The orthogonality properties of the transverse eigenmodes of optical resonators which have phase conjugate mirrors at both ends are derived. As in conventional resonators and also resonators with only one phase conjugate mirror, it is shown that the transverse eigenmodes are essentially biorthogonal, a relation which is satisfied between the set of modes propagating in one direction around the resonator and the adjoint set of modes propagating in the reverse direction

Cho, Che-hwang

Kim, Eun-soo

Park, Han-Kyu

Young, In-Eung

Caltech Authors - Main

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 12, DECEMBER 1987 2047 
Quantum  Electronics  Letters 
Orthogonality Properties of Transverse Eigenmodes of Phase Conjugate 
Optical Resonators 
EUN-SO0 KIM, CHE-HWANG CHO, IN-EUNG YOUNG, AND HAN-KYU PARK 
Abstract-The orthogonality properties of the transverse eigen- 
modes of optical resonators which  have  phase conjugate mirrors at  both 
ends are derived. As in conventional resonators and also resonators 
with only one phase conjugate mirror, it  is shown that the transverse 
eigenmodes are essentially biorthogonal, a relation which is satisfied 
between the set of modes propagating in one direction around the res- 
onator and the adjoint set of modes propagating in the reverse direc- 
tion. 
0 
I. INTRODUCTION 
NE OF  the main functions of phase-conjugate mir- 
rors (PCM’s) is to generate electromagnetic waves 
which retrace the original path of incident waves, with 
phase distributions which  are  the exact opposite of those 
of the incoming  waves [ l ] .  The reflected wave is,  there- 
fore,  a time-reversed replica of the incident field. 
Thus, it has  been  suggested  that  when  a  PCM is used 
to replace one  of the two  conventional mirrors that  form 
a laser cavity, it may also compensate for phase distor- 
tions contributed by poor intracavity optical components 
[2]. Whereas  a  conventional  resonator is stable  over  a  cer- 
tain range of mirror  curvatures  and  separations, the phase- 
conjugate  resonator (PCR),  due to the time-reversal prop- 
erties of the conjugate wave, is stable regardless of the 
cavity spacing and/or the conventional  mirror’s  curvature 
[3]. The  PCR is  the  least sensitive to internal phase fluc- 
tuations, corrects almost perfectly aberrations located near 
the  PCM, and is the resonator  with  the smallest diffraction 
losses  for  a  given  dimension [4]. Moreover  the  resonance 
frequencies are not sensitive to changes in the length of 
the resonator. Because of these properties, the use of 
PCR’s to attain  maximum  output  power in the fundamen- 
tal TEM,, mode  seems  promising [4]. 
It is known that complex paraxial elements such as 
Gaussian or hard-edged apertures must be added to the 
resonator to obtain a unique mode such as the TEMoo 
Manuscript received October 30, 1985; revised July 17,  1987. 
E.-S. Kim was  with the Department of Electronic  Engineering,  Kwang- 
woon University, Seoul, Korea. He is now with the Department of Elec- 
trical  Engineering,  California Institute of Technology,  Pasadena, CA 
91125. 
C.-H.  Cho  and  I.-E.  Young  are with the  Department of Electronic En- 
gineering,  Kwangwoon  University,  Seoul, Korea. 
H.-K. Park is with the Department of Electronic  Engineering, Yonsei 
University, Seoul, Korea. 
IEEE Log  Number  8717193. 
mode [5].  When the  resonator  contains  hard-edged  aper- 
tures or Gaussian  apertures,  numerical  procedures  are 
necessary to  derive  the  spatial distributions of the  mode. 
Most of these  procedures  are  based  on  the  Fox-Li  type 
calculation [6]. Knowledge of the orthogonality proper- 
ties of  the  transverse  modes,  however,  can  be  most help- 
ful for the computations  involved. For these reasons, 
Siegman  and  Hardy  have verified the orthogonality rela- 
tion for the conventional resonator and the PCR with a 
PCM at one end, respectively [7], [8]. In this letter, we 
drive for  the first time  the orthogonality relations of the 
eigenmodes  of  the  PCR  with  PCM’s at each  end. 
11. ORTHOGONALITY PROPERTIES OF TRANSVERSE 
EIGENMODES IN A PCR 
Let us consider  an  optical resonator which  has  PCM’s 
at both ends as shown in Fig. 1. The cavity contains an 
arbitrary set of paraxial optical elements represented by 
