Abstract

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

1. Introduction

The Talbot effect was first observed by H. F. Talbot in 1836 [1–3]. 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 zT=d2/λ. Here, d stands for the spatial period of the object and λ the light wavelength [4]. When the distances are rational multiples of zT (i.e.,z/zT=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 [5–8] 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 [16–18], 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 evanescently-coupled 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-Bloch–like solutions.

In this letter, we theoretically study the discrete behaviors of the Talbot effect in two-dimensional 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 one-dimensional waveguide array.

2. 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 (a1 and a2 in Fig. 1(a)), arbitrary waveguide element in the array can be easily addressed through a vector defined by

R=l1a1+l2a2=(l1,l2),
where l1 and l2 are integers. It is known that the coupling coefficient between two waveguides attenuates exponentially over their distance. Therefore, each waveguide element can only interact with its nearby elements. In order to simplify the theoretical analysis, we assume that (1) |a1|=|a2|; (2) the angle θ between a1 and a2 is less than or equal to 90°; and (3) the guided mode in a waveguide element can only couple into its nearest neighbors and the next nearest neighbors. The evolution of the electric field U(l1,l2) can be written as,
idU(l1,l2)dz+κ[U(l1+1,l2)+U(l11,l2)+U(l1,l2+1)+U(l1,l21)]+κ[α1(U(l1+1,l21)+U(l11,l2+1))+α2(U(l1+1,l2+1)+U(l11,l21))]=0,
where κ is the coupling coefficient bwteen the waveguides with a distance of |a1|( = |a2|). α1κ and α2κ stand for coupling coefficients between the adjacent waveguides along the directions of a1a2 and a1+a2, respectively. If we only consider the coupling between the nearest and the next nearest waveguides, the values of α1 and α2 can be easily written as

 

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 (k1,k2) to realize Talbot self-imaging with input periods of N1=N2=3.

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α1{>1θ<π3=1θ=π3<1θ>π3,α2{=α1θ=π2=0other.

Equation (2) has a Floquet-Bloch–like solution,

U(l1,l2)(z)=exp(iRK)exp(iλz),
where z is the propagation length and λ is an eigenvalue. The corresponding wave vectorK in the reciprocal space can be written as
K=k1b1+k2b2=(k1,k2),
where b1 and b2 are base vectors of the reciprocal lattice, k1 and k2 are the coefficients to decide K [Fig. 1(b)]. 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 a1 and a2 are N1 and N2, respectively, the input field satisfies
U(l1,l2)=U(l1+αN1,l2+βN2),
where α and β are arbitrary integers. Taking Eq. (6) into Eq. (4) and performing several simplifications, we can get
αN1k1+βN2k2=m,
where m is an integer. Equation (7) should be hold for arbitrary integers α and β. Therefore, k1 and k2 can’t take continuous values but some discrete points,
k1=d1N1,d1=0,1...N11k2=d2N2,d2=0,1...N21.
For example, the red points in Fig. 1(b) stand for the qualified vector K with the input periods of N1=N2=3.

By substituting Eq. (4) into Eq. (2), we can get the expression of the eigenvalue,

λ(k1,k2)=2κ[cos(2πk1)+cos(2πk2)+α1cos(2π(k1k2))+α2cos(2π(k1+k2))].
Equation (4) indicates that the Talbot self-image is possible at an interval of z if and only if λ(k1,k2)z=2πv, where v is an integer. Hence, the ratio of any two different nonzero eigenvalues must be a rational number, i.e.
λ(k1,k2)λ(k1',k2')=pq,
where p and q are two prime integers. In addition, one can easily prove that the ratio
λ(k1,k2)λ(0,0)λ(k1',k2')λ(0,0)=pq
must also be rational to realize self-imaging. Considering the case of k2=k2'=0, we can get
cos(2πk1)1cos(2πk1')1=pq,
where k1,k1'{1/N1,...,(N11)/N1}andk1k1'. One can easily conclude that cos(2πk1) and cos(2πk1') must both be rational for self-imaging. By using the Chebyshev polynomial, cos(2πk1)=cos(d1×2πN1) can be expanded as
cos(d12πN1)=Td1(cos(2πN1))=i=0[d12]ci(d1)(cos(2πN1))d12i
where [d12] is the integer part of d12 and the coefficients ci(d1) is given by
ci(d1)=d12(1)i(d1i1)!i!(d12i)!2d12i.
The Chebyshev coefficients ci(d1) are all integer numbers and the first one is decided by c0d1=2d11 . Clearly, cos(d12π/N1) is a rational number if cos(2π/N1) is also a rational number. For Talbot revivals to occur at certain intervals, it is necessary that cos(2π/N1) is a rational number. Now, the problem is to find N1 that satisfies this condition. By using the Chebyshev polynomial, we can expand the identical equation
cos(2π)=cos(2πN1N1)=TN1(cos(2πN1))=1
as a polynomial in cos(2π/N1),
2N11(cos(2πN1))N1+...+c[N1/2]N1cos(2πN1)an=0
where an=1 for an odd N1 and an=(1)N121 for an even N1. By applying the rational root theorem, one can find that the possible value of cos(2π/N1) belongs to ±{0,1,1/2,1/22,...,1/2N11}. Obviously, the possible N1 to realize Talbot self-imaging can only belong to {1,2,3,4,6}. Similarly, N2{1,2,3,4,6} can be deduced by considering k1=k1'=0 in Eq. (11).

