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A power conservation theorem for volume holograms
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
A P OWER CONSERVATION THEOREM FOR VOLUME HOLOGRAMS P.St.J. RUSSELL and L. SOLYMAR
Dept. EngineeringScience, Oxford University, Oxford, UK Received 23 April 1979
The two-beam coupled wave analysis of volume holograms given by Solymar, which is a two-dimensional generalisation of Kogelnik's formulation, is extended further to provide an alternative treatment of 'd.c.' dephasing effects, and the resulting coupled wave equations are shown to contain an implicit power conservation theorem.
Kogelnik's one-dimensional theory of volume holography [1 ] has been extended to two dimensions for both the lossless on-Bragg case [2] and again [3] including all the parameters he considered and in addition the possibility of absorption losses during hologram formation. However, that the two dimensional formulation contains an implicit power conservation theorem has been shown only for the lossless on-Bragg case [2] ; the present paper will illustrate that power conservation exists for the more general formulation in ref. [3]. In addition, a new treatment o f ' d . c . ' effects is presented. Both [2] and [3] neglect the dephasing effects caused by slow spatial variations in average dielectric constant which non-planar recording beams will produce in certain materials; recently it has been shown that these effects can be important [4] and in ref. [5] a possible way of incorporating them into the analysis was proposed. Presently we shall describe an alternative formulation which has certain advantages over that in ref. [5], and the new equations, differing from those of ref. [3] in only one term, are used to illustrate power conservation. We denote a Geometrical Optics (G.O.) beam by *
rio = aio(r ) exp(--j/3oPio),
(1)
where/3 o is the propagation constant and aio is a suitably normalised amplitude distribution related to the phase function Pio by the conditions of G.O. namely 4: Throughout the paper, barred quantities (e.g. F) may be complex.
V" (a2to VPio) = 0 ,
where (~7pio)2 = 1 .
(2)
The presence or absence of a 'o' in the subscripts denotes recording or reconstruction values of the parameters respectively. Two such beams enter the undeveloped hologram (e r = e o - jeo) from a medium of the same complex dielectric constant. This corresponds to the case treated in [3], which for simplicity we retain. Fig. la illustrates the situation. Their electric field variations in the xy plane are described by --
t
.
Pt
ffio =Aiorio e x p ( - ~ o P i o ) , i = 1 or 2 (3) where Aio is a constant scaling factor and the absorption constant s o is ,~/3o. Given that the development process produces a change in the complex dielectric constant proportional to the local irradiance, the final spatial dependence of-gr may be written -er = -~(r) + -emg cos(Kg),
(4)
where -gm
"~(r) =-ec + - - ~
A io
2 i,j~jo
a2o exp(-2o%Pio )
(5)
~-c is the overall 'base' value of the complex relative dielectric constant after development. All the ~- quantities are complex in the form (e' - je"). Also g = aloa2o exp {-o%(Plo + P2o)},
(6)
VKg =/3o(VPl o - VP2o)
(7)
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
choose the trial solution for eq. (9) in the form
E=Eo-jEo Wave
'~
E':E'°gi~ihic material o,o
E(r)=Rexp(-jf3pl)+Sexp(j(K-/3p2)),
(11)
where
10
K = Kg - f l ( P l - P 2 ) "
Although eq. (11) is not formally the same as the trial solution chosen in [3], presently it should become clear that it produces coupled wave equations which are equivalent to those in [3] and are in a form very convenient for illustrating power conservation./3 in eq. (11) is an average value equal to that in eq. (10); we assert that since the 'd.c.' variations of complex dielectric constant described by eq. (5) both are slow and are such that I(g(r) - ge)/ge I "¢ 1, this choice of [3 will be sufficiently close to the truth. Dephasing produced either by this 'd.c.' term or by errors in the phases guessed in eq. (11) will be reflected by the analysis into complexness of S-and K'. Entering eqs. (11) and (4) into eq. (9), neglecting all terms save those involving exp(-j/3pl) and exp(j(K - 13p2)}, equating the coefficients of these fast varying exponentials to zero, and assuming that S, R and K are slowly varying, we obtain as a result
(a)
N~
~=~"o-jEo
(12)
g: = ~(r) . ~ g cos k S
hologram
"v
Wove 1
(~7/~ + ½K'V). Vpl +/~(oe +JOdc) +jE.gS-= 0 ,
(13)
(b} Fig. 1. (a) Hologram recorded by waves lo and 20; (b) developed hologram illuminated by wave 1.
