulse box, (A/D) analog to digital converter,
(DIS) display, (AVE) averager, (COM) computer.
60
UJ
•**•
8
d5
Q-
QQ
ro
CVI
>
cr
T\1
J
E
o
CO
—*"
CVI
z
Q.
8
61
two lower frequency standards.
A one GHz signal is produced by multi-
plying up a 20 MHz oven controlled crystal oscillator.
This crystal has
-9
a frequency stability of 1 x 10
parts per day so that after multiplying
it up by 50, the 1 GHz should be stable to Hertz.
A variable frequency
oscillator (VFO) is also used for continuous frequency adjustment within
its range.
We use a 10 MHz to 480 MHz (in 5 bands) tube oscillator that
is stable to hundreds of Hertz over minutes.
This VFO is counted (COUNT)
by a standard seven digit frequency counter.
The mixer (Ml) mixes the
first harmonic of the VFO and the n
oscillator.
harmonic of the 1 GHz with the master
An IF frequency of 30 MHz is produced that is fed into the
frequency stabilizer.
This frequency stabilizer compares the IF frequency
with its own 30 MHz crystal and gives an error voltage whose magnitude is
proportional to the frequency difference and whose polarity depends on
whether the IF frequency is below or above 30 MHz.
When one inputs the
same frequency as the internal crystal, the error signal is zero.
If
the VFO's frequency is changed so that the mixed down frequency is no
longer 30 MHz, the frequency stabilizer will apply a voltage to the MO,
changing its output frequency so as to cause the mixed down signal to
once again be 30 MHz.
Now we can not only control the MO's frequency,
but by counting the VFO's frequency and by knowing which harmonic of the
1 GHz we are using, we can calculate the MO's frequency to hundreds of
Hertz.
The local oscillator (LO) is locked by another frequency stabilizer
to the master oscillator through mixer (M2). Part of the 30 MHz signal
that is fed into the second frequency stabilizer is split off to be used
as a phase coherent reference signal.
Both the master and local oscillator
are backward wave oscillators (BWO) and when locked, they have the
62
frequency stability of the VFO. BWO's have good bandwidth, a relatively
level power output and can be easily swept. This last feature is convenient when working with cavities, as the cavity resonant shape and position
are often adjusted. Cavity coupling information can also be obtained by
sweeping across the cavity resonance.
Since both the master and local oscillator are running continuously,
the pulse to the cavity is formed by opening PIN diode 1, (PIN 1) for a
time, t . This pulse of microwave energy, typically t_ = 6 ysec, goes
through the circulator (C), slide screw tuner (SST) and impinges on the
input coupling iris of the Fabry-Perot cavity. Any energy that is reflected from the cavity travels back up the waveguide and is routed to
a detector (DET) by the circulator.
The output of this diode rectifier
is fed into an oscilloscope which is triggered in synchronism with the
PIN diode pulses. By looking at the reflected power as displayed by the
oscilloscope, several things can be learned.
If the frequency of the MO is not the cavity resonant frequency,
all the power incident upon the cavity is reflected and the output of
the detector traces out the microwave pulse envelope. The PIN diodes
we use have rise times (10-90%) of 10 nsec and if one uses a fast detector such as a tunnel diode that can follow this rise time, the pulse envelope shape will almost be rectangular. When the MO or cavity resonant
frequency is changed so that one is driving the cavity on resonance, the
pulse envelope shape is different. Since the cavity energy builds up and
decays with a time constant x = 160 nsec, as given before, we can see
the process described in Chapter II, Section C. When the input wave hits
the iris, there is a reflected wave causing a large signal at the detector.
As the cavity energy builds up, this reflected wave is increasingly
63
cancelled by the fields coupled out of the cavity so that a spike with
an exponential decay is seen at the detector. The voltage that the decay
levels off to is indicative of the coupling.
When the input energy is
switched off, the input wave reflecting from the iris is gone, so that
only the wave coupled out is seen.
This causes another spike with an
exponential fall off as the energy dissipates from the cavity.
Energy during the pulse is coupled out the opposite mirror.
To
protect our detector mixer (M3), we have another PIN diode (PIN 2) that
blocks this energy.
diodes are shown m
The TTL trigger pulses that operate these two PIN
Figure 8.
The diodes conduct when the TTL level is
+5 volts and have a 80 dB isolation when the level is zero.
As seen in
Figure 8, PIN diode 2 reflects any power coupled out of the cavity when
PIN diode 1 is conducting.
The time t, is adjustable and the time t .
is equal to t_ plus another adjustable amount to make sure all of the
original power is dissipated.
The pulse box (PUL BOX) forms these pulses
and the pulse to the driver (DRV) that operates the molecular pulse valve.
The pulse length to the molecular valve, out of the driver, is about
t. = 3 msec.
This valve is a commercial m-line solenoid valve.
valves have been described in the literature
improve on ours.
