Russian Physics Journal, Vol. 48, No. 3, 2005
HOLOGRAPHIC REFLECTION DIFFRACTION GRATINGS IN ABSORBENT PHOTOPOLYMER MEDIA E. A. Dovol’nov and S. N. Sharangovich
UDC 535.33: 535.417: 773.93
Record and readout of holographic reflection gratings in absorbent photopolymer materials with a dye-sensitizer are theoretically investigated. An analytical model describing the spatiotemporal transformation of holographic grating field during record is constructed with allowance for the constant optical absorption and diffusion processes. It is demonstrated that the optical beam attenuation during record results in the spatial inhomogeneity of the grating profile and its growth kinetics. A degree of influence of the optical attenuation and contrast of interference pattern on the integral diffraction characteristics is evaluated based on numerical modeling. INTRODUCTION
Increasingly more and more works are devoted to the investigation of holographic photopolymer materials. The photopolymer materials are promising for the development of holographic optical elements based on diffraction gratings for various applications such as optical connectors [1], narrow-band (0.05 nm) holographic filters [2], optical disks [3], splitters and couplers selective to the wavelength [4, 5], input-output waveguide gratings [6], etc. In addition, the photopolymer materials have a number of advantages over other holographic materials, including long operating lifetime [7], high recording rate [8], relatively low cost [9], and absence of a wet treatment stage [10]. The main parts of holographic optical elements are transmission and reflection diffraction gratings. In this regard, record and readout of gratings of both types are of interest and call for adequate theoretical models. This affects the number of theoretical and experimental works in the given direction [1–16]. Worthy of mention is the model of transmission gratings recorded in an absorbent photopolymer material with a dye-sensitizer [15]. Dovol’nov et al. [15] demonstrated that the absorption results in a sharply inhomogeneous refractive index profile of the grating being recorded, which affects the diffraction efficiency and selective properties of the grating. Another theoretical work we are aware of is [16] in which the nonlocal model was generalized to consider the absorption. However, no analytical formulas were presented in [16]. Results obtained and conclusions drawn in [16] are in qualitative agreement with those presented in [15]. However, both these models are applicable only to transmission gratings. The present study is a continuation of [15] and is aimed at constructing an analytical model of record and readout of holographic reflection diffraction gratings in an absorbent photopolymer with a dye-sensitizer. In so doing, we consider the polymerization and diffusion mechanisms of record, dependence of the rate of diffusion processes on the integral polymerization level, and influence of the recording beam attenuation on the spatial structure of the holographic grating [17]. For a theoretical description of record and numerical simulation, we utilized the procedures and parameters described in [15]. INITIAL EQUATIONS
Let us consider two plane coherent monochromatic light waves with amplitudes E0 and E1 and wave vectors k0′ and k1′ propagating inside a plane absorbent photopolymer layer (0 ≤ y ≤ d) at angles θ0 and θ1, respectively. The spatial
Tomsk State University of Control Systems and Radio Electronics. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 62–70, March, 2005. Original article submitted August 5, 2004. 290
1064-8887/05/4803-0290 ©2005 Springer Science+Business Media, Inc.
x
x
E0 q
k1′ d
k·n(t) θ1
γ θ0
k·n(0)
E1 y
θ1 k0′
k1′(t) K(t)
θ0 k1′
k0′(t)
k0′
y
K(0) b
a
Fig. 1. Spatial geometry (a) and vector diagram (b) of holographic grating record in a photopolymer material.
