- Email: [email protected]
The double phase conjugate mirror is an oscillator
Optics Communications 90 (1992) 133-138 North-Holland
OPTICS COMMUNICATIONS
The double phase conjugate mirror is an oscillator K e n n e t h D. S h a w Nonlinear Optics Center of Technology, PhillipsLaboratory, PL/L TN, KirtlandAFB, NM 87117-6008, USA
Received 13 November 1991 It has recently been shown that in the context of a two-dimensional scalar theory, the double phase conjugate mirror does not function as an oscillator but as a convective amplifier. It is shown here that this conclusion results from the application of a scalar field theory to a problem in which the vectorial nature of the electromagnetic field plays a crucial role. The vector coupled wave equations in the undepleted pumps approximation are derived directly from Maxwell's equations, and it is found that solutions exist which satisfy the oscillator boundary conditions. The reasons for the failure of the scalar model are discussed, and a new physical picture of the operation of the double phase conjugate mirror in terms of coupled field components is presented.
The operation of the double phase conjugate mirror ( D P C M ) has in the past been described theoretically [ 1,2 ] through the use o f one-dimensional scalar coupled wave theory [ 2, 3 ]. Calculations o f measureable quantities, such as phase conjugate reflectivity, based on this approach have yielded results which agree quite well with experiments [ 4 ]. This I D scalar theory predicts that the D P C M works as an oscillator, with the phase conjugate beams building up from zero at the crystal boundaries (as a function the longitudinal spatial coordinate) when the coupling strength is above some threshold value. In the laboratory, the D P C M does indeed appear to have this property, since its operation is observed to depend critically on the angles of the p u m p beams inside the crystal with respect to each other and to the crystal's optic axis, parameters upon which the coupling strength depends. The phase conjugate beams appear to "turn on" suddenly to full intensity as the orientation o f the p u m p beams within the crystal is changed slightly. In a recent paper [ 5 ], however, Zozulya has shown that when the scalar theory is extended to two dimensions by the inclusion o f the transverse partial derivatives, the solution no longer satisfy the oscillator boundary conditions. That is, if the amplitudes o f the phase conjugate beams are forced to be zero at their respective boundaries, they will be zero everywhere, and the D P C M will not operate. Non-zero solutions are obtained, however, by assuming some small amount o f scattering o f radiation from the p u m p beams into the direction of the phase conjugate beams at the boundaries o f the interaction region. In this case, then, there is no threshold behavior: the D P C M works for any value o f coupling strength no matter how small, an a continuous variation of the phase conjugate intensity with coupling strength should be observed. U p until just recently, all D P C M experiments have been performed using photorefractive crystals like barium titanate and SBN, since the large electro-optic coefficients o f these materials result in large coupling strengths. In order to obtain the largest coupling possible, these experiments are done using radiation polarized in the plane o f incidence of the p u m p beams (TM polarization). Strictly speaking, then, the scalar theory (either one- or two-dimensional) is not capable of describing these experiments correctly, and erroneous conclusions might possibly be drawn by so doing. It will be shown in what follows that such is indeed the case, and that the vectorial nature o f the electromagnetic field is crucial to the operation o f the D P C M #1 #1
See next page.
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
133
Volume 90, number 1,2,3
OPTICS COMMUNICATIONS
1 June 1992
To derive the vectorial coupled wave equations for TM polarization, we will work in the set of coordinates for which the dielectric tensor of a photorefractive crystal with cubic symmetry is diagonal. While this is a somewhat artificial situation as far as experiments are concerned, it is the simplest situation possible, and will suffice to illustrate the point of this paper. More detailed calculations which can be more easily compared with experiments are in progress, and their results will be published elsewhere. In the approximation that the refractive index variation due to the photorefractive effect is much smaller than the background (unmodulated) refractive index, the diagonalized tensor takes the form
E=eo
(i °
1-rEo~
0
°0)
,
(1)
1+reo~
where eo is the background dielectric constant, r is the effective electro-optic coefficient, and g is the photorefractive space-charge field. With this dielectric tensor and TM polarization (magnetic field in the x-direction, plane o f incidence (y, z) ), the relevant Maxwell's equations are
OEz/Oy-OEy/Oz=ikoHx, OHffOz=-ikoDy, OHx/Oy=ikoDz.