Faster
91 92
'
and we should eventually
The time t. + t 2 is an important parameter, in this
duration, the mechanical valve has to open and the molecules have to travel
to the field regions of the cavity.
If you look at the molecular signal
as a function of this time interval, at short times you will see nothing,
then there will be a fast rise in signal amplitude and a fast fall to
about half height m
1.5 msec, then a slow tail for about 5-10 msec.
This delay, t. + t„, has to be adjusted for each different carrier gas
and many times for each gas mixture, to optimize the signal amplitude.
*3
+5 —
0 —
(I)
+5—
(2)
0 —
*2
<—:
>•
+5 —
(3)
0
Figure 8.
The timing sequence of PIN diode 1(1), PIN diode 2(2), and
the drive to the molecular valve (3).
en
65
The pulse to the molecular valve is repeated at a rate from one to
ten Hertz, depending on the carrier gas, back pressure and nozzle size.
The number of particles released was given by equation (III-8) and as an
18
example was calculated to be 4 x 10
particles per gas pulse.
ten inch diffusion pump that can pump around 4 x 10
19
We use a
particles per second,
hence the ten Hertz repetition rate.
The pulse sequence to the PIN diodes occurs at twice the solenoid
valve rate.
One microwave pulse interacts with the molecules in the free
expansion and the next microwave pulse interacts with the evacuated cavity.
If we call the frequency that the master oscillator is locked at, V,
then the local oscillator frequency is V - 30 MHz.
The pulse of microwave
energy creates in the expanding molecules the macroscopic polarization which
emits radiation at the rotational transition frequency, v . Both of the
frequencies, V and V , have to be within the bandwidth of the cavity, so
that |v -vj = A «
1 MHz.
Usually A is between 10 kHz and 600 kHz.
After
the power pulse dies away, PIN diode switch 2 opens and the molecular emission mixes with the local osciallator m
mixer (M3). The signal out of
mixer (M3) is at a frequency 30 MHz ± A, depending on whether v > V
V < V .
or
This is amplified and mixed down again in mixer (M4). The other
input to mixer (M4) is the 30 MHz signal we obtained before by mixing the
master and local oscillators.
If we did not use such a reference signal
here, the decaying emissions from the molecules in the cavity from pulse
to pulse would be arbitrarily phase shifted relative to one another.
means we could not average or add up all the molecular emissions.
This
The
signal out of mixer (M4) is at the frequency A, or the offset between the
master oscillator and the molecular transition frequency.
This signal is
filtered and amplified and then fed into our analog to digital converter
66
(A/D)•
The (A/D) is triggered by the pulse box to digitize a signal every
time the microwaves are pulsed.
The digitizer we currently use is only a
6 bit converter so that our dynamic range per pulse is only one in 64.
This will be shortly replaced by a 10 bit A/D that gives a one in 1024
dynamic range.
After the signal has been digitized, the data is transmitted to
an averager (AVE).
This averager adds and subtracts alternate pulses
so that all molecular emissions are added together and all alternate
background scans are subtracted.
As mentioned before, in the section on
resolution in Chapter IV, the digitizer we currently use has 256 points
and by using 0.5 ysec per point, the entire data scan takes 128 ysec.
Since the molecular pulse valve provides observable molecules in the cell
for times longer than this, we also have the option of taking n adds and
n subtracts in one molecular valve period.
After the averager memory
fills, the data is sent to the departmental VAX 11/780 computer which does
a fast Fourier transform and returns this data to the averager.
Either the
time domain or Fourier transformed data can be displayed from the averager.
Because of the way we mix down the molecular emission signal, the start of
the frequency domain data is at the master oscillator frequency.
By
counting the points over to a molecular peak we find A and therefore we
know the transition frequency V .
Searching for unknown molecular lines consists of stepping the cavity
mirror separation, hence its resonance frequency, and following along with
the microwave oscillators.
Since the cavity bandwidth is around 1 MHz,
the step size is 500 kHz or less.
Searches may easily involve going over
a GHz or more so that many, many steps are taken.
on ways to automate this stepping procedure.
We are presently working
67
C.
Design Considerations
The two mirrors used are solid aluminum type 6061 disks with a
spherical concave mirror surface.
It is very easy to machine a spherical
surface in a plate and mirror alignment is not critical.
The radius of
curvature of the mirror is found by making the Fresnel number unity for
the lowest frequency,
2
RX
lf
where a is the mirror radius, R the radius of curvature and X is the
wavelength.
This one requirement insures a good Q because the mirror then
captures over 95% of the wave amplitude at any point.
In Chapter II, Section A, we gave the complete expression for the
Gaussian standing wave.
We also learned that at the confocal arrangement
of the mirrors, R = SL, the beam waist, 2w is maximized.
o
So the worst
case to consider for diffraction losses is the confocal arrangement.