an ABCD matrix. In the  following discussion,, we  assume 
that the reflection from  the PCM is accomplished by 
means of degenerate  four-wave  mixing  (DFWM). If the 
nonlinear interaction in the  two  PCM’s is given by non- 
degenerate four-wave mixing, after several round trips, 
the  frequency of the  probe  would  walk off the gain spec- 
trum of both PCM’s. This means that oscillations be- 
tween  the  two PCM’s  are not possible unless the mirrors 
are  pumped by the  lasers  of  the  same  frequency.  We also 
assume  that  the reflection from  the  phase  conjugator 
PCM2 may have  an arbitrary transverse variation in  mag- 
nitude but not in  phase as described by 
E r  (x) = Ppcrn(x) E? (x) 
= I Ppcrn(x) I ~ X P  (jlc/,> E T  (x) (1) 
where Ei (x) is  the  incident field on PCM2, E, (x) is the 
phase  conjugated reflected field of the  PCM2, I ppcm (x)  I 
is the gain  or  loss of the  PCM2,  and,  whether  PCM2 is 
apertured or  ideal,  the phase $x is not a function of the 
transverse  coordinate x and y.  Similarly,  we  assume  the 
reflection from the  PCMl  at  the  other nd of the resonator 
may have an arbitrary transverse variation in magnitude 
but not in phase as  described by 
0018-9197/87/1200-2047$01 .OO 0 1987  IEEE 
2048 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 12, DECEMBER 1987 
I 
I 
i 
I 
I 
Fig. 1. Schematic  diagram of an optical resonator  with PCM’s at each end. 
where Ei ( y ) is  the incident field on  PCMl , E, ( y ) is  the 
phase conjugated reflected field of the  PCMl, I ppcm ( y )  I 
is the loss or gain of the  PCMl,  and, whether PCMl is 
apertured or ideal,  the  phase $y is not a function of the 
transverse  coordinate x and y .  
Let us assume that a transverse  eigenmode EPfm is  in- 
cident on PCM2 as shown in Fig. 1. The resulting field 
distribution Epcm ( y) just  in front of PCMl can  then  be 
written in  terms of a generalized  Huygens  integral [9] as 
where y1 is a complex constant representing the axial 
phase  shift,  x1  is  the  variable  of  PCM2, y1 is that of PCMl 
during  one round trip  in  the  resonator,  and k is  the wave- 
number. The field distribution EPc,(x) in front of the 
PCM2  can  be  likewise  written in terms  of E,,,( y)  as 
where the complex matrix DBCA for propagation in re- 
verse direction was used, y2 is a complex constant rep- 
resenting the axial phase shift, and x2 is the variable of 
PCM2 after  one round trip  around the resonator. By use 
of (3) and (4), we  obtain an eigenequation for E,,,,, 
namely 
Equation ( 5 )  describes the resulting field distribution 
just in front of PCM2  after  one  complete round trip around 
the  resonator. As we  shall  see  later,  when  we investigate 
whether or not the kernel of the eigenequation is Hermi- 
tian, it is noted that when variables x1 and x3 are inter- 
changed,  one must also interchange  the dummy variables 
y I  and y2 simultaneously, where y2 is  the variable of PCMl 
during the second round trip  and x3 is that of PCM2  after 
the second round trip.  We have need of an equation which 
contains the variable y2. That is, because y2  is  the variable 
of PCMl during the second round trip, we must obtain 
the resulting field distribution in  front  of  PCM2  after  the 
second round trip in the resonator. 
After the transverse  eigenmode E,,, has  made the sec- 
ond round trip  in  the resonator, the resulting field distri- 
butions  in  front of PCMl and  PCM2  are  given  as follows 
exp [ - - ( A x ;  - 2x2y2 + Dy:)] dx2 j k  
2B 
(7)  
We  can obtain, by use  of (6) and (7) in a procedure  similar 
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-23, NO. 12, DECEMBER 1987 2049 
to the one resulting in ( 5 ) ,  an  eigenequation representing 
the resulting field distribution in front of PCM2 after the G(x3, 
round trip around  the resonator. It is given by 
= 
* exp [k[lm (g) xi 
j k  A* 2x2y2 D* 
* exp 12 (B* - x 2  B* + G Y 2  - -
exp [-${($x$ - (i) A *  x? - i y 2 x 3  2 
- y2  + - A_ xi)] dx2 dy2. B 
2 2Y2x3 
2 
( 8 )  
Then  by  using ( 5 )  and (8), we  can  obtain  the resulting 
transverse field distribution  just in front of PCM2 after + (;)* X2Y2]] dYl dx2 dY2. 12) 
two  complete  round trips around the resonator. From this 
eigenequation we can derive the eigenequation which is Equation (12) is the desired equation which is needed 
