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Exact solution to photorefractive two-wave mixing with arbitrary modulation depth
15 January
1996
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
123 (1996) 201-206
Exact solution to photorefractive two-wave mixing with arbitrary modulation depth M.R. Belie a*b, D. TimotijeviC a, M. PetroviC a, M.V. JariC b a Institute of Physics, PO. Box 57, 11001 Belgrade, Yugoslavia b Physics Department, Texas A&M University, College Station lX 77843-4242, Received
15 June 1995; revised version
USA
received 27 September 1995; accepted 29 September 1995
Abstract
We obtain analytical solution to the slowly varying envelope wave equations describing two-wave mixing in photorefractive crystals with arbitrary dependence of the gain and absorption on the two-wave fringe modulation depth. We introduce a novel form of the correction function for the space charge field that offers an improved agreement with the experiment.
Motivation for this Communication comes from the interesting paper on two-wave mixing (2WM) in photorefractive (PR) crystals by Refregier, Solymar, Rajbenbach and Huignard [ 11. At one point in their paper the authors state: “What emerges from the solution [of the materials equations] is that ImEs [the imaginary part of the space charge field] instead of being proportional to m [the fringe modulation], becomes a function of m. Unfortunately, f(m), valid for the range 0 < m < 1 could only be found by solving numerically [the Kukhtarev] equations, which we have not done as yet. Alternatively, we can use a phenomenological approach which does not aim at solving the relevant equations in a formal sense but reflects the physics of the situation. We know that as fringe modulation increases, the modulation of the space charge field will increase at a less than linear rate. The problem is then to find a function which for small values of m is equal to m and [for] higher values of m will display slower growth”. While certainly it is worthwhile to determine an appropriate form of the function f(m) , we show that 0030-4018/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved SSDZOO30-4018(95)00620-6
in fact, the slowly varying envelope wave equations describing 2WM in PR crystals can be solved for any form of that function. Let us set the stage. The equations of interest are of theform [l]: 1; = __rs
az;
= -rs--
A(m) zlz2
---
a,, 9
Z
m fx(m)
m
1112
Z
+ffzz,
(la> (lb)
where It and I2 denote the (nearly) copropagating or counterpropagating laser beams impinging on the crystal, Z = Zr + Z2 is the total intensity, g = f is the parameter controlling the geometry of wave mixing (reflection or transmission) and Ts is the saturated coupling constant. Thus, we consider a PR crystal with pure nonlocal response and a real coupling constant. As mentioned, m is the fringe modulation,
(2) and we have added a subscript x to the function fX ( m) , to differentiate between various forms that have been
202
used thus far in the literature. We reserve the symbol f for another variable. Finally, a is the linear or nonlinear (NL) absorption in the crystal. The prime in Eqs. ( 1) denotes the derivative along the propagation (z ) direction. Steady-state, degenerate, and plane wave situation is assumed. If one compares Eqs. ( 1) with the 2WM equations in the standard notation [2], one notes that the coupling “constant” has acquired a more general form: T(pn) =rsfx(m)
m
.