geometry and the vector diagram of the process are shown in Fig. 1. Based on the superposition principle, the optical field inside of the photopolymer material can be represented as a sum of fields: E0 (t , r ) = e0 ⋅ E0 ⋅ exp [ −αy / cos θ0 ] exp [i (ω⋅ t − k0′ ⋅ r )] + c.c.,
(1)
E1 (t , r ) = e1 ⋅ E1 ⋅ exp [ −α(d − y ) / cos θ1 ] exp [i (ω⋅ t − k0′ ⋅ r )] + c.c.,
where e0,1 are polarization vectors, α is the photopolymer light absorption coefficient at the frequency ω, r is the radiusvector, k ′j = k ⋅ N j , k = nstω/с is the wave number, Nj is the wave normal, and nst is the refractive index. Let us write down the expression for the interference pattern of this light field: I ( r ) = I 0 ( y ) {1 + m( y ) cos ( Kr )} ,
(2)
where I 0 ( y ) = I 0 ( y ) + I 1 ( y ) , I 0 ( y ) = I 0 exp [ −αy / cos θ0 ] , I 1 ( y ) = I1 exp [ −α (d − y ) / cos θ1 ] , I 0 = E0 , I1 = E1 , 2
(
m( y ) = 2 I 0 ( y ) I 1 ( y ) ⋅ ( e1 ⋅ e0 ) / I 0 ( y ) + I 1 ( y )
)
2
is the local contrast of the interference pattern, K = Kq = k0 – k1 is the
grating vector, K = 2k⋅sin((π – θ0 – θ1)/2), and q is the unit vector. During record, the volume averaged refractive index increases (by <1%) [15], which causes changes in the absolute value and direction (for symmetric geometry, only in the absolute value) of the grating vector being recorded (Fig. 1b). However, this effect is disregarded below because of its smallness (<1%). The radical photopolymerization that occurs in the photopolymer material upon exposure to light causes local changes in the refractive index, that is, leads to the formation of a phase holographic grating in accordance with light field distribution (2). To describe the process of record in the photopolymer with a dye-sensitizer, we take advantage of the equations for the monomer concentration M and refractive index n [15] with allowance for the dependence of the monomer diffusion coefficient D on the polymerization degree [17]: ∂M = div ( D( M ) grad M ) − K g Kb−1/ 2 α 0β K τ0 I (r ) M , ∂t ⎛ ∂n M M = δn p K g Kb−1/ 2 α 0β K τ0 I (r ) + δni div ⎜ D( M ) grad ∂t Mn M ⎝ n
(3) ⎞ ⎟, ⎠
(4)
291
(
)
⎤, D( M ) = Dm exp ⎡ − s 1 − M M n ⎥⎦ ⎢⎣
(5)
where Kg and Kb are the coefficients of growth and breaking of the polymer chain; α0 is the dye absorption coefficient; is the volume averaged dye concentration; τ0 is the lifetime of the excited state of the dye molecule; Dm is the initial diffusion coefficient; Mn is the initial monomer concentration; δni and δnp are model parameters that describe changes in n caused by the polymerization and diffusion of the components of the material, respectively [19]; s is the model parameter that characterizes the rate of changes in the diffusion coefficient; and t is the recording time. We seek a solution of system of equations (3)–(5) in the form [15] M ( t , r ) = M 0 ( t , y ) + M1 ( t , y ) cos( Kr ) ,
(6)
n ( t , r ) = n0 ( t , y ) + n1 ( t , y ) cos( Kr ) ,
(7)
