Algorithm of Shaping Multiple-beam Braggs Acousto-optic Diffraction Laser Field Into 1D and 2D Patterns

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Algorithm of Shaping Multiple-beam Braggs Acousto-optic Diffraction Laser Field Into 1D and 2D Patterns To cite this article: S Zakharchenko and A Baturin 2015 J. Phys.: Conf. Ser Related content - Why Cambridge boasts the best minds George Ellis - X-ray diffraction and extended X-ray absorption fine structure study using synchrotron radiation of cobalt (II) complexes Ashotosh Mishra, Kritika Shukla, Jagrati Dwivedi et al. - On the Reflection of the X-Ray Spectrum of Palladium from Fluorspar H Pealing View the article online for updates and enhancements. This content was downloaded from IP address on 29/09/2018 at 19:10

2 Algorithm of Shaping Multiple-beam Braggs Acousto-optic Diffraction Laser Field Into 1D and 2D Patterns S. Zakharchenko, A. Baturin Moscow Institute of Physics and Technology; Russia, Moscow Region, Dolgoprudniy, Institutskiy pereulok, 9 Abstract. Algorithm of solving a direct problem of acousto-optic interaction between laser emission and acoustic signal consisting of a set of equidistant frequency components is proposed. An infinite system of coupled wave differential equations is reduced to eigenvalue problem. The contribution of the higher rediffraction orders is analyzed separately. Inverse problem of finding an optimal set of equidistant frequency components of a driving acoustic signal to form the objective diffraction pattern is also considered and a few optimization approaches are analyzed. A naïve heuristic method of splitting 2D pattern into subframes, each suitable for simultaneous projection by two acousto-optical deflectors driven by multifrequency composite signal, is developed. 1. Introduction Precise shaping of a laser beam has become of a great technological importance due to its possible applications in various fields of science and technology. For instance, the shaping a laser beam into confining potentials found its application for creation and subsequent manipulation of Bose-Einstein condensate [1]. Fiber-optic switch-multiplexer, based on high-effective multibeam Bragg acousto-optic diffraction, was shown [2] to have low switching time (less than 10µs), enhanced number of channels and the unique ability to transmit input signal to any group of channels simultaneously. All these applications require new rigorous algorithms for near realtime data processing with the delay between incoming pattern and calculated acoustic drive signal of up to a few seconds. A traveling acoustic wave creates a perturbation of refractive index in a form of periodic structure similar to phase diffraction grating. Propagating light scatters on this structure and forms a diffraction pattern in the far field [3]. Since a period of the grating and a modulation depth of the refractive index depends on a frequency and an amplitude of the acoustic wave, the propagating light could be shaped into a required pattern by means of applying the proper set of acoustic wave modes. The Bragg diffraction regime occurs when only one (principal) diffraction order is being produced, while others destructively interfere since they are not phase matched. If an acoustic field consists of N acoustic modes with a set of frequencies f 1 f N, the diffraction pattern becomes more complex: the output will have N principal diffraction orders corresponding to each acoustic mode. However, light from each principal beam could be rediffracted further by Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 another acoustic mode. This results in cross modulation and generates parasite intermodulation modes corresponding to various sum and difference frequencies [4]. An acousto-optic interaction is described by the following infinite system of differential equations [5]: dg n1 n 2 dg n1 n 2 n 3...n µ dg n1 dg 0 = j n 1 = j[ c n1 G n1 exp( jη 0 n1 x), c n 2 G n1 n 2 exp(jη n1 n 2 n 1 x) + c n 1 G 0 exp(jη 0 n1 x)], n 2 n 1 = j[ c n3 G n1 n 2 n 3 exp( jη n1 n 2 n 1 n 2 n 3 x) + c n2 G n1 exp( jη n1 n 2 n 1 x)], n 3 n 2 = j[ n µ+1 n µ c nµ+1 G n1 n 2 n 3...n µn µ+1 exp( jη n1 n 2 n 3...n µ n 1 n 2 n 3...n µn µ+1 x)+ + c nµ G n1 n 2 n 3...n µ 1 exp( jη n1 n 2 n 3...n µ n 1 n 2 n 3...n µ 1 x)], µ = 2L, L Z dg n1 n 2 n 3...n µn µ+1 = j[ c n µ+2 G n1 n 2 n 3...n µ+2 exp(jη n1 n 2 n 3...n µ+2 n 1 n 2 n 3...n µ+1 x)+ n µ+2 n µ+1 c n µ+1 G n1 n 2 n 3...n µ exp(jη n1 n 2 n 3...n µ n 1 n 2 n 3...n µ+1 x)], µ = 2L + 1, L Z, (1) where G n1 n 2 n 3...n µ (x) is the complex amplitude of the laser spatial mode n 1 n 2 n 3 n µ, which corresponds to all photons that have successively interacted with phonons from acoustic waves with the following frequencies f 1, f 2, f 3, f µ ; c n the complex amplitude of the acoustic wave with the frequency f n ; η n1 n 2 n 3...n µ n 1 n 2 n 3...n µn µ+1 is the phase mismatch between the corresponding modes (it depends on the acousto-optic interaction geometry, thus considered to be constant and known); j imaginary unit; c n complex conjugate of the amplitude. Without loss of generality, we assume the amplitude of input light is equal to 1, then the boundary conditions are { G 0 x=0 = 1; G n1...n µ x=0 = 0; µ Z, µ > 0; n 1,..., n µ [0, N]. Suppose the acoustic modes have equal frequency spacing between one another and f 0 f max is the frequency range, where diffraction losses are relatively small. Then, the frequencies of N acoustic modes are f n = f 0 + n f = f 0 + n f 0 f max. (3) N The angle through which the light beam corresponding to the spatial mode n 1 n 2 n 3 n µ is deviated depends on frequencies of acoustic waves it has interacted with. Suppose a linear approximation is valid, then (2) 2

