Discrete Talbot effect in two-dimensional waveguide arrays

Zhenhua Chen; Yong Zhang; Min Xiao

doi:10.1364/OE.23.014724

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 $z_{T} = d^{2} / λ$ . Here, $d$ stands for the spatial period of the object and $λ$ the light wavelength [4]. When the distances are rational multiples of $z_{T}$ (i.e., $z / z_{T} = p / q$ 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 ( ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ in Fig. 1(a)), arbitrary waveguide element in the array can be easily addressed through a vector defined by

\vec{R} = l_{1} {\vec{a}}_{1} + l_{2} {\vec{a}}_{2} = (l_{1}, l_{2}),

where

l_{1}

and

l_{2}

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)

| {\vec{a}}_{1} | = | {\vec{a}}_{2} |

; (2) the angle

θ

between

{\vec{a}}_{1}

and

{\vec{a}}_{2}

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_{(l_{1}, l_{2})}

can be written as,

\begin{array}{l} i \frac{d U_{(l_{1}, l_{2})}}{d z} + 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, \end{array}

where

κ

is the coupling coefficient bwteen the waveguides with a distance of

| {\vec{a}}_{1} |

( =

| {\vec{a}}_{2} |

).

α_{1} κ

and

α_{2} κ

stand for coupling coefficients between the adjacent waveguides along the directions of

{\vec{a}}_{1} - {\vec{a}}_{2}

and

{\vec{a}}_{1} + {\vec{a}}_{2}

, respectively. If we only consider the coupling between the nearest and the next nearest waveguides, the values of

α_{1}

and

α_{2}

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 $(k_{1}, k_{2})$ to realize Talbot self-imaging with input periods of $N_{1} = N_{2} = 3$ .

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α_{1} {\begin{matrix} > 1 θ < \frac{π}{3} \\ = 1 θ = \frac{π}{3} \\ < 1 θ > \frac{π}{3} \end{matrix}, α_{2} {\begin{matrix} = α_{1} θ = \frac{π}{2} \\ = 0 o t h e r \end{matrix} .

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

U_{(l_{1}, l_{2})} (z) = \exp (i \vec{R} \cdot \vec{K}) \exp (i λ z),

where z is the propagation length and

λ

is an eigenvalue. The corresponding wave vector

\vec{K}

in the reciprocal space can be written as

\vec{K} = k_{1} {\vec{b}}_{1} + k_{2} {\vec{b}}_{2} = (k_{1}, k_{2}),

where

{\vec{b}}_{1}

and

{\vec{b}}_{2}

are base vectors of the reciprocal lattice,

k_{1}

and

k_{2}

are the coefficients to decide

\vec{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

{\vec{a}}_{1}

and

{\vec{a}}_{2}

are

N_{1}

and

N_{2}

, respectively, the input field satisfies

U_{(l_{1}, l_{2})} = U_{(l_{1} + α N_{1}, l_{2} + β N_{2})},

where

α

and

β

are arbitrary integers. Taking Eq. (6) into Eq. (4) and performing several simplifications, we can get

α N_{1} k_{1} + β N_{2} k_{2} = 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,

\begin{array}{l} k_{1} = \frac{d_{1}}{N_{1}}, d_{1} = 0, 1... N_{1} - 1 \\ k_{2} = \frac{d_{2}}{N_{2}}, d_{2} = 0, 1... N_{2} - 1. \end{array}

For example, the red points in Fig. 1(b) stand for the qualified vector

\vec{K}

with the input periods of

N_{1} = N_{2} = 3

.

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

λ (k_{1}, k_{2}) = 2 κ [\cos (2 π k_{1}) + \cos (2 π k_{2}) + α_{1} \cos (2 π (k_{1} - k_{2})) + α_{2} \cos (2 π (k_{1} + k_{2}))] .

Equation (4) indicates that the Talbot self-image is possible at an interval of z if and only if

λ (k_{1}, k_{2}) z = 2 π v

, where v is an integer. Hence, the ratio of any two different nonzero eigenvalues must be a rational number, i.e.

