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Double-mode Two-photon Absorption and Enhanced Photon Antibunching Due to Interference.

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Annalen der Physik. 7. Folge, Band 38, Heft 2, 1981, S. 123-136
J. A. Barth, Leipzig
Double-mode Two-photon Absorption and Enhanced Photon
Antibunching Due to Interference
By A. BANDILLA
and H.-H. RITZE
Zentralinstitut fur Optik und Spektroskopie der Akademie der Wissenschaften der DDR, BerlinAdlershof
Dedicated to Prof. Dr. &Slav Richter om t h Occu&m of th.e 70th Anniveraay of his Birthday
Abstract. Inspired by results of interfering signal and idler from a nondegenerate parametric
amplifier we investigate the photon statistice of the resulting field after interference of two components subjected to double-mode two-photon absorption. This absorption process leads to a strong
correlation of the participating modes, which can be used to generate fields with photon antibunching
in interference experiments. I n addition the photon number can be made small, which produoea
enhanced antibunching.
Zwei-Photonen-Absorptionaus zwei Moden und durch Interferenz
verstiirktes photon antibunching
Inhaltsubersicht. Die quantenmechanische Betrachtung der Interferenz fiihrt zu neuen Ergebnissen, wenn Felder ohne klassisches Analogon betrachtet werden. Insbesondere ergibt sich durch
die Reduktion der Photonenzahl d m h Interferenz eine effektive Verstiirkung des Photon Antibunching, wie von den Verfaasern in vorhergehenden Arbeiten gezeigt wurde. Die vorliegende Untemuchung
betrachtet die Interferenz von zwei korrelierten Maden, wobei die Komlation durch Zwei-PhotonenAbsorption am den beiden Moden zustande kommt. In jeder einzelnen Mode e@bt sich lediglich ein
gewisses Bunching, wenn man mit kohiirentem Licht in beiden Moden beginnt. Es wird die Interferenz
der Feldstiirke-Komponenten in bestimmten Polarierttionsrichtungenuntersucht. Zur Vereinfachung
wird in den betrachteten Moden die gleiche Anfangsphotonenzahl vorausgesetzt und der Analymtor
auf minimale Transmittanz gebracht. Das eigentliche Signal entsteht dann durch Einfiihrung einer
endlichen Phasenverschiebung zwischen den beiden Moden. Dieaes Signal migt Antibunching und
kann in seiner Intensitiit beliebig variiert werden, was wegen des (l/<n>)-Charakteredes Antibunching
zu seiner Verstiirkung fiihrt. Ferner wird gezeigt, daB die zuniichst fur zwei linear polarisierte Moden
durchgefiihrte Rechnung auf zwei zirkulare Moden sowie auf zwei gegenliiufige Strahlen bei der dopplerfreien Zwei-Quanten-Absorptionubertragen werden kann. Die Ergebnisse werden durch numerische
Rechnungen gestutzt und schlieSlich durch approximative Methoden reproduziert und erweitert.
1. Introduction
The time dependence of the density matrix elements of two light modes undergoing
double-mode two-photon absorption (DMTPA) is now well-known [l- 31'). The results
show a certain equivalence t o those obtained for the nondegenerate parametric amplifier
(see, for example, [4]). I n both cases due to the intraction the two modes are strongly
I)
9'
In the literature this process is also called double-beam two-photon absorption (DBTPA).
A. BAND-
124
and H.-H. RITZE
correlated.This correlation leads to the breakdown of the P representation for both modes
[4]. I n a single absorption act of DMTPA only one photon from each mode is removed in
contrast t o single-mode two-photon absorption (SMTYA), whci e two photons are
simultaneously taken away from the same mode. Therefore we cannot expect photon
antibunching in one mode during DMTPA. The calculation gives even a small bunching
for each mode alone, if we start with coherent states in both modes. This is connected
with the fluctuation transfer discussed in [2].