where M0(t, y), n0(t, y), and M1(t, y), n1(t, y) are the zero and first harmonics of the monomer concentration and refractive index of the grating, respectively. Moreover, the dependence of Mj(t, y) and nj(t, y) on the spatial coordinate y (grating depths) is caused by the influence of the light wave attenuation in the photopolymer material. ANALYTIC SOLUTION
To solve the above-formulated problem, we take into account the monomer diffusion at distances of about the grating period and neglect its diffusion at longer distances and the diffusion of polymer chains because of the small characteristic polymerization time [15]. To solve Eqs. (3) and (5), we take advantage of the expansion of the nonlinear function I (r ) = I 0 ( y ) [1 + m( y ) cos( Kr ) ] in a Taylor series and consider only the first three terms of the series: ⎡ m( y ) ⎤ m2 ( y) I (r ) ≈ I 0 ( y )0.5 ⎢1 + cos( Kr ) − cos 2 ( Kr ) ⎥ . 2 8 ⎣ ⎦
(8)
The error in approximating Eq. (8) for 0.75 < m(y) < 1 is within 1.5–3%; it is less than 1.5% for m(y) < 0.75. Substituting Eqs. (2) and (5)–(8) into Eqs. (3) and (4), assuming M1 << M0 and n1 << n0, and using the property of orthogonality, we obtain the kinetic equations for the zero harmonics: −
∂M 0 (t , y ) 2 = M 0 (t , y ) T py ∂t
⎛ m 2 ( y ) ⎞ ∂n0 (t , y ) M (t , y ) 2 = δn p 0 ⎜⎜ 1 − ⎟⎟ , 16 ⎠ M n Tpy ∂t ⎝
⎛ m2 ( y) ⎞ ⎜⎜1 − ⎟, 16 ⎟⎠ ⎝
where T py = T p ( y ) = 2 Kb / K g 2 α 0βτ0 K I 0 ( y ) is the local polymerization time. Integrating the kinetic equations for the zero harmonics with the initial conditions M ( t = 0, y ) = M n , n ( t = 0, y ) = nst
and going over to the relative time τ = t/Tm, where Tm = 1/K2Dm is the diffusion time, we derive the following expressions for M0(τ, y) and n0(τ, y): M 0 ( τ, y ) = M n ⋅ p (τ, y ) , n0 ( τ, y ) = nst + δn p {1 − p (τ, y )} ,
292
(9)
(
)
where p(τ, y ) = exp ⎡⎣ − 2 ⋅ τ 1 − m y 2 /16 / by ⎤⎦ , m y = m( y ) , and by = Tpy/Tm. Then, taking advantage of Eqs. (5)–(9), the introduced designations, and the property of orthogonality of the harmonics, from system of equations (3)–(4) we derive the following equations for М1(τ, y) and n1(τ, y): −
⎧⎪ ⎫⎪ ∂M1 (τ, y ) 2 2 = M1 (τ, y ) ⎨exp [ − s (1 − p (τ, y ) )] + m2 y ⎬ + M 0 (τ, y ) my , ∂τ 2 b b y y ⎩⎪ ⎭⎪
⎤ M (τ, y ) ∂n1 (τ, y ) 2 ⎡ M 0 (τ, y ) m y M1 (τ, y ) exp [ − s (1 − p(τ, y ) )] + δn p m2 y ⎥ , = −δni 1 + ⎢ 2 Mn by ⎣ M n Mn ∂τ ⎦
(
(10)
(11)
)
where m2 y = 1 − 3 ⋅ m y 2 / 32 . After substitution of Eq. (9) into Eq. (10) and integration, we obtain the final expression describing the behavior of the monomer grating М1(τ, y): M1 ( τ, y ) = − M n ⋅ f ( τ, y ) ,
(12)
τ ⎛ 2m y 2 ⎞ exp ∫ ⎜⎜ 32b τ′ − ∫ exp [ − s (1 − p(τ′′, y ) )] d τ′′ ⎟⎟ d τ′ . y τ′ 0 ⎝ ⎠ From Eq. (11), with allowance for Eqs. (9) and (12), we obtain the spatiotemporal distribution of the refractive index amplitude of the grating being recorded (0
where f (τ, y ) =
⎡ 2 ⎤ 2 my m2 y ⋅ τ ⎥ exp ⎢ − by 2 ⎣⎢ by ⎦⎥
τ
n1 (τ, y ) = n1 p (τ, y ) + n1i (τ, y ) , n1 p (τ, y ) = δn p
⎞ 2 τ ⎛ my p (τ′, y ) − m2 y f (τ′, y ) ⎟ d τ′ , ⎜ ∫ by 0 ⎝ 2 ⎠ τ
n1 i (τ, y ) = δni ∫ f (τ′, y ) exp [ − s (1 − p (τ′, y ) )] d τ′ .