\frac{λ (k_{_{1}}, k_{2})}{λ (k_{1}^{'}, k_{2}^{'})} = \frac{p}{q},

where p and q are two prime integers. In addition, one can easily prove that the ratio

\frac{λ (k_{1}, k_{2}) - λ (0, 0)}{λ (k_{1}^{'}, k_{2}^{'}) - λ (0, 0)} = \frac{p}{q}

must also be rational to realize self-imaging. Considering the case of

k_{2} = k_{2}^{'} = 0

, we can get

\frac{\cos (2 π k_{1}) - 1}{\cos (2 π k_{1}^{'}) - 1} = \frac{p}{q},

where

k_{1}, k_{1}^{'} \in {1 / N_{1}, ..., (N_{1} - 1) / N_{1}} and k_{1} \neq k_{1}^{'}

. One can easily conclude that

\cos (2 π k_{1})

and

\cos (2 π k_{1}^{'})

must both be rational for self-imaging. By using the Chebyshev polynomial,

\cos (2 π k_{1}) = \cos (d_{1} \times \frac{2 π}{N_{1}})

can be expanded as

\cos (d_{1} \frac{2 π}{N_{1}}) = T_{d_{1}} (\cos (\frac{2 π}{N_{1}})) = \sum_{i = 0}^{[\frac{d_{1}}{2}]} c_{i}^{(d_{1})} {(\cos (\frac{2 π}{N_{1}}))}^{d_{1} - 2 i}

where

[\frac{d_{1}}{2}]

is the integer part of

\frac{d_{1}}{2}

and the coefficients

c_{i}^{(d_{1})}

is given by

c_{i}^{(d_{1})} = \frac{d_{1}}{2} {(- 1)}^{i} \frac{(d_{1} - i - 1)!}{i! (d_{1} - 2 i)!} 2^{d_{1} - 2 i} .

The Chebyshev coefficients

c_{i}^{(d_{1})}

are all integer numbers and the first one is decided by

c_{0}^{d_{1}} = 2^{d_{1} - 1}

. Clearly,

\cos (d_{1} 2 π / N_{1})

is a rational number if

\cos (2 π / N_{1})

is also a rational number. For Talbot revivals to occur at certain intervals, it is necessary that

\cos (2 π / N_{1})

is a rational number. Now, the problem is to find

N_{1}

that satisfies this condition. By using the Chebyshev polynomial, we can expand the identical equation

\cos (2 π) = \cos (\frac{2 π}{N_{1}} N_{1}) = T_{N_{1}} (\cos (\frac{2 π}{N_{1}})) = 1

as a polynomial in

\cos (2 π / N_{1})

,

2^{N_{1} - 1} {(\cos (\frac{2 π}{N_{1}}))}^{N_{1}} + ... + c_{[N_{1} / 2]}^{N_{1}} \cos (\frac{2 π}{N_{1}}) - a n = 0

where

a n = 1

for an odd

N_{1}

and

a n = {(- 1)}^{\frac{N_{1}}{2}} - 1

for an even

N_{1}

. By applying the rational root theorem, one can find that the possible value of

\cos (2 π / N_{1})

belongs to

\pm {0, 1, 1 / 2, 1 / 2^{2}, ..., 1 / 2^{N_{1} - 1}}

. Obviously, the possible

N_{1}

to realize Talbot self-imaging can only belong to

{1, 2, 3, 4, 6}

. Similarly,

N_{2} \in {1, 2, 3, 4, 6}

can be deduced by considering

k_{1} = k_{1}^{'} = 0

in Eq. (11).

From Eq. (10), the ratio $λ (k_{1}, 0) / λ (0, 0)$ is also a rational number, which can be satisfied only when $α_{1}$ and $α_{2}$ 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 $α_{1}$ and $α_{2}$ 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, $\cos (2 π (k_{1} \pm k_{2}))$ 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 $\cos (2 π (\frac{1}{3} \pm \frac{1}{4}))$ and $\cos (2 π (\frac{1}{4} \pm \frac{1}{6}))$ 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 ratios $α_{1}$ and $α_{2}$ 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 $(l_{1}, l_{2})$ is the superposition of a set of Floquet-Bloch–like solutions

U_{(l_{1}, l_{2})} (z) = \sum c_{(k_{1}, k_{2})} \exp (2 π i (l_{1} k_{1} + l_{2} k_{2})) \exp (i λ_{(k_{1}, k_{2})} z) .