(2)
Here ko is the vacuum wavenumber of the incident radiation, and D + eE, e in this case being given by eq. ( 1 ). We label the left and right p u m p beams (entering the crystal faces at z = 0 and z = l , respectively) 4 and 2, and the left and right phase conjugate beams l and 3, respectively. Then, the total electric and magnetic field components in the crystal can be written as sums o f the field of the four beams: 4
Ey(y, z) = ~ Vj(y, z) exp(ikj.r) ,
(3a)
j=l 4
E~(y, z ) = ~ Wj(y, z) e x p ( i k j . r ) ,
(3b)
j=l 4
Hx(y, z)= ~ Uj(y, z) exp(ikj.r) .
(3c)
j=l
The space-charge field term in eq. ( 1 ) contains the information on the modulation o f the refractive index, and for the D P C M (transmission grating) is given by
rg= (7/Io) [ ( V~V$+ V~ V3) + ( W~ W $ + W~ W3) ] ,
(4)
where 7 is the amplitude o f the space-charge field as derived from Kukhtarev's equation [7 ] multiplied by the effective electro-optic coefficient. Eqs. (1), (3) and (4) are then substituted into Maxwell's equations (2), and phase-matched terms satisfying k~ + k2 = k3 q-k4 = 0 are retained. This results in twelve coupled partial differential equations for the unknown quantities Uj, Vj and Wj. By invoking the undepleted pumps approximation [2], the number o f equations is reduced to six. We then apply a Fourier transform with respect to y to reduce these to differential equations in the longitudinal coordinate z only. The two equations for the z-
~ Operation of the DPCM in cubic crystals such as BSO and indium phosphide has also been reported recently; see ref. [6]. In the case of indium phosphide, TE polarization (perpendicular to the incident plane) is used. It will be shown in results soon to be published that the DPCM will operate for TE polarization, but as an amplifier rather than an oscillator, in agreement with the conclusion of ref. [ 5 I. Thus operation of the DPCM in this case is fundamentally different from the TM case discussed here. The situation for BSO is more complicated because of its strong optical activity. Operation of the DPCM in such a case is probably a combination of the TE and TM modes. 134
Volume90, number1,2,3
OPTICSCOMMUNICATIONS
1June 1992
components of the fields WI and W3 involve derivatives with respect to y only, and so they reduce to two algebraic equations under the Fourier transformation. They can then be used to eliminate W~ and W3 from the remaining equations, ultimately yielding a set of four coupled first-order linear differential equations. Denoting the Fourier transformed quantities by a tilde and the transform variable by Q, and defining W24- I W212+ I W412,we finally have
(k2y+Q)2( 1+ ~t°r IW=lZ) -ko2to( '1+ o , )~-~o l W2,
OzOP,(z,Q)=i ( 1+ E°7-~oW24) - ' { ~ [
0,(z,Q)
[to7(k2y+~.~)(k4y+Q)W~W41 ~.?3(z,Q) -~-[k2z(~-[-2I~~24)-[-2I~(k2y--[-~2)V~4]~r~(Z~)--~-[2/~ (k2y--~-~-~)W4V~] ~r3(Z~Q)}~ 1 koeo 1_21o
)l{
0z
~o w2,
l[
( .o,
(5a)
- kotoI_2Io
)
(.o,)]
+~-~o (k4r+Q)2 1+2-~o1W412 -k°Zt° 1+~0W24
03(z'Q)
] f,(z,a)+ [ k,~( l+i~oW~, to, )to:,(k,~+a)V~W2]~,(z,a)}, +[t°Y(k,,+a)w2v: +~7~o L2Io
(Sb)
] ) (~'0~){[(IEO~5 (k2y..t_~.~)V4W~Ol(Z,Q k2z l + ~-o W24)tO' -- ~o
0
-'
Oz O, (z, o9) = i 1+ -~o WE4 ~ ~ ~0 ~
L2Io (k4y+Q) W~ V41 03(z, Q)
0 03(z, Q) +i( l + t ~° 'w 2 , ) Oz
' [ l
{ - ~o t°'(k2y+Q)W~V2~]l(Z'Q)
+[k4z(l+~I~oW24)-t°Y't" ~,~ +Q'tV'/W.1 , 2' 230,(z,a)
[,o, ]
[,o,
+koto ~og2V'~ Vt(z,Q)-koto l+~o(W24-lg212)
]
}
V3(z,Q) ,
(5d)
where Io is the total pump intensity, and kjy(z) is the y (or z) component of the wavevector of the jth beam. Eqs. (5) can be written compactly in a matrix form by defining the column vector U(z, Q) such that
V(z,n)
-- 03 p,
.