(13)
(14)
(15)
0
Solutions (13)–(15) are functions of the spatial and temporal coordinates y and τ. Thus, the spatial structure of the grating being recorded is inhomogeneous and is transformed with time. Moreover, the kinetics of changes in the grating amplitude at each spatial point differs. It is determined by contributions of the polymerization and diffusion processes of the components of the material proportional to δnp and δni, respectively. These expressions at α ≈ 0 can be used to calculate the refractive index profile of the transmission grating in nonabsorbent photopolymer materials. If we set α ≈ 0 and m = 1, expressions (13)–(15) will coincide with those presented in [15] to within approximation (8) whose error has already been estimated above. Having the expression for the spatial distribution of the grating refractive index, we now consider the diffraction characteristics during readout or monitoring that are determined by solving system of equations (16) for coupled waves in the first approximation of the method of slowly varying amplitudes [20]: α ⎧ dE0 (τ, y ) − E0 (τ, y ) = −i ⋅ G0 ⋅ n1 (τ, y ) E1 (τ, y )ei⋅∆K ⋅ y , ⎪ dy cos( θ ) 0 ⎪ ⎨ α ⎪ dE1 (τ, y ) − E1 (τ, y ) = +i ⋅ G1 ⋅ n1 (τ, y ) E0 (τ, y )e −i⋅∆K ⋅ y , ⎪ dy cos( θ ) ⎩ 1
(16)
293
x
nst⋅2π /λ
nst⋅2π /(λ+∆λ) θ0
θ1 ∆θ1
k1′
k1
k0′
y
∆θ0 k0
K ∆K
K
Fig. 2. Vector diagram of the readout process. where E0(τ, y) and E1(τ, y) are the amplitudes of transmitted and diffracted light waves, respectively; λ is the light wavelength in vacuum; G0,1 = k/(2cosθ0,1); n1(τ, y) is the grating profile calculated from Eq. (13); ∆K = |∆K | = |k0 – k1 – K| is the modulus of the phase mismatch vector linearly related to the deviations of the angle ∆θ0 and wavelength ∆λ of the readout beam from Bragg’s conditions [20] (Fig. 2). The inclusion of the absorption in the stage of readout is not so important as in the stage of record and can be omitted, since the two methods are mainly used for readout of gratings, namely: monitoring of the grating by nonactinic radiation during record or readout after the termination of grating record and, as a rule, heat treatment aimed at utilization of the remaining amounts of the monomer and dye. Thus, in both cases the absorption during readout is negligible; therefore, setting α = 0 in Eq. (16), taking into account the smallness of the amplitude n1(τ, y) ∼ δnp << nst in Eq. (13), and using the perturbation method [18] with the boundary conditions for the light wave amplitudes during readout E0(τ, y = 0) = Е0 and E1(τ, y = d) = 0, we obtain the solution of the diffraction problem in terms of the diffraction efficiency ηd defined as a ratio of the diffracted to the incident radiation intensities: 2
ηd (τ, ∆ ) =
E1 (τ, 0) , E0 (τ, 0)
(17)
where ∞ ⎛ δn p ⎞ E0 (τ, y ) = ∑ E0,2⋅k (τ, y ) ⎜ ⎟ ⎝ nst ⎠ k =0
⎛ δn p ⎞ E1 (τ, y ) = ∑ E1,2⋅k +1 (τ, y ) ⎜ ⎟ ⎝ nst ⎠ k =0 ∞
2⋅k
d−y
, E0,2 k ( y, τ) = G0 ∫ n1′ (τ, y ′) E1,2 k −1 (τ, y ′)ei⋅∆⋅ y′ / d dy ′ , 0
2⋅k +1
,
E1,2k +1 (τ, y ) = G1
d−y
−i⋅∆⋅ y ′ / d dy ′ , ∫ n1′ (τ, y ′) E0,2k (τ, y ′)e
0
E0,0 (τ, y ) = E0 , n1′ (τ, y ) = n1 (τ, y ) / δn p , and ∆ = ∆Kd is generalized Bragg’s mismatch.
When Bragg’s condition is satisfied (∆ = 0), Eq. (17) is transformed into Eq. (18) that can be derived from system of equations (16) by direct setting ∆ = 0 and α = 0: ηd (τ) =
d G0 ⎪⎧ ⎪⎫ tanh 2 ⎨ G0G1 ∫ n1 (τ, y ′)dy ′⎬ . G1 ⎩⎪ ⎭⎪ 0
(18)
Expression (18) for a homogeneous grating profile coincides with the Kogel’nik formula that determines the diffraction efficiency of the 3D reflection grating given that Bragg’s conditions are satisfied [20].
