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Efficiency of volume phase reflection holograms recorded in an attenuating medium
Volume 34, number 3
OPTICS COMMUNICATIONS
September 1980
EFFICIENCY OF VOLUME PHASE REFLECTION HOLOGRAMS RECORDED IN AN ATTENUATING MEDIUM M.P. OWEN and L. SOLYMAR Holographic Group, Departmwt of Eugineeriug Science, University of Oxford, Oxford, OX1 3PJ, UK Received 9 April 1980 Revised manuscript received 29 May 1980
The cffcct of attenuation when recording a volume phase hologram has been treated by a number of authors ncglcct the cffcct of the varying avcragc pcrmittivity. It is shown here that such a variation, and the consequent of the Bragg conditions, lcads to lower efficiency, the effect being more detrimental for greater attenuation and beam ratio. The mathematical solution is obtained with the aid of a generalisation of the coupled wave approach which a set of numerical results is derived,
1. Introduction The materials used for volume holography (e.g. photographic emulsions or dichromated gelatin) may exhibit appreciable losses at the recording process Icading to a varying modulation depth and average (bias) term. For absorbtion holograms the relevant quant i t its arc the amplitude of the absorbtion modulation and the average absorbtion, for phase holograms we are concerned with the amplitude of the permittivity variation and with the average permittivity, whereas for the mixed holograms we need to take into account the variation of all four quantities. For recording with plane waves the effects of varying modulation depth and/or varying average absorbtion have been analyscd by a number of authors [l-7] but all the analysts assumed a constant average permittivity. Kubota argues for example that the change in the avcragc pcrmittivity need to be taken into account bccausc it is small in comparison with the constant term. The purpose of the present communication is to nnalysc for the case of reflection type volume phase holograms the effect of this varying average permittivity, a direct consequence of absorbtion at the recording stage. A symmetric reflection regime is investigated to give constant modulation and to disengage the ef-
who violation greater from
fects of varying average permittivity from varying modulation.
2. Recording We shall assume that two monochromatic recording plane waves lo and 2, of frequency ~3~ are incident symmetrically at an angle 8 1o upon a lossy holographic material from media of the same permittivity as shown schematically in fig. la. Taking the polarisation of the electric field in both waves in the z-direction, and noting further that the paths of the light rays in the lossy medium are xkos 8 1o and (Cl- X)/COS0 10 respectively, we may describe the variation of the electric field in the holographic material by the equations l:‘,o=~oA,oexp
( G%;)
exp( --j&P 10)
(0
and fYzo = EoA2, exp [- !+Jsi+J
exp(-jflo~9ZO),
(2)
where E. is a constant, A 1o and AZ0 are the relative 321
uuuw 34. numlsr io\
September 1980
OPTK’S COMMUNICATIONS
3
SECOROING __ ___ --_-----
THE
t4OLOGRAM
** f. -
E, =E,l?
= Erl [cosh(2ax - c) + cos(2/30x cos 8 IO)],
(4)
where erl = 2kE~AloA2,
_)\
*10
X
c = ad + log,r,
y = A,,/A,,
(5)
and k is a proportionality constant depending on the exposure time and on the properties of the material. Thus after development the permittivity of the holographic material may be written in the form
RECORDING MATERIAL (COEFFICIENT = a)
OF ABSORBTION
a = cv/cos 0 1~,
exp(--ad),
% = 6,s + crf9
(6)
where ‘c’
RECOXSTRUCTION
E, =
L
OF-. THE
HOLOGRAM
f rs = Er() + q1 c osh(2ax -- c),
E,>
Crf = erl cos(2p()x cos 0 I()),
9
*
X
(7)
and E,.~and erf represent the slowly and fast varying parts of the permittivity. It is worth noting an intcrcsting feature of the chosen physical configuration, different from all previous studies, that the amplitude of the fast varying pcrmittivity modulation is constant whereas the average perniittivity is a slowly varying function of space.