From Eq. (10), the ratio λ(k1,0)/λ(0,0) is also a rational number, which can be satisfied only when α1 and α2 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 α1 and α2 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, cos(2π(k1±k2)) 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 N1 and N2 are qualified to realize two-dimensional discrete Talbot revivals: if N1 = 4, N2 can’t be 3 or 6 because cos(2π(13±14)) and cos(2π(14±16)) 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 two-dimensional waveguide array: (1) both N1 and N2 belong to {1,2,3,4,6}; (2) the ratiosα1 and α2 of the coupling coefficients must be rational; (3) the combinations of N1 & N2 cannot be 3 & 4, or 4 & 6.

Next, we calculate the Talbot distance. The electric-field amplitude in the waveguide element (l1,l2) is the superposition of a set of Floquet-Bloch–like solutions

U(l1,l2)(z)=c(k1,k2)exp(2πi(l1k1+l2k2))exp(iλ(k1,k2)z).
The distribution of the light intensity can be written as
I(l1,l2)(z)=c0+c1cos((λ1λ2)z)+...+cmcos((λiλj)z)+...,
where ij and λiλj. The Talbot distance is decided by
zT=F(2π|λiλj|),
where the function F means to find the least common multiple of all possible2π|λiλj|.

3. 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.

3.1. Hexagonal waveguide arrays

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., α1=1andα2=0 in Eq. (2)). The hexagonal waveguide array is shown in Fig. 2. The coupling coefficient between the nearest neighbors is set to be κ=π/20mm−1. First, we choose the input periods of N1 = 2 and N2 = 4 [Fig. 1(a)], which can satisfy the requirement to realize discrete Talbot self-imaging. The Talbot images atzT/4, zT/3, zT/2, andzT are shown in Figs. 1(b)-1(f), respectively. Here, zT=10mm. The general characteristics of the Talbot effect can be observed [24,25]. For example, self-image is realized at z=zT [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. 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

{I(0,0)=164[30+24cos(4κz)+10cos(8κz)]I(0,1)=I(0,3)=I(1,1)=I(1,3)=132[1cos(8κz)]I(0,2)=I(1,2)=164[68cos(4κz)+2cos(8κz)]I(1,0)=164[148cos(4κz)6cos(8κz)].
From Eq. (20), the corresponding Talbot length can be deduced to be zT=2π/(4κ)=10mm, which is well consistent with the numerical simulations.

 

Fig. 2 The first row is the simulated intensity patterns with input periods of N1 = 2 and N2 = 4. (a) shows the structure of the waveguide array and the input fields. (b)-(e) are the corresponding Talbot images at zT/4, zT/3, zT/2, and zT planes. The second row is the simulated patterns with input periods of N1 = N2 = 3. (f) presents the array structure and the input periods. (g)-(j) are the the corresponding Talbot images at zT/4, zT/3, zT/2, andzTplanes. The insets in (a) and (f) show the base vectors of the hexagonal array.