In this paper we investigate the two correlated modes emerging from a DMTPA
in an interference scheme, especially the resulting photon statistics. I n order t o avoid
additional beating we assume the same frequency for both modes. In the following we
will consider the case where the two modes are represented by the orthogonal components
of a linearly polarized single light beam. Furthermore there should be a given phase
difference between these components. Then we will show that due to the interference
the resulting field exhibits photon antibunching, and owing to the phase difference
mentioned we can adjust the resulting photon number in order t o enhance this antibunching effect. To demonstrate these properties clearly we have also to include numerical calculations. There is a far-reaching equivalence between DMTPA and the interaction in a parametric amplifier or attenuator. The Manley-Rowe relation is of particular
importance, which secures that the difference of the photon numbers remains constant.
I n the following section we reproduce the solution of the master equation for DMTPA.
Here we restrict ourselves to initially coherent light in both modes.
After that we consider the interference of the correlated modes. Especially we are
interested in a field state which effectively does not depend on the initial photon number
and which builds up before the field reaches the stationary state.
We call this state a quasi-asymptotic one. In section 4 we investigate the photon
statistics of the resulting field. I n the “coherent” part, which is due t o a finite phase
difference between the two modes, we can observe enhanced photon antibunching.
In section 5 we present a n approximate procedure of DMTPA in connection with
interference which treats the whole time region exept the stationary state. Finally i t
will be shown that our calculations are also applicable for DMTPA from two circularly
polarized modes (0 ++ 0 transition) and from two light beams (Doppler-free two-photon
absorption).
2. Solution of the Master Equation
We start with the following master equation for the reduced density operator
the light field [l,51
de _
+
+
e of
(2.1)
(a 1+ a2+ a1 a2@ @ a ~ a $ a 1 a 2 )
ala@afa$
dT - where the coupling constant y is included in “time” T = 2yt (with t = normal time) and
at,a$ are the creation operators of the two absorbed modes (1)and (2). Eq. (2.1) describes a n absorption process, in which simultaneously a photon from the mode (1) and a
photon from the mode (2) are taken away.
Introducing the matrix elements
I
en,m+r;n+”,m(T)
and the substitution
= a(nl
l(m
9
+ I4 e m Im> +
11%
y)27
p9 y
20
(2.2)
Double-mode Two-photon Absorption and Enhanced Photon Antibunching
125
we obtain from (2.1)
d~n.m+a;n+v,m
- 1
-
+ P ) + m(n + v)1
+ (m+ 1)(n + 1)
-"n(m
2
dT
Yn,m+p;n+v,m
~n+~,m+l+p;n+l+~,m+l.
(2.4)
Note that the introduction of the off-diagonality in (2.2) and (2.3) is a little bit different
in comparison with eq. (5) from [3]. This is due t o the fact that we calculated the offdiagonal matrix elements for an interference experiment becoming aware of [3] only when
we had already finished our calculation.
From (2.4) follows, that s = n - m is a constant of iiiotion [3,6], we obtain therefore
two subsets of equations for
Y n , m + p;n+ v,m
-
j Ym+e.m+p:m+e+v,m
Ym(S,
Yn(s, ~7
lYn,n-s+p;n+v,n-e
T),
8 2 0
V , TI, I
0.
P* V,
(2.5)
Defining the generating function
00
FP"(Y,T ) =
2~
m=O
T)
~ ~ m p( ) Y,
s 7
we have t o solve the following partial differential. equation
which can be done by a separation ansatz
00
Ff,'(y,
T )=
2
bmj,(y)
e-amT.
m=O
The j m ( y ) are given by the Jacobi polynomials [7]
and
am = 172 (m + p+y2
The other generating function
(2.8) by the substitutions
s+--s7pLLv)
+ + p1 p
8)
.
(2.10)
containing the yn(s,p , v , T ) (s 5 0) is obtained from
(2.11)
as can be proved easily.
For the calculation of the bm in (2.8) we restrict ourselves to coherent light with
equal photon numbers in both modes, where these coherent states are generated by
single-mode excitation, illustrated in Fig. 1:
(2.12)
A. BAND-
126
and H.-H. RITZE
(2.13)
IzvF)
where I,(lar
is the modified Bessel function and DC = Ioc 1 eh. Now we can use the
following relation [8]
(2.14)
Analyzer
Fig. 1. Spatial orientation of the field vectors of the two modes
Comparing (2.8) with (2.14) follows
(2.15)
In this way we have determined the generating function for our initial conditions and
are ready t o calculate arbitrary expectation values.