(6)
The matrix equation is then OU/Oz=AU,
(7) 135
V o l u m e 90, n u m b e r 1,2,3
OPTICS COMMUNICATIONS
1 J u n e 1992
where A is the matrix of coefficients and g2 is now just a parameter. The solution to (7) is obtained by finding the eigenvalues and eigenvectors of A. Unfortunately, the complexity of the quartic equation for the eigenvalues necessitates a numerical solution. The eigenvectors can be solved for analytically in a formal sense, however, and the boundary value equations can be written down in terms of the four eigenvalues 2j which are obtained from the numerical solution of the quartic equation. For simplicity, index matching at the crystal boundaries is assumed so that Fresnel reflections may be neglected. Normalizing all the eigenvectors to the y-component of the electric field (the choice of normalization is arbitrary), the boundary value equations are 4
E,y(0,.(2) = 0 = ~ aj,
(8a)
j=l 4
E3y(l, 1 2 ) = 0 = ~ aiF:exp(2jl ) ,
(8b)
j=l 4
H,x(O, g2)=0= ~ ajGj,
(8c)
)=1 4
n3x(l, g 2 ) = 0 = ~ ajK2exp(2jl ) .
(8d)
j=l
Here, the quantities Fj, Gj, and Kj are the coefficients which give the components of the jth eigenvector with respect to the normalization specified above, and the ass are coefficients depending on the coupling and geometrical parameters which must be determined so that eqs. (8) are satisfied. These are the oscillator boundary conditions for the DPCM: if there exists, for a given set of incident angles for the pump beams, a value of coupling strength for which eqs. (8) are satisfied, then the DPCM is an oscillator, and the coupling strength thus obtained is the threshold for the device. The occurrence of such a value is analogous to the situation which occurs in the 1D scalar theory of the DPCM in the undepleted pumps approximation. In that case, there is one value of coupling strength ( IFII = 2) for which the reflectivity becomes infinite, indicating the buildup of the phase conjugate field through self-oscillation. When pump depletion is included, it is found that solutions exist for all values of coupling above this threshold value. To determine if there is such a solution, a global minimization routine is used to search the complex coupling strength plane for a value of coupling such that eqs. (8) are satisfied in an interval of f2 centered about f2= 0 (£2 is related to the deviation of the phase conjugate beams from the direction counterpropagating to the pump beams). The coupling strength 1-'1is related to the parameter X which has been used here by Fl= yl(ko ~~3/2) )i 4i. It is found that there are indeed solutions, depending, of course, on the incident angles of the pumps. That is, there are some angles for which there are no solutions. Table 1 lists several angular configurations and their associated coupling thresholds. It must be stressed here that these threshold values are not related to the coupling which actually occurs in the crystal for the particular angular configuration. For a given geometry, the coupling in the crystal may or may not be above the threshold value, so that the values in the table do not give any information as to whether the DPCM will operate for that geometry. The information they do provide is that if the coupling for the given geometry is above the associated threshold, then the DPCM will operate, and that it operates as a photorefractive oscillator. It should also be stressed that one should avoid the temptation Table 1 P u m p b e a m angles a n d a s s o c i a t e d c o u p l i n g t h r e s h o l d s
136
04
02
IFlhhreshold
20 ° 45 °
--12 ° 40 °
0.013 40.01
20 °
5°
no solution
1 June 1992
OPTICS COMMUNICATIONS
Volume 90, number 1,2,3
to compare these numbers with experimental results, since, as mentioned earlier, the choice of coordinate system makes such comparisons meaningless. The important point here is that there are solutions to the oscillator boundary conditions, as opposed to the scalar case in which there are none. A calculation similar to the one just discussed has also been performed for the case of TE polarization, and it is found that no solutions to the oscillator boundary value equations exist, a result in agreement with that in ref. [ 5 ]. In order to understand the difference between the two cases, it is necessary to examine the TE equations analogous to (5). These are
0__~'l(z, 12)=ikoO,(z, 12)+ikEz~'l(Z, t2)
(9a)
0 V3(z, 12)=ikoUa(z, I2)+ik4= V3(z, 12) 0z
(9b)
0z
[ko~o( ,o,), [ 0 03(z, t2)=ik,~O3(z,Q)- [~ko~o(,o,) Ho VzV4.]
