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Optical bistability in four-wave mixing with photorefractive crystals
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
Optical bistability in four-wave mixing with photorefractive crystals M i n g u n Z h a o a n d Yulin Li Xian Institute of optics and PrecisionMechanics, Academia Sinica, Xian 710068, PR China Received 22 August 1990; revised manuscript received 8 February 1991
Based on our study of multi-wavemixing in photorefractive crystal, this paper describes a strictly analytically demonstration that the multi-gratingfour-wavemixing coupling equations have a bistability solution when the coupling constant are real. This solution is discussed under different conditions.
1. Introduction The effect of optical bistability has become of growing interest in different nonlinear devices where the output signal may have two stable states for the same input signal. Recently, bistability was studied in photorefractive nonlinear optics, such as passive phase conjugation mirror [ 1 ] four-wave mixing [2 ], etc. Shaw [ 3 ] e t al. demonstrated that optical bistability behavior existed in photorefractive four-wave mixing with transmission grating model, but from this theory, the probe wave ratio (to the pumpwave) is quite small ( 1 . 0 × 10-6), this magnitude would be most likely much difficult to achieve in practice. We have reported the observation of a multi-grating in photorefractive four-wave mixing [4] (such as both transmission and reflection gratings), that paper proposed a strictly analytically demonstration that bistability exists in this multi-grating four-wave mixing with photorefractive crystal only for the case of the coupling constant is real.
ESf z=0
E2 -...... z=L
Fig. 1. Schematicdiagram illustrating the FWM in PRC.
ff~j=Ejexp[i(cot-kjr)]+c.c.,
j=l,2, 3,4.
(1)
The four waves of equal frequency and of the same polarization are through the photorefractive medium. The write intensity of the four-wave results in the refractive index variation of the photorefractive medium through the electro-optic effect. Then the index change as follows n=no+
nt exp ( i ~ i ) ETE3 + E2E~ 2 I exp(iKir) +c.c.
+ n2exp(igbn) E1E~ +E~E3 exp(iKiir) +c.c. 2 I 2. Theory The basic interaction geometry of four-wave mixing in photorefractive ( P R ) crystal is illustrated in fig. 1, where/~l,/~2; J~3; J~4; are counterpropagating respectively, and
+
naexp (ig~m) EIE~ 2 T exp(iKmr) +c.c.
+ n4exp(i~lv) E~E4 -exp(iKivr) + c . c . , 2 I where
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
265
Volume83, number3,4 4
j~l
(2)
/j is the intensity of beam wave j, and in the following all intensities are normalized. Through this normalization by I we anticipate that the coupling constants yj in the PR effect are approximately independent of total intensity L Also Kx= K 3 - Kt =K2-K4; KH = KI - K4 = K 3 - K2; Kin= 2K~(KI-K2=0); K~v= 2Ka (K3-K4=0); and KI expresses the transimission grating vector; Kn the reflection grating vector; KIII and Kiv are the 0 grating vectors. By using the standard slowly varying field approximation, the Maxwell equation VZE+K2E=O
1June 1991
A3 0(]~)3/0z= -- (Yl
4
I = E 6 = )-'. IEjl. j~l
OPTICSCOMMUNICATIONS
(3)
2
+y2)A1A2A4c 05 ¢I~
2
2 -- (YIA 1+Y2A2)A3 -y4A4A3,
(6c)
A4 0~4/0z= (Yl + y2)AIA2A3COs + (yl A2 +y2A2)A4 +y4A2A4,
(6d)
and OAl/ Oz= (Yl + y2)A2A3A4sin ~ ,
(7a)
OA2/ Oz= - (Yl + y2 )AiA3A4sin ~ ,
(7b)
OA3/Oz= - (Yl + y2)AiA2A4sin q~ ,
(7c)
OA4/Oz= (Yl +y2)AiA2A3sin ~ ,
(7d)
where ~ = l~o++"l" 1~3 -- I~2 -- 4~1 .
(8)
Eqs. (7) can be combined to obtain
divided into the four-coupled wave equation along the z direction,
OA2/Oz=-OA~/Oz=-OA]/Oz=OAE/Oz.