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We change the input periods to be N1 = N2 = 3 as shown in Fig. 1(f). The numerical simulations atzT/4, zT/3, zT/2, andzT are shown in Figs. 1(g)-1(j), respectively. Here, zT=13.33mm. Classic performance of Talbot self-imaging with a hexagonal input period, such as pattern rotation at 1/3 Talbot plane [Fig. 1(h)] [19, 21], can be easily found. Similarly, the analytical solutions of the output intensity in the unit cell can be written as

{I(0,0)=181[41+12cos(6κz)+4cos(9κz)+24cos(3κz)]I(0,1)=I(0,2)=I(1,2)=I(1,0)=I(2,0)=I(2,1)=281[1cos(9κz)]I(1,1)=I(2,2)=181[14+4cos(9κz)6cos(6κz)12cos(3κz)]
The Talbot length can also be deduced from Eq. (21) to be zT=2π/(3κ)=13.33mm.

3.2. Square waveguide arrays

The structure of the square waveguide array is shown in Fig. 3(a). We set the coupling coefficient to be κ=π/20mm−1. The analytical solutions of the output intensity in the unit cell [Fig. 3(a)] can be written as

 

Fig. 3 The first row is the simulated patterns withα1=α2=0. (a) shows the array structure, and the input periods. (b)-(e) are the corresponding Talbot images at zT/4, zT/3, zT/2, and zT planes. The second row shows the case of α1=1/10 and α2=0. (f) is the array structure. (g)-(j) are the corresponding Talbot images at zT/4, zT/3, zT/2, and zT planes. The input periods are N1 = N2 = 3. The insets in (a) and (f) show the base vectors of the square array.

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{I(0,0)=181[33+8cos((3+6α1)κz)+8cos((6+3α1)κz)+32cos((33α1)κz)]I(0,1)=I(1,0)=I(0,2)=I(2,0)=181[6+2cos((3+6α1)κz)4cos((6+3α1)κz)4cos((33α1)κz)]I(1,1)=I(2,2)=I(1,2)=I(2,1)=181[64cos((3+6α1)κz)+2cos((6+3α1)κz)4cos((33α1)κz)]

Here, we investigate the changes in the Talbot self-images induced by considering the coupling with the next nearest waveguides. First, we set α1=α2=0, i.e. only the coupling between the nearest waveguides is considered. The input periods of N1 = N2 = 3 are chosen. The Talbot length is zT=2π/(3κ)=40/3mm. The evolution of the Talbot images along the propagation direction [Figs. 3(b)-3(e)] is quite similar to its free-space counterpart [18,21]. For example, the period at 1/3 Talbot plane become 1/3 of the input field [Fig. 3(c)].

When the coupling with the next nearest waveguides is considered (α1=1/10), the Talbot length can be calculated from Eqs. (19) and (23) to be zT=44.444mm. The existence of α1 makes the periodic light fields revive at a longer distance. The fractional Talbot images as shown in Figs. 3(g)-3(j) are very different from the case of α1=0. For example, the image at 1/3 Talbot plane [Fig. 3(h)] becomes very like the input field except that the intensity of the bright waveguide is a little below the input one and the background is not completely dark. The coefficient α1 introduces novel performances in the Talbot self-images. Our numerical simulations show that the coupling between the next nearest waveguides has no significant contribution only if α1<105 in this case.

3.3. Irregular waveguide arrays

Besides the hexagonal and square waveguide arrays, we also study more general case, i.e. irregular structures with θ<π/3 and π/3θ<π/2. The input period is selected to be N1 = N2 = 3. We assume that the coupling coefficient between the neighbor waveguides along a1 or a2 is κ=π/20mm−1. The simulated intensity patterns at different propagation distances are shown in Fig. 4. The analytical solutions can be written as

 

Fig. 4 The first row shows the simulated patterns with θ<π/3. (a) is the array structure. (b)-(e) are the corresponding Talbot images at zT/4, zT/3, zT/2, and zT planes. The second row is the calculated patterns with π/3θ<π/2. (f) is the structure of waveguide array. (g)-(j) are the corresponding Talbot images at zT/4, zT/3, zT/2, andzTplanes. The input periods are N1 = N2 = 3. The insets in (a) and (f) show the base vectors of the irregular arrays.