~--O,(z, g 2 ) g2)=ik2~O, + i O(z, z 0z
1-~oI4
-~(kz,-12)
]
[ V~V4] , (k,y-t2) z] P3(z, t2)
z V,(z, 1 2 ) - ikoeo
P , + i ko*o 1-2-~0/2
-
go
V3 ,
'
(9c)
(9d)
where Vj is now the x-component (perpendicular to the incident plane) of the electric field of the jth beam, and ~ is the corresponding y-component of the magnetic field. If eqs. (5a) and (5c) are evaluated at z = 0 , and (5b) and (5d) at z=l, and the oscillator boundary conditions are required to hold, we see that the derivatives of the phase conjugate fields are non-zero at the boundaries where the corresponding fields themselves are zero, and that these derivatives depend on the fields themselves of the other phase conjugate beam at the boundary. Since in the steady state these fields are non-zero, the derivatives at the boundaries will also be nonzero, and it can be shown that this results in the buildup of the phase conjugate fields from zero if the second derivatives of the phase conjugate intensities have the proper signs. In the time-dependent picture, all that is needed for this to occur is an infinitesimal amount of scattered radiation to initiate the formation of the photorefractive grating. It is easy to see, then, how changing the incident angles of the pumps determines whether or not the solutions to (5) will satisfy the oscillator boundary conditions (8): the signs of the pump field components and the wavevector components must be such that the condition on the second derivatives is fulfilled in order for the boundary value equations (8) to be satisfied. Examination of the TE equations (9a) and (9b) will show that the coupling between the opposite phase conjugate fields is missing: if the fields are zero at the boundary, then their derivatives are zero there also, and it can be shown that this results in both the first and second derivatives of the phase conjugate intensities being zero at their respective boundaries. Hence, self-buildup of the phase conjugate fields cannot occur for TE polarization. In the TM case, these important coupling terms arise from the z-components of the fields, and enter eqs. (5) through the algebraic equations for ff'~ and I~3. Hence, independent of the tensorial character of the effective electro-optic coefficient, the operation of the DPCM depends in a fundamental way on the vectorial nature of the electromagnetic field. I would like to thank P. Peterson and M.P. Sharma for many interesting discussions, and the National Research Council for its support of this work through an NRC Associateship.
References [ 1] S. Weiss, S. Sternklar and B. Fischer, Optics Lett. 12 ( 1987) 114. [2l M. Cronin-Golomb,B. Fischer, J.O. White and A. Yariv, IEEE J. Quant. Electron. QE-20 (1984) 12. [3] H. Kogelnik, Bell. Sys. Tech. J. 9 (1969) 2909. 137
Volume 90, number 1,2,3
OPTICS COMMUNICATIONS
[4] B. Fischer, S. Sternklar and S. Weiss, IEEE J. Quant. Electron 25 (1989) 550. [5] A.A. Zozulya, Optics Lett. 16 ( 1991 ) 545. [6 ] N. Wolffer, P. Gravey, J.Y. Moisan, C. Laulan and J.C. Launay, Optics Comm. 73 ( 1989 ) 351. N. Wolffer, P. Gravey, G. Picoli and V. Vieux, OSA Technical Digest 14 ( 1991 ) 310. [7] N.V. Kukhtarev, V. Markov, S. Odulov, M. Soskin and V. Vinetskii, Ferroelectrics 22 (1979) 949.
