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Angular selectivity of volume holograms recorded in photorefractive crystals; An analytical treatment
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
ANGULAR SELECTIVITY OF VOLUME HOLOGRAMS R E C O R D E D I N P H O T O R E F R A C T I V E CRYSTALS; A N A N A L Y T I C A L T R E A T M E N T J. G O L T Z and T. T S C H U D I Technische HochschuleDarmstadt, Hochschutstr. 2, 6100 Darmstadt, Fed. Rep. Germany Received 10 February 1988
Previous examinations of the angular selectivity of volume holograms recorded in photorefractive crystals are not fully analytical. These examinations have shown that the minima of the diffraction efficiencyversus angular mismatch curve are nonzero for volume holograms attenuated along the axis perpendicular to the grating vector. The periodic variation of this curve becomes obscure as the coupling strength increases and for strong coupling the curve becomes monotonic. We can describe this behaviour with an analytical solution which is based on the undepleted pumps approximation. Furthermore, our solution shows that all minima of the order n < N vanish, where N increases with increasing coupling strength.
1. Introduction
2. Theory
Holograms in photorefractive crystals are characterized by the nonuniform index grating profile. Hologram gratings attenuated along the direction perpendicular to the grating vector have been examined in ref. [1]. Theoretical and experimental studies of the angular selectivity of volume holograms recorded in photorefractive crystals are presented in refs. [2,3]. These studies show that the nonuniform grating profile leads to nonzero diffraction m i n i m a and the periodic variation o f the diffraction efficiency with the angular mismatch becomes obscure as the coupling strength increases. In order to obtain a fully analytical solution we use the undepleted p u m p s approximation. With this we get an easy expression which describes the above mentioned behaviour of the angular selectivity. Our solution shows that by the periodic variation o f the diffraction efficiency with the angular mismatch all minima of the order n < N vanish. Then we present an analytical calculation o f N. This calculation shows that N increases with increasing coupling strength.
From ref. [ 2 ] we get the differential equation for hologram readout which for undepleted pumps in our notation is written as ds _ c~O e x p ( R e cz) exp(i~z) P , dz A
where s is the amplitude of the reconstructed signal, p is the amplitude o f the readout pump, • and/* are the amplitudes o f the writing beams, the index " 0 " indicates the boundary z = 0 , c is the coupling coefficient and Re means real part. The off-Bragg parameter is defined in the same way as in ref. [ 2 ], see fig. 1,
Ks+(O,O,~)-Kp=K,+(O,O, Imc)-K~,
(2a)
where Im means imaginary part. F r o m eq. (2a) we get ~ = I m c + K [ 1 - (sin 0, + s i n 0/~ - s i n 0p) 2 ] 1/2 + K ( - c o s a~ + c o s 0 ~ - c o s
0p) .
(2b)
In photorefractive crystals c is much smaller than the wave number K = 2n/2 and with this even a smaller angular misalignment leads to a low diffraction efficiency. Thus eq. (2b) becomes ~ = I m c + K (sin 0~+ sin 0/~)(0p-0p) .
164
( 1)
(2c)
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
(0,0,6)
k-;,
ol
05
(0,0, Im c) m-Z
Fig. 1. Diagram of the wave vectors. Definition of the off-Braggparameter 6. Solution of eq. ( 1 ) is mp s= --c A
exp(Re cz+i6z)- 1 Re c + i 6
(3)
From (3) we calculate the angular dependence of the diffraction efficiency r/= Is l 2/IPl 2, 0
~(u) =,7(0) 1+exp(-2y)-2 X
exp(-y)
[l_exp(_y)]2[l+(u/7)2
cos u ]
,
(4)
2
4
" "
Fig. 2, Normalized diffraction efficiency (log) versus normalized deviation from Bragg-angle. Illustration of eq. (3): (a) F=4.5. (b) 7=3.5, N= 5.2, (c) F=3, N= 3.1, (d) F=2.5, N= 1.85. (e) y= 1.5, N=0.6, (f) 7=0.1.
where we have combined y:=tRecz[,
u:=lSz[.
For weak coupling ( y - . 0 ) , eq. (4) reduces to r/(u) = r / ( 0 ) s i n c 2 ( u / 2 ) .