HOLOGRAM /SC
ARSOHBTION
1
recording,
and (b) recon-
plitudes of the ikaves. PO = ~~~~~~~~~~)li2, p. and eO are the free space permeability and permittivity respectively. Q-, is the relative permittivity of the medium. Q is the attenuation coefficient of the holographic medium,
j73(’ =
x cos
0
-1’
sin 9 1o
63) x. _\‘,f are rectangular coordinates and d is the thickness of the hologram. Assuming further that at the end of the development process the permittivity of the holographic material will change by an amount proportional to the square 0i rhe recording electric field. we have lo
+
3. Reconstruction, derivation of the coupled wave equations Following the method outlined in ref. [8] we shall solve the problem of reconstruction in two steps. Firstly we shall look at the influence of the slowly varying term in the permittivity, i.e. find the solution of the wave equation in a medium in which the permittivity variation is given by ers. For simplicity we shall disregarr! refraction at t!re input boundary which we can do if WCassume that under reconstruction the pcrmittivity of mcdiun1 1 c). (tig. 1b) is always qua1 to qO + crl cosh(2ax Our aim is now to find the propagation of waves, which are close to the recording waves, in this Ilypothetical medium. If crl = 0 and a plane wave is incident at at1 angle 0 1 (close to 0 1o) then the solutions we are intcrestcd in are of the form I:’- cxp [ --jpo(x cos 0 I + y sin 0 , )I .
(W
OPTICS COMMUNICATIONS
Volume 34, uumber 3
When erl # 0 the permittivity varies in the x-direction hence the phase of the wave in the x-direction is bound to be affected. We shall therefore assume the solutions in the form Ej=exp[--jyj(x,y)],
i= 1,2,
Septcmbcr 1980
exp(-jyl ) (2j cos 0 1 dA 1/dx +(ErlP lDrOM2
exPC-jG)l
+ exp(-jy2){-2j
cos 6, dA2/dx
(9)
where
+ (ErlP1/2Ero)A 1 ex~ (iG)1
71 =P1(xcosQ
+v sin 0,) +g(x),
y2 = &(--x cos 8,
+_I’
sin 01) --g(x),
(10)
+ (ErlPl /2ErO){A1 expL--j(y, + 200x cos fl lo)]
(11)
+ A2 exp [---j(y2 -- 2flox cos 8 lo)]} = 0,
112 01 = “1 (C1(JeOCr()), w1 is the frequency of the reconstructing wave close to ~0 and the function g(x) represents a small perturbation. Substituting eq. (8) into the wave equation
(171
where G = y2 -- y1 f 20,~ cos 8 1o = ---2(tx + g(x)) < = Ap cos 0 l~ ---poAO sin 0 lo,
v2E + O:/JOEOCrsE= 0,
(12) op=P1
the function g(x) may be found as
‘rl‘!---sinll(3ax
g(x)-
_____._
4r,C)a cos 0 1
M
-
33)
C)*
Having determined the propagation of waves in the medium with permittivity variation crs we may now proceed to the second step in which the effect of the fast varying term in the permittivity is considered. We shall now claim that the waves described by eq. (9) will be coupled to each other due to the periodic pcrmittivity modulation. The wave equation for the complete problem is V2E f &e,/e,o)l:’
= 0,
(14)
where er is given by cq. (6), and its solution will be assumed in the form 2
I:’ = K.
2
exp( -jri).
dA,/dx - j&I l exp(jG) = 0,
(20)
where K’= frlP1/4Er() cos Ol*
(21)
The relevant boundary conditions are and
A2=OonxZd.
(22)
(15) Expressing A, from eq. (20) and substituting it into eq. (19) we obtain a differential equation in A2
cq . (
where
1 2 ), and 2 -4 p 1 CL4 1 /(IX
(1’))
Lllld
d2A2/dx’
Jdx
(18)
d.Al /dx + jK’A2 exp(--jG) = 0,
A, = 1011x=0, Ai
A0 = 0, .- Olo.
We shall now use the approximation inherent in all “two-wave” theories (see for example rei’. [9] ) that the higher order modes may be neglected, i.e. the third line in eq. (17) may be disregarded In that case eq, (17) may be satisfied by setting the expressions in both curly brackets equal to zero, obtaining the coupled wave equations
Substituting cq. (15) into (14) WCobtain a rather lengthy cxpressi:?:? but it can be siw.p!ifi.Qtdby considering that cxp( jyi) is an approximate solution of
‘A4
--PO,
illId
g(x)
< p,
co!3 0 1.
- K”&
6 = [ + @/dx
(W
and
With the above simplifications cq. (14) may be written in the form
t 2jb dA,/dx
the boundary conditions are
A2 = 0 at x =tl, and
=
0,
(23)
(24)
\‘olrtmc
34.
ri.4 z /dx =
numtwr3 j&l \ (0)
OPTICS COMMUNICATIONS
at x = 0.
tsp(jG(0))
(25)
if tltt’ *&lx term ws ahetlt clearly s the parameter. d off-Bragg the lea w&e eq. the (26)
e’t~r.
shall that
rendition
5= t’
(I
introduce
the
when
is is
off-Bragg average b zero. {
g(O)].