294
a) b = 0.25
b) b= 5 Fig. 3
NUMERICAL SIMULATION
To obtain the most general results of numerical simulation, we take advantage of the generalized parameters introduced in [15] including the ratio of polymerization to diffusion times b = Tp/Tm, ratio of contributions of polymerization and diffusion recording mechanisms Cn = δni/δnp, and absorption of the photopolymer material αd = αd, where
(
)
T p = 2 K b / K g α 0β K τ0 ( I 0 + I1 ) . We have also introduced the ratio of the recording beam intensities m0 = I1/I0. In
this case, the spatial dependence of the contrast interference patterns mу and by in Eqs. (2) and (9) assumes the form my =
2 m0 exp [ −0,5α d ( y /(d cos θ0 ) + ( y − 1) /(d cos θ1 ) )] ( e1 ⋅ e0 )
by = b
exp [ −α d y /(d cos θ0 ) ] + m0 exp [ −α d ( y − 1) /(d cos θ1 ) ]
exp [ −α d y /(d cos(θ0 )) ] + m0 exp [ α d ( y − 1) /(d cos(θ1 )) ] . 1 + m0
,
(19)
(20)
In this section, we study the influence of the absorption α, ratio of the recording beam intensities m0 = E12/E02, and parameter b = Tp/Tm on the kinetics of the refractive index n1(τ, y) and on the diffraction characteristics of the recorded diffraction grating during its readout by nonactinic radiation. As demonstrated in [15], the parameter b influences significantly the kinetics of the refractive index profile. In general, two regions with b < 1 and b > 1 can be identified for which changes in b do not lead to changes in the general behavior of the curve and affect only its temporal scale. Therefore, below the results of simulation for two points b = 0.25 and 5 and conclusions are generalized to the entire characteristic regions with b < 1 and b > 1, respectively. Figure 3 shows the spatiotemporal grating profiles calculated for Cn = δni/δnp = 1, s = 1, m0 = 1, and αd = 4 Np for two values of the parameter b = Tp/Tm = 0.25 and 5. In actual practice, the parameter b can be varied by changing the internal parameters of the photopolymer material or the external conditions of record (I0 + I1 and θ0 + θ1), which is more suitable for practice. As can be seen from expressions for Tp and Tm, a decrease in the intensity and an increase in the convergence angle of the recording beams cause an increase in the parameter b and vice versa. Owing to the absorption, the spatial grating profiles become sharply inhomogeneous with depth and are transformed with time (Fig. 3). For small recording times, the spatial refractive index profile copies the distribution of the local contrast interference pattern with y, which characterizes the competitive behavior of the zero and first harmonics
295
0.75
1
ηd ( τ ), rel. units
ηd ( τ ), rel. units
0.12
2
0.08
3
0.04 4 5 0.00
0
1 2 τ = t /Tm , rel. units
3
a) b = 0.25
1 2
0.50 0.25 0.00
3
5 4 0
10 20 30 40 τ = t /Tm , rel. units
50
b) b = 5 Fig. 4
during monomer polymerization, that is, a portion of the monomer participating in the formation of the first harmonic decreases with the local contrast at the expense of the zero harmonic, and vice versa. For both regions of b, the maximum growth of the grating is observed in the central part 0.4 < у/d < 0.6, which is caused by two factors. First, contrast (19) takes its maximum value close to unity, and second, the total recording field intensity minimizes due to the beam attenuation, and hence polymerization time (20) maximizes. Due to the increased polymerization time, the monomer has additional time to diffuse from dark regions of the interference pattern to light regions where it is polymerized thereby increasing the grating amplitude but for a longer time, which can be seen from the time of achieving the maximum that increases with у/d starting from у/d = 0 to у/d = 0.5 and then decreases with further increase in у/d to у/d = 1. As demonstrated in [15], an increase in the ratio of polymerization to diffusion times (b = Tp/Tm) leads to an increase in the portion of the monomer that diffuses toward light regions of the diffraction pattern in which it polymerizes and increases the grating amplitude instead of the polymerization in dark region that would decrease the grating amplitude. However, the absorption for b > 1 causes only a stretch of the kinetics described by Eq. (20) and a decrease in the average grating amplitude at the expense of contrast (19) changing with depth. In practice, the kinetics of the grating profile is determined by that of the diffraction efficiency ηd(τ) which, being an integral characteristic, takes into account the inhomogeneity of the refractive index profile described by Eqs. (17) and (18). Figure 4 shows the dependences ηd(τ) calculated from Eq. (18) for the following parameters: Cn = δni/δnp = 1, s = 1, and δnp = 0.01. Here αd ≈ 0 (curves 1 and 4), 2 (curve 2), and 4 Np (curves 3 and 5) and m0 = 1 (curves 1–3) and 0.1 (curves 4 and 5). The increase of the absorption for b < 1 at m0 = 1, as can be seen from Fig. 4a (curves 1–3), results in the increased time of achieving the maximum diffraction efficiency of the grating being recorded and of its stationary value, whereas the value of the maximum diffraction efficiency decreases. The decrease of the maximum value is caused by the reduction of the local contrast my leading to a decrease in the grating amplitude. The increase of the stationary value can be explained by the increase in bу leading to the stretch of the kinetics and the increase of the monomer diffusion contribution to the polymerization process. The monomer diffusion for b > 1 occurs sufficiently rapidly; therefore, the increase of the absorption results in further increase of the local parameter by > b, which slows down the polymerization process. The reduction of the local contrast my causes the diffraction efficiency ηd to decrease (curves 1–3 in Fig. 4b). However, the absorption for m0 = 0.1 (curve 5 in Fig. 4b), on the contrary, leads to an increase in ηd. As can be seen from Fig. 4 (curves 4), at m0 = 0.1 the diffraction efficiency without absorption decreases for both values of b, which is explained by the fact that a decrease in m0 causes the contrast of the interference pattern to decrease almost by one half, thereby decreasing the amplitude of the first harmonic of the refractive index grating and hence ηd. In the presence of absorption at b = 0.25 (Fig. 4a), not only the stationary value of ηd (curve 4), as for m0 = 1 (curves 2 and 3), but also the maximum value of ηd (curve 5) increases, whereas for m0 = 1, it decreases (curves 2 and 3).