(VS-+ ½S~7) • Vp2 + S-(o~+ JOdc +jtg) +j•" where t
It
I
II
I
ec, em, e m ,~ e c .
--
t
(15)
for 'd.c.' term dephasing, K = e-ml3/4e'c
(16)
Kogelnik's coupling constant, (9)
where /32 = (2zr/X)2e'c ,
tl
(8)
The next step is to attempt the solution of the scalar wave equation
V2ff +/32(-g(r)/e'c)ff= 0
.
0 d c = Odc -- JOdc = fl(~(r) -- ec)/2e c
the grating vector, and
gR = 0, (14)
(10)
and X is the vacuum wavelength, in the form of wave (1) provided by the reference beam during reconstruction and wave (2) resulting from Bragg diffraction (fig. lb); no other waves are considered to be present in significant quantities in the grating - i.e. higher order modes are neglected. For most cases this is a good approximation provided the reference beam is incident well within a Bragg regime [6], and so we
0 = VK. Vp2
(17)
for dephasing caused by inexact fulfillment of the Bragg condition. For 0dc = 0, the equations (13) and (14) reduce to those in [3] by way of the transformations = A 2 P 2 exp(j(K - / 3 p 2 ) -
ap2},
_R = A l P 1 exp{(j/3 - a ) p l } . For infinite plane beams throughout the analysis, a 0 = 0 and Odc= 0, eqs. (13) and (14) reduce directly to Kogelnik's equations [1 ] to the first order of accuracy.
Volume 30, number 1
OPTICS COMMUNICATIONS
Eqs. (13) and (14) are in a form which is very convenient for demonstrating that coupled wave theory in two dimensions contains an implicit power conservation theorem. Multiply (13) by R* and (14) by S*, and add the sum of the resulting two equations to its complex conjugate to give V- (ISI2Vp2 +
IKI2Vpl)~/e'c/Zo
July 1979
velope of the microscopic behaviour of the fields over many grating periods. The two-beam coupled wave equations of ref. [3] for TE polarisation have been generalised further to include 'd.c.' effects and have been presented in a form which has made it easy to illustrate that an implicit power conservation theorem exists for their solutions.
+ {2(o~ + Oac)(ISI 2 + IK'I2) -jg(k--k-*)[R
"S* +S.R*])X/e'c/Zo=O.
(18)
In order to give eq. (18) the correct dimensions it has been multiplied by ~/e'c/Z0 where Z 0 is the wave impedance of free space. The first term in eq. (18) may be regarded as the divergence of an intuitional power flow vector. The remaining terms therefore should correspond to the energy loss in the grating: i.e. should equal ol E" 12 where o = (27r/XZo)e r is the conductivity and er(r) is given by eq. (4). It is easy to show that this is so provided eq. (18) is integrated over a volume of the grating large enough to ensure that certain rapidly sinusoidal terms in ol E'I2 which are unaccounted for in eq. (18) integrate to zero. This is a reflection of the approximations of the model - we are interested only in slow average changes in/~ and S which are the en-
Acknowledgement Mr. P.St.J. Russell would like to thank Oriel College, Oxford for a Research Fellowship. The authors wish to acknowledge the support of the Science Research Council.
References [1] [2] [3] [4] [5] [6]
H. Kogelnik, B.S.T.J. 48 (1969) 2909. L. Solymax, Elect. Lett. 12 (1976) 606. L. Solymar, Appl. Phys. Lett. 31 (1977) 820. M.P. Jordan and L. Solymar, Elect. Lett. 14 (1978) 271. L. Solymar, Optics Commun. 26 (1978) 158. R.S. Chu and T. Tamir, IEEE Trans. MTT-18 (1970) 486.