The distribution of the light intensity can be written as

I_{(l_{1}, l_{2})} (z) = c_{0} + c_{1} \cos ((λ_{1} - λ_{2}) z) + ... + c_{m} \cos ((λ_{i} - λ_{j}) z) + ...,

where

i \neq j

and

λ_{i} \neq λ_{j}

. The Talbot distance is decided by

z_{T} = F (\frac{2 π}{| λ_{i} - λ_{j} |}),

where the function F means to find the least common multiple of all possible

\frac{2 π}{| λ_{i} - λ_{j} |}

.

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., $α_{1} = 1 and α_{2} = 0$ in Eq. (2)). The hexagonal waveguide array is shown in Fig. 2. The coupling coefficient between the nearest neighbors is set to be $κ = π / 20$ mm⁻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 at $z_{T} / 4$ , $z_{T} / 3$ , $z_{T} / 2$ , and $z_{T}$ are shown in Figs. 1(b)-1(f), respectively. Here, $z_{T} = 10$ mm. The general characteristics of the Talbot effect can be observed [24,25]. For example, self-image is realized at $z = z_{T}$ [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

{\begin{cases} I_{(0, 0)} = \frac{1}{64} [30 + 24 \cos (4 κ z) + 10 \cos (8 κ z)] \\ I_{(0, 1)} = I_{(0, 3)} = I_{(1, 1)} = I_{(1, 3)} = \frac{1}{32} [1 - \cos (8 κ z)] \\ I_{(0, 2)} = I_{(1, 2)} = \frac{1}{64} [6 - 8 \cos (4 κ z) + 2 \cos (8 κ z)] \\ I_{(1, 0)} = \frac{1}{64} [14 - 8 \cos (4 κ z) - 6 \cos (8 κ z)] \end{cases} .

From Eq. (20), the corresponding Talbot length can be deduced to be

z_{T} = 2 π / (4 κ) = 10

mm, 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 $z_{T} / 4$ , $z_{T} / 3$ , $z_{T} / 2$ , 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.

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We change the input periods to be N₁ = N₂ = 3 as shown in Fig. 1(f). The numerical simulations at $z_{T} / 4$ , $z_{T} / 3$ , $z_{T} / 2$ , and $z_{T}$ are shown in Figs. 1(g)-1(j), respectively. Here, $z_{T} = 13.33$ mm. 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

{\begin{cases} I_{(0, 0)} = \frac{1}{81} [41 + 12 \cos (6 κ z) + 4 \cos (9 κ z) + 24 \cos (3 κ z)] \\ I_{(0, 1)} = I_{(0, 2)} = I_{(1, 2)} = I_{(1, 0)} = I_{(2, 0)} = I_{(2, 1)} = \frac{2}{81} [1 - \cos (9 κ z)] \\ I_{(1, 1)} = I_{(2, 2)} = \frac{1}{81} [14 + 4 \cos (9 κ z) - 6 \cos (6 κ z) - 12 \cos (3 κ z)] \end{cases}

The Talbot length can also be deduced from Eq. (21) to be

z_{T} = 2 π / (3 κ) = 13.33

mm.

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 $κ = π / 20$ mm⁻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 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.

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{\begin{cases} I_{(0, 0)} = \frac{1}{81} [33 + 8 \cos ((3 + 6 α_{1}) κ z) + 8 \cos ((6 + 3 α_{1}) κ z) + 32 \cos ((3 - 3 α_{1}) κ z)] \\ I_{(0, 1)} = I_{(1, 0)} = I_{(0, 2)} = I_{(2, 0)} \\ = \frac{1}{81} [6 + 2 \cos ((3 + 6 α_{1}) κ z) - 4 \cos ((6 + 3 α_{1}) κ z) - 4 \cos ((3 - 3 α_{1}) κ z)] \\ I_{(1, 1)} = I_{(2, 2)} = I_{(1, 2)} = I_{(2, 1)} \\ = \frac{1}{81} [6 - 4 \cos ((3 + 6 α_{1}) κ z) + 2 \cos ((6 + 3 α_{1}) κ z) - 4 \cos ((3 - 3 α_{1}) κ z)] \end{cases}