296
2∆ 0.5
ηd_SL
-20 -40 a -60 -20
-10
0 10 ∆ , rel. units
5
8
3
7
b
2
6 5
20
-10
9
ηd_SL ( τ ), dB
ηd_SL
2 ∆0.5( τ), rel. units
ηdn ( τ , ∆ ), dB
0
1 4 0
1 2 τ = t /Tm , rel. units
4
-20
2 5
-30 3
-40 -50
3
1
0
c
1
2
τ = t / Tm , rel. units
3
Fig. 5. Diffraction characteristics of the grating at b = 0.25.
2∆ 0.5
ηd_SL
-20 -40 a -60 -20
-10
0 10 ∆ , rel. units
20
9
3
8 7
5
6
4
5
0
-10 1
2 1 b
10 20 30 40 50 τ = t /Tm , rel. units
η d_SL( τ ), dB
ηd_SL
2 ∆0.5( τ ), rel. units
ηdn ( τ,∆ ), dB
0
2
-20 -30
5
3 c
-40 -50
4
0
10
20
30
40
τ = t / Tm , rel. units
50
Fig. 6. Diffraction characteristics of the grating at b = 5.
Based on the results of our numerical analysis carried out with the help of expressions (13)–(15), we conclude that for m0 < 0.2 the absorption affects the diffraction efficiency ηd in the same way as for transmission gratings [15], in which the presence of the absorption increases ηd (especially for b < 1), since the reflection grating profile becomes asymmetric for m0 < 0.2 and approaches the corresponding profile of the transmission grating. However, the absorption for the reflection gratings causes the diffraction efficiency to decrease for m0 > 0.5, especially for b > 1, due to its greater influence on the local contrast of the interference pattern being recorded. To consider the influence of the absorption on the reflection gratings in every detail, we now analyze their selective properties ηd(∆) with the help of Eq. (17). By analogy with the transmission gratings, we consider the bandwidth 2∆0.5 or the generalized selectivity at a level of –3 dB and the level of the first side lobe ηd SL [dB] as the main parameters characterizing the selective properties. Figures 5 (for b = 0.25) and 6 (for b = 5) illustrate the dependences ηdn(τ, ∆) = ηd(τ, ∆)/ηd(τ, 0) (a), 2∆0.5(τ) (b), and ηd SL(τ) (c) that characterize the influence of the attenuation on the selective properties of gratings and their dynamics for the following parameters: Cn = δni/δnp = 1, s = 1, and δnp = 0.01. In Figs. 5a and 6a, dashed curves show the results of calculations for αd = 4 Np, m0 = 1, and τ = 0.125 (b = 0.25) and 0.5 (b =5); solid curves show the results of calculations for τ = 3 (b = 0.25) and 50 (b = 5). Dot-and-dash curves present the results of calculations for αd = 0, m0 = 1, and τ = 3 (b = 0.25) and 50 (b = 5). Curves shown in Figs. 5b and c and 6b and c were calculated for the same parameters as those shown in Fig. 4. For b < 1 (Fig. 5), all changes of these characteristics are caused primarily by the inhomogeneity of the refractive index profile, since the influence of the grating amplitude and hence ηd(τ, 0) on these characteristics, as can be seen from the dependences shown by curves 1 and 4 in Fig. 5b and c (for a homogeneous profile and αd = 0) is negligible in comparison with the influence of the inhomogeneity of the refractive index profile. From this we conclude that the inhomogeneity of the refractive index profile significantly increases the bandwidth and decreases the level of the first side lobe. The former is caused by a decrease in the effective grating length due to the decrease in the grating amplitude starting
297
from y/d = 0.5 toward y/d = 0 and from y/d = 0.5 toward у/d = 1, which can be characterized as a lower rate of growth of the amplitude on the grating periphery reducing, in turn, the side lobe levels. A discontinuity in the dependence of the side lobe for αd = 4 Np and m0 = 1 (curves 3 in Figs. 5c and 6c) is caused by the fact that the amplitude and width of the first lobe decrease under the influence of the extending principal lobe given that the second lobe remains unchanged (solid curves in Figs. 5a and 6а). At a certain moment, the second side lobe turns out to be the first lobe, as can be seen from a comparison