Vol. 25, No. 20 | 2 Oct 2017 | OPTICS EXPRESS 24780
Efficient tensor approach for simulating paraxial propagation of arbitrary partially coherent beams JUN CHEN,1,3 ENXIN ZHANG,1 XIAOFENG PENG,2 AND YANGJIAN CAI2,4 1College
of Optical and electronic technology, China Jiliang University, Hangzhou 310018, China of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
[email protected] [email protected] 2College
Abstract: Complicated partially coherent beams (PCBs) are useful in many applications, such as free-space optical communications, particle trapping and optical imaging, while usually it is hard to derive analytical propagation formulae for such beams, and one has to fall back on numerical methods. The conventional numerical methods have some intrinsic drawbacks. In this paper, we introduce an efficient tensor approach (ETA) for simulating paraxial propagation of arbitrary PCBs. The ETA is a direct reconstruction of the propagated PCB without aliasing and rippling problems, and the algorithm is simple and robust with a tensor/matrix multiplication as the main calculation. The validity of ETA is verified through comparing simulation results with analytical results, numerical integration results and experimental results, respectively. The ETA provides a fast and reliable way for simulating paraxial propagation of arbitrary PCBs. © 2017 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (350.5500) Propagation.
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#305896 Journal © 2017
https://doi.org/10.1364/OE.25.024780 Received 29 Aug 2017; accepted 24 Sep 2017; published 28 Sep 2017
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1. Introduction Partially coherent beams (PCBs) attract intensive attentions due to their wide applications in free-space optical communications [1], holography [2], lithography [3], optical imaging [4,5], second-harmonic generation [6], particle scattering [7], optical trapping [8], atom cooling [9], remote sensing [10, 11]. Gaussian Schell-model (GSM) beam is a conventional partially
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coherent beam whose intensity and degree of coherence are of Gaussian distributions [12, 13]. Complicated partially coherent beams, such as partially coherent beam with nonconventional correlation function (i.e., degree of coherence) [14], partially coherent beam with twist phase or vortex phase [15, 16] and vector partially coherent beam [17], display many unique properties and have advantages over conventional GSM beams in many applications [18–20]. In practical applications, one has to treat the propagation of PCBs. Up to now, numerous analytical and numerical methods/approaches have been proposed to treat the propagation of PCBs. Analytic methods, such as coherent-mode decomposition [21–23], Wigner distribution function [24], tensor ABCD law [25], angular spectrum representation [26] and complex Gaussian functions [27–29], give exact solutions for the propagation, but not applicable for every PCB. For some complicated partially coherent beams, such as scalar and electromagnetic nonuniformly correlated beams [30, 31], scalar and electromagnetic Gaussian Schell-model vortex beam [32, 33], it is hard to derive analytical propagation formulae, and one has to fall back on time-consuming numerical methods. Some numerical methods, such as numerical integration and Discrete Fourier transform (DFT), have been developed to simulate the propagation of PCB. Numerical integration method using adaptive Simpson algorithm is time-consuming [34]. DFT and its related algorithms such as the fast Fourier transform (FFT) have advantages in computational efficiency. However, the DFT/FFT-based algorithms have some intrinsic drawbacks [35], which result in distortions in the reconstructed optical filed. These drawbacks can be summarized as follows: the finite length of the sampling window which results in a loss of the high frequency information, an improper sampling interval which would cause the problem of aliasing, and the sameness of the number of sampling points in the input and output planes which