(3)
Based on this fact, we advance the following argument involving absorption in PR crystals. In general, for LYconstant, we have linear absorption, and for cr a function of intensities (or m) , we have NL PR absorption. In principle, the dependence of LYon m could involve a function other than fX (m). However, there is no reason to presume that the saturation mechanism in PR gain is fund~ent~ly different from the one in PR loss. Thus, for the NL absorption we also assume: a(m)
=ffs-,
f&n)
(4)
m
even though (as we shall show) the wave equation can be solved in quadratures for arbitrary dependence Ly(m) .Other models (in which absorption is assumed to arise through absorption gratings from the beginning) are possible [ 31. We will also consider the linear case cu( m) = cr = const. We mention that for (T = +, active is the reflection geometry (RG), in which the beams are incident on the opposite faces of the crystal, and for CT= -, active is the transmission geometry (TG), in which the beams are incident on the same (z = 0) face of the crystal. We treat both geometries exactly. The simplest choice for fn (m) is: f](m)
= i
m $3(m)
=1+
(7)
based purely on ~lculation~ convenience. Here b is the fitting parameter, function~ly similar to the parameter a in Eq. (6). This form of correction function allows for the solution of wave mixing equations in TG. Nowever, the paper deserves some further comment. ’ At one point the authors claim: “Although [Eq. (6)] gives a good explanation of experimental results for large signal effects, it is not possible to obtain analytical solution for two-wave mixing gain with such a functional form of fz(m>. Refregier et al. obtained an analytical solution by just using an undepleted pump approximation. So far, to the best of our knowledge, there has been no analytical treatment for the two-wave couplings including large modulation effects and pump depletions in a PR material which are valid for the whole range of m.” This statement is wrong on both counts. First, it is possible to obtain an analytical solution to 2WM with any functional form of fX (m) , as we shall show below. Second, there exists a number of analytical treatments [ 71 of the two-wave couplings including large modulation effects and pump depletion, only they are not known under such a name; they are known as the two-wave mixing with intensity dependent couplings. While one may argue that couplings dependent on the modulation depth stem from different physics than the couplings dependent on the intensities, as far as the solution of wave equations goes, that doesn’t matter. A typical model for the coupling used in the intensity dependent theories is of the form [ 71:
(5)
=m,
which is valid for small modulation. One then obtains the standard Kukhtarev equations [ 41, that have been solved exactly in many ways [ 2,5]. Refregier et al. have chosen {“after some attempts”, they say) the function: fi(m)
However, they only solved the undepleted pump case. While convenient for fitting, this function is impractical for solving the wave equations. Recently, another paper [6] appeared on the very same subject. In that paper the authors introduce another form for the correction function:
[l - exp(-am)]
.
(6)
I‘ z-(Z) = z-y”-IfC that is also based on phenomenological arguments. In the simplest approxima~on, C is chosen to be a constant. However, it need not be a constant, and if it i The notation of angles in the equations and in some of the figures in this paper are not consistent with each other.
MR. BeliC et al. /Optics
Communications
is chosen as C = 2b( II I,) ‘12, then the model of Kwak et al. [ 61 is recovered. We show that such a choice for Z(Z) , which leads to the following form of fX (m) : f4(m)
=
~ f
F(z) = sech-‘(m) given by: Zi = f exp(F)
mf +Cm'
203
123 (1996) 201-206
,
and f(z).
The intensities
,
12 = f exp(-F)
are
(1.9
and the 2WM gain is:
where by definition: 12L
f = 2(Z&)l’2,
(10)
results in a model that not only can be solved analytically, but offers a better agreement with the experimental data. We assume C to be a constant. The procedure for solving Eqs. ( 1) is as follows. We first treat TG. This case is simple, since boundary conditions are given on the one side of the crystal: Cr = Zr(O), C2 = 12(O). When the equations (with u = -) are rewritten using the combinations u = Zr + Z2 and u = It - 12, it is seen that there exists a convenient set of dependent variables f and F (instead of Zr and 12) in terms of which Eqs. (1) decouple. The definition is [ 51: Zr +Z2=
Zr -Z2=
fcoshF,
and the equations
fsinhF,
(11)
become:
f’=
ZS -fX(m) 12
F’=
-$f,(m)coshF.
sinhF
- a(m)
1 f,
J
ny, f,(m)-’
dm
m0
(16)
’
where p = Zr/Z2 = exp( 2F) is the beam intensity ratio and L denotes the thickness of the crystal. These are the general expressions. Of the four correction functions mentioned, only the integral for f2 can not be expressed in terms of elementary functions. For the function f4 (m) we introduced, one obtains: rsz =$-{exp[-2c(F-Fo)]
-l}-2(F-Fo). (17)
This completes the solution procedure. In TG, there is no need to fit boundary conditions, as they are given explicitly: tanhFo=s,
1
2
(18)
However, in RG the fitting has to be done separately from the solution process. Before we present our results, let us go through the RG solution procedure. In RG (a = +) the procedure is less explicit. One introduces the same set of variables, and the equations to be solved become: f’=--fX(m>fcoshF, 2 F’ = $fX(m)
sinhF
- a(m).
(19)
(13)
where c = czS/JIS. For linear absorption, the dependence of f on m (and z ) is even simpler: f = fomexp(--az).
c
rs
cash F. -, cash F
m
=2
PL ( -PO >
mo = sech FO.
