c Allerton Press, Inc., 2010. ISSN 1541-308X, Physics of Wave Phenomena, 2010, Vol. 18, No. 4, pp. 289–293.
WAVES IN SUPERSTRUCTURES
Transfer Functions of Nonuniform Transmission Photonic Structures in Polymer-Dispersed Liquid-Crystal Materials S. V. Ustyuzhanin, B. F. Nozdrevatykh, and S. N. Sharangovich* Tomsk State University of Control Systems and Radioelectronics, pr. Lenina 40, Tomsk, 634050 Russia Received September 2, 2010
Abstract—The transfer functions of nonuniform transmission photonic structures holographically formed in polymer-dispersed liquid-crystal materials are calculated. The diffraction properties of these structures are numerically calculated for different types of inhomogeneity, coupling parameters, and light beam divergences. DOI: 10.3103/S1541308X10040102
Since the diffraction properties of photonic structures (PSs) in polymer-dispersed liquid-crystal (PDLC) materials can be electrically controlled, these structures are considered as basic elements for dynamic spectral-selective and commutation elements for photonic devices [1]. The holographic technique for forming PSs in absorbing PDLC materials is most widespread. Illumination of PDLC materials leads to the formation of LC domains [2] with characteristic sizes less than 100 nm [1] and one-dimensional periodic structures, where dynamic apodization of the amplitude profiles of refractive-index (RI) spatial harmonics [3] is observed.
To find the angular spectra and spatial distributions of the diffraction field in the far-field zone we used the analytical model [4], which determines the spatial distribution of light beam profiles E0m (ξ0 ) and E1m (ξ1 ) (see Fig. 1) in the aperture coordinates ξ0 and ξ1 . Model [4] allows one to investigate numerically the diffraction properties of PSs in PDLC materials with allowance for the inhomogeneous character of RI profile, divergence of the reading light beam, statistics of distribution of LC molecules in capsules, and the orientational effect of electric field.
A mathematical model of diffraction of divergent light beams from inhomogeneous PSs in PDLC materials (Fig. 1) in the near-field zone, with allowance for the inhomogeneity of the amplitude profile of the RI first harmonic n1 (y) was developed in [4]. This model makes it possible to study such structures as active elements for optical communication and data processing systems. Our purpose was to develop an analytical model of interaction of light beams with an inhomogeneous, electrically controlled one-dimensional PDLC PS. This model allows one to describe the angular spectra and spatial distributions of the diffraction field in the far-field zone using the transfer-function formalism, and numerical simulation of diffraction properties of such structures at different RI profiles and light beam divergences. *
Fig. 1. Schematic of light diffraction from a PS in a PDLC material.
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Using the relationship between the spatial distributions Ejm (ξj ) at the PS output, which were obtained in [4], and the angular spectra Ejm (θ, E), ∞ m θm 2πn j j Ejm (θ) = Ejm (ξj ) exp i ξj dξj , (1) λ cos βjm −∞
one can find that Ejm (θ, E) = Hjm (θ, E) E m (θ).
(2)
θ = θjm
Here, the angle characterizes the direction of the plane-wave components of the angular spectra
2
H0m (θ, E)
bm (E) A =1− 0 2
Ejm (θ) with respect to the normals Nm j (see Fig. 1); m m nj are the refractive indices; βj are the walkoff angles; j = 0, 1 is the diffraction order; m = o, e is the index denoting ordinary and extraordinary waves, respectively; λ is the light wavelength in vacuum; E m (θ) are the angular spectra of the incident beam E m (ξ0 ); and E is the applied field strength. The PDLC PS transfer functions Hjm (θ, E) in formula (2) characterize transformation of the angular spectra Ejm (θ):
+1 cs(1 + q) ΔK m L (1 − q) sinh exp i 2 F1 (1−α, 1+α; 2; w) dq, 2 2
(3)
−1
H1m (θ, E)
bm (E) = −i 1 2
+1 ΔK m L s(1 − q) −1 (1 − q) cosh − t 2 F1 (−α, α; 1; w) dq. exp −i c 2 2
−1
Here, 2 F1 (a, b; c, z) is the Gauss hypergeometric m m m m function; α = bm j (E), bj (E) = LCj (E)/ ν1 ν0 are m the coupling parameters; νjm = cos ϕm j , ϕj are the m angles between the group normals Nrj and the axis y (see Fig. 1); m m m m −1 (E) = 0.25ω(em C0,1 1 Δε(E)e0 )(cc n1,0 cos β1,0 ) ;
cc is the speed of light; em j are polarizations; Δε(E) is the statistically averaged perturbation of the dielectric tensor [4], which characterize the orientational effect of the control electric field E [2]; sinh[cs(1 − q)/2] sinh[cs(1 + q)/2] , cosh(ct) cosh[c(s − t)] 1 , A = cs cosh(ct) cosh[c(s − t)] w =
(4)
(5)
frequency ω, and the angle θ (see Fig. 1) and can be approximated by the linear relation ΔK m = ΔK m (E) + ΔK m (θ) + ΔK m (ω).