3. Interference of the ! h o Correlated Modes (1) and (2)
If we observe the photon statistics of the modes (1)and (2) behind the analyzer owing
to interference we have t o write for the resulting field (Fig. 1)
B + ( T )= e-Zid al+(!f?)COB 8 - az+(T)sin 8 .
(3.1)
I n the mode (1)we introduced an additional phase displacement e-2is, which is important
for the resulting field in an analyzer (Fig. l),as will be seen in the discussion of the photon
statistics uf the light field behind the analyzer.
Double-mode Two-photonAbsorption and Enhanced Photon Antibunching
127
Now we let 8 = n/4, which gives
1
e-2i6 - a g ( T ) ] .
B + ( T )= - [ a $ ( T )
(3.la)
1/2
If we would also have 6 = 0, then (3.1a) would niean the crossed configuration and we
would expect only bunched light behind the analyzer (compare, e.g. [9]). The =me is
shown in [lo] for nondegenemte parametric interaction. The finite phase difference 6
gives us a “coherent” signal with an arbitrary intensity.
With (3.1s) the photon number is
)+ (az+(T)az(T’)>
1
@+B> = ;z- { < a W )%(T)
- e-2i6 <al+(T)az(T)>- e2is<al(T)a z f ( T ) ) } .
(3.2)
Expression (3.2) will be calcuhted as illustrative exainple how to deal with all other
averages occuring later. From the generating function (2.6) we obtain (cf. [ 2 ] )
and (as a definition)
(3.4)
Due to (2.5) we must also define n(’)(s) containing the y,,(s, p , v, T ) and this is because
of (2.11)
(3.5)
%(‘)(a)= d ) ( - s ) .
Then we have
8-
- m n=O
-1
m
=
C m(l)(a) + 2
s=o
c
[n(’)(s)- sn(o)(s)].
6=--8
-8
From (2.4) follows, that
&+s,k;k+,,k(T)
is a constant of motion. Therefore the last
k=O
term of (3.6), that is
-1
-
1
2
s=- w
mco)(s) =
8
w
6=O
s=O
w
C sl’(o)(--s)= c 2
@m+6,fil;m+8,tn(0)
m=O
(3-7)
gives the stationary value of (3.6), because all other terms in (3.6) are damped out by
the exponentials. According to our initial conditions we find from (2.13) for (3.7)
m
2
s=o
m
c
m=o
W
em+s,m;m+s,m(0)
= e-’‘’l
c
r=O
816(l&
la)
128
.
For la
l2
A. BAND-
aad €I.-H. RITZE
9 1 we can use a n approximation for (3.7a) and get
Thus the stationary photon number in one mode is of the order of the square root of
the initial photon number if we start with coherent light. The other parts of (3.6) are
found from (2.8) with (2.9), (-2.10) and (2.15) by means of relation (3.4) to be
-1
W
2 m'"(s) + e =2
e=O
-00
00
n("(s) = 2
2 m(')(s)- m(')(O)
6=0
If we use a uniform asymptotic expansion [ll]for the Bessel functions we obtain for
(2m
+
3)
< la12
(3.10)
+
Now, from (3.10) we see that (2m s) la1 is permissible without violating the condition
(2m 8)
Ia12.
Therefore from the double series (3.9) follows that for T > 0 the sum over s is cut off
almost only by the Bessel functions and not by the exponentials exp (--msT) in contrast
t o the sum over m, where the exponentials exp (-m2T) quickly terminate the series.
The lattei' property leads to the existence of an asymptotic state during single-mode
two-photon absorption (SMTPA), whereas in the ease of DMTPA there will be only a
quasi-asymptotic state appearing in the interniediate T-region because the stationary
state depends on the initial photon number again (cf. (3.8)).
Now, because of the Manley-Rowe-relation [2] and
(at(0) a1 (0)) = <az+(O)
azW> (cf. (2.12))
we have
(3.11)
(4-(T)a,(T)) = (4(T)a,(T)>*
For the last two terms in (3.2) we find
+
<
(3.12)
(3.13)
(3.14)
Double-mode Two-photon Absorption and Enhanced Photon Antibunching
129
Findly
(3.16)
m
m
(3.16)
+ 2 c + 1) 18+1(\a e-+-lal8
W
x e-m(m+l)T + e-14'1 ,(la 12)
(8
a=l
12)
= (a: (T)a l ( T ) )- (at(!!') a2(T)) + 2 sineS(al+(T) a2(T)).