This set of equations is easily solved in quadratures. The variable F is simply connected with the modulation depth, m = sech F, hence one can find f as a function of F (or m) and m (or F) as a function of z:
rsz
l+Po l+PL
fo=2(c,c2)“2,
(12)
f =foexp[2c(F-Fdl
,=c2=-
(14)
Thus, for an arbitrary function f X(m) , one first evaluates the quadrature Tsz in Eq. ( 13) and then finds
They are similar to the corresponding equations for TG, but the solution is more involved. Again, f can be expressed in terms of F and F can be expressed (implicitly) in terms of z, In
(
+
where
>
=Z,(F,Fo),
rSz = Z2(EFo),
(20)
M.R. Beli6 et al. /Optics
204
Communicarions
123 (1996) 201-206
.O
Fig. 1. Two-wave mixing gain y in the transmission geometry, as a function of the crystal thickness z, for different values of the input beam intensity ratio &I = Cl/& Dashed curves correspond to the fl model, dotted curves correspond to the f3 model, and full curves correspond to our f4 model. The parameters are: C=lO mW/cm*, cxs=1.6 cm-‘, rs=14 cm-’ for our model, and b=1.7, ~0.52 cm-‘, Ts=18.3 cm-’ for the other models, as in Ref. [6]. (a) &20, 5, (b) /3n=2000,500.
I
00
0.5
0.0
I
I
1.0
1~5Z(crrl)2~0
fig. 2. (a) Fringe’modulation depth m. (b) Nonlinear absorption (Y in TG, as functions of the crystal thickness z, for different values of the input beams ratio PO. Our form of the correction function f4 is used. The parameters are the same as in Fig. 1.
F
Zl(EFo)
=
J
dF
(21)
tanhF-2c’
Fo F
z2(F; F”) =
J
fx(sechF)
Ilo = Cl, ZzL = C2. Thus, one has to solve a system of two NL algebraic equations for Fo and FL: Z~(FL,FO)
=rJ,
2dF (sinh F - 2ccosh F) ’
Fo+FL+ZI(FL,FO)
=ln
(22)
FO
and the integrals are to be found in the table of integrals or evaluated numerically. A bit more complicated part is the fitting of boundary conditions. The conditions are now given on the opposite sides of the crystal:
This is easily accomplished on computer. Once FO and FL are determined, one uses Eq. (20) to find F(z ) and f(F) . For reasonable values of Cl and CZ, unique solutions are found.
M.R. BeliC et al. /Optics Communications 123 (1996) 201-206
205
1.0 -
m
-
0.8-
0.5
1000
0.2
I
-
0.001 0.0
0.0
0.2
1 0.4
, 0.6
,
,
I
0.8
1.0
a.0
, 0.0
0.2
L
/ 0.6
0.4
a
I 0.8
z(cm>
i
,
1.0 dcm>
Fig. 3. The same as Fig. 2, only for RG. The parameters are: C=lO mW/cm2, as=2 cm -I, I’s=5cm -l, Cz=l, which is the unit of intensity. (a) Modulation depth m as a function of z. (b) Nonlinear absorption (Yas a function of z, for different values of the pump intensity Cr.
Some of our results are presented in Figs. l-4. They should be compared with the corresponding results of Kwak et al. [6] and Refregier et al. [l] Two-wave mixing gain for different models is presented in Fig. 1, as a function of the crystal thickness z. Contrary to the belief expressed elsewhere [6], the value of the saturated gain depends on the model function J”~( m) . Fig. 2 depicts the modulation depth m and the NL absorption LYas functions of z, for our form f4 of the correction function, and in TG. It is interesting to note that at some value of z within the crystal, the modulation depth reaches the m~imum value of 1, and then decreases. After that point the signal saturates or even deamplifies (in the presence of absorption). In other words, there exists an optimal value of the crystal thickness at which the maximum amplification of the signal is achieved. The figure also points out to the inadequacy of the Kukhtarev model with small modulation, once the depletion of pump is taken into account. Fig. 3 presents the same quantities as Fig. 2, only in RG. Instead of /3e, the relevant parameter now is Ct, the intensity of the first beam at z = 0. The intensity of the other beam, which is incident on the z = L face of the crystal, is fixed at C2 = 1. The dependence of m on .z now displays a monotonous change. In Fig. 4, the comparison of different models with the experimental results is represented. Such comparisons are the most relevant tests for any of the phe-
IO4
____-------I~,r.~__“_
..,_-..-
,*
,’
/.“‘-
r’
,/
,’
<‘
,d
,’
,‘
103
/’
~
I
IO2
10'
IO0 100
102
lo4
IO6
PO
‘OS
Eig. 4. Two-wave mixing gain y as a function of the input beam ratio @a,for different models. The parameters and the notation are the same as in Fig. 1, and tbe crystal thickness is set to L = 0.5 cm. The circles represent experimental values from Ref. [ 61.