- idEl/dz = Yt ( E l E$ + E[E4)E3
Connecting (8) with eq. (6), we get the following equation:
+y2(EIE$+E~E3)E4+y3(E~E~)E2,
(4a)
O+ _(AIA2A3 Oz -- \ A4
idE2/ dz= yl ( ET E3 + E2 E$ ) E4 +y2(ETE4 +E2E$)E3 +ya(ETE2)Et,
(4b)
- idEa/dz=yl (ETE3 + E2E$)EI +y2 (ETE4 +E2E$)E2 +y4(E$E3)E4,
(4d)
where yj (j= 1, 2, 3, 4) are assumed to be the real coupling constant, for this condition is more familiar to the degenerate four-wave mixing in the Kerr medium [ 6 ]. In general, Ej is a complex amplitude, so that, Ej=Ajexp(i~j), j = 1, 2, 3, 4.
(5)
Substituting ( 5 ) into (4), and from their real part and imaginary part, we can get a cross couple equation for both amplitude and phase:
+y2A2)AI -y3A2AI,
(6a)
A2 0~2/0z--- (Yx + y2 )AI A3A4cos + (ylA 2 +y2A2)A2 +y3A2tA2, 266
X ( y l + y 3 ) c o s ~ - [ ( y 2 - y l ) ( A 2-A3-A2+Ai)]2
2
2 (10)
Now, using eqs. (7) and (10), we can get the following equation
O(AIA2A3A4cos (I~)
Yl --Y2 +Y'
0Z
4(yl +Y2)
X (A ]-A]-A22+A~) 0 ( A 2 - A 2 - A I + A ~ ) 0z
(ll)
where Y3=Y4=Y', using boundary conditions
A2(z)
-----130+ I 4 0 - A ~ ( z ) ,
A21(z) =110 --140 +A2(z) , A2(z) =I2L--A](z).
(12)
In general, we mostly studied the conjugation wave
AI 0 ~ l / 0 z = - (Yl +y2)A2A3A4 cOS - - (yl A 2
A2A3A4~ ~ ]
+ [y3(A ~-A~,) +y4(A~ - A ~ ) ] (4c)
idE4/dz=yl (El E$ +E$E4 ) E2 +y2(EIE~ +E~E3)EI +y4(E4E~)E3,
AIA2A4 AIA3A4 A~ + A~
(9)
A 4 ( z ) , therefore, connecting (Td) with (10) as well
as (12), the fourth wave A4 satisfied 2dA4/dz= (r/A6 +rlDlA~ +D2A 2 +D3)1/2,
(6b)
where
(13)
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
¥
q= 16#2--422 , DI =I1o--I2L--I3o--2/4O, 0.8
Y=Y2 --Yl +Y',
I
D3 = 16f1212L(1304140) (110 --140),
o Sj' I
0.4
fl=2(yl +Y2) ,
D2 = 16fl21140(I30 4140) -I30(I10 --I2L) --I40(I10
212L) -- I1012L ]
--
0.2
__y2{ (11o --I2L) [I10 --I2L
Eq. ( 13) may be recognized as a cubic polynomial in A 41, but solving it is much complicated. Our purpose is to prove that (1 3) has a bistable behavior solution. Therefore, we can simplify it.
3. Conclusion and discussion If we let t/=0 in eq. 913), then it can be written by
-DoA 2] 1/2, (14)
where
+I2o -- 412L/4O.
Do= (llo+I2L)E+2130(llo--lEL)
0.6
0.8
1.0
Fig. 3. PC reflection output showing hysteresis and bistability, u = 1.35.
--2(•30 +2•40) ] +414o(13o +14o) +I2o} •
dA4/dz--fl[412L(I30 4140) (110 --140)
0.4
Solving (14), A2(z) can he expressed as follows
A~(z) =412L(I30 +14o) X (I,o--14o)/Dosin2[flx/~o(L--z)] .