1
The terms besides the one proportional to sin2 6 in (3.16) give the photon number which
ie transmitted in the crossed configuration because the analyzer is perpendicular t o the
polarization of the incident beam (Fig. 1). Thus this photon intensity is due to the
distortion of the coherence of first order by DMTPA. The term proportional t o sina b
is the coherent part of (3.16) which can show enhanced antibunching as we will see later.
--g-39.89
/
1o - ~
10-2
10-1
T
Fig. 2. Time dependenceof the photonnumber(af(5") al(T)>and of the difference <a$(T)a(lT)> <a$ (2') a#')), which is transmitted in the crossed configuration for 6 = 0
A. BANDILLA
and H.-H. RITZE
130
The stationary value of (3.16) is
(3.17)
The leading term in (3.17) does not depend on 6 and represents therefore depolarized
light (compare also [12]). We will later see that this light is bunched.
Fig. 2 illustrates the magnitude of the various terms in (3.16), where we used (3.10)
for the calcuhtion. We see fromFig. 2 that the disturbing photon number ( a t (T)al(T))< a f ( T )a2(T))(the "incoherent" part) is not noticeable until the field is near the stationary state. In a semiclassicaltreatment the photon number ( a t ( T )a l ( T ) )shows the
same dependence except in the stationary region where it goes to zero.
4. Photon Statistics of the Resulting Field
As a measure of the photon statistics of the resulting field behind the analyzer we
consider the quantity
Taking into account our initial conditions ((2.12) and (3.la))we have for the numerator
of (4.1)2)
1
(B+2Ba) - (B+B)2 = - ((at2af)- (a$al)2) (ajalaJaz)
2
+
+ 1 ( a t 2 a f )cos 46 - 2 ( ~ l + ~ a cos
~ a 28
~)
(4.2)
1
<al+a1>2
+ [< a t a l >- 1 <a2a2>cos 26 - <a?az> cos 28 - 2
1
a
2
The calculation of (4.2) is analogous t o that of (3.2) and therefore we will give only the
results :
<B+2B2)- <B+B)2 = -4(al+a2) sin2 6
+ ( 1 - sin26)
- (1 - sin2 8 ) <a,+a2)
<a1+2alaz)
<ata2>
(alfa2)
3
(4.3)
2)
In order to simplify the expressions we will not write down the T-dependenceexplicitely.
Double-mode Two-photon Absorption and Enhanced Photon Antibunching
131
where we introduced the last abbreviations in order to simplify further notations. Note
that 2 sin2 G(afae> represents in a very good approximation the transmitted photon
number (cf. (3.2), (3.16) and Fig. 2) as long as the stationary state (cf. (3.17)) is not 'reached. Fig. 3 shows the time dependenceof 2{1} and 2[{1} - {ZZ}]. The curve 2{1} corresponds t o the case where 6 and sina 6 are very small and therefore the transmitted
photon number also. Neglecting the influence of {ZZZ}we get a large photon antibunching
due to the smallness of the photon number. This means an enhancement of the antibunching effect.
t
I
I
c
I
10-3
10-2
T
Fig.3. Calculated dependence of 2{I} and
2[{I} - {11}]for the =me initial photon
number aa in Fig. 2 (I a 12 = 10')
Fig. 4. (111) is plotted in the same T-region
3 (I a 12 = 1@)
as Figs. 2 and
t
30 -
I
10-2
I
lo-'
m
T
Fig. 6. Photon number (a$(!/')a,(!/')>- (nf(T)
az(!Z')>and photon statistics (<(An)') - (n>)/(n>=
(III}/(<af(F)a,(T)>- ( a t ( ! / ' aa(T)>)
)
in the crossed configuration (I.]? = 104)
A. BANDILLA
and H.-H. R ~ Z E
132
For 6 = 7212 we find no enhancement but the plausible result, that ((An)2)/(n)
approaches the value 2 / 3 . This is shown in the curve 2 [ { I } - { I I } ] . Note that (4.3)
gives ((An)2)- (n). Fig. 3 is calculated with the help of (3.10). The same is true for
Fig. 4, which confirms the smallness of { I I I } for the region where (3.10) is applicable
except for the stationary region. Finally we also calculated the photon statistics in the
crossed configuration (Fig. 5). As long as the photon number is small in comparison with
the stationary photon number, the radiation is bunched stronger than in chaotic light.