leads to an inflexibility of the sampling and the observation One may ask whether there is a method which can simulate the propagation of arbitrary PCBs efficiently in a fast way without the drawbacks of DFT/FFT. In a recent paper [36], Shakir et al. presented a matrix approach for numerically reconstructing a propagated laser beam in a direct way. However, this work is limited to the case in which the light beams are assumed to be completely coherent, and the spatial coherence of the light field isn’t taken into consideration. Regarding the importance of the spatial coherence and the wide applications of PCBs, in this paper, we introduce an efficient tensor approach (ETA) for simulating paraxial propagation of arbitrary PCBs. Our method can avoid drawbacks of DFT/FFT, and its validity is verified through comparing simulation results with analytical results, numerical integration results and experimental results 2. Tensor approach for simulating paraxial propagation of arbitrary partially coherent beams In this section, we will introduce an efficient tensor approach to simulate paraxial propagation of arbitrary PCBs. Paraxial propagation of the cross-spectral density (CSD) function of a PCB through a stigmatic ABCD optical system in free space can be treated by the following generalized Collins formula [25, 37] W (u1 , u2 , v1 , v2 ) =
1 | λ B |2
∞ ∞ ∞ ∞
∫ ∫ ∫ ∫ W (x , x , y , y ) 0
1
2
1
2
−∞ −∞ −∞ −∞
iπ iπ × exp − * ( A* x12 − 2 x1u1 + D*u12 ) + ( Ax22 − 2 x2 u2 + Du22 ) , (1) λB λB iπ iπ × exp − * ( A* y12 − 2 y1v1 + D*v12 ) + ( Ay22 − 2 y2 v2 + Dv22 ) dx1dx2 dy1dy2 λB λB
where xi, yi and ui, vi (i = 1,2) are the position coordinates in the input and output planes, A, B, C, D are transfer matrix elements of the optical system, the superscript “*” means complex
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conjugate, λ is the wavelength. W0 ( x1 , x2 , y1 , y2 ) and W (u1 , u2 , v1 , v2 ) are the CSD functions of an arbitrary PCB in the input plane and output planes, respectively. For some complicated PCBs, it is hard to derive analytical propagation formulae for their CSD functions. Here we introduce a tensor approach to treat the propagation of such beams. We reconstruct Eq. (1) with a tensor (multidimensional array) representation W = H y T H x T W0 H x H y ,
(2)
where W0 ≡ {W0 ( x j1 , x j2 , yk1 , yk2 )} with j1,2 , k1,2 = 1 N1 is the discrete form of the input CSD function
which
is
a
tensor
of
N1 × N1 × N1 × N1 . W ≡ {W (um1 , um2 , vn1 , vn2 )} with
m1,2 , n1,2 = 1 N 2 is the discrete form of the output CSD function which is a tensor of N 2 × N 2 × N 2 × N 2 . Here N1 and N2 are integers, representing the numbers of sampling points in the source plane and the observation plane respectively. Without loss of generality, we have assumed that the x and y transverse grid points of the input/output plane are similar. The grid (sampling) separations in the input and output planes are ∆1 and ∆ 2 respectively. As a
result, to denote the spatial points of a given plane, we have used x j1= j1∆1 , yk1= k1∆1 , um= m1∆ 2 , vn1= n1∆ 2 , while x j2 , yk2 , um2 and vn2 can be defined similarly. H x and H y in 1
with j 1= N1 , m 1 N 2 and Eq. (2) are N1 × N 2 matrices, i.e., H x ≡ {H x ( x j , um )} = with k 1= H y ≡ {H y ( yk , vn )} = N1 , n 1 N 2 . The superscript “T” represents the matrix complex conjugate transpose operation. H x and H y are the discrete form of the impulse response functions of the stigmatic ABCD system. [ H x ] jm= ≡ H x ( x j , um )
wj iλ B
exp[i
π ( Ax 2j − 2 x j um + Dum2 )], λB
(3)
where w j is a weight coefficient and would be given latter. [Hy]kn ≡ Hy(yk,vn) can be obtained in a similar form of Eq. (3) just by replacing x j and um with yk and vn , respectively. To calculate a single element of the tensor W , one can use
[W ]m m n n 1
2 1 2
N1
N1
N1
N1
j1
j2
k1
k2
= ∑∑∑∑ [ H yT ]n1k1 [H xT ]m1 j1 [W0 ] j1 j2 k1k2 [ H x ] j2 m2 [H y ]k2 n2 .
(4)
It is evident that Eq. (4) is the discrete formulae of Eq. (1). Recalling the paraxial approximation which implies the grid (sampling) separation in the source plane should satisfy ∆1