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{I(0,0)=181[25+8cos((3+3α1)κz)+4cos(6κz)+4cos((6+3α1)κz)+16cos((33α1)κz)+16cos(3κz)+8cos(3α1κz)]I(0,1)=I(1,0)=I(0,2)=I(2,0)=181[2cos((3+3α1)κz)2cos(6κz)2cos((6+3α1)κz)2cos((33α1)κz)2cos(3κz)+2cos(3α1κz)+4]I(1,1)=I(2,2)=181[104cos((3+3α1)κz)2cos(6κz)+4cos((6+3α1)κz)+4cos((33α1)κz)8cos(3κz)4cos(3α1κz)]I(1,2)=I(2,1)=181[104cos((3+3α1)κz)+4cos(6κz)2cos((6+3α1)κz)8cos((33α1)κz)+4cos(3κz)4cos(3α1κz)]

For the θ<π/3 case, the coefficient α1 is bigger than 1 because the waveguides along the a1a2 direction are closer than those along the a1 direction. Here, we assume α1=5. From Eq. (23), the Talbot distances can be written as zT=2π/(3κ)=40/3mm. For the π/3θ<π/2 case, we set α1=1/5 and the corresponding Talbot length is zT=2π/(0.6κ)=66.66mm. These results are well in agreement with the simulations in Fig. 4.

4. 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 one-dimensional case.

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). [CrossRef]  

4. L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881). [CrossRef]  

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). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

13. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

15. J. F. Clauser and S. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994). [CrossRef]   [PubMed]  

16. Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]  

22. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

References

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  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).
    [Crossref]
  4. L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  13. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  15. J. F. Clauser and S. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
    [Crossref] [PubMed]
  16. Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref]
  22. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]

2014 (1)

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).
[Crossref] [PubMed]

2013 (1)

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).
[Crossref]

2012 (2)

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

2011 (1)

2010 (1)

Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010).
[Crossref] [PubMed]

2008 (1)

2007 (1)

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).
[Crossref]

2005 (2)

J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. 30(3), 227–229 (2005).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

2001 (1)

1999 (1)

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).
[Crossref]

1995 (1)

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).
[Crossref] [PubMed]

1994 (1)

J. F. Clauser and S. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
[Crossref] [PubMed]

1991 (1)

1989 (2)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55(4), 334 (1989).
[Crossref]

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttg.) 79, 41–45 (1988).

1986 (1)

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).

1981 (1)

1881 (1)

L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Antyukhov, V. V.

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).

Azaña, J.

Bernabeu, E.

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).
[Crossref]

Chapman, M. S.

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).
[Crossref] [PubMed]

Chen, Z.

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Christodoulides, D. N.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Clark, C. W.

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).
[Crossref]

Clauser, J. F.

J. F. Clauser and S. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
[Crossref] [PubMed]

Deng, L.

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).
[Crossref]

Denschlag, J.

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).
[Crossref]

Edwards, M.

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).
[Crossref]

Ekstrom, C. R.

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).
[Crossref] [PubMed]

Gavrielides, A.

Glova, A. F.

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).

Guo, C. S.

Hagley, E. W.

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).
[Crossref]

Hammond, T. D.

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).
[Crossref] [PubMed]

Helmerson, K.

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).
[Crossref]

Hong, Z.

Hu, X.

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).
[Crossref] [PubMed]

Iwanow, R.

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).
[Crossref] [PubMed]

Jannson, J.

Jannson, T.

Kachurin, O. R.

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).

Kottos, T.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref] [PubMed]

Kovanis, V.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref] [PubMed]

Lebedev, F. V.

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).

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Leger, J. R.

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55(4), 334 (1989).
[Crossref]

Li, S.

J. F. Clauser and S. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
[Crossref] [PubMed]

Likhanskii, V. V.

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).

Liu, D.

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Lohmann, A. W.

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttg.) 79, 41–45 (1988).

May-Arrioja, D. A.

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).
[Crossref] [PubMed]

Mehuys, D.

Min, Y.

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).
[Crossref] [PubMed]

Napartovich, A. P.

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).

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).

Peterson, P.

Phillips, W. D.

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).
[Crossref]

Pismennyi, V. D.

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).

Pritchard, D. E.

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).
[Crossref] [PubMed]

Ramezani, H.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref] [PubMed]

Rayleigh, L.

L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Rolston, S. L.

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).
[Crossref]

Sanchez-Brea, L. M.

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).
[Crossref]

Schmiedmayer, J.

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).
[Crossref] [PubMed]

Sharma, M.

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Simsarian, J. E.

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).
[Crossref]

Sohler, W.

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).
[Crossref] [PubMed]

Stegeman, G. I.

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).
[Crossref] [PubMed]

Streifer, W.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Tannian, B. E.

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).
[Crossref] [PubMed]

Torcal-Milla, F. J.