138
1 June 1992
OPTICS COMMUNICATIONS
The double phase conjugate mirror is an oscillator K e n n e t h D. S h a w Nonlinear Optics Center of Technology, PhillipsLaboratory, PL/L TN, KirtlandAFB, NM 87117-6008, USA
Received 13 November 1991 It has recently been shown that in the context of a two-dimensional scalar theory, the double phase conjugate mirror does not function as an oscillator but as a convective amplifier. It is shown here that this conclusion results from the application of a scalar field theory to a problem in which the vectorial nature of the electromagnetic field plays a crucial role. The vector coupled wave equations in the undepleted pumps approximation are derived directly from Maxwell's equations, and it is found that solutions exist which satisfy the oscillator boundary conditions. The reasons for the failure of the scalar model are discussed, and a new physical picture of the operation of the double phase conjugate mirror in terms of coupled field components is presented.
The operation of the double phase conjugate mirror ( D P C M ) has in the past been described theoretically [ 1,2 ] through the use o f one-dimensional scalar coupled wave theory [ 2, 3 ]. Calculations o f measureable quantities, such as phase conjugate reflectivity, based on this approach have yielded results which agree quite well with experiments [ 4 ]. This I D scalar theory predicts that the D P C M works as an oscillator, with the phase conjugate beams building up from zero at the crystal boundaries (as a function the longitudinal spatial coordinate) when the coupling strength is above some threshold value. In the laboratory, the D P C M does indeed appear to have this property, since its operation is observed to depend critically on the angles of the p u m p beams inside the crystal with respect to each other and to the crystal's optic axis, parameters upon which the coupling strength depends. The phase conjugate beams appear to "turn on" suddenly to full intensity as the orientation o f the p u m p beams within the crystal is changed slightly. In a recent paper [ 5 ], however, Zozulya has shown that when the scalar theory is extended to two dimensions by the inclusion o f the transverse partial derivatives, the solution no longer satisfy the oscillator boundary conditions. That is, if the amplitudes o f the phase conjugate beams are forced to be zero at their respective boundaries, they will be zero everywhere, and the D P C M will not operate. Non-zero solutions are obtained, however, by assuming some small amount o f scattering o f radiation from the p u m p beams into the direction of the phase conjugate beams at the boundaries o f the interaction region. In this case, then, there is no threshold behavior: the D P C M works for any value o f coupling strength no matter how small, an a continuous variation of the phase conjugate intensity with coupling strength should be observed. U p until just recently, all D P C M experiments have been performed using photorefractive crystals like barium titanate and SBN, since the large electro-optic coefficients o f these materials result in large coupling strengths. In order to obtain the largest coupling possible, these experiments are done using radiation polarized in the plane o f incidence of the p u m p beams (TM polarization). Strictly speaking, then, the scalar theory (either one- or two-dimensional) is not capable of describing these experiments correctly, and erroneous conclusions might possibly be drawn by so doing. It will be shown in what follows that such is indeed the case, and that the vectorial nature o f the electromagnetic field is crucial to the operation o f the D P C M #1 #1
See next page.
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
133
Volume 90, number 1,2,3
OPTICS COMMUNICATIONS
1 June 1992
To derive the vectorial coupled wave equations for TM polarization, we will work in the set of coordinates for which the dielectric tensor of a photorefractive crystal with cubic symmetry is diagonal. While this is a somewhat artificial situation as far as experiments are concerned, it is the simplest situation possible, and will suffice to illustrate the point of this paper. More detailed calculations which can be more easily compared with experiments are in progress, and their results will be published elsewhere. In the approximation that the refractive index variation due to the photorefractive effect is much smaller than the background (unmodulated) refractive index, the diagonalized tensor takes the form
E=eo
(i °
1-rEo~
0
°0)
,
(1)
1+reo~
where eo is the background dielectric constant, r is the effective electro-optic coefficient, and g is the photorefractive space-charge field. With this dielectric tensor and TM polarization (magnetic field in the x-direction, plane o f incidence (y, z) ), the relevant Maxwell's equations are
OEz/Oy-OEy/Oz=ikoHx, OHffOz=-ikoDy, OHx/Oy=ikoDz.