(15)
Further, we will discuss the backward conjugation waveA4(z) at the exit face z = 0 , that is I4o=A42o (0), under symmetry (I~o=I2L=Ip). From (15), we can get following equation:
y= 4 ( q -l _Y Y) ()1_-q 2 sin2[ux/4( l _ Y)_uZ] ' (16) where Y=I4o/Ip is the conversion efficiency ( Y< 1 ); u=flLIp is the pump excitation; q=I3o/Ip is the ratio of input signal to pump wave. The curve of Y versus u is shown in fig. 2, where q= 1, is a fixed parameter, and it is clear that when u exceeds approximately 2.1, the conversion efficiency jumps discontinuously to a low state. Therefore, by controlling the pump excitation, we can obtain the conversion efficiency having multistable behavior. Further, if we chose a u that exceeds the threshold of oscillation such as u = 1.35, the output phase conjugation reflection has a hystersis and the bistability versus the input signal, is shown in fig. 3; here Ip is a fixed parameter [ 5 ]. Now we have solved exactly for the special case of multi-grating four-wave mixing in a photorefractive crystal, but the stability analysis of the steady-state solution is not described; further studying as well as discussion are currently in progress.
Acknowledgements Y
We acknowledge the support of the National Science Foundation of China.
1.0
•
/ !
e
/
p /
1
/
,¢
/ /
/
.
o ,
4
/ ,P
•
i
/i
t I
- %.
•
/
I
t • ~J
.-
References •
. t
2
4
6
8
U
Fig. 2. Conversion efficiency versus pump excitation, q= 1.
[1] P. Gunter, E. Voitmand and M.Z. Zha, Optics Comm. 55 (1985)210. [2] A.E. Kaplan and C.T. Law, IEEE J. Quantum Ele~ron. 17 (1988) 1893.
267
Volume 83, number 3,4
OPTICS COMMUNICATIONS
[3] ICD. Shaw and M. Cronin-Golomb, Optics Comm. 65 (1988) 301. [4] Zhao Mingjun and Li Yulin, Optics Comm. 70 (1989) 67. [ 5 ] K. Tajiwa and H. Hsu, Optical and Quantum Electronics 17 (1985) 149.
268
1 June 1991
[6 ] C. Pard, M. Pichd and P.A. Belanger, J. Opt. Soc. Am. B 5 (1988) 679.
OPTICS COMMUNICATIONS
1 June 1991
Optical bistability in four-wave mixing with photorefractive crystals M i n g u n Z h a o a n d Yulin Li Xian Institute of optics and PrecisionMechanics, Academia Sinica, Xian 710068, PR China Received 22 August 1990; revised manuscript received 8 February 1991
Based on our study of multi-wavemixing in photorefractive crystal, this paper describes a strictly analytically demonstration that the multi-gratingfour-wavemixing coupling equations have a bistability solution when the coupling constant are real. This solution is discussed under different conditions.
1. Introduction The effect of optical bistability has become of growing interest in different nonlinear devices where the output signal may have two stable states for the same input signal. Recently, bistability was studied in photorefractive nonlinear optics, such as passive phase conjugation mirror [ 1 ] four-wave mixing [2 ], etc. Shaw [ 3 ] e t al. demonstrated that optical bistability behavior existed in photorefractive four-wave mixing with transmission grating model, but from this theory, the probe wave ratio (to the pumpwave) is quite small ( 1 . 0 × 10-6), this magnitude would be most likely much difficult to achieve in practice. We have reported the observation of a multi-grating in photorefractive four-wave mixing [4] (such as both transmission and reflection gratings), that paper proposed a strictly analytically demonstration that bistability exists in this multi-grating four-wave mixing with photorefractive crystal only for the case of the coupling constant is real.
ESf z=0
E2 -...... z=L
Fig. 1. Schematicdiagram illustrating the FWM in PRC.
ff~j=Ejexp[i(cot-kjr)]+c.c.,
j=l,2, 3,4.
(1)
The four waves of equal frequency and of the same polarization are through the photorefractive medium. The write intensity of the four-wave results in the refractive index variation of the photorefractive medium through the electro-optic effect. Then the index change as follows n=no+
nt exp ( i ~ i ) ETE3 + E2E~ 2 I exp(iKir) +c.c.
+ n2exp(igbn) E1E~ +E~E3 exp(iKiir) +c.c. 2 I 2. Theory The basic interaction geometry of four-wave mixing in photorefractive ( P R ) crystal is illustrated in fig. 1, where/~l,/~2; J~3; J~4; are counterpropagating respectively, and
+
naexp (ig~m) EIE~ 2 T exp(iKmr) +c.c.