Thus new average value E, instead of
the -
4.
Eqs. ( 19 ) and (20)
solutions solution the equar~? tither the condiarc _civen different surfaces. One possibility is to LISAthe iterative method outlined in reference [lo). Ttdid indeed yield a solution for I’d up to ,abcsut0.6 but with the available computer accuracy the iterations did not converge beyond that. However we reatised that if we assumed the boundary conditions
2*
A, =.-I and‘42 =Oonx=d
(2%
a&l built up the solution moving backwards from Y= tl to x = 0 by means of the fourth-order RungeKutta-Xystrom method we would end up with (say) .4,(O)= Qantifl~\“, 1 A rn’ = C and then c:oki claim that an input of unity at x = 0 would yield a reflected wave CIutput at .x = 0 equal to C/B. Hence t!le hologram sfficienCy is simply
323
September 1980
5. Numerical results The question we wish to ask is how much we are worse off if the holographic material exhibits losses a; the recording stage. If we keep the recording beam amplitudes constant then higher losses will, naturally, lead to lower permittivity modulation and hence to lower efficiency. In most cases however we would be able to increase the power in the input beams and restore the permittivity modulation to its original value. We would therefore like to ask the slightly different question, that of how will the hologram efficiency depend on loss for a fixed value of K ‘d? The other parameters of interest are the amount of loss, expressed by the value of exp(-&), the beam r’atio Y, and the Bragg parameter {. First we wish to note that the curves for a beam ratio of l/r are the same as for Y,i.c. it makes no difference which of the two beams is the stronger one. This was borne out by the numerical results and may also be proven mathematically from our equations. The efficiency as a function of { is plotted in figs. 2a-f for I’d = 0.5 and 2, exp( -WI)= 1, f and $, and Y= 1,2anti4. For larger values of Ythe efficiency may appreciably decrease when absorbtion loss was present as may be seen in figs . 2b, c, c anct f. This may bc cxplained by considering that the modulation depends on the product of the recording amplitudes so that for unequal beams, E,, is mainly determined by the stronger beam. Thus for large beam ratio the variation of erS is large in comparison with the modulation and hence there is considerable dephasing leading to reduction in efficiency. It may be seen that the reduction in efficiency increases with increasing beam ratio and increasing loss, and it is more noticeable at larger values of I’d. It is therefore of interest to show the variation of the maximum available efficiency (occurring at the optimum of {) against rc’d with exp( --arc/)and Yas parameters. For Y= 1 there is so little decrease in efficiency that the curves are not worth showing. For I’= 2 and y = 4 however the results arc plotted in figs. 3a and b. All the curves display the characteristic behaviour of reflection holograms, namely the efficiency tends to unity as K ‘cl + 00 but for r = 4 and exp( --ml) = $ the reduction in efficiency is quite appreciable.
Volume 34, number 3
OPTICS COMMUNICATIONS
., , _d
Sep tcmbcr 1980
I
4
--_---_
,~___ P
i
I
-_
----_ -_-..----_I_--____
g”
\’\
-I*
-_ _-.. al
1’ ’ I
;,’
I .____
A--1----..
September
OPTICS COMMUNICATIONS
1980
I _
5
L)
K'd
K’d
I-‘&. 3. Plots of maximum cfficicncy
6. Conchsions .-!;tenuation at hc recording stage is bound to be 6~rrnful because for a given input power permittivity xl~~~ulrlticw dwlitws. However, even if the input power is ‘sufficiently large to produce the required permittivity modulation, the attenuation still leads to deleterious effects by producing a slowly varying permittivity whiA means that the Bragg conditions are not general11’satisfied. It has been shown that this effect is relatively insignificant for unity beam ratio but becomes increasingly important as the beam ratio and the attenuation increase.
.~cknow!edwilents 3
The authors would like to thank Dr. NJ. Philips
7) versus
I’d
for (a) r = 2, (b) r = 4.
of Lou&borough University for helpful and stimulati~~gdiscussions and the U.K. Science Research Council for financial support.