nomenologic~ models. It is evident that our model offers the best agreement with the experiment. In summary, we have shown that the slowly varying envelope wave equations for 2WM in PR crystals, in either TG or RG, can be solved exactly (up to a quadrature) for an arbitrary dependence of gain and abso~tion on the fringe m~ulation depth. We intro-
206
MR. Belid et al. /Optics
Communications
duced a new form of the correction function for the space charge field, that offers an improved agreement with the experiment.
This work has been in part supported by the United States National Science Foundation, grant No. DMR 9215231. References [ 11 P Refregier, L. Solymar, H. Rajbenbach and J.-P Huignard, J. Appl. Phys. 58 (1985) 45. [2] l? Yeh, IEEE J. Quantum Electron. QE-25 (1989) 484.
I23 (1996) 201-206
131 C.H. Kwak, S.Y. Park, H.K. Lee and E.-H. Lee, Optics
Comm. 79 ( 1990) 349; M. Belie, unpublished. [4] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, MS. Soskin and V.L. Vinetski, Ferroelectrics 22 ( 1979) 949. [5i M. Belie, Opt. Quan~m Electron. 16 (1984) 551. [6] C.H. Kwak, S.Y. Park, J.S. Jeong, HII. Suh and E.-H. Lee, Optics Comm. 105 (1994) 353. 171 R.L. Townsend and J.T. I..ah%acchia,J. Appl. Phys. 41(1970) 5188; Y.H. Ja, Opt. Quantum Electron. 17 (1985) 291; G.A. Broost, K.A. Motes and J.R. Rotge, J. Opt. Sot. Am. B 5 (1988) 1879: P. Tayebati and D. Mahgerefteh, J. Opt. Sot. Am. B 8 (1991) 1053; D. ‘Emotijevic, M. Belici and R.W. Boyd, IEEE J. Quantum Electron. QE-28 (1992) 1915
1996
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
123 (1996) 201-206
Exact solution to photorefractive two-wave mixing with arbitrary modulation depth M.R. Belie a*b, D. TimotijeviC a, M. PetroviC a, M.V. JariC b a Institute of Physics, PO. Box 57, 11001 Belgrade, Yugoslavia b Physics Department, Texas A&M University, College Station lX 77843-4242, Received
15 June 1995; revised version
USA
received 27 September 1995; accepted 29 September 1995
Abstract
We obtain analytical solution to the slowly varying envelope wave equations describing two-wave mixing in photorefractive crystals with arbitrary dependence of the gain and absorption on the two-wave fringe modulation depth. We introduce a novel form of the correction function for the space charge field that offers an improved agreement with the experiment.
Motivation for this Communication comes from the interesting paper on two-wave mixing (2WM) in photorefractive (PR) crystals by Refregier, Solymar, Rajbenbach and Huignard [ 11. At one point in their paper the authors state: “What emerges from the solution [of the materials equations] is that ImEs [the imaginary part of the space charge field] instead of being proportional to m [the fringe modulation], becomes a function of m. Unfortunately, f(m), valid for the range 0 < m < 1 could only be found by solving numerically [the Kukhtarev] equations, which we have not done as yet. Alternatively, we can use a phenomenological approach which does not aim at solving the relevant equations in a formal sense but reflects the physics of the situation. We know that as fringe modulation increases, the modulation of the space charge field will increase at a less than linear rate. The problem is then to find a function which for small values of m is equal to m and [for] higher values of m will display slower growth”. While certainly it is worthwhile to determine an appropriate form of the function f(m) , we show that 0030-4018/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved SSDZOO30-4018(95)00620-6
in fact, the slowly varying envelope wave equations describing 2WM in PR crystals can be solved for any form of that function. Let us set the stage. The equations of interest are of theform [l]: 1; = __rs
az;
= -rs--
A(m) zlz2
---
a,, 9
Z
m fx(m)
m
1112
Z
+ffzz,
(la> (lb)
where It and I2 denote the (nearly) copropagating or counterpropagating laser beams impinging on the crystal, Z = Zr + Z2 is the total intensity, g = f is the parameter controlling the geometry of wave mixing (reflection or transmission) and Ts is the saturated coupling constant. Thus, we consider a PR crystal with pure nonlocal response and a real coupling constant. As mentioned, m is the fringe modulation,
(2) and we have added a subscript x to the function fX ( m) , to differentiate between various forms that have been
202
used thus far in the literature. We reserve the symbol f for another variable. Finally, a is the linear or nonlinear (NL) absorption in the crystal. The prime in Eqs. ( 1) denotes the derivative along the propagation (z ) direction. Steady-state, degenerate, and plane wave situation is assumed. If one compares Eqs. ( 1) with the 2WM equations in the standard notation [2], one notes that the coupling “constant” has acquired a more general form: T(pn) =rsfx(m)
m
.