+ n4exp(i~lv) E~E4 -exp(iKivr) + c . c . , 2 I where
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
265
Volume83, number3,4 4
j~l
(2)
/j is the intensity of beam wave j, and in the following all intensities are normalized. Through this normalization by I we anticipate that the coupling constants yj in the PR effect are approximately independent of total intensity L Also Kx= K 3 - Kt =K2-K4; KH = KI - K4 = K 3 - K2; Kin= 2K~(KI-K2=0); K~v= 2Ka (K3-K4=0); and KI expresses the transimission grating vector; Kn the reflection grating vector; KIII and Kiv are the 0 grating vectors. By using the standard slowly varying field approximation, the Maxwell equation VZE+K2E=O
1June 1991
A3 0(]~)3/0z= -- (Yl
4
I = E 6 = )-'. IEjl. j~l
OPTICSCOMMUNICATIONS
(3)
2
+y2)A1A2A4c 05 ¢I~
2
2 -- (YIA 1+Y2A2)A3 -y4A4A3,
(6c)
A4 0~4/0z= (Yl + y2)AIA2A3COs + (yl A2 +y2A2)A4 +y4A2A4,
(6d)
and OAl/ Oz= (Yl + y2)A2A3A4sin ~ ,
(7a)
OA2/ Oz= - (Yl + y2 )AiA3A4sin ~ ,
(7b)
OA3/Oz= - (Yl + y2)AiA2A4sin q~ ,
(7c)
OA4/Oz= (Yl +y2)AiA2A3sin ~ ,
(7d)
where ~ = l~o++"l" 1~3 -- I~2 -- 4~1 .
(8)
Eqs. (7) can be combined to obtain
divided into the four-coupled wave equation along the z direction,
OA2/Oz=-OA~/Oz=-OA]/Oz=OAE/Oz.
- idEl/dz = Yt ( E l E$ + E[E4)E3
Connecting (8) with eq. (6), we get the following equation:
+y2(EIE$+E~E3)E4+y3(E~E~)E2,
(4a)
O+ _(AIA2A3 Oz -- \ A4
idE2/ dz= yl ( ET E3 + E2 E$ ) E4 +y2(ETE4 +E2E$)E3 +ya(ETE2)Et,
(4b)
- idEa/dz=yl (ETE3 + E2E$)EI +y2 (ETE4 +E2E$)E2 +y4(E$E3)E4,
(4d)
where yj (j= 1, 2, 3, 4) are assumed to be the real coupling constant, for this condition is more familiar to the degenerate four-wave mixing in the Kerr medium [ 6 ]. In general, Ej is a complex amplitude, so that, Ej=Ajexp(i~j), j = 1, 2, 3, 4.
(5)
Substituting ( 5 ) into (4), and from their real part and imaginary part, we can get a cross couple equation for both amplitude and phase:
+y2A2)AI -y3A2AI,
(6a)
A2 0~2/0z--- (Yx + y2 )AI A3A4cos + (ylA 2 +y2A2)A2 +y3A2tA2, 266
X ( y l + y 3 ) c o s ~ - [ ( y 2 - y l ) ( A 2-A3-A2+Ai)]2
2
2 (10)
Now, using eqs. (7) and (10), we can get the following equation
O(AIA2A3A4cos (I~)
Yl --Y2 +Y'
0Z
4(yl +Y2)
X (A ]-A]-A22+A~) 0 ( A 2 - A 2 - A I + A ~ ) 0z
(ll)
where Y3=Y4=Y', using boundary conditions
A2(z)
-----130+ I 4 0 - A ~ ( z ) ,
A21(z) =110 --140 +A2(z) , A2(z) =I2L--A](z).