Here, we investigate the changes in the Talbot self-images induced by considering the coupling with the next nearest waveguides. First, we set $α_{1} = α_{2} = 0$ , i.e. only the coupling between the nearest waveguides is considered. The input periods of N₁ = N₂ = 3 are chosen. The Talbot length is $z_{T} = 2 π / (3 κ) = 40 / 3$ mm. 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 ( $α_{1} = 1 / 10$ ), the Talbot length can be calculated from Eqs. (19) and (23) to be $z_{T} = 44.444$ mm. The existence of $α_{1}$ 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 $α_{1} = 0$ . 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 $α_{1}$ 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 $α_{1} < 10^{- 5}$ 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 $π / 3 \leq θ < π / 2$ . The input period is selected to be N₁ = N₂ = 3. We assume that the coupling coefficient between the neighbor waveguides along ${\vec{a}}_{1}$ or ${\vec{a}}_{2}$ is $κ = π / 20$ mm⁻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 $z_{T} / 4$ , $z_{T} / 3$ , $z_{T} / 2$ , and $z_{T}$ planes. The second row is the calculated patterns with $π / 3 \leq θ < π / 2$ . (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.

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{\begin{cases} I_{(0, 0)} = \frac{1}{81} [\begin{array}{l} 25 + 8 \cos ((3 + 3 α_{1}) κ z) + 4 \cos (6 κ z) + 4 \cos ((6 + 3 α_{1}) κ z) \\ + 16 \cos ((3 - 3 α_{1}) κ z) + 16 \cos (3 κ z) + 8 \cos (3 α_{1} κ z) \end{array}] \\ I_{(0, 1)} = I_{(1, 0)} = I_{(0, 2)} = I_{(2, 0)} = \frac{1}{81} [\begin{array}{l} 2 \cos ((3 + 3 α_{1}) κ z) - 2 \cos (6 κ z) - 2 \cos ((6 + 3 α_{1}) κ z) \\ - 2 \cos ((3 - 3 α_{1}) κ z) - 2 \cos (3 κ z) + 2 \cos (3 α_{1} κ z) + 4 \end{array}] \\ I_{(1, 1)} = I_{(2, 2)} = \frac{1}{81} [\begin{array}{l} 10 - 4 \cos ((3 + 3 α_{1}) κ z) - 2 \cos (6 κ z) + 4 \cos ((6 + 3 α_{1}) κ z) \\ + 4 \cos ((3 - 3 α_{1}) κ z) - 8 \cos (3 κ z) - 4 \cos (3 α_{1} κ z) \end{array}] \\ I_{(1, 2)} = I_{(2, 1)} = \frac{1}{81} [\begin{array}{l} 10 - 4 \cos ((3 + 3 α_{1}) κ z) + 4 \cos (6 κ z) - 2 \cos ((6 + 3 α_{1}) κ z) \\ - 8 \cos ((3 - 3 α_{1}) κ z) + 4 \cos (3 κ z) - 4 \cos (3 α_{1} κ z) \end{array}] \end{cases}

For the $θ < π / 3$ case, the coefficient $α_{1}$ is bigger than 1 because the waveguides along the ${\vec{a}}_{1} - {\vec{a}}_{2}$ direction are closer than those along the ${\vec{a}}_{1}$ direction. Here, we assume $α_{1} = 5$ . From Eq. (23), the Talbot distances can be written as $z_{T} = 2 π / (3 κ) = 40 / 3$ mm. For the $π / 3 \leq θ < π / 2$ case, we set $α_{1} = 1 / 5$ and the corresponding Talbot length is $z_{T} = 2 π / (0.6 κ) = 66.66$ mm. 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.

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Discrete Talbot effect in two-dimensional waveguide arrays

Abstract

1. Introduction

2. Theory

3. Numerical simulation and analysis

3.1. Hexagonal waveguide arrays

3.2. Square waveguide arrays

3.3. Irregular waveguide arrays

4. Conclusions

References and links

Cited By

Figures (4)

Equations (23)

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