(2)
Here ko is the vacuum wavenumber of the incident radiation, and D + eE, e in this case being given by eq. ( 1 ). We label the left and right p u m p beams (entering the crystal faces at z = 0 and z = l , respectively) 4 and 2, and the left and right phase conjugate beams l and 3, respectively. Then, the total electric and magnetic field components in the crystal can be written as sums o f the field of the four beams: 4
Ey(y, z) = ~ Vj(y, z) exp(ikj.r) ,
(3a)
j=l 4
E~(y, z ) = ~ Wj(y, z) e x p ( i k j . r ) ,
(3b)
j=l 4
Hx(y, z)= ~ Uj(y, z) exp(ikj.r) .
(3c)
j=l
The space-charge field term in eq. ( 1 ) contains the information on the modulation o f the refractive index, and for the D P C M (transmission grating) is given by
rg= (7/Io) [ ( V~V$+ V~ V3) + ( W~ W $ + W~ W3) ] ,
(4)
where 7 is the amplitude o f the space-charge field as derived from Kukhtarev's equation [7 ] multiplied by the effective electro-optic coefficient. Eqs. (1), (3) and (4) are then substituted into Maxwell's equations (2), and phase-matched terms satisfying k~ + k2 = k3 q-k4 = 0 are retained. This results in twelve coupled partial differential equations for the unknown quantities Uj, Vj and Wj. By invoking the undepleted pumps approximation [2], the number o f equations is reduced to six. We then apply a Fourier transform with respect to y to reduce these to differential equations in the longitudinal coordinate z only. The two equations for the z-
~ Operation of the DPCM in cubic crystals such as BSO and indium phosphide has also been reported recently; see ref. [6]. In the case of indium phosphide, TE polarization (perpendicular to the incident plane) is used. It will be shown in results soon to be published that the DPCM will operate for TE polarization, but as an amplifier rather than an oscillator, in agreement with the conclusion of ref. [ 5 I. Thus operation of the DPCM in this case is fundamentally different from the TM case discussed here. The situation for BSO is more complicated because of its strong optical activity. Operation of the DPCM in such a case is probably a combination of the TE and TM modes. 134
Volume90, number1,2,3
OPTICSCOMMUNICATIONS
1June 1992
components of the fields WI and W3 involve derivatives with respect to y only, and so they reduce to two algebraic equations under the Fourier transformation. They can then be used to eliminate W~ and W3 from the remaining equations, ultimately yielding a set of four coupled first-order linear differential equations. Denoting the Fourier transformed quantities by a tilde and the transform variable by Q, and defining W24- I W212+ I W412,we finally have
(k2y+Q)2( 1+ ~t°r IW=lZ) -ko2to( '1+ o , )~-~o l W2,
OzOP,(z,Q)=i ( 1+ E°7-~oW24) - ' { ~ [
0,(z,Q)
[to7(k2y+~.~)(k4y+Q)W~W41 ~.?3(z,Q) -~-[k2z(~-[-2I~~24)-[-2I~(k2y--[-~2)V~4]~r~(Z~)--~-[2/~ (k2y--~-~-~)W4V~] ~r3(Z~Q)}~ 1 koeo 1_21o
)l{
0z
~o w2,
l[
( .o,
(5a)
- kotoI_2Io
)
(.o,)]
+~-~o (k4r+Q)2 1+2-~o1W412 -k°Zt° 1+~0W24
03(z'Q)
] f,(z,a)+ [ k,~( l+i~oW~, to, )to:,(k,~+a)V~W2]~,(z,a)}, +[t°Y(k,,+a)w2v: +~7~o L2Io
(Sb)
] ) (~'0~){[(IEO~5 (k2y..t_~.~)V4W~Ol(Z,Q k2z l + ~-o W24)tO' -- ~o
0
-'
Oz O, (z, o9) = i 1+ -~o WE4 ~ ~ ~0 ~
L2Io (k4y+Q) W~ V41 03(z, Q)
0 03(z, Q) +i( l + t ~° 'w 2 , ) Oz
' [ l
{ - ~o t°'(k2y+Q)W~V2~]l(Z'Q)
+[k4z(l+~I~oW24)-t°Y't" ~,~ +Q'tV'/W.1 , 2' 230,(z,a)
[,o, ]
[,o,
+koto ~og2V'~ Vt(z,Q)-koto l+~o(W24-lg212)
]
}
V3(z,Q) ,
(5d)
where Io is the total pump intensity, and kjy(z) is the y (or z) component of the wavevector of the jth beam. Eqs. (5) can be written compactly in a matrix form by defining the column vector U(z, Q) such that
V(z,n)
-- 03 p,
.