The dependence ΔK m (E) in (7) is due to the change in the diffraction geometry at variation in the control field E; ΔK m (ω) = (C−AD/B)ω; ΔK m (θ) = (D/B)θ. The coefficients A, B, C, and D were determined in [5]. Based on formula (4), we numerically investigated the dependences of the squared modulus of the transfer function H = |H1e (ΔK ∗, be )|2 and the phase distribution arg(H1e (ΔK ∗, be )) on the effective coupling parameter be and generalized phase mismatch
(6)
and L is the PDLC PS sample thickness. The parameters c, s, and t in (4) determine the shape of the inhomogeneity profile n1 (y) of the PS RI, which is set by the model function n1 (y, c, s, t) = cosh−1 [c(sy−t)]. We analyzed the following characteristic versions of inhomogeneity profiles n1 (y) of PS RI, which are formed during dynamic apodization (Fig. 2): uniform (curve 1), descending (curve 2), dome-shaped (curve 3), and ascending (curve 4) profiles [4]. The phase mismatch parameter ΔK m generally depends on the control electric field E, incident light
(7)
Fig. 2. Dependence n1 (y), see text for explanation. PHYSICS OF WAVE PHENOMENA
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Fig. 3. Dependences of H and arg(H1e ) on be and ΔK ∗ .
ΔK ∗ = ΔK e L. The effective coupling parameter characterizes the interaction region of light beam and PS and is determined by the expression [5] 1 . be (E) = be1 (E) c(s−t) (2/cs)[arctan(e )− arctan(e−ct )] (8) The dependences H = |H1e (ΔK ∗ , be )|2 are shown in Figs. 3(a)−3(c) for (a) uniform, (b) descending, and (c) dome-shaped PS RI profiles. A comparison of the results obtained suggests that the PS with a uniform RI profile exhibits the best selectivity and a strong dependence of the lateral-lobe levels on be . The PS with a descending RI profile has a wide bandwidth. The PS with a dome-shaped RI profile is characterized by a narrow bandwidth, a minimum level of lateral lobes, and their weak dependence on be . PHYSICS OF WAVE PHENOMENA
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The dependences arg(H1e (ΔK ∗ )) are shown in Fig. 3(d) at be = 1.57 for the (1) uniform, (2) descending, and (3) dome-shaped profiles. It can be seen that the phase characteristic of the PS transfer function for uniform and dome-shaped RI profiles depends linearly on ΔK ∗ , whereas for the descending profile this dependence is nonlinear. Hence, asymmetric RI profiles exhibit a dependence of the spatial distributions of diffracted beams at the output of PDLC PS on the coupling parameter be , and, specifically, asymmetry of their profiles and different spatial displacement. The corresponding changes in the distribution of light beam intensities Ij = Ije (ξj , be ) = |Eje (ξj , be )|2 of zero (j = 0) and first (j = 1) diffraction orders in the near-field zone for (a, d) uniform, (b, e) descending, and (c, f) dome-shaped RI profiles and strongly divergent incident Gaussian light beam are shown in
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Fig. 4. Dependences of the intensity profiles on the coupling parameter be at g = 1.68 for (a, b, c) diffracted and (d, e, f) transmitted beams.
