Discrete Talbot effect in two-dimensional waveguide arrays

We theoretically study discrete Talbot self-imaging in hexagonal, square, and irregular two-dimensional waveguide arrays. Different from its counterpart in a continuous system, the periods of the input fields must belong to {1, 2, 3, 4, 6} for Talbot self-imaging. Also, the combinations of the input periods cannot be 3 & 4, or 4 & 6 along two different directions, which distinguishes itself from the one-dimensional discrete Talbot effect. ©2015 Optical Society of America OCIS codes: (070.6760) Talbot and self-imaging effects; (230.7370) Waveguides. References and links 1. H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836). 2. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989). 3. J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013). 4. L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881). 5. V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, and V. D. Pismennyi, “Effective phase locking of an array of lasers,” JETP Lett. 44, 78 (1986). 6. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttg.) 79, 41–45 (1988). 7. J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55(4), 334 (1989). 8. D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. 16(11), 823–825 (1991). 9. T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373– 1376 (1981). 10. J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. 30(3), 227–229 (2005). 11. P. Peterson, A. Gavrielides, and M. Sharma, “Extraction characteristics of a one dimensional Talbot cavity with stochastic propagation phase,” Opt. Express 8(12), 670–681 (2001). 12. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot Effect in Waveguide Arrays,” Phys. Rev. Lett. 95(5), 053902 (2005). 13. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012). 14. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51(1), R14– R17 (1995). 15. J. F. Clauser and S. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994). 16. Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010). 17. J. Wen, Y. Zhang, S. N. Zhu, and M. Xiao, “Theory of nonlinear Talbot effect,” J. Opt. Soc. Am. B 28(2), 275– 280 (2011). 18. D. Liu, Y. Zhang, J. Wen, Z. Chen, D. Wei, X. Hu, G. Zhao, S. N. Zhu, and M. Xiao, “Diffraction interference induced superfocusing in nonlinear Talbot effect,” Sci. Rep. 4, 6134 (2014). 19. L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, Matter-Wave-Dispersion Talbot Effect,” Phys. Rev. Lett. 83(26), 5407–5411 (1999). 20. J. M. Cowley, Diffraction Physics (Elsevier, 1995). 21. L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Talbot effect in metallic gratings under Gaussian illumination,” Opt. Commun. 278(1), 23–27 (2007). #238432 $15.00 USD Received 20 Apr 2015; revised 19 May 2015; accepted 20 May 2015; published 27 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.014724 | OPTICS EXPRESS 14724 22. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). 23. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976). 24. Z. Chen, D. Liu, Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Fractional second-harmonic Talbot effect,” Opt. Lett. 37(4), 689–691 (2012). 25. L. W. Zhu, X. Yin, Z. Hong, and C. S. Guo, “Reciprocal vector theory for diffractive self-imaging,” J. Opt. Soc. Am. A 25(1), 203–210 (2008).