(5)
For strong coupling (exp y>> 1 ), eq. (4) reduces to 1 -exp(-y) cos u ~/(u) = r/(0) 1+ (u/7) 2 '
(6a)
and for very strong coupling (exp y--, oo ), eq. (4) reduces to 1
t/(u) =~/(0) 1 + ( u / y ) 2"
(6b)
As our solution shows, with increasing coupling strength the sharpness o f the periodic variation of the r/versus u := 16z I curve decreases until the curve becomes monotonic. This result is illustrated in fig. 2. In the next section we calculate the condition for vanishing extrema.
As our solution shows, only for strong coupling the extrema of the r/versus u := 16zl curve vanish. Thus for the discussion we consider eq. (6a). The requirement drl/du = 0 leads to ( y 2 + u 2 ) sin u = u [ e x p ( 7 ) - 2 cos u] .
(7)
Due to strong coupling 2 cos u can be neglected in favour of exp(7), and sin u is smaller than unity. Thus condition (7) can be fulfilled only for 1/2
U> exp(y......~)2+ ( e x p ~ 2 y , _ 7 2 )
(8,
Due to the periodicity of the cosine function the position of the nth m i n i m u m is approximately given by (9)
u=2rm .
Combination of eqs. (8) and (9) leads to the resultthat all m i n i m a of the order n < N : = exp(?) + [ e x p ( 2 r ) - 4 y 2 ] '/2
(10)
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OPTICS COMMUNICATIONS
vanish, see fig. 2. E.g. in curve (d) with N = 1.85 we see that the first minimum ( n = 1 < 1.85) vanishes while the other minima remain. From (10) we calculate that for y< 2 all minima of the r/versus 8z curve remain.
1 July 1988
approximative solution which allows to discuss the angular selectivity without any numerical evaluation. We have shown that by the ~/versus 8z curve for y> 2 all minima of the order n < N vanish, where N increases with increasing coupling strength y.
References 3. Conclusion To support previous examinations of the diffraction efficiency of volume holograms recorded in photorefractive materials, we have presented an easy
166
[ 1 ] N. Uchida, J. Opt. Soc. Am. 63 (1973) 280. [2] J.M. Heaton, P.A. Mills, E.G.S. Paige, L. Solymar and T. Wilson, Optica Acta 31 (1984) 885. [3 ] L. Arizmendi, M,J. Kliewer and R.C. Powell, J. Appl. Phys. 61 (1987) 1682.
OPTICS COMMUNICATIONS
1 July 1988
ANGULAR SELECTIVITY OF VOLUME HOLOGRAMS R E C O R D E D I N P H O T O R E F R A C T I V E CRYSTALS; A N A N A L Y T I C A L T R E A T M E N T J. G O L T Z and T. T S C H U D I Technische HochschuleDarmstadt, Hochschutstr. 2, 6100 Darmstadt, Fed. Rep. Germany Received 10 February 1988
Previous examinations of the angular selectivity of volume holograms recorded in photorefractive crystals are not fully analytical. These examinations have shown that the minima of the diffraction efficiencyversus angular mismatch curve are nonzero for volume holograms attenuated along the axis perpendicular to the grating vector. The periodic variation of this curve becomes obscure as the coupling strength increases and for strong coupling the curve becomes monotonic. We can describe this behaviour with an analytical solution which is based on the undepleted pumps approximation. Furthermore, our solution shows that all minima of the order n < N vanish, where N increases with increasing coupling strength.
1. Introduction
2. Theory
Holograms in photorefractive crystals are characterized by the nonuniform index grating profile. Hologram gratings attenuated along the direction perpendicular to the grating vector have been examined in ref. [1]. Theoretical and experimental studies of the angular selectivity of volume holograms recorded in photorefractive crystals are presented in refs. [2,3]. These studies show that the nonuniform grating profile leads to nonzero diffraction m i n i m a and the periodic variation o f the diffraction efficiency with the angular mismatch becomes obscure as the coupling strength increases. In order to obtain a fully analytical solution we use the undepleted p u m p s approximation. With this we get an easy expression which describes the above mentioned behaviour of the angular selectivity. Our solution shows that by the periodic variation o f the diffraction efficiency with the angular mismatch all minima of the order n < N vanish. Then we present an analytical calculation o f N. This calculation shows that N increases with increasing coupling strength.