(3)
Based on this fact, we advance the following argument involving absorption in PR crystals. In general, for LYconstant, we have linear absorption, and for cr a function of intensities (or m) , we have NL PR absorption. In principle, the dependence of LYon m could involve a function other than fX (m). However, there is no reason to presume that the saturation mechanism in PR gain is fund~ent~ly different from the one in PR loss. Thus, for the NL absorption we also assume: a(m)
=ffs-,
f&n)
(4)
m
even though (as we shall show) the wave equation can be solved in quadratures for arbitrary dependence Ly(m) .Other models (in which absorption is assumed to arise through absorption gratings from the beginning) are possible [ 31. We will also consider the linear case cu( m) = cr = const. We mention that for (T = +, active is the reflection geometry (RG), in which the beams are incident on the opposite faces of the crystal, and for CT= -, active is the transmission geometry (TG), in which the beams are incident on the same (z = 0) face of the crystal. We treat both geometries exactly. The simplest choice for fn (m) is: f](m)
= i
m $3(m)
=1+
(7)
based purely on ~lculation~ convenience. Here b is the fitting parameter, function~ly similar to the parameter a in Eq. (6). This form of correction function allows for the solution of wave mixing equations in TG. Nowever, the paper deserves some further comment. ’ At one point the authors claim: “Although [Eq. (6)] gives a good explanation of experimental results for large signal effects, it is not possible to obtain analytical solution for two-wave mixing gain with such a functional form of fz(m>. Refregier et al. obtained an analytical solution by just using an undepleted pump approximation. So far, to the best of our knowledge, there has been no analytical treatment for the two-wave couplings including large modulation effects and pump depletions in a PR material which are valid for the whole range of m.” This statement is wrong on both counts. First, it is possible to obtain an analytical solution to 2WM with any functional form of fX (m) , as we shall show below. Second, there exists a number of analytical treatments [ 71 of the two-wave couplings including large modulation effects and pump depletion, only they are not known under such a name; they are known as the two-wave mixing with intensity dependent couplings. While one may argue that couplings dependent on the modulation depth stem from different physics than the couplings dependent on the intensities, as far as the solution of wave equations goes, that doesn’t matter. A typical model for the coupling used in the intensity dependent theories is of the form [ 71:
(5)
=m,
which is valid for small modulation. One then obtains the standard Kukhtarev equations [ 41, that have been solved exactly in many ways [ 2,5]. Refregier et al. have chosen {“after some attempts”, they say) the function: fi(m)
However, they only solved the undepleted pump case. While convenient for fitting, this function is impractical for solving the wave equations. Recently, another paper [6] appeared on the very same subject. In that paper the authors introduce another form for the correction function:
[l - exp(-am)]
.
(6)
I‘ z-(Z) = z-y”-IfC that is also based on phenomenological arguments. In the simplest approxima~on, C is chosen to be a constant. However, it need not be a constant, and if it i The notation of angles in the equations and in some of the figures in this paper are not consistent with each other.
MR. BeliC et al. /Optics
Communications
is chosen as C = 2b( II I,) ‘12, then the model of Kwak et al. [ 61 is recovered. We show that such a choice for Z(Z) , which leads to the following form of fX (m) : f4(m)
=
~ f
F(z) = sech-‘(m) given by: Zi = f exp(F)
mf +Cm'
203
123 (1996) 201-206
,
and f(z).