It is also interesting t o study the case where the detector measures the light field in
the absence of the analyzer. Then the corresponding photon number can be expressed
by N = .?a1
a2+a2.It is possible t o show that the factorial moments are approximately the same as for B+B (6 = n l 2 ) if we are sufficiently far away from the stationdry
state ((B+B)a,ol<B+B)a=x/2
< 1).
+
6. Approximate Solution of the Equations of Motion
As we know from the treatment of SMTPA it is possible to obtain approximate
solutions of the equations of motion if we start with large initial photon numbers in
coherent states [ 1 3 ] . I n a similar way we can treat the case of DMTPA. The solution
for DMTPA greatly simplifies if we assume
( a t ( O )al(0)>= (az+(O)a2(0)>= no =
I& l2
2 '
a s was also done in (2.12). If we use the abbreviations
n1 = a,+al, n2 = az+a2,
we obtain the following system of equations for the averages appearing in (4.1) (cf.
( 2 . 4 ) ):
(5.1)
d
- (a,f2nya;>.
dT
The essence of our approximate solution consists in the assumption of a certain amplitude stabilization of the field [ 1 4 ] , especially the relation ( ( ~ l n ~ , ~(n1,2)2
) ~ ) must be
fulfilled.
So we cannot expect to give a correct description of the stationary state showing
bunching (cf. Fig. 5), where the photon number is of the order
as we can see from
( 3 . 6 ) , ( 3 . 7 a ) , ( 3 . 8 ) and (3.17). Therefore it is reasonable t o assume
-(a,'2nla~>= -2(a,+2nln&
<
VG,
<nl(m>
(n,(T)) P V G
( ( n , ( T ) )= <nz(T)))*
(5.2)
Double-mode Two-photon Absorption and Enhanced Photon Antibunching
-<
133
With ( n l ( T ) )
1/T we conclude, that our results are valid for
noT2 1.
Then we can write approximately
(5.3)
+ <nl) [((dn1)2>+ 2<(Anl) (An2)>]
(~t'"nl~za9
=
(<atkn2& + <.f"l&)
- <nd2<at"$> + ((An,) (An,)) <w4>
k = 1,2.
<u,'kn;ug)
(2<al+%la;)(a,'"$))
+ ((A%)2> <w4>
< n W = <nd3
<n1>
= (n1)
(n1)
Note that
(5-4)
(n,(T)>= (nz(T)>
but
*
( n l m nz(T)) <n?(T)>.
Inserting (5.4) into (5.1) and substituting
+
<
5 = (1 n0T)-l, "6' ta5 1
we obtain after some algebraic manipulations
(5.5)
<a1
+ 2a2>
2 = n:t2
+ no [
1
-7
+t
- P - t3/3
+ ("2 + 0 (31n0
Expressions (5.5) are now used to determine the properties of the resulting field behind
the analyzer corresponding to (3.2)and (4.2). We easily obtain (n = B+B,An = B f B -
<B+B>)
We distinguish two important cases :
A. B m m and
~ H.-H. RITZE
134
a) Small absorption region
Here we have nOT 1 and can therefore restrict ourselves to the first power in T.
Then we obtain from (5.6) and (5.7)
<
( n ) E ( B + B ) = 2n0(l - noT)sin2 6
(5.8)
and
((An)2)- ( n ) - ( B f 2 P )- (B+B)2
-
(A>
<B+B>
- -noT.
-
(5.9)
Eq. (5.9) shows, that there is photon antibunching from the beginning. Further we see
that the quantity (5.9) is independent of 6, i.e., varying ( n ) (by changing of 6) ((An)2)/
( n ) is conserved.
b) Quasi-asymptotic region characterized by 1 noT
This approximation corresponds to (3.10). The absorption is well advanced but far
from the stationary state. Then we find
<
<&
(,?.lo)
and
(5.11)
x
where we have further assumed sin26 9 n 0 P . If we let 6 + - we conclude from (5.11)
in a good approximation
2
(5.12)
in agreement with the numerical results, plotted in Fig. 3. On the other hand (5.11)
gives a decrease of ((An)S)/(n) with decreasing 6 ((B+B) 9 n o T ) :
((An)*> - ( n )
(n>
1
d ~+
o -1.