Fig. 5. Dependences ηd (be ) and ηd (E).
Fig. 4. Hereinafter, the divergence of the incident light beam is characterized by the geometric parameter g = L sin(ϕ0 + ϕ1 )/2W cos ϕ1 , where 2W is the width of incident Gaussian beam at the PS input (at y = 0). The plots in Fig. 4 suggest that the amplitude apodization for the PS characterized by a symmetric dome-shaped profile allows one to reduce the distortions of the spatial structure of diffraction field at almost any attainable values of the coupling parameter be . To determine the external electric field, controlling the diffraction properties of PDLC PS, it is necessary
to estimate the dependence of the integral diffraction efficiency ηd on the effective coupling parameter be and applied field E. These dependences at ΔK e = 0 and different RI profiles are shown in Fig. 5. Curves 1 and 2 describe the efficiency of interaction between light with a geometric parameter g = 0.34 and a PS with a uniform RI profile; curve 3 demonstrates the interaction between light with g = 1.68 and a PS with a uniform RI profile; and curves 4 and 5 show the interaction between light with g = 1.68 and PSs with descending and bell-shaped profiles, respectively (Ec is the critical voltage). PHYSICS OF WAVE PHENOMENA
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The plots in Fig. 5 indicate that the energy characteristics of diffracted field depend strongly on the degree of inhomogeneity of amplitude RI profile. The results obtained are explained within the transfer-function formalism due to the dependence of arg(H) on be and the parameters c, s, and t, which determine the RI profile. These effects have a geometric interpretation: the spatial arrangement of the diffraction efficiency maximum for the diffracted beam is determined by the region of intersection of the incident light beam with the maximum of amplitude RI profile. Thus, we can conclude that the transfer functions (3) and (4) of inhomogeneous PSs formed in a PDLC material make it possible to determine the diffraction and selective properties as well as the spatial distributions and energy characteristic of the diffracted field. The results of numerical simulation show that the effect of dynamic apodization of amplitude PS profiles, which arises during holographic recording, allows one to obtain better energy characteristics of PSs used as electrically controlled spectral-selective commutation elements of light fields with wide frequencyangular spectra. ACKNOWLEDGMENTS This study was supported by the Program “Development of the Higher School Scientific Potential” (Project No. RNP.2.1.1.429) and the Federal Target Program “Scientific and Scientific-Pedagogical
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Personnel of Innovative Russia” (State Contract No. 02.740.11.0553). REFERENCES 1. N. Paras, Prasad Nanophotonics (Wiley, Hoboken, New Jersey, 2004). 2. L. Richard, “Polarization and Switching Properties of Holographic Polymer-Dispersed Liquid-Crystal Gratings. I. Theoretical Model,” J. Opt. Soc. Am. B. 19(12), 2995 (2002). 3. E. A. Dovolnov, S. V. Ustyuzhanin, and S. N. Sharangovich, “Formation of Holographic Transmission and Reflecting Gratings in Photopolymers Under LightInduced Absorption, ” Russ. Phys. J. 49(10), 1129 (2006). 4. S. V. Ustyuzhanin and S. N. Sharangovich, “Diffraction of Light Beam by Not Uniform Electro Control One-Dimensional Photon PDLC Structure,” in Proceedings 6th International Conference of Young Scientists and Specialists “Optics-2009” (St. Petersburg, 19−23 October, 2009); B. F. Nozdrevatykh, S. V. Ustyuzhanin, and S. N. Sharangovich, “Diffraction Properties of Non-Uniform Photonic Structures in Transmittance Polymer Dispersed Liquid Crystals,” in Proceedings of Tomsk State University of Control Systems and Radioelectronics (Izd. TUSUR, Tomsk, 2010), Iss. 1(21), Pt. 2, p. 109. 5. S. N. Sharangovich, “Transfer Function of Strong Acoustooptic Interaction in Amplitude and Phase Nonuniform Acoustic Fields,” Sov.-Phys. Tech. Phys. 65(1), 107 (1995).