OPTICS COMMUNICATIONS
July 1979
A P OWER CONSERVATION THEOREM FOR VOLUME HOLOGRAMS P.St.J. RUSSELL and L. SOLYMAR
Dept. EngineeringScience, Oxford University, Oxford, UK Received 23 April 1979
The two-beam coupled wave analysis of volume holograms given by Solymar, which is a two-dimensional generalisation of Kogelnik's formulation, is extended further to provide an alternative treatment of 'd.c.' dephasing effects, and the resulting coupled wave equations are shown to contain an implicit power conservation theorem.
Kogelnik's one-dimensional theory of volume holography [1 ] has been extended to two dimensions for both the lossless on-Bragg case [2] and again [3] including all the parameters he considered and in addition the possibility of absorption losses during hologram formation. However, that the two dimensional formulation contains an implicit power conservation theorem has been shown only for the lossless on-Bragg case [2] ; the present paper will illustrate that power conservation exists for the more general formulation in ref. [3]. In addition, a new treatment o f ' d . c . ' effects is presented. Both [2] and [3] neglect the dephasing effects caused by slow spatial variations in average dielectric constant which non-planar recording beams will produce in certain materials; recently it has been shown that these effects can be important [4] and in ref. [5] a possible way of incorporating them into the analysis was proposed. Presently we shall describe an alternative formulation which has certain advantages over that in ref. [5], and the new equations, differing from those of ref. [3] in only one term, are used to illustrate power conservation. We denote a Geometrical Optics (G.O.) beam by *
rio = aio(r ) exp(--j/3oPio),
(1)
where/3 o is the propagation constant and aio is a suitably normalised amplitude distribution related to the phase function Pio by the conditions of G.O. namely 4: Throughout the paper, barred quantities (e.g. F) may be complex.
V" (a2to VPio) = 0 ,
where (~7pio)2 = 1 .
(2)
The presence or absence of a 'o' in the subscripts denotes recording or reconstruction values of the parameters respectively. Two such beams enter the undeveloped hologram (e r = e o - jeo) from a medium of the same complex dielectric constant. This corresponds to the case treated in [3], which for simplicity we retain. Fig. la illustrates the situation. Their electric field variations in the xy plane are described by --
t
.
Pt
ffio =Aiorio e x p ( - ~ o P i o ) , i = 1 or 2 (3) where Aio is a constant scaling factor and the absorption constant s o is ,~/3o. Given that the development process produces a change in the complex dielectric constant proportional to the local irradiance, the final spatial dependence of-gr may be written -er = -~(r) + -emg cos(Kg),
(4)
where -gm
"~(r) =-ec + - - ~
A io
2 i,j~jo
a2o exp(-2o%Pio )
(5)
~-c is the overall 'base' value of the complex relative dielectric constant after development. All the ~- quantities are complex in the form (e' - je"). Also g = aloa2o exp {-o%(Plo + P2o)},
(6)
VKg =/3o(VPl o - VP2o)
(7)
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
choose the trial solution for eq. (9) in the form
E=Eo-jEo Wave
'~
E':E'°gi~ihic material o,o
E(r)=Rexp(-jf3pl)+Sexp(j(K-/3p2)),
(11)
where
10
K = Kg - f l ( P l - P 2 ) "
Although eq. (11) is not formally the same as the trial solution chosen in [3], presently it should become clear that it produces coupled wave equations which are equivalent to those in [3] and are in a form very convenient for illustrating power conservation./3 in eq. (11) is an average value equal to that in eq. (10); we assert that since the 'd.c.' variations of complex dielectric constant described by eq. (5) both are slow and are such that I(g(r) - ge)/ge I "¢ 1, this choice of [3 will be sufficiently close to the truth. Dephasing produced either by this 'd.c.' term or by errors in the phases guessed in eq. (11) will be reflected by the analysis into complexness of S-and K'. Entering eqs. (11) and (4) into eq. (9), neglecting all terms save those involving exp(-j/3pl) and exp(j(K - 13p2)}, equating the coefficients of these fast varying exponentials to zero, and assuming that S, R and K are slowly varying, we obtain as a result
(a)
N~
~=~"o-jEo
(12)
g: = ~(r) . ~ g cos k S
hologram
"v
Wove 1
(~7/~ + ½K'V). Vpl +/~(oe +JOdc) +jE.gS-= 0 ,
(13)
(b} Fig. 1. (a) Hologram recorded by waves lo and 20; (b) developed hologram illuminated by wave 1.