(6)
The matrix equation is then OU/Oz=AU,
(7) 135
V o l u m e 90, n u m b e r 1,2,3
OPTICS COMMUNICATIONS
1 J u n e 1992
where A is the matrix of coefficients and g2 is now just a parameter. The solution to (7) is obtained by finding the eigenvalues and eigenvectors of A. Unfortunately, the complexity of the quartic equation for the eigenvalues necessitates a numerical solution. The eigenvectors can be solved for analytically in a formal sense, however, and the boundary value equations can be written down in terms of the four eigenvalues 2j which are obtained from the numerical solution of the quartic equation. For simplicity, index matching at the crystal boundaries is assumed so that Fresnel reflections may be neglected. Normalizing all the eigenvectors to the y-component of the electric field (the choice of normalization is arbitrary), the boundary value equations are 4
E,y(0,.(2) = 0 = ~ aj,
(8a)
j=l 4
E3y(l, 1 2 ) = 0 = ~ aiF:exp(2jl ) ,
(8b)
j=l 4
H,x(O, g2)=0= ~ ajGj,
(8c)
)=1 4
n3x(l, g 2 ) = 0 = ~ ajK2exp(2jl ) .
(8d)
j=l
Here, the quantities Fj, Gj, and Kj are the coefficients which give the components of the jth eigenvector with respect to the normalization specified above, and the ass are coefficients depending on the coupling and geometrical parameters which must be determined so that eqs. (8) are satisfied. These are the oscillator boundary conditions for the DPCM: if there exists, for a given set of incident angles for the pump beams, a value of coupling strength for which eqs. (8) are satisfied, then the DPCM is an oscillator, and the coupling strength thus obtained is the threshold for the device. The occurrence of such a value is analogous to the situation which occurs in the 1D scalar theory of the DPCM in the undepleted pumps approximation. In that case, there is one value of coupling strength ( IFII = 2) for which the reflectivity becomes infinite, indicating the buildup of the phase conjugate field through self-oscillation. When pump depletion is included, it is found that solutions exist for all values of coupling above this threshold value. To determine if there is such a solution, a global minimization routine is used to search the complex coupling strength plane for a value of coupling such that eqs. (8) are satisfied in an interval of f2 centered about f2= 0 (£2 is related to the deviation of the phase conjugate beams from the direction counterpropagating to the pump beams). The coupling strength 1-'1is related to the parameter X which has been used here by Fl= yl(ko ~~3/2) )i 4i. It is found that there are indeed solutions, depending, of course, on the incident angles of the pumps. That is, there are some angles for which there are no solutions. Table 1 lists several angular configurations and their associated coupling thresholds. It must be stressed here that these threshold values are not related to the coupling which actually occurs in the crystal for the particular angular configuration. For a given geometry, the coupling in the crystal may or may not be above the threshold value, so that the values in the table do not give any information as to whether the DPCM will operate for that geometry. The information they do provide is that if the coupling for the given geometry is above the associated threshold, then the DPCM will operate, and that it operates as a photorefractive oscillator. It should also be stressed that one should avoid the temptation Table 1 P u m p b e a m angles a n d a s s o c i a t e d c o u p l i n g t h r e s h o l d s
136
04
02
IFlhhreshold
20 ° 45 °
--12 ° 40 °
0.013 40.01
20 °
5°
no solution
1 June 1992
OPTICS COMMUNICATIONS
Volume 90, number 1,2,3
to compare these numbers with experimental results, since, as mentioned earlier, the choice of coordinate system makes such comparisons meaningless. The important point here is that there are solutions to the oscillator boundary conditions, as opposed to the scalar case in which there are none. A calculation similar to the one just discussed has also been performed for the case of TE polarization, and it is found that no solutions to the oscillator boundary value equations exist, a result in agreement with that in ref. [ 5 ]. In order to understand the difference between the two cases, it is necessary to examine the TE equations analogous to (5). These are
0__~'l(z, 12)=ikoO,(z, 12)+ikEz~'l(Z, t2)
(9a)
0 V3(z, 12)=ikoUa(z, I2)+ik4= V3(z, 12) 0z
(9b)
0z
[ko~o( ,o,), [ 0 03(z, t2)=ik,~O3(z,Q)- [~ko~o(,o,) Ho VzV4.]