4 θ n1 n 2 n 3...n µ = θ Br + κ θ 0 + κ L n 2i κ L L n 2i+1 = θbr; k k = L 1 n 2i κ n 2i+1 = θ0; l l = L n 2i L n 2i+1, µ = 2L + 1, L Z; L L n 2i n 2i+1, µ = 2L, L Z; where κ is the constant defined by the interaction geometry, θ Br = θbr 0 diffraction angle of the principal beam produced by the acoustic wave f = f 0, θ 0 = θ0 0 incidence angle of the undiffracted light beam and upper indices l and k are normalized angle. Henceforth, upper indices l and k are to be referred as normalized angles in zeroth and Bragg s field respectively. Different spatial modes traveling at the same angle are inseparable; thus, there is no possibility to measure their individual amplitudes experimentally. Instead, we focus on overall beam intensities from Bragg s and zeroth-order fields I(θBr k ) and I(θl 0 ). They have more practical value since they define the diffraction pattern intensity distribution, which could be captured by an array of photo detectors. Summing amplitudes of all spectral modes from single beam at the specific angle θbr k or θl 0, we get µ [µ max/2] G k Br =, (5) G l 0 = µ µ [µ max/2] µ {n 1,n 2...n 2µ+1 }: µ n 2i µ n 2i+1=k {n 1,n 2...n 2µ }: µ n 2i µ 1 n 2i+1=k (4) G n1 n 2 n 3...n 2µ+1 G n1 n 2 n 3...n 2µ, (6) where µ max is the maximum number of successive photon-phonon interaction that will be taken into the model and 0 n i N, i. The phase mismatch η n1 n 2 n 3...n µ n 1 n 2 n 3...n µn µ+1 is defined by the interaction geometry, so if the spatial mode n 1 n 2 n 3 n µ has the same angle as n 1 n 2 n 3 n µ and the spatial mode n 1 n 2 n 3 n µ, n µ+1 has the same angle as n 1 n 2 n 3 n µ, n µ+1, then η n1 n 2 n 3...n µ n 1 n 2 n 3...n µn µ+1 = η n 1 n 2 n 3...n µ n 1 n 2 n 3...n µn = µ+1 = η θl, µ = 2 L, L Z; (7) 0 θk Br Substituting (7), (5), (6) in (1), we obtain: dg l 0 = j[ N dg k Br and the boundary conditions are n=0 = j[ N n=0 c n3 G l+n Br c n3 G k n Br G 0 0 x=0 = 1; G l 0 x=0 = 0, l Z; G k Br x=0 = 0, k Z. exp( jη θ x)], (8) 0 l θl+n Br exp(jη θ k Br θ k n x)], (9) 0 (10) 3

5 Normalized angle θ min 0.1% zone θ max 0.1% zone θ min 1% zone θ max 1% zone θ min 5% zone θ max 5% zone Number or acoustic modes Figure 1. Diffraction region, where all intensities are greater then the specific threshold. 2. Spectral algorithm Let us truncate the system (8) by limiting the normalized angle coefficients L l L and L k L. The stiffness ratio of the truncated system is relatively high ( 1000). Because of that, let us build a spectral method by writing the solution as the following linear combination of base functions: G k Br (x) = s V s g s Br,k exp(jxξs Br,k ) (11) G l 0 (x) = s V s g0,l s exp(jxξs 0,l ), (12) where ξbr,k s is a spatial frequency; s, V s, g0,l s and gs Br,k are unknown coefficients; L l L and L k L; k, l, s Z. Substituting (11), (12) in (8), we obtain ξ0,l s = ξs 0,l+n η θ0 l θl+n Br (13) ξbr,k s = ξs Br,k n + η θ. Br k θk n 0 (14) Substituting (13), (14), (11), (12) in (8), we get eigenvalue equation g s 0, L [ ( M ξ s 0,0 E )] g s 0,L gbr, L s = 0, (15) gbr,l s 4