Introduction
The Talbot effect was first observed by H. F. Talbot in 1836 [1][2][3].It is a near-field diffraction phenomenon in which the structure of a periodic grating illuminated with a coherent light can periodically replicate itself at certain propagation distances.A few decades later, Rayleigh theoretically proved that any periodic one-dimensional object can be used to achieve such remarkable effect along the propagation direction of the incident light at even integer multiples of the Talbot distance defined by 2 / Here, d stands for the spatial period of the object and λ the light wavelength [4].When the distances are rational multiples of T z (i.e., / / T z z p q = where p and q are both prime integers), the fractional Talbot effect can be observed.Nowadays, the simplicity and beauty of the Talbot effect still attract many researchers.Its applications have been extended from spatial domain [5][6][7][8] to temporal domains [9,10].Besides, Talbot effects are also realized in many other fields including coupled lasers [11], waveguide arrays [12,13], atom optics [14,15], nonlinear systems [16][17][18], and Bose-Einstein condensates [19].Talbot effect is not only an optical curiosity for physicists, but also leads to a variety of applications, such as imaging processing and synthesis, microcopy [20], optical testing [21], optical computing [2] and photolithography [14], and so on.
Recently, discrete structures attract growing interests because wave propagation in such system presents unique characteristics [22].One typical discrete system is the evanescentlycoupled waveguide array.In contrast to the case in a homogeneous medium, the light behavior in a waveguide array is quite different because the transverse coordinate is discrete and the waves propagate through evanescent coupling between the waveguides.In a weakly coupled system, it is usually assumed that only adjacent waveguide elements interact with each other.Such discrete structures have a lot in common with crystal lattice.For example, the optical discrete systems have forbidden Floquet-Bloch gaps and allowed bands.Besides, the tight binding approximation is also applicable [23].The field evolution in waveguide array can be described by a set of coupled differential equations with periodic Floquet-Blochlike solutions.
In this letter, we theoretically study the discrete behaviors of the Talbot effect in twodimensional waveguide arrays.Our theoretical analysis shows that in order to realize Talbot self-imaging, the periods of the input fields along different directions can only be special combinations of a few integers, which is much stricter comparing to the case in a onedimensional waveguide array.

Theory
To study such two-dimensional discrete Talbot effect, let's hypothetically consider an infinite waveguide array with identical periodic elements [Fig.1(a)].All the elements are homogeneous and lossless.By setting two non-collinear base vectors ( 1 a  and 2 a  in Fig. 1(a)), arbitrary waveguide element in the array can be easily addressed through a vector defined by ( ) ( ) where κ is the coupling coefficient bwteen the waveguides with a distance of 1 a  ( = 2 a  ).α can be easily written as Equation ( 2) has a Floquet-Bloch-like solution, , exp exp , where z is the propagation length and λ is an eigenvalue.The corresponding wave vector K  in the reciprocal space can be written as  In order to achieve the Talbot effect, the input field distribution on the two-dimensional waveguide array must be periodic.Assuming that the periods along 1 a  and 2 a  are 1 N and 2 N , respectively, the input field satisfies ( ) ( ) , where α and β are arbitrary integers.Taking Eq. ( 6) into Eq.( 4) and performing several simplifications, we can get where m is an integer.Equation ( 7) should be hold for arbitrary integers α and β .
Therefore, k 1 and k 2 can't take continuous values but some discrete points, For example, the red points in Fig. 1(b) stand for the qualified vector K  with the input periods of 1 By substituting Eq. ( 4) into Eq.( 2), we can get the expression of the eigenvalue, ( Equation ( 4) indicates that the Talbot self-image is possible at an interval of z if and only if ( ) , where v is an integer.Hence, the ratio of any two different nonzero eigenvalues must be a rational number, i.e.
( ) ( ) where p and q are two prime integers.In addition, one can easily prove that the ratio , 0 ,0 , 0 ,0 must also be rational to realize self-imaging.Considering the case of cos 2 k π must both be rational for self-imaging.By using the Chebyshev polynomial, ( ) can be expanded as where 1  2 is the integer part of 1 2 d and the coefficients ( ) ) The Chebyshev coefficients ( ) are all integer numbers and the first one is decided by π is also a rational number.For Talbot revivals to occur at certain intervals, it is necessary that ( ) π is a rational number.Now, the problem is to find 1 N that satisfies this condition.By using the Chebyshev polynomial, we can expand the identical equation ( ) as a polynomial in ( ) where 1 an = for an odd 1 N and ( ) From Eq. ( 10), the ratio ( ) ( ) is also a rational number, which can be satisfied only when 1 α and 2 α are rational.It should be noted that this conclusion is obtained only when considering the coupling between the nearest and the next nearest waveguides (i.e. the values of 1 α and 2 α are decided by Eq. ( 3)).This assumption is usually valid because the coupling coefficients between waveguides exponentially decay as the distance between them increases.As a result, ( ) ( ) must also be rational numbers, which can be understood as the requirement of the periodic distribution along different non-base-vector directions.Therefore, not all the combinations of N 1 and N 2 are qualified to realize two- are irrational numbers, and vice versa.These further restrictions on the coupling coefficients and the periods of the input fields distinguish the two-dimensional Talbot revivals from the one-dimensional case [12].Now we obtain the necessary condition to realize strict Talbot self-imaging in a twodimensional waveguide array: (1) both N 1 and N 2 belong to { } , l l is the superposition of a set of Floquet-Bloch-like solutions ( ) The distribution of the light intensity can be written as ( ) where i j ≠ and i j λ λ ≠ .The Talbot distance is decided by 2 , where the function F means to find the least common multiple of all possible 2 i j π λ λ − .

