1. Introduction

The Talbot effect refers to the self-imaging phenomenon occurring at a certain distance after plane light incident on grating structures, which was discovered and proposed by Talbot in 1836 [1]. Since the Talbot effect does not involve a lens in its self-imaging process, it is also referred as lens-less imaging. After more than twenty years, Rayleigh made a theoretical analysis of the Talbot effect in 1881, proposed that it was an interference diffraction effect of coherent plane light, and provided a mathematical formula for one-dimensional Talbot distance [2]. In later studies, it was also found that the Talbot effect has self-healing properties, that is, defects in periodic structures can be suppressed to a certain extent during self-imaging [3,4].

In recent years, the study of Talbot effect has been continuously developed. Angular Talbot effect [5], nonlinear Talbot effect [6,7], Airy-Talbot effect [8], Quantum Talbot effect [9], temporal Talbot effect [10–12] and many other theories related with the Talbot effect have been proposed. In addition, Talbot effect has also been found in Bose-Einstein condensation [13], PT-symmetric optical systems [14–17], and waveguide arrays [18–22]. Specially, Talbot effect has also been found in reconfigurable non-Hermitian photonic graphene, which is used to exhibit the properties of a lattice switching [23]. With the deepening of research, the Talbot effect has been widely used in many fields, such as image processing [24,25], optical measurement [26], and X-ray imaging [27–33].

In the previous studies, it was generally assumed that the Talbot effect can only occur in periodic structures. In 1967, Montgomery published a paper on self-imaging objects with infinite aperture [34]. He divided self-imaging into weak imaging and strong imaging, and considered doubly-periodic and non-periodic structures in the second case. Different researchers have tried to explore the possibility of self-imaging phenomenon in quasi-periodic structures and aperiodic structures, but only approximate Talbot effects are achieved [35,36]. So far, whether aperiodic structures can realize perfect self-imaging is still lacking in research.

On the basis of these studies, we continue to study the Talbot effect in one-dimensional aperiodic gratings, and propose a class of aperiodic structures which can realize perfect Talbot effect under certain circumstances. The detailed self-imaging property of the proposed structures is deduced theoretically and then verified by numerical simulations. Our discovery removes the periodicity requirement of the Talbot effect, which expands the field of classical Talbot effect theory and has a high application prospect for the subsequent research fields such as optical imaging and measurement.

2. Method

For one-dimensional periodic gratings, the Talbot distance can be expressed as

We consider a grating whose initial complex amplitude distribution is a superposition of several sine terms:

For the case of a superposition of two sine terms, Eq. (2) is reduced to the form of $u({x,z = 0} )= \sin \frac{{2\pi x}}{T} + \sin \frac{{2\pi x}}{{CT}}$, where $C > 1$. Obviously, when C is rational, it is a periodic structure; And when C is irrational, it is an aperiodic structure.

According to Fresnel’s formula of diffraction distribution, the complex amplitude distribution at the distance of z can be expressed as

The expression of light intensity distribution can be obtained according to Eq. (3) as

It can be seen from Eq. (4) that, in the expression of light intensity distribution, only the cosine term in the third term is related to the propagation distance z, whose value requires to be 1 in order to realize self-repetition of light intensity. Thus, we can obtain

This structure can realize self-imaging at Talbot distance determined by Eq. (5) no matter C is rational or not. However, only light intensity can be restored to its initial value here, and whether phase can be restored is still not certain. To ensure the restoration of phase, it is necessary to consider from the perspective of complex amplitude. By comparing Eq. (2) and Eq. (3), we can see that complex amplitude distribution at the distance of z has two more phase terms than initial complex amplitude distribution, both of which should be 1 to achieve both restoration of light intensity and phase (restoration of complex amplitude). So, we can obtain

Perfect Talbot effect in a grating with two sine terms superposed as its initial complex amplitude distribution is discussed above, and the conclusions can also be extended to more complex structures, with both similarities and differences.