For this spacing, the formula for w(z=£/2) and w
w
o
simplify to,
= (RX/2ir)1/2
w(z=V2)= /2"w .
o
" 2
The electric field falls off as exp - (r/w(z)) as a function of z,
the cavity axis and r, the radial direction.
z = %/2.
At the mirror surface
2
By using the requirement a /RX = 1, one is requiring,
a
= RX ,
or
a = J^RT = W Q / 2 ? = w(z=V2)/F.
68
Since r = a at the mirror edge and z = Si/2,
exp-(r/w(z))
2
= exp(-ir).
The field amplitude is 0.043 times what it is on the cavity axis.
Ex-
perimentally when the field amplitude is greater than 0.13, one can
notice effects of the field spilling out of the mirrors and reflecting
around the vacuum chamber.
We have used a set of mirrors designed for 8 GHz from 4.5 to 18 GHz.
The mirrors could be used at even higher frequencies but other microwave
components of the spectrometer will not operate there.
Flat mirrors have
also been tried but due to the very critical parallelism adjustment, high
Q's were never achieved.
The coupling iris is a round hole centered along the waveguide axis.
Part of the back of the mirrors are machined out so that the waveguide can
butt up against an area that is from ten to twenty thousands of an inch
from the front mirror surface.
The coupling hole is centered in this
area and centered in the front mirror surface.
The inside dimensions
of X-band (8-12 GHz) waveguide are 0.4" x 0.9" and the coupling hole is
about 0.38" in diameter.
For lower frequencies, a coaxial connector
device replaces the waveguide section.
Then antennas are inserted from
the inside of the cavity into this coaxial connector.
Simple straight
wire configurations with the wire coming out of the connector and bending
90° along the mirror surface work well.
The length of the wire and its
distance from the mirror surface are adjusted to control the coupling,
just as the circular diameter size controls the coupling in the waveguide
feed mentioned first.
The mirrors are 14 inches in diameter and are suspended on four oneinch diameter stainless steel rods.
The rods run the entire length of the
vacuum chamber, which is an 18-inch outside diameter tube, 41 inches long.
69
The rods connect two end plates that are bolted onto the main tube.
There
is a ten-inch port coming off at the middle of the main tube that a teninch diffusion pump is connected to.
Directly above the ten-inch port is
a six-inch flange where the pulse valve and associated electrical and
gas feedthroughs are located.
The mirrors ride on the four rods and are
connected to the outside by waveguides that feed through end caps of the
18-inch tube.
One mirror is held fixed while the position of the other
mirror is adjusted by a rack and pinion and gear reduction mechanism.
This mirror translation mechanism has to be made with some care
m
order to step the cavity resonant frequency 500 kHz as explained before.
Considering the dominant mode, the resonant frequency was given as
V = (c/2A) [ (q+1) + (l/7r)cos"1(l-VR)].
The difference between two closely space frequencies Av = V.-V„ is given
by
Av = (c/2)(l/£1-l/£2)(q+1).
The movement of the mirrors AS. = &2~^-\ ^s
2
^1^2 ~ ^ ^
AO. - 2Ava2
* ~ c(q+l) '
A
where & is the distance between the mirrors and q + 1 is the number of
half wavelengths between the mirrors.
As an example, for % = 70 cm,
X = 3.5 cm, so that q + 1 = 40 and Av = 500 kHz, the mirror movement is
41 ym.
This points to another concern:
mechanical vibration can vary
the cavity resonant frequency and thereby can amplitude modulate the
output signal.
An amplitude modulated signal is demodulated in the mixer
70
and contributes to low frequency noise in the receiver.
The diffusion
pump with its boiling silicone oil contributes to noise in this way.
The filter amplifier after mixer (M4) has a low frequency roll-off of
about 1 kHz and blocks most of this noise.
D.
Molecular Spectra
The first van der Waals molecule to be seen was Ar*HCl which had
93 94
been assigned before. '
The only technique for studying rotational
transitions in van der Waals molecules up to the development of this
spectrometer was the molecular beam electric resonance spectrometer.
This
technique is severely limited in the number of possible transitions it can
observe because of the use of Stark focusing.
While only one of the quad-
rupole components of the J=2 to J=3 transitions were observed with the
electric resonance machine, we could easily see the full seven-line multiplet.
The next van der Waals complex to be observed was Kr*HCl.
molecule had never been assigned before.
This
The complete structure and
molecular constants of this molecule were published along with an introduc39
tion to this new method.
Within two months, two more molecular complexes were observed and assigned. The rotational spectra, molecular
constants and equilibrium structure of Ar«HBr and Kr*HBr was the subject
of this next paper.
95
A number of rotational lines of other complexes
had also been observed but never assigned.
These include the van der
Waals molecules Kr'CHJF, Ar«CH3F, Xe'HBr and Ar«CF,Br.