References [ 1] D. Kcrmisch, J. Opt. Sot. Am. 59 ( 1969) 1409. [ 21 N. Uchida, J. Opt. Sot. Am. 63 (1973) 280. [ 3) R. Howarschik, Optica Acta 23 (1976) 1039. (4 1 S. Morozumi, Jap. J. Appl. Phys. 15 ( 1976) 1929. [51 T. Kubotn, Optics Comm. 16 (1976) 347. [6] U. Killat, Optics Comm. 2 1 (1977) I 10. 171 T. Kubota, Optica Acta 25 (1978) 1035. [S] L. Solymar, Optics Comm. 26 (1978) 158. 191 H. Kopclnik, Bell Syst. Tech. J. 48 (1969) 2909. [lo! W.1:. Parry and I.. So!y!nar. Opt. (hl!lnf I+hp++ 0/ \‘lQ77r y I” . . . . .r.ws,,. 1/ I I , 527.’
OPTICS COMMUNICATIONS
September 1980
EFFICIENCY OF VOLUME PHASE REFLECTION HOLOGRAMS RECORDED IN AN ATTENUATING MEDIUM M.P. OWEN and L. SOLYMAR Holographic Group, Departmwt of Eugineeriug Science, University of Oxford, Oxford, OX1 3PJ, UK Received 9 April 1980 Revised manuscript received 29 May 1980
The cffcct of attenuation when recording a volume phase hologram has been treated by a number of authors ncglcct the cffcct of the varying avcragc pcrmittivity. It is shown here that such a variation, and the consequent of the Bragg conditions, lcads to lower efficiency, the effect being more detrimental for greater attenuation and beam ratio. The mathematical solution is obtained with the aid of a generalisation of the coupled wave approach which a set of numerical results is derived,
1. Introduction The materials used for volume holography (e.g. photographic emulsions or dichromated gelatin) may exhibit appreciable losses at the recording process Icading to a varying modulation depth and average (bias) term. For absorbtion holograms the relevant quant i t its arc the amplitude of the absorbtion modulation and the average absorbtion, for phase holograms we are concerned with the amplitude of the permittivity variation and with the average permittivity, whereas for the mixed holograms we need to take into account the variation of all four quantities. For recording with plane waves the effects of varying modulation depth and/or varying average absorbtion have been analyscd by a number of authors [l-7] but all the analysts assumed a constant average permittivity. Kubota argues for example that the change in the avcragc pcrmittivity need to be taken into account bccausc it is small in comparison with the constant term. The purpose of the present communication is to nnalysc for the case of reflection type volume phase holograms the effect of this varying average permittivity, a direct consequence of absorbtion at the recording stage. A symmetric reflection regime is investigated to give constant modulation and to disengage the ef-
who violation greater from
fects of varying average permittivity from varying modulation.
2. Recording We shall assume that two monochromatic recording plane waves lo and 2, of frequency ~3~ are incident symmetrically at an angle 8 1o upon a lossy holographic material from media of the same permittivity as shown schematically in fig. la. Taking the polarisation of the electric field in both waves in the z-direction, and noting further that the paths of the light rays in the lossy medium are xkos 8 1o and (Cl- X)/COS0 10 respectively, we may describe the variation of the electric field in the holographic material by the equations l:‘,o=~oA,oexp
( G%;)
exp( --j&P 10)
(0
and fYzo = EoA2, exp [- !+Jsi+J
exp(-jflo~9ZO),
(2)
where E. is a constant, A 1o and AZ0 are the relative 321
uuuw 34. numlsr io\
September 1980
OPTK’S COMMUNICATIONS
3
SECOROING __ ___ --_-----
THE
t4OLOGRAM
** f. -
E, =E,l?
= Erl [cosh(2ax - c) + cos(2/30x cos 8 IO)],
(4)
where erl = 2kE~AloA2,
_)\
*10
X
c = ad + log,r,
y = A,,/A,,
(5)
and k is a proportionality constant depending on the exposure time and on the properties of the material. Thus after development the permittivity of the holographic material may be written in the form
RECORDING MATERIAL (COEFFICIENT = a)
OF ABSORBTION
a = cv/cos 0 1~,
exp(--ad),
% = 6,s + crf9
(6)
where ‘c’
RECOXSTRUCTION
E, =
L
OF-. THE
HOLOGRAM
f rs = Er() + q1 c osh(2ax -- c),
E,>
Crf = erl cos(2p()x cos 0 I()),
9
*
X
(7)
and E,.~and erf represent the slowly and fast varying parts of the permittivity. It is worth noting an intcrcsting feature of the chosen physical configuration, different from all previous studies, that the amplitude of the fast varying pcrmittivity modulation is constant whereas the average perniittivity is a slowly varying function of space.