6 Figure 2. The number of spectral algorithm executions done by various optimization methods during minimization of the residual (18), averaged from 50 trials with random objective pattern I. where M is some hermitian matrix 1 and E is identity matrix. Solving (15), we obtain spatial frequencies ξ0,0 s from eigenvalues and {gs 0,l }, {gs Br,k } from eigenvalues. Substituting (11), (12) in 10, we get the system of equations for calculating V s : s V s g0 0 x=0 = 1; s V s g0,l s x=0 = 0, l Z; (16) s V s gbr,l s x=0 = 0, k Z. This completes solving direct problem by means of spectral algorithm. However, by truncating the original system (8) we have artificially constrained which optical modes could exist and which could not. To make this assumption reasonable an amount of energy that could be transfered into the forbidden modes must be estimated. Let us select a neglectable intensity level I n and start increasing the normalized angle cut-off coefficient L until θ min = min ( {k G k Br (Gk Br ) < I n } ) and θ max = max ( {k G k Br (Gk Br ) < I n } ) stop changing. According to fig. 1, the diffraction region [θ min θ max within which all the intensities are greater then the specific threshold, depends linearly on the number of acoustic modes. Thus, the following rough estimation is valid: L 2.5N. (17) 3. Reverse problem A reverse problem is the optimization problem of finding complex amplitudes c = {c i } which produce the diffraction pattern I = {I(θBr k )} with the least residual f({c i}) = I I to the objective pattern I = {I (θbr k )}. We have compared a performance of different optimization algorithms applied to this problem (Fig. 2). Benchmarking was conducted with the following parameters of the model: the number of acoustic modes N = 5, the normalized angle limitation L = 50, the objective function is f({c i }) = L ( I(θ k Br ) I (θbr k )) 2. (18) k= L 1 For brevity sake, we omit exact form of this matrix since it is trivial to calculate it 5

7 4. 2D patterns All the discussion above was related to the shaping of a light into 1D pattern. Let us consider attaching second successive acousto-optical deflector to the output of the first one. Suppose it is splitting each beam from the incoming 1D pattern in the orthogonal direction, thus forming a 2D pattern. It is easily shown that such setup is capable of projecting only specific class of 2D patterns. We say that 2d pattern defined by N M binary matrix G is projectable iif it has rank-one factorization into outer product of some binary vectors C and R: S = CR = c 1 c 2 c N (r 1, r 2,, r M ). (19) Therefore, we have a problem of decomposing an arbitrary 2d pattern defined by N M binary matrix G into the sum of the least number of projectable 2d patterns S i. Detailed discussion of this problem is the object of another paper, so we shall limit ourselves to the consideration of iterative naïve greedy heuristics. Let us consider k-th step of the algorithm; let G k be the input for this step and G 1 for the first step be equal to G. Suppose r i, 1 i N is a row vector of the matrix G k. Let us introduce matrix H k, whose elements are the normalized inner product of the corresponding rows: h ij = ri rj r i r, if r i r i 0; 0 otherwise. Consider the row r i w that has the biggest value of r w r w 0. Then, find a set J k, such as J k = {j h wj η}, where 0 < η < 1 is a parameter of heuristics. Finally, the elements of R and C from (19) are r i = r j i ; c i = 1 Jk (i), (20) j J k where 1 Jk (x) is the indicator function. Substituting (20) in (19), we get projectable 2D pattern S k. If the residual G k+1 = G k S k is a zero matrix, the algorithm finished the decomposition; otherwise repeat the procedure above for G k Conclusion The spectral algorithm for solving direct problem of acousto-optical interaction is proposed. This approach is more favorable than explicit and implicit methods because it doesn t suffer from high stiffness of the governing equations, doesn t accumulate errors and provides a convenient way to calculate partial derivatives of the intensity distribution via matrix perturbation. The reverse problem of finding the acoustic field that produce diffraction pattern close to the objective pattern is discussed. A few optimization approaches was compared. Acknowledgments This work was performed using the equipment provided by MIPT Center of Collective Usage (CCU MIPT) with the financial support from the Ministry of Education and Science of Russian Federation. (Grant ID RFMEFI59414X0009) 6. References [1] Trypogeorgos D, Harte T, Bonnin A and Foot C 2013 Opt Express. 21(21): [2] Vainer A, Antonov S, Proklov V and Rezvov Y 2009 Applied Optics 48(7) [3] Korpel A 1988 Acousto-optics / Adrian Korpel (M. Dekker New York) ISBN [4] Hecht D, Petrie G and Wofford S 1979 Multifrequency acousto-optic diffraction in optically birefringent media 1979 Ultrasonics Symposium pp [5] Gazalet M, Kastelik J, Bruneel C, Bazzi O and Bridoux E 1993 Appl. Opt