(VS-+ ½S~7) • Vp2 + S-(o~+ JOdc +jtg) +j•" where t
It
I
II
I
ec, em, e m ,~ e c .
--
t
(15)
for 'd.c.' term dephasing, K = e-ml3/4e'c
(16)
Kogelnik's coupling constant, (9)
where /32 = (2zr/X)2e'c ,
tl
(8)
The next step is to attempt the solution of the scalar wave equation
V2ff +/32(-g(r)/e'c)ff= 0
.
0 d c = Odc -- JOdc = fl(~(r) -- ec)/2e c
the grating vector, and
gR = 0, (14)
(10)
and X is the vacuum wavelength, in the form of wave (1) provided by the reference beam during reconstruction and wave (2) resulting from Bragg diffraction (fig. lb); no other waves are considered to be present in significant quantities in the grating - i.e. higher order modes are neglected. For most cases this is a good approximation provided the reference beam is incident well within a Bragg regime [6], and so we
0 = VK. Vp2
(17)
for dephasing caused by inexact fulfillment of the Bragg condition. For 0dc = 0, the equations (13) and (14) reduce to those in [3] by way of the transformations = A 2 P 2 exp(j(K - / 3 p 2 ) -
ap2},
_R = A l P 1 exp{(j/3 - a ) p l } . For infinite plane beams throughout the analysis, a 0 = 0 and Odc= 0, eqs. (13) and (14) reduce directly to Kogelnik's equations [1 ] to the first order of accuracy.
Volume 30, number 1
OPTICS COMMUNICATIONS
Eqs. (13) and (14) are in a form which is very convenient for demonstrating that coupled wave theory in two dimensions contains an implicit power conservation theorem. Multiply (13) by R* and (14) by S*, and add the sum of the resulting two equations to its complex conjugate to give V- (ISI2Vp2 +
IKI2Vpl)~/e'c/Zo
July 1979
velope of the microscopic behaviour of the fields over many grating periods. The two-beam coupled wave equations of ref. [3] for TE polarisation have been generalised further to include 'd.c.' effects and have been presented in a form which has made it easy to illustrate that an implicit power conservation theorem exists for their solutions.
+ {2(o~ + Oac)(ISI 2 + IK'I2) -jg(k--k-*)[R
"S* +S.R*])X/e'c/Zo=O.
(18)
In order to give eq. (18) the correct dimensions it has been multiplied by ~/e'c/Z0 where Z 0 is the wave impedance of free space. The first term in eq. (18) may be regarded as the divergence of an intuitional power flow vector. The remaining terms therefore should correspond to the energy loss in the grating: i.e. should equal ol E" 12 where o = (27r/XZo)e r is the conductivity and er(r) is given by eq. (4). It is easy to show that this is so provided eq. (18) is integrated over a volume of the grating large enough to ensure that certain rapidly sinusoidal terms in ol E'I2 which are unaccounted for in eq. (18) integrate to zero. This is a reflection of the approximations of the model - we are interested only in slow average changes in/~ and S which are the en-
Acknowledgement Mr. P.St.J. Russell would like to thank Oriel College, Oxford for a Research Fellowship. The authors wish to acknowledge the support of the Science Research Council.
References [1] [2] [3] [4] [5] [6]
H. Kogelnik, B.S.T.J. 48 (1969) 2909. L. Solymax, Elect. Lett. 12 (1976) 606. L. Solymar, Appl. Phys. Lett. 31 (1977) 820. M.P. Jordan and L. Solymar, Elect. Lett. 14 (1978) 271. L. Solymar, Optics Commun. 26 (1978) 158. R.S. Chu and T. Tamir, IEEE Trans. MTT-18 (1970) 486.