The next paper
96
van der Waals bond.
attempts to g a m insight into the nature of the
In this paper, the Kr*HF complex was assigned and
for the first time the quadrupole splitting caused by a rare gas atom was
71
observed.
spins.
The 82, 84 and 86 isotopes of the Kr atom have zero nuclear
The fluorine and hydrogen atoms have nuclear spins of 1/2 and
therefore have no quadrupole splitting.
The 83 isotope of Kr has a
nuclear spin of 9/2 and so will couple to the rotational angular momentum
to give a quadrupole multiplet.
This multiplet splitting is proportional
to the coupling constant which in turn is proportional to the product of
the nuclear quadrupole moment (a known constant) and the field gradient
at the nucleus due to the electrons.
In the free Kr atom this field
gradient is zero and therefore the coupling constant is zero.
But, due
to the bonding in the complex, the field gradient at the nucleus is distorted and this results in a measureable quadrupole coupling constant.
What this changed field gradient means in terms of bonding is discussed.
Further work done in this connection is presented in a paper just submitted
97
for publication.
Here the two quadrupole complex
1 "^1
Xe'HCl is analyzed.
The series of complexes, CO bonded to HF, HC1 and HBr has also been
observed and assigned.
98
These molecules are interesting because it is
the carbon of the carbon monoxide that binds to the HX as OC"**HX. Although the hydrogen has a vibrationally averaged position not colmear
with the OC---X, its potential minimum is colinear.
Work is continuing
on these three complexes to completely elucidate their structure.
Other molecules that are being studied included HF'HCN and Ar-HCN,
as well as others.
E.
Potential of the Method
One point that I think should be made clear is that all the work
described in the previous section has been done in less than one year.
There is considerable work to be done on just refining and developing
72
new instrumental techniques to make this method even faster, easier and
more versatile.
The major contribution of this spectrometer so far and
for some time to come is in the structure and bonding of weak molecular
complexes.
The vibrationally averaged bond distances and angles are
obtained from the rotational constants.
fitted to a distortion constant.
The rotational lines are also
Since all the spectra are taken in the
lowest vibrational level, the harmonic approximation to the potential is
reasonable.
In this approximation, the distortion constant can easily
be related to the vibrational stretching frequency of the van der Waals
bond.
Using this stretching frequency and the reduced mass, an effective
force constant can be calculated.
Assuming a form for the potential and
these parameters, one can get an estimation of the well depth of the complex.
39 95
'
The measured quadrupole multiplet and coupling constant gives
structural and bonding information.
Other interactions such as spin-spin
and spin-rotation also give information into the nature of the bond.
As well as dimer complexes of van der Waals or hydrogen bonded molecules, one could look for trimers or n-mers.
seen by laser fluorescence techniques
99
Higher complexes have been
and in mass spectroscopy.
Although we have concentrated on molecular complexes, monomers can
obviously be studied.
Due to the sensitivity, one can observe most
isotopes in natural abundance.
Although one would want to discourage
dimer formation, by using He or Ne, use of a seeded beam still provides
a larger signal for some cases.
It has been shown experimentally,
for
a beam expansion employing pure molecules, the beam temperature was
measured to be 30°K.
For an expansion using 5% molecules seeded in Ar,
the beam temperature was measured to be 3°K.
If we use these two tempera-
tures for a diatomic molecule with a rotational constant of 1200 MHz, we
73
can compare the values of AN
the two cases.
and therefore the signal strength in
For a J=0 to J=l rotational transition, we gain a factor
of 20 in the number density due to the pure molecular expansion, but
we lose by a factor of 100 in the partition function times the Boltzmann
factors due to the higher temperature.
The temperature effects nearly
cancel for a J=9 to J=10 transition.
Spectroscopy on heavy molecules that have low vapor pressures could
be done by vaporizing them in a high temperature oven that could easily
be placed in the vacuum chamber.
Another possibility that is being
pursued is to use a high temperature nozzle source to form rare gas metal
dimers.
Some of these types of complexes have been observed by molecular
beam magnetic resonance spectroscopy
and by laser fluorescence.
By crossing the nozzle expansion with some excitation source such
as a laser, electron beam or plasma, one might be able to see rotational
transitions in excited states.
As mentioned before, for the weaker com-
plexes no rotational lines have been seen in even a vibrationally excited
state.
Other types of nozzle sources might allow one to observe combus-
tion or explosion products.
Molecular radicals or ions would also be
interesting species to study m
the large cell.
A Start cell to measure dipole moments would be a useful addition to
the cavity spectrometer.
Due to the large extent of the fields, plates
cannot be placed very near to one another to insure a high homogeneous
electric field.
expansion.
Another problem is the large divergence of the nozzle
A solution to the first problem is to use a combined parallel
plate, Fabry-Perot cavity.