(12)
In general, we mostly studied the conjugation wave
AI 0 ~ l / 0 z = - (Yl +y2)A2A3A4 cOS - - (yl A 2
A2A3A4~ ~ ]
+ [y3(A ~-A~,) +y4(A~ - A ~ ) ] (4c)
idE4/dz=yl (El E$ +E$E4 ) E2 +y2(EIE~ +E~E3)EI +y4(E4E~)E3,
AIA2A4 AIA3A4 A~ + A~
(9)
A 4 ( z ) , therefore, connecting (Td) with (10) as well
as (12), the fourth wave A4 satisfied 2dA4/dz= (r/A6 +rlDlA~ +D2A 2 +D3)1/2,
(6b)
where
(13)
Volume 83, number 3,4
OPTICS COMMUNICATIONS
1 June 1991
¥
q= 16#2--422 , DI =I1o--I2L--I3o--2/4O, 0.8
Y=Y2 --Yl +Y',
I
D3 = 16f1212L(1304140) (110 --140),
o Sj' I
0.4
fl=2(yl +Y2) ,
D2 = 16fl21140(I30 4140) -I30(I10 --I2L) --I40(I10
212L) -- I1012L ]
--
0.2
__y2{ (11o --I2L) [I10 --I2L
Eq. ( 13) may be recognized as a cubic polynomial in A 41, but solving it is much complicated. Our purpose is to prove that (1 3) has a bistable behavior solution. Therefore, we can simplify it.
3. Conclusion and discussion If we let t/=0 in eq. 913), then it can be written by
-DoA 2] 1/2, (14)
where
+I2o -- 412L/4O.
Do= (llo+I2L)E+2130(llo--lEL)
0.6
0.8
1.0
Fig. 3. PC reflection output showing hysteresis and bistability, u = 1.35.
--2(•30 +2•40) ] +414o(13o +14o) +I2o} •
dA4/dz--fl[412L(I30 4140) (110 --140)
0.4
Solving (14), A2(z) can he expressed as follows
A~(z) =412L(I30 +14o) X (I,o--14o)/Dosin2[flx/~o(L--z)] .
(15)
Further, we will discuss the backward conjugation waveA4(z) at the exit face z = 0 , that is I4o=A42o (0), under symmetry (I~o=I2L=Ip). From (15), we can get following equation:
y= 4 ( q -l _Y Y) ()1_-q 2 sin2[ux/4( l _ Y)_uZ] ' (16) where Y=I4o/Ip is the conversion efficiency ( Y< 1 ); u=flLIp is the pump excitation; q=I3o/Ip is the ratio of input signal to pump wave. The curve of Y versus u is shown in fig. 2, where q= 1, is a fixed parameter, and it is clear that when u exceeds approximately 2.1, the conversion efficiency jumps discontinuously to a low state. Therefore, by controlling the pump excitation, we can obtain the conversion efficiency having multistable behavior. Further, if we chose a u that exceeds the threshold of oscillation such as u = 1.35, the output phase conjugation reflection has a hystersis and the bistability versus the input signal, is shown in fig. 3; here Ip is a fixed parameter [ 5 ]. Now we have solved exactly for the special case of multi-grating four-wave mixing in a photorefractive crystal, but the stability analysis of the steady-state solution is not described; further studying as well as discussion are currently in progress.
Acknowledgements Y
We acknowledge the support of the National Science Foundation of China.
1.0
•
/ !
e
/
p /
1
/
,¢
/ /
/
.
o ,
4
/ ,P
•
i
/i
t I
- %.
•
/
I
t • ~J
.-
References •
. t
2
4
6
8
U
Fig. 2. Conversion efficiency versus pump excitation, q= 1.
[1] P. Gunter, E. Voitmand and M.Z. Zha, Optics Comm. 55 (1985)210. [2] A.E. Kaplan and C.T. Law, IEEE J. Quantum Ele~ron. 17 (1988) 1893.
267
Volume 83, number 3,4
OPTICS COMMUNICATIONS
[3] ICD. Shaw and M. Cronin-Golomb, Optics Comm. 65 (1988) 301. [4] Zhao Mingjun and Li Yulin, Optics Comm. 70 (1989) 67. [ 5 ] K. Tajiwa and H. Hsu, Optical and Quantum Electronics 17 (1985) 149.
268
1 June 1991
[6 ] C. Pard, M. Pichd and P.A. Belanger, J. Opt. Soc. Am. B 5 (1988) 679.