The intensities
,
12 = f exp(-F)
are
(1.9
and the 2WM gain is:
where by definition: 12L
f = 2(Z&)l’2,
(10)
results in a model that not only can be solved analytically, but offers a better agreement with the experimental data. We assume C to be a constant. The procedure for solving Eqs. ( 1) is as follows. We first treat TG. This case is simple, since boundary conditions are given on the one side of the crystal: Cr = Zr(O), C2 = 12(O). When the equations (with u = -) are rewritten using the combinations u = Zr + Z2 and u = It - 12, it is seen that there exists a convenient set of dependent variables f and F (instead of Zr and 12) in terms of which Eqs. (1) decouple. The definition is [ 51: Zr +Z2=
Zr -Z2=
fcoshF,
and the equations
fsinhF,
(11)
become:
f’=
ZS -fX(m) 12
F’=
-$f,(m)coshF.
sinhF
- a(m)
1 f,
J
ny, f,(m)-’
dm
m0
(16)
’
where p = Zr/Z2 = exp( 2F) is the beam intensity ratio and L denotes the thickness of the crystal. These are the general expressions. Of the four correction functions mentioned, only the integral for f2 can not be expressed in terms of elementary functions. For the function f4 (m) we introduced, one obtains: rsz =$-{exp[-2c(F-Fo)]
-l}-2(F-Fo). (17)
This completes the solution procedure. In TG, there is no need to fit boundary conditions, as they are given explicitly: tanhFo=s,
1
2
(18)
However, in RG the fitting has to be done separately from the solution process. Before we present our results, let us go through the RG solution procedure. In RG (a = +) the procedure is less explicit. One introduces the same set of variables, and the equations to be solved become: f’=--fX(m>fcoshF, 2 F’ = $fX(m)
sinhF
- a(m).
(19)
(13)
where c = czS/JIS. For linear absorption, the dependence of f on m (and z ) is even simpler: f = fomexp(--az).
c
rs
cash F. -, cash F
m
=2
PL ( -PO >
mo = sech FO.
This set of equations is easily solved in quadratures. The variable F is simply connected with the modulation depth, m = sech F, hence one can find f as a function of F (or m) and m (or F) as a function of z:
rsz
l+Po l+PL
fo=2(c,c2)“2,
(12)
f =foexp[2c(F-Fdl
,=c2=-
(14)
Thus, for an arbitrary function f X(m) , one first evaluates the quadrature Tsz in Eq. ( 13) and then finds
They are similar to the corresponding equations for TG, but the solution is more involved. Again, f can be expressed in terms of F and F can be expressed (implicitly) in terms of z, In
(
+
where
>
=Z,(F,Fo),
rSz = Z2(EFo),
(20)
M.R. Beli6 et al. /Optics
204
Communicarions
123 (1996) 201-206
.O
Fig. 1. Two-wave mixing gain y in the transmission geometry, as a function of the crystal thickness z, for different values of the input beam intensity ratio &I = Cl/& Dashed curves correspond to the fl model, dotted curves correspond to the f3 model, and full curves correspond to our f4 model. The parameters are: C=lO mW/cm*, cxs=1.6 cm-‘, rs=14 cm-’ for our model, and b=1.7, ~0.52 cm-‘, Ts=18.3 cm-’ for the other models, as in Ref. [6]. (a) &20, 5, (b) /3n=2000,500.
I
00
0.5
0.0
I
I
1.0
1~5Z(crrl)2~0
fig. 2. (a) Fringe’modulation depth m. (b) Nonlinear absorption (Y in TG, as functions of the crystal thickness z, for different values of the input beams ratio PO. Our form of the correction function f4 is used. The parameters are the same as in Fig. 1.
F
Zl(EFo)
=
J
dF
(21)
tanhF-2c’
Fo F
z2(F; F”) =
J
fx(sechF)
Ilo = Cl, ZzL = C2. Thus, one has to solve a system of two NL algebraic equations for Fo and FL: Z~(FL,FO)
=rJ,
2dF (sinh F - 2ccosh F) ’
Fo+FL+ZI(FL,FO)
=ln
(22)
FO
and the integrals are to be found in the table of integrals or evaluated numerically. A bit more complicated part is the fitting of boundary conditions. The conditions are now given on the opposite sides of the crystal:
This is easily accomplished on computer. Once FO and FL are determined, one uses Eq. (20) to find F(z ) and f(F) . For reasonable values of Cl and CZ, unique solutions are found.