This type of cavity m
skimmed beam would be worth trying.
in our laboratory.
conjunction with a
A Zeeman cell is under construction
We have purchased a superconducting solenoid magnet
74
with a 12-inch diameter bore.
A Fabry-Perot cavity and nozzle source will
be placed in this bore with a diffusion pump underneath.
The nozzle source
will pulse molecules along the axis of the bore while the cavity axis will
be perpendicular to the bore.
The electric field polarization vector of
the microwaves can then be rotated either parallel or perpendicular to
the magnetic field lines to observe all AM transitions.
Double Fabry-Perot cells have been constructed to do microwavemicrowave double resonance
and double resonance modulation.
Other
types of double resonance spectroscopy could also be done combining the
microwaves with a laser or a radio frequency field.
sequences used in NMR
tried in the cavity.
78
Multiple pulse
and in microwave spectroscopy have never been
And non-linear effects such as two-photon transient
spectroscopy which has been done in optics
and in NMR
should be
observed in the microwave region.
Another area that this new method will hopefully contribute to is
better understanding of the gas dynamics of nozzle expansions.
It was
shown in Chapter IV, Section E-2, within the approximations given, how
the time domain signal can be used to find the power of the cosine distribution.
Finally, knowledge about condensation or dimerization
processes may be discerned.
Once the reproducibility of the pulse
nozzle is improved so that relative concentrations can be inferred from
signal intensities, three component mixtures could be studied.
The
signal intensities from two van der Waals complexes, obtained from a
single rare gas and two polar molecules, could be observed.
By varying
the concentration of the two polar molecules, one might learn about the
competing kinetic processes that form the dimers.
75
F.
Discussion
The last section will go over the main features of the three elements that make up the spectrometer, the Fabry-Perot cavity, pulsed
Fourier transform, and the pulsed nozzle source.
The advantages and
disadvantages mentioned are not exclusive to this method but they all
contribute to this method.
The real advantages of the Fabry-Perot cavity are its high Q, and
therefore its high field strengths and its large open structure.
The
large electric fields ensure that one can polarize a band of frequencies
that is at least as large as the cavity bandwidth.
lend to studying non-lmear effects.
The high fields also
The large open structure provides
a large volume for the nozzle expansion.
There is room in the cavity to
place pulse valves, hot ovens, other mirrors or whatever, in an area
readily accessible to the field-molecule interaction region.
The disad-
vantages of the Fabry-Perot are the non-uniform complex field distribution
and the necessity to operate in a high mode to have the large volume. Even
though the Q and the bandwidth cannot be separately adjusted, the 1 MHz
window we have to operate in is a disadvantage.
Finally, as with all
cavity spectrometers, mechanical vibrations have to be isolated to reduce noise.
The classic advantage to using pulse Fourier transform techniques
is in the signal to noise enhancement of the square root of the ratio
of the polarization bandwidth to the line width.
For the cavity, this
polarization bandwidth is limited by the cavity bandwidth.
By working in
the time domain, it is very easy and fast to add and subtract successive
scans, thus simulating phase sensitive detection.
With this type of
microwave set up, it is easy to use superheterodyne detection.
A
76
superheterodyne detector eliminates 1/f noise by converting the microwave
signal to the IF frequency where this noise is small.
frequency is easy to amplify and filter.
The signal at this
Finally, because the IF is the
result of the mix between the molecular signal and the LO signal, much
weaker molecular signals are observable.
Since the frequency is fixed
when the spectra are recorded, the frequency dependence of the microwave
components is less trouble.
The fact that one is observing the signal
at different times than when one is exciting the sample, like a radar,
eliminates having to deal with the exciting power.
You observe a signal
with theoretically only thermal noise in the background instead of trying
to observe a small change on top of a much larger microwave background.
Also, since you are observing the molecules when the only field they experience is their own emitted field, there are no power saturation problems
or Stark shifts.
The m a m disadvantage to the pulsed Fourier transform
method is that it is expensive.
One has to have two oscillators with lock
boxes, switches and timing controls, several mixers, a digitizer, and a
computer, as well as all the microwave plumbing.
The advantage, in signal to noise, of using a pulsed nozzle source
as compared to a continuous source is now considered.
There are a fixed
number of molecules that can be pumped through the cell per unit time.
This fixed number of molecules can either be pulsed all at once into the
cavity or be strung out over the unit of time and be observed a number of
times, m. The same number of molecules are observed in both situations so
the signal produced is the same (assuming nozzle dynamics are the same).
But every time a scan is taken, noise proportional to vST is also recorded.
The signal to noise ratio for the pulsed experiment is proportional to the
number of molecules, N, but the signal to noise ratio for the continuous
77
flow is N/v^n.
The number of shots, m to see all the molecules would
be,
m = t/t = d/2w ,
w
' o'
where t is the unit of time, one second, t is the time it takes the
w
molecules to travel through the beam waist diameter 2w and d is the
total distance traveled by the molecules in the unit of time.