For the case of a superposition of k ($k \ge 3$) sine terms, complex amplitude distribution and light intensity distribution at the distance of z can be obtained similarly as

In the expression of light intensity distribution (Eq. (8)), there are $\frac{{k({k - 1} )}}{2}$ cosine terms related to the propagation distance z, whose values should all be 1 in order to realize self-imaging. In practice, however, we only need to consider the first $k - 1$ cosine terms, the other $\frac{{({k - 1} )({k - 2} )}}{2}$ cosine terms will automatically become 1 when these $k - 1$ terms value 1. The process of proving this conclusion is elaborated in Section 3 of the Supplemental Document. Similar to Eq. (5), we can obtain

Similar to the case of the superposition of two sine terms, in the above discussion of the self-imaging of a grating whose initial complex amplitude distribution is a superposition of k ($k \ge 3$) sine terms, only light intensity is restored to its initial value, while phase is not necessarily restored, and the condition to restore both light intensity and phase can be obtained as

3. Results

For the grating whose initial complex amplitude distribution is superimposed by two sine terms, three cases can be divided according to the value of the period ratio of the two terms:

Firstly, when C is rational, this is a periodic structure. When the distance meets the condition of Eq. (5), self-imaging can be realized in this structure, but with only light intensity restored, while both light intensity and phase can be restored when the distance meets the condition of Eq. (6), which is different from the general case (a sine grating). In addition, the periods of the two superposition terms are commensurable in this case, and a corresponding Talbot distance (commensurable Talbot distance) can be obtained according to traditional Talbot theory, which is consistent with Eq. (6) and different from Eq. (5), and there is a multiple relation between them. If we express C as an irreducible fraction $C = r/w$ with positive integers r and w, and $r > w$, then this multiple relation can be expressed as ${r^2} - {w^2}$. See Section 4 of the Supplemental Document for specific derivation process.

Secondly, when C is irrational but ${C^2}$ is rational, this is an aperiodic structure. When the distance satisfies the condition of Eq. (5), this structure can realize self-imaging with only light intensity restored, and this self-imaging phenomenon can be regarded as “aperiodic Talbot effect”. Similar to the first case, the distance to restore both light intensity and phase needs to meet the condition of Eq. (6). Although there is no commensurable Talbot distance in this case, there is still a multiple relation between the other two Talbot distances. If we express ${C^2}$ as an irreducible fraction ${C^2} = p/q$ with positive integers p and q, and $p > q$, then this multiple relation can be expressed as $p - q$. See Section 4 of the Supplemental Document for specific derivation process.

The above conclusions will be illustrated using the following example where $C = \sqrt 3 $. The first Talbot distance under this circumstance can be calculated by Eq. (5), and the answer is 600 µm. As can be seen from Fig. 1(a) and 1(c), the restoration parameter of light intensity is 0 and perfect self-imaging is realized at the distance of 600 µm, while phase does not restore to its initial value in Fig. 1(b) and the phase difference at this distance is π in Fig. 1(d). At the distance of 1200 µm calculated by Eq. (6), light intensity and phase are both restored, which can also be seen in Fig. 1, that the restoration parameters of light intensity and phase are both 0 at this distance. The distance for both light intensity and phase to be restored is twice the distance of self-imaging.

 

Fig. 1. Image of (a) light intensity and (b) phase, (c) restoration degree of light intensity and (d) phase when $C = \sqrt 3 ,\textrm{ }T = 10{\ \mathrm{\mu} \mathrm{m}},\textrm{ }\lambda = 500\textrm{ nm}$. The restoration degree parameter valuing 0 corresponds to the complete restoration of light intensity or phase (see Section 5 in the Supplemental Document).

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Thirdly, when ${C^2}$ is irrational, this is still an aperiodic structure. When the distance meets the condition of Eq. (5), the structure can still realize self-imaging, so as to say “aperiodic Talbot effect”. In this case, Eq. (6) cannot be established, which means light intensity and phase cannot both be restored.

The above conclusions will be illustrated using the following example where $C = \sqrt \pi $. The first Talbot distance in this case can be calculated by Eq. (5), and the answer is 587 µm. As can be seen from Fig. 2(a) and 2(c), the restoration parameter of light intensity is 0 and perfect self-imaging is realized at the distance of 587 µm, while phase does not restore to its initial value in Fig. 2(b) and 2(d). Under this circumstance, Eq. (6) cannot be established, and the restoration of both light intensity and phase cannot be realized, which is consistent with the restoration of phase never reaching 0 in Fig. 2(d).

 

Fig. 2. Image of (a) light intensity and (b) phase, (c) restoration degree of light intensity and (d) phase when $C = \sqrt \pi ,\textrm{ }T = 10{\ \mathrm{\mu} \mathrm{m}},\textrm{ }\lambda = 500\textrm{ nm}$. The restoration degree parameter valuing 0 corresponds to the complete restoration of light intensity or phase.