needed to  investigate  the orthogonality relations between to verify the orthogonality o f  the  transverse  eigenmodes 
the set of  transverse  eigenmodes incident on the  PCM  and of a PCR with PChl's at both ends. Now let define the 
the set of the reflected transverse  eigenmodes.  Such  a re- modified eigenfunction  and kernel 
lation is described by u (x> = P ~ C  (x)E p c m  (x> 
where  Im ( A / B )  and  Im ( D / B )  are  the imaginary  parts 
of A / B  and D / B ,  respectively. If we assume that the 
structure  and properties of the  two PCM's are  identical, 
then we can obtain the relation 1c;, = $J~. Therefore, by 
use  of ( l ) ,  ( 2 ) ,  and (9), we  can  obtain  the representation 
P,*,m (x1 ) P,*,m (x2 ) Ppcm ( ~ 1 )  Ppcm ( ~ 2 )  
= IPpcm(x1)I  I ~ p c m ( x 2 ) I  l P p c m ( Y 1 ) l  I ~ p c m ( ~ 2 1 1 -  
( 101 
Then, by use  of (9) and (lo), we  obtain  the  following 
eigenequation: 
01 = 7 7 7 2 ) .  
It is now easily verified by  means  of (12) and (14) that 
the kernel v(x3, x1 ) is Hermitian; that is, it satisfies 
4x3,  X I )  = y*(x1, x3). (16) 
When variables x1 and x3 are interchanged, one must 
also interchange  variables yI  and y2 for PCMl to  obtain 
(16). Since  the kernel u(x3, x]) of (15) is Hermitian, its 
eigenfunctions u (x) form  a  complete  set of orthonomal 
functions [lo] in the  sense that 
s .:(x) u,(x) dx = I!&,. 
By introducing (13) and (1) into (17), and  after  proper 
normalization, one obtains the following representation 
which describes the orthogonality relations between the 
transverse eigenmodes of the resonator. 
(17) 
where the kernel G(x3, x1 ) is given by This orthogonality relationship between  any  forward 
2050 IEEE JOURNAL OF QUANTUM  ELECTRONICS. VOL. QE-23, NO. 12, DECEMBER 1987 
traveling transverse  eigenmode Es and  any backward trav- 
eling transverse eigenmode Eb is satisfied for all planes 
perpendicular  to  the  optical axis of the resonator.  Equa- 
tion (18) is essentially the  same  as orthogonality relation 
obtained by Siegman for conventional optical resonators 
and by Hardy et al.  for  the PCR  which has one  PCM at 
one end [ 111. 
111. CONCLUSION 
We  have  derived the relationship for the orthogonality 
of the transverse eigenmodes of PCR’s terminated by 
identical PCM’s which are of the degenerate four-wave 
mixing sort. As in  conventional  resonators and in optical 
resonators which have only one PCM, we have shown 
that the eigenmodes which propagate  in  one  direction are 
orthogonal  to  the modes which propagate in the  opposite 
direction  over any plane  perpendicular to the  optical  axis 
of the resonator 
REFERENCES 
[ l ]  D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng., 
vol. 21, p. 155, 1982. 
[2] C. R. Giuliano, “Applications of optical phase conjugation,’’ Phys. 
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131 J .  AuYeung, D. Fekede, D. M. Pepper,  and A. Yariv,  “A  theoretical 
and  experimental  investigation of the  modes of optical resonators  with 
phase conjugate mirrors,” ZEEE J Quuntum Electron., vol. QE-15, 
p .  1180, 1979. 
[4] P. A. Belanger, “Phase conjugation and optical resonators,” Opt. 
Eng., vol. 21, p. 266, 1982. 
[ 5 ]  P. A. Belanger, A. Hardy,  and  Siegman,  “Resonant  modes of optical 
cavities with phase-conjugate mirrors,” Appl. Opt . ,  vol. 19, p. 602, 
1980. 
[6] A. G .  Fox and T. Li, “Resonant modes in a maser interferometer,” 
Bell Syst. Tech. J . ,  vol. 40, p. 435, 1961. 
171 A.  Hardy, P. A. Belanger,  and  A. F. Siegman,  “Orthogonality prop- 
erties of phase-conjugate o,ptical resonators,” Appl. Opt., vol.  21,  p. 
1122, 1982. 
[8] A. E. Siegman,  “Orthogonality  properties of optical resonator eigen- 
modes,” Opt. Commun., vol. 31, p. 369, 1979. 
[9] S .  A. Collins, Jr . ,  “Lens-system diffraction integral written in terms 
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demic, 1970, p.  743. 
[ l l ]  A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators 
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Orthogonality properties of transverse eigenmodes of phase conjugate optical resonators

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Orthogonality properties of transverse eigenmodes of phase conjugate optical resonators

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