M.R. BeliC et al. /Optics Communications 123 (1996) 201-206
205
1.0 -
m
-
0.8-
0.5
1000
0.2
I
-
0.001 0.0
0.0
0.2
1 0.4
, 0.6
,
,
I
0.8
1.0
a.0
, 0.0
0.2
L
/ 0.6
0.4
a
I 0.8
z(cm>
i
,
1.0 dcm>
Fig. 3. The same as Fig. 2, only for RG. The parameters are: C=lO mW/cm2, as=2 cm -I, I’s=5cm -l, Cz=l, which is the unit of intensity. (a) Modulation depth m as a function of z. (b) Nonlinear absorption (Yas a function of z, for different values of the pump intensity Cr.
Some of our results are presented in Figs. l-4. They should be compared with the corresponding results of Kwak et al. [6] and Refregier et al. [l] Two-wave mixing gain for different models is presented in Fig. 1, as a function of the crystal thickness z. Contrary to the belief expressed elsewhere [6], the value of the saturated gain depends on the model function J”~( m) . Fig. 2 depicts the modulation depth m and the NL absorption LYas functions of z, for our form f4 of the correction function, and in TG. It is interesting to note that at some value of z within the crystal, the modulation depth reaches the m~imum value of 1, and then decreases. After that point the signal saturates or even deamplifies (in the presence of absorption). In other words, there exists an optimal value of the crystal thickness at which the maximum amplification of the signal is achieved. The figure also points out to the inadequacy of the Kukhtarev model with small modulation, once the depletion of pump is taken into account. Fig. 3 presents the same quantities as Fig. 2, only in RG. Instead of /3e, the relevant parameter now is Ct, the intensity of the first beam at z = 0. The intensity of the other beam, which is incident on the z = L face of the crystal, is fixed at C2 = 1. The dependence of m on .z now displays a monotonous change. In Fig. 4, the comparison of different models with the experimental results is represented. Such comparisons are the most relevant tests for any of the phe-
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Eig. 4. Two-wave mixing gain y as a function of the input beam ratio @a,for different models. The parameters and the notation are the same as in Fig. 1, and tbe crystal thickness is set to L = 0.5 cm. The circles represent experimental values from Ref. [ 61.
nomenologic~ models. It is evident that our model offers the best agreement with the experiment. In summary, we have shown that the slowly varying envelope wave equations for 2WM in PR crystals, in either TG or RG, can be solved exactly (up to a quadrature) for an arbitrary dependence of gain and abso~tion on the fringe m~ulation depth. We intro-
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Communications
duced a new form of the correction function for the space charge field, that offers an improved agreement with the experiment.
This work has been in part supported by the United States National Science Foundation, grant No. DMR 9215231. References [ 11 P Refregier, L. Solymar, H. Rajbenbach and J.-P Huignard, J. Appl. Phys. 58 (1985) 45. [2] l? Yeh, IEEE J. Quantum Electron. QE-25 (1989) 484.
I23 (1996) 201-206
131 C.H. Kwak, S.Y. Park, H.K. Lee and E.-H. Lee, Optics
Comm. 79 ( 1990) 349; M. Belie, unpublished. [4] N.V. Kukhtarev, V.B. Markov, S.G. Odulov, MS. Soskin and V.L. Vinetski, Ferroelectrics 22 ( 1979) 949. [5i M. Belie, Opt. Quan~m Electron. 16 (1984) 551. [6] C.H. Kwak, S.Y. Park, J.S. Jeong, HII. Suh and E.-H. Lee, Optics Comm. 105 (1994) 353. 171 R.L. Townsend and J.T. I..ah%acchia,J. Appl. Phys. 41(1970) 5188; Y.H. Ja, Opt. Quantum Electron. 17 (1985) 291; G.A. Broost, K.A. Motes and J.R. Rotge, J. Opt. Sot. Am. B 5 (1988) 1879: P. Tayebati and D. Mahgerefteh, J. Opt. Sot. Am. B 8 (1991) 1053; D. ‘Emotijevic, M. Belici and R.W. Boyd, IEEE J. Quantum Electron. QE-28 (1992) 1915