HOLOGRAM /SC
ARSOHBTION
1
recording,
and (b) recon-
plitudes of the ikaves. PO = ~~~~~~~~~~)li2, p. and eO are the free space permeability and permittivity respectively. Q-, is the relative permittivity of the medium. Q is the attenuation coefficient of the holographic medium,
j73(’ =
x cos
0
-1’
sin 9 1o
63) x. _\‘,f are rectangular coordinates and d is the thickness of the hologram. Assuming further that at the end of the development process the permittivity of the holographic material will change by an amount proportional to the square 0i rhe recording electric field. we have lo
+
3. Reconstruction, derivation of the coupled wave equations Following the method outlined in ref. [8] we shall solve the problem of reconstruction in two steps. Firstly we shall look at the influence of the slowly varying term in the permittivity, i.e. find the solution of the wave equation in a medium in which the permittivity variation is given by ers. For simplicity we shall disregarr! refraction at t!re input boundary which we can do if WCassume that under reconstruction the pcrmittivity of mcdiun1 1 c). (tig. 1b) is always qua1 to qO + crl cosh(2ax Our aim is now to find the propagation of waves, which are close to the recording waves, in this Ilypothetical medium. If crl = 0 and a plane wave is incident at at1 angle 0 1 (close to 0 1o) then the solutions we are intcrestcd in are of the form I:’- cxp [ --jpo(x cos 0 I + y sin 0 , )I .
(W
OPTICS COMMUNICATIONS
Volume 34, uumber 3
When erl # 0 the permittivity varies in the x-direction hence the phase of the wave in the x-direction is bound to be affected. We shall therefore assume the solutions in the form Ej=exp[--jyj(x,y)],
i= 1,2,
Septcmbcr 1980
exp(-jyl ) (2j cos 0 1 dA 1/dx +(ErlP lDrOM2
exPC-jG)l
+ exp(-jy2){-2j
cos 6, dA2/dx
(9)
where
+ (ErlP1/2Ero)A 1 ex~ (iG)1
71 =P1(xcosQ
+v sin 0,) +g(x),
y2 = &(--x cos 8,
+_I’
sin 01) --g(x),
(10)
+ (ErlPl /2ErO){A1 expL--j(y, + 200x cos fl lo)]
(11)
+ A2 exp [---j(y2 -- 2flox cos 8 lo)]} = 0,
112 01 = “1 (C1(JeOCr()), w1 is the frequency of the reconstructing wave close to ~0 and the function g(x) represents a small perturbation. Substituting eq. (8) into the wave equation
(171
where G = y2 -- y1 f 20,~ cos 8 1o = ---2(tx + g(x)) < = Ap cos 0 l~ ---poAO sin 0 lo,
v2E + O:/JOEOCrsE= 0,
(12) op=P1
the function g(x) may be found as
‘rl‘!---sinll(3ax
g(x)-
_____._
4r,C)a cos 0 1
M
-
33)
C)*
Having determined the propagation of waves in the medium with permittivity variation crs we may now proceed to the second step in which the effect of the fast varying term in the permittivity is considered. We shall now claim that the waves described by eq. (9) will be coupled to each other due to the periodic pcrmittivity modulation. The wave equation for the complete problem is V2E f &e,/e,o)l:’
= 0,
(14)
where er is given by cq. (6), and its solution will be assumed in the form 2
I:’ = K.
2
exp( -jri).
dA,/dx - j&I l exp(jG) = 0,
(20)
where K’= frlP1/4Er() cos Ol*
(21)
The relevant boundary conditions are and
A2=OonxZd.
(22)
(15) Expressing A, from eq. (20) and substituting it into eq. (19) we obtain a differential equation in A2
cq . (
where
1 2 ), and 2 -4 p 1 CL4 1 /(IX
(1’))
Lllld
d2A2/dx’
Jdx
(18)
d.Al /dx + jK’A2 exp(--jG) = 0,
A, = 1011x=0, Ai
A0 = 0, .- Olo.
We shall now use the approximation inherent in all “two-wave” theories (see for example rei’. [9] ) that the higher order modes may be neglected, i.e. the third line in eq. (17) may be disregarded In that case eq, (17) may be satisfied by setting the expressions in both curly brackets equal to zero, obtaining the coupled wave equations
Substituting cq. (15) into (14) WCobtain a rather lengthy cxpressi:?:? but it can be siw.p!ifi.Qtdby considering that cxp( jyi) is an approximate solution of
‘A4
--PO,
illId
g(x)
< p,
co!3 0 1.