For Kr
4
-4
where v. = 3.8 x 10 cm/sec and 2w = 12 cm, t = 3 x 10
sec so that
t
o
w
m is around 3.3 x 10
or the S/N is down by a factor 58 after 1 second.
The rotational cooling, as could be seen by the difference it made
in the calculated values of AN is a great advantage for looking at low
rotational levels.
species.
The cooling is crucial to the observation of dimer
Finally, once any molecule or complex makes it through the
region of the expansion where all the energy exchange takes place, and
passes into the more or less collionless flow, the complexes or molecules
are available for observation for a long time.
The disadvantages of the nozzle expansion are that the beam is too
cold, that it is hard to control the temperature, and that the expansion
dynamics are not completely understood.
The expansion dynamics are not
understood in the sense that experimental conditions such as source temperature, pressure, nozzle diameter or geometry are not known beforehand
to achieve a maximum dimer concentration.
Finally, the manifestation of
the Doppler effect that doubles the number of lines and causes the signal
to decay more rapidly is a definite disadvantage.
78
VI.
A.
A JOSEPHSON JUNCTION MIXER
Introduction
We currently use superheterodyne detection in a balanced mixer using
a backward wave local oscillator and the difference frequency of 30 MHz.
The balanced mixer has a noise figure of 6.5 db relative to 300°K which
is equivalent to a mixer noise power generation at about T
= 1000°K
(T = (F - 1)T ) . There are two ways to reduce the effects of this noisy
e
o
balanced mixer.
First, we could amplify our transient microwave signal
(from the molecular coherent emission) before the mixer.
This is expen-
sive, but more importantly, wideband low noise amplifiers in the 4-12.4 GHz
range currently have noise figures around 5dB so not much is gained.
second alternative is to develop a low noise mixer system.
The
A Josephson
Junction (JJ) metal-metal oxide superconducting detector could replace our
balanced mixer described above.
The noise figure of the JJ detector should
be equivalent to noise power in the 50-100 K temperature range, thereby
improving greatly the sensitivity of our system.
The JJ detector will
work effectively in low power applications which is ideal for our superheterodyne detection system used here in the Fourier transform cavity system.
B.
Josephson Junction Detector-Mixer
The mixing and detecting properties of point contact Josephson Junctions have been investigated since the late sixties.
We would like to
79
utilize the low noise temperature (as low as 50°K), high sensitivity
-14
(10
-15
-10
12
watt/ Hz NEP), and large bandwidths (10
cal detection system for microwave spectroscopy.
Hz) in a practi-
Josephson devices are
typically one of three basic configurations, thin films, sold blobs, or
point contacts (or arrays of point contacts).
Thin films and point con-
tact devices have been used to detect microwave radiation, although the
theory is the same for all types.
A Josephson Junction is a "weak link" between similar or dissimilar
107
metals with each in the superconducting state.
The Ginzberg-Landau complex wavefunction for a superconductor is of the form lp =
n = |I(J|
n e
, where
is the density of Cooper pairs and 9 is the phase, in general a
function of space and time.
The Cooper pairs comprise the superconducting
media arising through phonon interaction between electrons of opposite
spin and different wavevectors.
As the metal becomes superconducting, an
energy gap appears, A, symmetrical about the Fermi level, which is a function of the temperature.
The gap is zero at T , the transition temperature,
and is a maximum at 0°K.
The value of the energy gap is dependent on the
particular material.
The bound Cooper pairs are lower in energy by an
amount 2A from the normal electrons (quasiparticles).
2A is 3.05 meV for
Nb, which has the largest gap and therefore the largest T
conducting elements.
tacts.
of all super-
Nb is the material most commonly used for point con-
When two superconductors are brought close to one another but not
touching, ~ 10 A, phase coherence can take place that lowers the free
energy sufficiently to exceed the thermal fluctuations, thus weakened superconducting order extends across the gap.
In a regular superconductor there
is long-range order that fixes the phase over the entire macroscopic length
of the material.
Due to this fixed relative phase or order, if one tries
to establish a voltage drop across a section of superconducting wire, a
80
supercurrent will be induced.
If the voltage is of such a magnitude as
to cause a current that qualitatively corresponds to a kinetic energy
equal to 2A, the superconductivity is destroyed and the wire acts as a
normal conductor.
The value of this current is called the critical current
and it plays an important part in the Josephson theory.
The critical cur-
rent for a typical Josephson point contact is in the range of 1-500 y amps.
The sharpened end of a point contact is on the order of microns in diameter
although the exact geometry of a point contact is open to question.
The
critical current of a superconducting wire 1 ym in diameter is ~ 3 y amps
-7
2
assuming a critical current density of 10
amps/cm .