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).
[Crossref]

Vitebskiy, I.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-Symmetric Talbot Effects,” Phys. Rev. Lett. 109(3), 033902 (2012).
[Crossref] [PubMed]

Waarts, R. G.

Wehinger, S.

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).
[Crossref] [PubMed]

Wei, D.

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).
[Crossref] [PubMed]

Welch, D. F.

Wen, J.

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).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref] [PubMed]

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).

Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010).
[Crossref] [PubMed]

Xiao, M.

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).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref] [PubMed]

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).

Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010).
[Crossref] [PubMed]

Yin, X.

Zhang, Y.

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).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref] [PubMed]

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).

Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010).
[Crossref] [PubMed]

Zhao, G.

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).
[Crossref] [PubMed]

Zhu, L. W.

Zhu, S. N.

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

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).

Y. Zhang, J. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104(18), 183901 (2010).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

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).
[Crossref]

Appl. Phys. Lett. (1)

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55(4), 334 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

JETP Lett. (1)

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).

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

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).
[Crossref]

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Figures (4)

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 ( k 1 , k 2 ) to realize Talbot self-imaging with input periods of N 1 = N 2 =3 .
Fig. 2
Fig. 2 The first row is the simulated intensity patterns with input periods of N1 = 2 and N2 = 4. (a) shows the structure of the waveguide array and the input fields. (b)-(e) are the corresponding Talbot images at z T /4 , z T /3 , z T /2 , and z T planes. The second row is the simulated patterns with input periods of N1 = N2 = 3. (f) presents the array structure and the input periods. (g)-(j) are the the corresponding Talbot images at z T /4 , z T /3 , z T /2 , and z T planes. The insets in (a) and (f) show the base vectors of the hexagonal array.
Fig. 3
Fig. 3 The first row is the simulated patterns with α 1 = α 2 =0 . (a) shows the array structure, and the input periods. (b)-(e) are the corresponding Talbot images at z T /4 , z T /3 , z T /2 , and z T planes. The second row shows the case of α 1 =1/10 and α 2 =0 . (f) is the array structure. (g)-(j) are the corresponding Talbot images at z T /4 , z T /3 , z T /2 , and z T planes. The input periods are N1 = N2 = 3. The insets in (a) and (f) show the base vectors of the square array.
Fig. 4
Fig. 4 The first row shows the simulated patterns with θ<π/3 . (a) is the array structure. (b)-(e) are the corresponding Talbot images at z T /4 , z T /3 , z T /2 , and z T planes. The second row is the calculated patterns with π/3θ<π/2 . (f) is the structure of waveguide array. (g)-(j) are the corresponding Talbot images at z T /4 , z T /3 , z T /2 , and z T planes. The input periods are N1 = N2 = 3. The insets in (a) and (f) show the base vectors of the irregular arrays.

Equations (23)