From ref. [ 2 ] we get the differential equation for hologram readout which for undepleted pumps in our notation is written as ds _ c~O e x p ( R e cz) exp(i~z) P , dz A
where s is the amplitude of the reconstructed signal, p is the amplitude o f the readout pump, • and/* are the amplitudes o f the writing beams, the index " 0 " indicates the boundary z = 0 , c is the coupling coefficient and Re means real part. The off-Bragg parameter is defined in the same way as in ref. [ 2 ], see fig. 1,
Ks+(O,O,~)-Kp=K,+(O,O, Imc)-K~,
(2a)
where Im means imaginary part. F r o m eq. (2a) we get ~ = I m c + K [ 1 - (sin 0, + s i n 0/~ - s i n 0p) 2 ] 1/2 + K ( - c o s a~ + c o s 0 ~ - c o s
0p) .
(2b)
In photorefractive crystals c is much smaller than the wave number K = 2n/2 and with this even a smaller angular misalignment leads to a low diffraction efficiency. Thus eq. (2b) becomes ~ = I m c + K (sin 0~+ sin 0/~)(0p-0p) .
164
( 1)
(2c)
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
(0,0,6)
k-;,
ol
05
(0,0, Im c) m-Z
Fig. 1. Diagram of the wave vectors. Definition of the off-Braggparameter 6. Solution of eq. ( 1 ) is mp s= --c A
exp(Re cz+i6z)- 1 Re c + i 6
(3)
From (3) we calculate the angular dependence of the diffraction efficiency r/= Is l 2/IPl 2, 0
~(u) =,7(0) 1+exp(-2y)-2 X
exp(-y)
[l_exp(_y)]2[l+(u/7)2
cos u ]
,
(4)
2
4
" "
Fig. 2, Normalized diffraction efficiency (log) versus normalized deviation from Bragg-angle. Illustration of eq. (3): (a) F=4.5. (b) 7=3.5, N= 5.2, (c) F=3, N= 3.1, (d) F=2.5, N= 1.85. (e) y= 1.5, N=0.6, (f) 7=0.1.
where we have combined y:=tRecz[,
u:=lSz[.
For weak coupling ( y - . 0 ) , eq. (4) reduces to r/(u) = r / ( 0 ) s i n c 2 ( u / 2 ) .
(5)
For strong coupling (exp y>> 1 ), eq. (4) reduces to 1 -exp(-y) cos u ~/(u) = r/(0) 1+ (u/7) 2 '
(6a)
and for very strong coupling (exp y--, oo ), eq. (4) reduces to 1
t/(u) =~/(0) 1 + ( u / y ) 2"
(6b)
As our solution shows, with increasing coupling strength the sharpness o f the periodic variation of the r/versus u := 16z I curve decreases until the curve becomes monotonic. This result is illustrated in fig. 2. In the next section we calculate the condition for vanishing extrema.
As our solution shows, only for strong coupling the extrema of the r/versus u := 16zl curve vanish. Thus for the discussion we consider eq. (6a). The requirement drl/du = 0 leads to ( y 2 + u 2 ) sin u = u [ e x p ( 7 ) - 2 cos u] .
(7)
Due to strong coupling 2 cos u can be neglected in favour of exp(7), and sin u is smaller than unity. Thus condition (7) can be fulfilled only for 1/2
U> exp(y......~)2+ ( e x p ~ 2 y , _ 7 2 )
(8,
Due to the periodicity of the cosine function the position of the nth m i n i m u m is approximately given by (9)
u=2rm .
Combination of eqs. (8) and (9) leads to the resultthat all m i n i m a of the order n < N : = exp(?) + [ e x p ( 2 r ) - 4 y 2 ] '/2
(10)
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Volume 67, number 3
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vanish, see fig. 2. E.g. in curve (d) with N = 1.85 we see that the first minimum ( n = 1 < 1.85) vanishes while the other minima remain. From (10) we calculate that for y< 2 all minima of the r/versus 8z curve remain.
1 July 1988
approximative solution which allows to discuss the angular selectivity without any numerical evaluation. We have shown that by the ~/versus 8z curve for y> 2 all minima of the order n < N vanish, where N increases with increasing coupling strength y.
References 3. Conclusion To support previous examinations of the diffraction efficiency of volume holograms recorded in photorefractive materials, we have presented an easy
166
[ 1 ] N. Uchida, J. Opt. Soc. Am. 63 (1973) 280. [2] J.M. Heaton, P.A. Mills, E.G.S. Paige, L. Solymar and T. Wilson, Optica Acta 31 (1984) 885. [3 ] L. Arizmendi, M,J. Kliewer and R.C. Powell, J. Appl. Phys. 61 (1987) 1682.