(5.13)
Expression (5.13) corresponds to 2{1) in Fig. 3, where the tendency expressed in (5.13)
is confirmed.
From (5.G), (5.10) and (5.11) it follows that the quantity
is independent of 6 for ( n ) >> (noP)-l. Reducing the photon number ( n ) of the interference field by decreasing the phase difference 6 (starting with 6 = n/2),the quantum
fluctuations decrease in a similar way like those of a classical fluctuating field
(( (An)2)= O((n)”) during one-photon absorption (conserving ((An)8)/(n)2).This
behaviour is different from that for the interference arrangement using SMTPA where
an increase of the absorption path length in the asymptotic region does not cause an
additional enhancement of photon antibunching [13].
6. Equivalence with Other Interference Arrahgements
Hitherto we considered DMTPA, where in one absorption act simultaneously two
linearly polarized photons were absorbed as illustrated in Fig. 1. The atomic system,
which “works” in such a way can be prepared, e.g., by a transversal magnetic field.
Now we start with linearly polarized light and let the atoms absorb simultaneously
only a left-hand and a right-hand circularly polarized photon. Such an absorption process
Double-mode Two-photon Absorption and Enhanced Photon Antibunching
135
Initial polarization plane
Fig. 6. Linearly polarized light, its circular components a$(!!')and a$(!!') and the position of the
analyzer
appears for two-photon transitions between levels of momentum quantum number = 0
(0 t+ 0 transition [12]). This situation is outlined in Jlig. 6 (cf. [12]). The analyzer is
rotated by an angle x out of the initial polarization plane. If we denote by a$ (a$) the
creation operator of the right-(left-)hand circularly polarized field mode we obtain for
the field behind the analyzer
1
Bf
(a$e-ix + a,'e'x) =Fek+i=(al+e-eil-i=
-at).
(6.1)
=+
+
Whereas the first phase factor in (6.1), exp (ix in)common to both modes, is meaningless we find the equivalence to (3.la) if we set
26 = 2% n.
(6.2)
The expectation values of products of field operators are the same as calculated above
for DMTPA, the photon number and the statistics of the detectedfield as functions of the
position x of the analyzer are described by (3.16), (4.2) and (5.6), (6.7). The enhancement
of photon antibunching with increasing x is connected with the fact that the 0 ++ 0-two
photon interaction changes the state of polarization of initially linearly polarized light
[12].
Finally we want to point out the possibility to use two counterpropagating light
beams of the same frequency to realize DMTPA in dilute gases. In the center of such
two-photon transition atoms moving not perpendicularly to the lightbeams absorb only
one photon from each beam. Enhanced photon antibunching can be obtained due t o
the interference of the beams behind a beamsplitter of equal reflectivity and transmittance. Changing the position of the beam-splitter the detected intensity can be varied.
The equivalence is the same aa discussed in [13].
+
References
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J. Phys. A Math. Nual. Gen. 8, 1140 (1975).
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S
Opt. Commun. 31, 21 (1979).
[4] B. R. MOLLOWand R. J. GLAWER,Phys. Rev. 160,1076 (1967); 160,1097 (1967).
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136
A. BANDILLA
and H.-H. R ~ Z E
[6] H. D. S~iruuuvand R. LOUDON,
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and I. STEOUN,
Handbook of mathematical functions. New York: Dover Pubilcations 1965.
[12] H.-H. RITZEand A. BANDIILA,
Phys. Lett. 78 A, 447 (1980).
[13] A. BANDILLA
and H.-H. RITZE,Opt. Commun. 33,195 (1980).
[14] H. PAUL,
U. Mom and W. BRUNNER,
Opt. Commun. 17, 145 (1976).
Bei der Redaktion eingegangen am 3. September 1980.
h c h r . d. Verf.: Dr. A. BANDILLA
und Dr. H.-H. RITZE
Zentralinstitut fur Optik und Spektroskopie
der Akademie der Wissemchaften der DDR
DDR-1199 Berlin-Adlershof
Rudower Chauasee 6
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