Josephson derived the relations that govern the behavior of loosely
108
coupled superconductors in his original papers.
The required set of
equations for a flat barrier in the z plane are:
3H
_^_
3x
3H
. „ *„,.,
4TTJ
X_4TTC3VjtL=_J.z.
9y
c
t
c
where H is the tengential magnetic field in the barrier, C is the selfcapacitance, V is the potential difference across the barrier, and j
is
z
the current density through the barrier.
|i-fv<t, .
(VX-2,
where <|> is the change in phase of the wavefunctions across the barrier,
(J> = 8
- 9 , and 2e is the charge of a Cooper pair.
M = 2ed
3x ~ ilc
y
(VI-3)
M = _ 2ed
3y
he z
where d is the effective thickness of magnetic field in the barrier.
81
J
z
where j
= j sin<|>(r,t)
c
,
is the critical current density.
(VI-4)
Since we will not be considering
any magnetic phenomenon, Eqs. (VI-2) and (VI-4) are sufficient to describe
the Josephson junction used as a detector-mixer.
Eq. (VI-4) is a simplified
form neglecting contributions from quasiparticle currents.
Eq. (VI-2) can
be rewritten by setting 3<j)/3t = to; then h.C0 = 2eV(t) and the Cooper pair can
radiate a photon of energy hto as it falls through the potential V(t).
radiation has been detected experimentally.
'
This
At small values of
impressed potential across the barrier, the junction acts as a single superconductor and there is a current through the barrier with no potential difference across it.
When the external potential is sufficient for the cur-
rent to exceed j , a potential difference develops across the barrier with
c
the two metals remaining superconducting.
for V(t) = V
By solving Eq. (VI-2) and (VI-4)
(a constant) gives, considering the current instead of the
current density,
1=1
sin
en
or
1 = 1 s i n to t
c
D
,
where
2eV
w
j
=
DC
-F-
is called the Josephson frequency.
exceed and V
=
3cj>
at
Therefore, when the critical current is
is nonzero, an oscillating current of 483.6 MHz/y volt (i.e.,
2e/2l) flows in the circuit.
This relation is exact and forms the basis for
a new measurement of the constant
lefr.111
82
112
The Resistively Shunted Junction (RSJ) model
considers an ideal
JJ shunted by a real resistance R, as in figure 9 where I__ = I_ + I_
DC
R
J
and V D C = <V(t)> = (h/2e)<d<j>/dt). Then for I D C < I c , V D C = 0 and for
T
DC
> I
V
C DC
= R(I
2
2 1/2
J >
" C
' I n t h S l i m i t ° fIDC > > IC '
DC
the slope
of the current voltage (I-V) plot goes to 1/R as in figure 10. Many
point contact systems can usefully be described by this approach.
Considering now incident microwave radiation on a DC biased point contact, Josephsen has given for a R.F. voltage source,
V(t>
"V D C +
V
BPcoa(WRPt)
or
«
(to1 -/'
t))
ua
3t = tr
h v(V
DC + V
RF l-cos
^RF
Solving for I gives
I = I 0 s m [ f VDCt + g ^ - . smfco^t + * )]
RF
or
1 = 1
c
/2SVRFA
2 J 1-T1 sin
n \ hco„„/
m
n=-°°
\
RF/
to. + ntor,_)t + r <H •
jJ
RF
o
This predicts that there will be DC spikes when co = nto
J
, where n i s
RF
an integer. Usually the impedance of a JJ is smaller than that of the RF
source so the junction sees more nearly a current source than a voltage
source. Taking account of the shunt resistance, current steps at constant
voltage instead of spikes should result.
Experimentally 113 if a fixed microwave signal is applied to a JJ as
one varies the D.C. bias, current steps will occur in the I-V plot whenever
the bias voltage produces Josepheson oscillation at co. which satisfies
V
= n&co /2e or co. = 2eV „fa = nto . The slope of the line between
83
!
Figure 9.
R+
I,}
The resistively shunted junction model.
84
Figure 10.
The current versus voltage plot of a Josephson junction with
no applied field.
I
is the critical current.
85
current steps is related to the dynamic resistance, R^ = (dl/dV)
as
in figure 11.
The first step height varies from I to zero as the microwave power
c
is increased.
With high power this and all other steps are smoothed out,
giving a straight line I-V plot.
The JJ used as a video detector utilizes this change in step height.
Consider a junction biased at a point of high R^ (point A on the figure)
with a constant current DC bias.
change from V
to V
The voltage through the circuit will
as the bias point changes from A to B with the appli-
cation of microwave power that depresses the I-V plot.
114
Kanter and Vernon
See figure 12.
3
have calculated the current response to be ~ 3 x 10
amps/watt at 10 GHz and the NEP ~ 5 x 10
watt/ Hz at 80 GHz with point
contacts.