Numerical simulation and analysis
To intuitively and quantitatively understand the two-dimensional discrete Talbot effect, we built a model to simulate the light propagation in different waveguide arrays.In order to reduce the influence of the boundary effect, the model consists of more than 600 waveguide elements and only the central part of the model is used for analysis.We also give the analytical solutions of the output intensity for comparison with the simulated results.In a hexagonal waveguide array, the next nearest neighbor of a waveguide is twice as far away as the nearest one.Hence, we consider only the coupling of the nearest elements (i.e.,  1(d)].Because of the periodicity, the evolution of the light in a unit cell (the marked area in Fig. 1(a)) can be extended to the entire imaging plane.In this hexagonal case, each unit cell includes 8 waveguides.Only the waveguide element (0, 0) is illuminated at the input plane.The analytical solutions of the output intensity in the unit cell can be written as

Conclusions
In this paper, we theoretically and numerically analyzed the discrete Talbot effect in hexagonal, square, and irregular two-dimensional waveguide arrays.The periods of the input fields must belong to { } 1, 2,3, 4, 6 .Because of the requirements of the periodicity along the non-base-vector directions, the period combinations of the input fields along the base vectors cannot be 3 and 4, or 4 and 6.The ratio of the coupling coefficients along different directions must be rational to achieve the Talbot effect.Our theoretical work shows that it is much more difficult to realize discrete two-dimensional Talbot self-imaging comparing to the onedimensional case.

1 α κ and 2 α κ stand for coupling coefficients between the adjacent waveguides along the directions of 1
respectively.If we only consider the coupling between the nearest and the next nearest waveguides, the values of 1 α and 2

Fig. 1 .
Fig. 1.Sketches of a waveguide array structure in real space (a) and in reciprocal space (b).In our model, only the nearest and next nearest neighbors of a waveguide are considered as marked in the red polygon in (a).The red points in (b) stand for all possible ( ) 1 2 , k k to

where 1 b  and 2 b
 are base vectors of the reciprocal lattice, 1 k and 2 k are the coefficients to decide K  [Fig.1(b)].

#Fig. 2 .
Fig. 2. The first row is the simulated intensity patterns with input periods of N 1 = 2 and N 2 = 4. (a) shows the structure of the waveguide array and the input fields.(b)-(e) are the corresponding Talbot images at / 4 T z . (2)).The hexagonal waveguide array is shown in Fig. 2. The coupling coefficient between the nearest neighbors is set to be = / 20 κ π mm −1 .First, we choose the input periods of N 1 = 2 and N 2 = 4 [Fig.1(a)], which can satisfy the requirement to realize discrete Talbot self-imaging.The Talbot images at / 4 in Figs.1(b)-1(f), respectively.Here, 10 T z = mm.The general characteristics of the Talbot effect can be observed [24,25].For example, self-image is realized at T z z = [Fig.1(e)].At 1/2 Talbot plane, the period becomes half of the input one and the intensity of each bright waveguide is 1/4 of the original one [Fig.
are closer than those along the 1 a  direction.Here, we assume 1 =5 α .From Eq. (23), the Talbot distances can be written as results are well in agreement with the simulations in Fig.4.