- K”&
6 = [ + @/dx
(W
and
With the above simplifications cq. (14) may be written in the form
t 2jb dA,/dx
the boundary conditions are
A2 = 0 at x =tl, and
=
0,
(23)
(24)
\‘olrtmc
34.
ri.4 z /dx =
numtwr3 j&l \ (0)
OPTICS COMMUNICATIONS
at x = 0.
tsp(jG(0))
(25)
if tltt’ *&lx term ws ahetlt clearly s the parameter. d off-Bragg the lea w&e eq. the (26)
e’t~r.
shall that
rendition
5= t’
(I
introduce
the
when
is is
off-Bragg average b zero. {
g(O)].
Thus new average value E, instead of
the -
4.
Eqs. ( 19 ) and (20)
solutions solution the equar~? tither the condiarc _civen different surfaces. One possibility is to LISAthe iterative method outlined in reference [lo). Ttdid indeed yield a solution for I’d up to ,abcsut0.6 but with the available computer accuracy the iterations did not converge beyond that. However we reatised that if we assumed the boundary conditions
2*
A, =.-I and‘42 =Oonx=d
(2%
a&l built up the solution moving backwards from Y= tl to x = 0 by means of the fourth-order RungeKutta-Xystrom method we would end up with (say) .4,(O)= Qantifl~\“, 1 A rn’ = C and then c:oki claim that an input of unity at x = 0 would yield a reflected wave CIutput at .x = 0 equal to C/B. Hence t!le hologram sfficienCy is simply
323
September 1980
5. Numerical results The question we wish to ask is how much we are worse off if the holographic material exhibits losses a; the recording stage. If we keep the recording beam amplitudes constant then higher losses will, naturally, lead to lower permittivity modulation and hence to lower efficiency. In most cases however we would be able to increase the power in the input beams and restore the permittivity modulation to its original value. We would therefore like to ask the slightly different question, that of how will the hologram efficiency depend on loss for a fixed value of K ‘d? The other parameters of interest are the amount of loss, expressed by the value of exp(-&), the beam r’atio Y, and the Bragg parameter {. First we wish to note that the curves for a beam ratio of l/r are the same as for Y,i.c. it makes no difference which of the two beams is the stronger one. This was borne out by the numerical results and may also be proven mathematically from our equations. The efficiency as a function of { is plotted in figs. 2a-f for I’d = 0.5 and 2, exp( -WI)= 1, f and $, and Y= 1,2anti4. For larger values of Ythe efficiency may appreciably decrease when absorbtion loss was present as may be seen in figs . 2b, c, c anct f. This may bc cxplained by considering that the modulation depends on the product of the recording amplitudes so that for unequal beams, E,, is mainly determined by the stronger beam. Thus for large beam ratio the variation of erS is large in comparison with the modulation and hence there is considerable dephasing leading to reduction in efficiency. It may be seen that the reduction in efficiency increases with increasing beam ratio and increasing loss, and it is more noticeable at larger values of I’d. It is therefore of interest to show the variation of the maximum available efficiency (occurring at the optimum of {) against rc’d with exp( --arc/)and Yas parameters. For Y= 1 there is so little decrease in efficiency that the curves are not worth showing. For I’= 2 and y = 4 however the results arc plotted in figs. 3a and b. All the curves display the characteristic behaviour of reflection holograms, namely the efficiency tends to unity as K ‘cl + 00 but for r = 4 and exp( --ml) = $ the reduction in efficiency is quite appreciable.
Volume 34, number 3
OPTICS COMMUNICATIONS
., , _d
Sep tcmbcr 1980
I
4
--_---_
,~___ P
i
I
-_
----_ -_-..----_I_--____
g”
\’\
-I*
-_ _-.. al
1’ ’ I
;,’
I .____
A--1----..
September
OPTICS COMMUNICATIONS
1980
I _
5
L)
K'd
K’d
I-‘&. 3. Plots of maximum cfficicncy
6. Conchsions .-!;tenuation at hc recording stage is bound to be 6~rrnful because for a given input power permittivity xl~~~ulrlticw dwlitws. However, even if the input power is ‘sufficiently large to produce the required permittivity modulation, the attenuation still leads to deleterious effects by producing a slowly varying permittivity whiA means that the Bragg conditions are not general11’satisfied. It has been shown that this effect is relatively insignificant for unity beam ratio but becomes increasingly important as the beam ratio and the attenuation increase.
.~cknow!edwilents 3
The authors would like to thank Dr. NJ. Philips
7) versus
I’d
for (a) r = 2, (b) r = 4.
of Lou&borough University for helpful and stimulati~~gdiscussions and the U.K. Science Research Council for financial support.
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