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For the grating whose initial complex amplitude distribution is superimposed by k ($k \ge 3$) sine terms, take the second term and the j-th term from Eq. (9), and the following equation can be obtained after deformation:

If $C_2^2$ is chosen in advance, when ${s_2}$ takes different values, different numbers of ${s_j}$ can be obtained to meet the conditions in the brackets of Eq. (11). Each ${s_j}$ can correspond to a $C_j^2$ respectively. When there’s $k - 2$ ${s_j}$’s available, the $k - 2$ $C_j^2$’s obtained can then form a grating with k sine terms superimposed, and this grating will be able to realize perfect Talbot effect.

A series of parameters are listed in Table 1 to illustrate this conclusion, where $C_2^2 = \pi $. When ${s_2} = 1,2$, no eligible ${s_j}$ exists. When ${s_2} = 3,4$, there’s one eligible ${s_j}$ respectively, which also corresponds to one $C_j^2$ and one rational ${A_j}$. In this case, a grating with three sine terms superimposed that can realize aperiodic Talbot effect will be obtained. When ${s_2} = 5$, there’s two eligible ${s_j}$’s, which also correspond to two $C_j^2$’s and two rational ${A_j}$’s. In this case, a grating with four sine terms superimposed that can realize aperiodic Talbot effect will be obtained. When ${s_2}$ take a larger value, there will be more eligible ${s_j}$’s, $C_j^2$’s and ${A_j}$’s existing, which means that a grating with more sine terms superimposed that can realize aperiodic Talbot effect will be obtained. Besides, Section 6 of the Supplemental Document lists a series of parameters where $C_2^2 = 2$ for referrence.

Table 1. Values of parameters to realize Talbot effect, where $C_2^2 = \pi $

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Therefore, four cases can be divided according to the values of the period ratios of the k ($k \ge 3$) terms:

Firstly, when ${C_j}$’s ($2 \le j \le k$) are all rational, this is a periodic structure. When the distance meets the condition of Eq. (9), self-imaging can be realized, but only light intensity is restored, and both light intensity and phase can be restored when the distance meets the condition of Eq. (10).

Secondly, when not all ${C_j}$’s ($2 \le j \le k$) are rational, but $C_j^2$’s ($2 \le j \le k$) are all rational, this is an aperiodic structure. When the distance satisfies the condition of Eq. (9), this structure can realize self-imaging and this self-imaging phenomenon can be regarded as “aperiodic Talbot effect”. Similar to the first case, the distance to restore both light intensity and phase needs to meet the condition of Eq. (10).

Thirdly, when not all $C_j^2$’s ($2 \le j \le k$) are rational, but ${A_j}$’s ($3 \le j \le k$) are all rational, this is still an aperiodic structure. When the distance meets the condition of Eq. (9), the structure can still realize self-imaging, so as to say “aperiodic Talbot effect”. In this case, Eq. (10) cannot be established, and restoration of both light intensity and phase cannot be achieved.

Fourthly, when not all ${A_j}$’s ($3 \le j \le k$) are rational, neither Eq. (9) nor Eq. (10) is valid, and this structure cannot achieve self-imaging.

All the above conclusions can be summarized in Table 2.

Table 2. Self-imaging properties of gratings whose initial complex amplitude distribution is a superposition of k sine terms (it’s a sine grating when $k = 1$, its initial complex amplitude distribution is composed by a sine term plus a constant)

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4. Conclusions

In summary, a class of one-dimensional aperiodic grating structure with several sine terms has been studied. It is found that perfect Talbot effect can be achieved under certain conditions. For the superposition of two sine terms, the structure can achieve perfect self-imaging with any period ratios. When the ratio is rational, it is a periodic structure and there is a multiple relation between the commensurable Talbot distance and the Talbot distance of light intensity. When the ratio is irrational, it is an aperiodic structure, and the self-imaging phenomenon can be called “aperiodic Talbot effect”. Restoration of both light intensity and phase requires that the square of the period ratio be rational. For the superposition of three or more sine terms, the realization of self-imaging requires certain conditions to be satisfied between various period ratios, but it does not require period ratios or the square of period ratios to be rational, in which case, “aperiodic Talbot effect” can still be realized. These conclusions can be further promoted to the case of several rectangular function terms superposed with similarities and differences. The aperiodic Talbot effect proposed in this paper can be considered as an extension of the basic theory of traditional Talbot effect, and the practical realization of the aperiodic Talbot effect is similar to the traditional Talbot effect, except that a class of aperiodic structure is used instead of the periodic ones. The concept of aperiodic Talbot effect is of universal significance. In addition to optical systems, it may also be applicable to acoustic systems, atomic systems, nano systems, etc.