The application that we are primarily interested in would be to use
a JJ device as a mixer with an external local oscillator (LO). Grimes
113
and Shapiro
analyzed the mixing of microwave signals at 23 and 72 GHz.
Proceeding in an analogous manner to the video detector for an R.F.
voltage source,
V(t) = V D C + V cos (co t+6) + V2cos(co.t+62)
where to and co. a r e t h e two microwave f r e q u e n c i e s ,
00
°°
/2eV.\
/2eV2\
+ £(to 2 t+9 2 )
where k and & are integers, <f> is the initial phase difference across the
junction, and 9. and 9
are phase factors for the incidence radiations.
86
Figure 11.
The current versus voltage plot of a Josephson junction with an
applied microwave signal.
87
V
Figure 12.
A
V
B
The current versus voltage plot of a Josephson junction showing
the change in the step height of one step as the applied microwave power is varied.
88
Current steps are observed at co. = kco. + &C0,,. The DC current that
flows in the step at zero voltage should then go as I J I -r——J J j -r-
jsin
The intermediate frequency (IF) current at Aco = to -to would go as
1
WW
As pointed out by Grimes and Shapiro for the case of to
~ CO- » to.,-CO
and the amplitude of one signal much larger than the other, the signals
would add to produce a beat.
This is equivalent to a signal which is ampli-
tude modulated at the I.F. frequency.
In this case the detection mechanism
is essentially the same as for the video detector, only now the bias point
will oscillate from A to B at the I.F. frequency.
in the detection circuit.
This is thus the signal
I.F. frequencies as high as 9.2 GHz have been
used.
The JJ used as a mixer has advantages over its use as a video detector.
Since the mechanism of detection is the same both should have similar
sensitivities but for the mixer, not only can the bias current be optimized
but the L.O. power also.
The output at an I.F. instead of DC also allows
for less noise and more sensitive amplification.
We have constructed a
point contact Josephson junction mixer and have observed mixing.
But, the
point contact that was built proved to be too unstable and unreliable to
be used routinely.
89
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VIII. VITA
Terrill Joseph Balle was born on October 22, 1947 in Eugene, Oregon.
After graduating from high school in 1965 he attended the University of
Oregon for 2 years majoring in sculpture. He enlisted in the U.S. Army
in 1967 and spent a year in Viet Nam. After separation from the army
in 1970 he worked several years as a welder and millwright for Pierce Corporation in Eugene, Oregon.
In 1971 he re-enrolled at the University of
Oregon and received a B.S. in chemistry in 1975. He started graduate
school in 1975 and will receive his Ph.D. in Physical Chemistry in 1980.
He is co-author of the following publications:
J. S. Wieczorek, T. Koemg, T. Balle, The He (I) Photoelectron Spectra
of Amine N-Oxides, J. Electron Spect. S Rel. Phenon. , 6^, 215 (1975) .
T. Koenig, R. Wielesek, W. Snell, T. Balle, Helium(I) Photoelectron Spectrum of p-Qumodimethane, J. Amer. Chem. Soc, 97_, 3225 (1975).
T. Koenig, T. Balle, W. Snell, Helium(I) Photoelectron Spectra of Organic
Radicals, J. Amer. Chem. Soc, 97_, 662 (1975).
T. Balle, E. Campbell, M. Keenan, W. H. Flygare, A New Method for Observing
the Rotational Spectra of Weak Molecular Complexes; KrHCl, J. Chem. Phys.,
71, 2723 (1979) (Communication)
T. Balle, E. Campbell, M. Keenan, W. H. Flygare, A. New Method for Observing the Rotational Spectra of Weak Molecular Complexes; KrHCl, J. Chem.
Phys. 72, 922 (1980).
K. F. Gebhardt, P. D. Soper, J. Merski, T. J. Balle, W. H. Flygare, Conductivity of a-Silver Iodide in the Microwave Range, J. Chem. Phys. 72,
272 (1980).
M. R. Keenan, E. J. Campbell, T. J. Balle, L. W. Buxton, T. K. Minton, P. D.
Soper, W. H. Flygare, Rotational Spectra and Molecular Structure of ArHBr
and KrHBr, J. Chem. Phys. 72, 3070 (1980).
96
E. J. Campbell, M. R. Keenan, L. W. Buxton, T. J. Balle, P. D. Soper, A. C.
Legon, W. H. Flygare, Chem. Phys. Lett. 70/ 420 (1980).
A. C. Legon, P. D. Soper, M. R. Keenan, T. K. Minton, T. J. Balle, W. H.
Flygare, to be published J. Chem. Phys.
M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J. Balle, W. H. Flygare,
submitted J. Chem. Phys.
E. J. Campbell, L. W. Buxton, T. J. Balle, W. H. Flygare, submitted
Chem. Phys.
T. J. Balle, W. H. Flygare, submitted Rev. Sci. Inst.
J.
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