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R = l 1 a 1 + l 2 a 2 =( l 1 , l 2 ),
i d U ( l 1 , l 2 ) dz +κ[ U ( l 1 +1, l 2 ) + U ( l 1 1, l 2 ) + U ( l 1 , l 2 +1 ) + U ( l 1 , l 2 1 ) ]+κ [ α 1 ( U ( l 1 +1, l 2 1 ) + U ( l 1 1, l 2 +1 ) )+ α 2 ( U ( l 1 +1, l 2 +1 ) + U ( l 1 1, l 2 1 ) ) ]=0,
α 1 { >1 θ< π 3 =1 θ= π 3 <1 θ> π 3 , α 2 { = α 1 θ= π 2 =0 other .
U ( l 1 , l 2 ) ( z )=exp( i R K )exp( iλz ),
K = k 1 b 1 + k 2 b 2 =( k 1 , k 2 ),
U ( l 1 , l 2 ) = U ( l 1 +α N 1 , l 2 +β N 2 ) ,
α N 1 k 1 +β N 2 k 2 =m,
k 1 = d 1 N 1 , d 1 =0,1... N 1 1 k 2 = d 2 N 2 , d 2 =0,1... N 2 1.
λ( k 1 , k 2 )=2κ[ cos( 2π k 1 )+cos( 2π k 2 )+ α 1 cos( 2π( k 1 k 2 ) )+ α 2 cos( 2π( k 1 + k 2 ) ) ].
λ( k 1 , k 2 ) λ( k 1 ' , k 2 ' ) = p q ,
λ( k 1 , k 2 )λ( 0,0 ) λ( k 1 ' , k 2 ' )λ( 0,0 ) = p q
cos( 2π k 1 )1 cos( 2π k 1 ' )1 = p q ,
cos( d 1 2π N 1 )= T d 1 ( cos( 2π N 1 ) )= i=0 [ d 1 2 ] c i ( d 1 ) ( cos( 2π N 1 ) ) d 1 2i
c i ( d 1 ) = d 1 2 ( 1 ) i ( d 1 i1 )! i!( d 1 2i )! 2 d 1 2i .
cos( 2π )=cos( 2π N 1 N 1 )= T N 1 ( cos( 2π N 1 ) )=1
2 N 1 1 ( cos( 2π N 1 ) ) N 1 +...+ c [ N 1 /2 ] N 1 cos( 2π N 1 )an=0
U ( l 1 , l 2 ) ( z )= c ( k 1 , k 2 ) exp( 2πi( l 1 k 1 + l 2 k 2 ) ) exp( i λ ( k 1 , k 2 ) z ).
I ( l 1 , l 2 ) ( z )= c 0 + c 1 cos( ( λ 1 λ 2 )z )+...+ c m cos( ( λ i λ j )z )+...,
z T =F( 2π | λ i λ j | ),
{ I ( 0,0 ) = 1 64 [ 30+24cos( 4κz )+10cos( 8κz ) ] I ( 0,1 ) = I ( 0,3 ) = I ( 1,1 ) = I ( 1,3 ) = 1 32 [ 1cos( 8κz ) ] I ( 0,2 ) = I ( 1,2 ) = 1 64 [ 68cos( 4κz )+2cos( 8κz ) ] I ( 1,0 ) = 1 64 [ 148cos( 4κz )6cos( 8κz ) ] .
{ I ( 0,0 ) = 1 81 [ 41+12cos( 6κz )+4cos( 9κz )+24cos( 3κz ) ] I ( 0,1 ) = I ( 0,2 ) = I ( 1,2 ) = I ( 1,0 ) = I ( 2,0 ) = I ( 2,1 ) = 2 81 [ 1cos( 9κz ) ] I ( 1,1 ) = I ( 2,2 ) = 1 81 [ 14+4cos( 9κz )6cos( 6κz )12cos( 3κz ) ]
{ I ( 0,0 ) = 1 81 [ 33+8cos( ( 3+6 α 1 )κz )+8cos( ( 6+3 α 1 )κz )+32cos( ( 33 α 1 )κz ) ] I ( 0,1 ) = I ( 1,0 ) = I ( 0,2 ) = I ( 2,0 ) = 1 81 [ 6+2cos( ( 3+6 α 1 )κz )4cos( ( 6+3 α 1 )κz )4cos( ( 33 α 1 )κz ) ] I ( 1,1 ) = I ( 2,2 ) = I ( 1,2 ) = I ( 2,1 ) = 1 81 [ 64cos( ( 3+6 α 1 )κz )+2cos( ( 6+3 α 1 )κz )4cos( ( 33 α 1 )κz ) ]
{ I ( 0,0 ) = 1 81 [ 25+8cos( ( 3+3 α 1 )κz )+4cos( 6κz )+4cos( ( 6+3 α 1 )κz ) +16cos( ( 33 α 1 )κz )+16cos( 3κz )+8cos( 3 α 1 κz ) ] I ( 0,1 ) = I ( 1,0 ) = I ( 0,2 ) = I ( 2,0 ) = 1 81 [ 2cos( ( 3+3 α 1 )κz )2cos( 6κz )2cos( ( 6+3 α 1 )κz ) 2cos( ( 33 α 1 )κz )2cos( 3κz )+2cos( 3 α 1 κz )+4 ] I ( 1,1 ) = I ( 2,2 ) = 1 81 [ 104cos( ( 3+3 α 1 )κz )2cos( 6κz )+4cos( ( 6+3 α 1 )κz ) +4cos( ( 33 α 1 )κz )8cos( 3κz )4cos( 3 α 1 κz ) ] I ( 1,2 ) = I ( 2,1 ) = 1 81 [ 104cos( ( 3+3 α 1 )κz )+4cos( 6κz )2cos( ( 6+3 α 1 )κz ) 8cos( ( 33 α 1 )κz )+4cos( 3κz )4cos( 3 α 1 κz ) ]

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