~--O,(z, g 2 ) g2)=ik2~O, + i O(z, z 0z
1-~oI4
-~(kz,-12)
]
[ V~V4] , (k,y-t2) z] P3(z, t2)
z V,(z, 1 2 ) - ikoeo
P , + i ko*o 1-2-~0/2
-
go
V3 ,
'
(9c)
(9d)
where Vj is now the x-component (perpendicular to the incident plane) of the electric field of the jth beam, and ~ is the corresponding y-component of the magnetic field. If eqs. (5a) and (5c) are evaluated at z = 0 , and (5b) and (5d) at z=l, and the oscillator boundary conditions are required to hold, we see that the derivatives of the phase conjugate fields are non-zero at the boundaries where the corresponding fields themselves are zero, and that these derivatives depend on the fields themselves of the other phase conjugate beam at the boundary. Since in the steady state these fields are non-zero, the derivatives at the boundaries will also be nonzero, and it can be shown that this results in the buildup of the phase conjugate fields from zero if the second derivatives of the phase conjugate intensities have the proper signs. In the time-dependent picture, all that is needed for this to occur is an infinitesimal amount of scattered radiation to initiate the formation of the photorefractive grating. It is easy to see, then, how changing the incident angles of the pumps determines whether or not the solutions to (5) will satisfy the oscillator boundary conditions (8): the signs of the pump field components and the wavevector components must be such that the condition on the second derivatives is fulfilled in order for the boundary value equations (8) to be satisfied. Examination of the TE equations (9a) and (9b) will show that the coupling between the opposite phase conjugate fields is missing: if the fields are zero at the boundary, then their derivatives are zero there also, and it can be shown that this results in both the first and second derivatives of the phase conjugate intensities being zero at their respective boundaries. Hence, self-buildup of the phase conjugate fields cannot occur for TE polarization. In the TM case, these important coupling terms arise from the z-components of the fields, and enter eqs. (5) through the algebraic equations for ff'~ and I~3. Hence, independent of the tensorial character of the effective electro-optic coefficient, the operation of the DPCM depends in a fundamental way on the vectorial nature of the electromagnetic field. I would like to thank P. Peterson and M.P. Sharma for many interesting discussions, and the National Research Council for its support of this work through an NRC Associateship.
References [ 1] S. Weiss, S. Sternklar and B. Fischer, Optics Lett. 12 ( 1987) 114. [2l M. Cronin-Golomb,B. Fischer, J.O. White and A. Yariv, IEEE J. Quant. Electron. QE-20 (1984) 12. [3] H. Kogelnik, Bell. Sys. Tech. J. 9 (1969) 2909. 137
Volume 90, number 1,2,3
OPTICS COMMUNICATIONS
[4] B. Fischer, S. Sternklar and S. Weiss, IEEE J. Quant. Electron 25 (1989) 550. [5] A.A. Zozulya, Optics Lett. 16 ( 1991 ) 545. [6 ] N. Wolffer, P. Gravey, J.Y. Moisan, C. Laulan and J.C. Launay, Optics Comm. 73 ( 1989 ) 351. N. Wolffer, P. Gravey, G. Picoli and V. Vieux, OSA Technical Digest 14 ( 1991 ) 310. [7] N.V. Kukhtarev, V. Markov, S. Odulov, M. Soskin and V. Vinetskii, Ferroelectrics 22 (1979) 949.
138
1 June 1992