of the behavior of the solid and dashed curves in Fig. 6a. Then the minima of the characteristic are smeared (the solid curve in Fig. 6а). However, from a comparison of the solid curves in Figs. 5a and 6a it follows that the resultant characteristics differ significantly due to the fact that the second process only begins at τ = 43 for b = 5, while it has already been well developed at τ = 2 for b = 0.25. For m0 = 0.1 and αd = 4 Np, the above-described discontinuities are observed 3 times for b = 0.25 (curve 5 in Fig. 5c); they are observed 2 times for b = 5 (curve 5 in Fig. 6c). In this case, the second lobe turns out to be the first lobe, then the third lobe turns out to be the first lobe, and so on. This occurs because of smearing of the minima which in this case occurs faster, and a decrease in the side lobe amplitude has not enough time to be manifested. The influence of the grating amplitude on its selective properties can be traced without absorption for b > 1 (curves 1 in Fig. 6b and c), since the spatial profile in this case is homogeneous. A comparison of curves 2 and 3 with curve 1 and of curve 5 with curve 4 in Fig. 6b and c demonstrates that the absorption causes the bandwidth to increase further and the first side lobe level to decrease, which can be connected with the localization of the grating profile for 0.3 < y/d < 0.7 (see Fig. 3b) leading to a decrease in the effective length of light interaction with the grating. This influence of the absorption on the selective properties is similar to that for b < 1, since the effective interaction length remains virtually unchanged for any b. CONCLUSIONS
In this work, results of theoretical investigations of record and readout of the spatially inhomogeneous holographic reflection gratings in absorbent photopolymer materials have been presented. The analytical models describing the spatiotemporal transformation of the holographic grating field in the stage of recording were constructed with allowance for the optical absorption and diffusion processes. The degree of influence of the absorption on the inhomogeneity of the profiles of the gratings being recorded and their diffraction characteristics was demonstrated using numerical simulation for two cases with different ratios of the polymerization Тр to monomer diffusion times Tm. It was demonstrated that spatial grating profiles became sharply inhomogeneous with depth and were transformed with time. An increase in the absorption for Тр/Тт < 1 leads to a stretch of the kinetics and an increase in the stationary level of the diffraction efficiency ηd and in the bandwidth given that the recording beam intensities remain unchanged (m0 = 1), whereas the maximum diffraction efficiency and the level of the first side lobe decrease. For Тр/Тт > 1 and m0 = 1, an increase in the absorption causes the diffraction efficiency and the side lobe level to decrease and the bandwidth to increase. A decrease in the ratio of the recording beam intensities causes the diffraction efficiency of the gratings being recorded to decrease. For gratings formed by beams whose intensity ratio exceeds five (m0 < 0.2), the absorption causes the diffraction efficiency and the bandwidth to increase and the first side lobe level to decrease. To refine further the model suggested in this work, light-induced changes in the absorption caused by the dye discoloration during grating record must be taken into account. This work was supported in part by the Federal Education Agency under Project No. 711 “Investigation of Nanodimensional Dynamically Controlled Diffraction Periodical Structures in Liquid-Crystal Composite Materials” (grant of the Program “Development of Scientific Potential of Higher School” for 2005).
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