1. Introduction

Sampled fiber Bragg gratings (SFBGs) have attracted considerable attention due to their versatile applications in multi-channel filters [1–10], dispersion compensators [10–12], and optical code division multiple access (OCDMA) technique. As for dense multi-channel filters, the so-called spectral Talbot effect (i.e., spectral self-imaging effect) is a promising choice. The spectral Talbot effect, a member of Talbot effect family, is able to create exact images of the original object in specific spectral (or wavelength) domain. Via “strongly chirped SFBGs” [2], this effect was first used for channel multiplication. Then the concept “spectral Talbot effect” for SFBGs was definitely proposed in Ref [4]. Based on their work concerning temporal Talbot effect [13], Azaña, Chen, and Wang have conducted systematized and comprehensive investigations about the spectral Talbot effect [4–6]. Further, this effect can be obtained by the combination of chirp effect and phase shifts as well [7, 12].

Usually, the phase profiles of aforementioned approaches vary continuously along the whole grating; i.e., the profiles are non-periodic. Recently, we have proposed a periodically chirped SFBG consisting of many identical, repeating super-periodic structures for implementing spectral Talbot effect [8]. Also, the multiple-phase-shift (MPS) technique [9] can be regarded as one kind of super-periodic structures.

After the introduction of “super-periodic structure,” it is necessary to further carry out a general and systematized investigation on SFBGs with such structures, which is similar to the research on periodic Talbot filters for pulse multiplication [14]. For this purpose, we propose a general configuration for SFBGs with super-periodic structures, and this general configuration allows a large degree of freedom to obtain spectral Talbot effect. The rest of this paper is organized as follows. In section 2, a general phase condition for spectral Talbot effect is derived and analyzed, which provides theoretical origin for periodic phase profile. Then, in Section 3, we are devoted to derive equivalent phase transition points for super-periodic structures. In order to get transition points, the parity of parameters involved in phase profile should be specifically designed. Then detailed analysis and discussion on phase profile, phase increment (phase condition) and spectral Talbot phenomena are given out. Finally, we draw a compact conclusion in section 4.

2. General phase condition for spectral Talbot effect

For an arbitrary FBG, we can express its refractive-index modulation profile Δn(z) by

where Δn is the “dc” index modulation (the fringe visibility is specified as 1), u(z) and θ(z) represent the sampling function and the phase profile, β ₀=2π/Λ₀ is the central spatial frequency, and Λ₀ is the central grating period. As for Talbot effect, the phase profile θ(z) is always the decisive factor. Aided by Taylor expansion formula, θ(z) can be written as

where θ_r is the r-th derivative of θ(z) at z = 0 . Meanwhile, we can get a general phase condition for spectral Talbot effect. In Ref. [13], authors have already analyzed phase conditions for integer and fractional Talbot effect, respectively. According to those phase conditions, we introduce a phase increment “θ(k) to express a general phase condition for both integer and fractional effect:

where N_k is an integer determined by k, m (a positive integer) is the multiplying factor of Talbot effect, and P is the sampling period. If m = 1, Eq. (3) corresponds to integer Talbot effect; if m = 2, 3, 4, ⋯, it stands for fractional effect. Then we return to the phase profile, Eq. (2). There are so many terms on its right side that it is hard to derive each θ_r if we treat all terms as an entity. For simplicity, by means of treating each term as an independent part to satisfy Eq. (3) respectively, θ_r can be easily obtained as:

where N_k,r is an integer determined by k and r As a coefficient of Taylor series, each θ_r should be a constant along z-axis. Namely θ_r is required to be independent of the variation of k. To eliminate the influence of k, we replace N_k,r with a fixed number N_k,r ^́. Consequently, θ_r can be expressed by Eq. (4).

Prior to our work, related investigation on high-order phase terms (θ ₃ and θ ₄) has been released [5]. Those high-order phase terms are restricted as an integral multiple of 2π, and as a result N_k,r ^́ is only set as an integral multiple of m. Obviously, it is just a specific example of Eq. (4) because N_k,r ^́ can be an arbitrary integer here, as long as (N_k,r ^́/m) is a rational number. In general, θ ₂ is likely to be involved in most cases [4, 6–9, 12], and the following derivation and analysis are concentrated on quadratic phase term.

3. Super-periodic structures for implementing spectral Talbot effect

In Section 2, we have derived a general phase condition and corresponding Taylor coefficients for spectral Talbot effect. They enable us to design conventional chirped-SFBGs conveniently. However, when it comes to the SFBGs with super-periodic structures, a series of extra equivalent transition points are needed to achieve spectral Talbot effect.

3.1 Phase transition points for super-periodic structures

First of all, we have to state an equivalent transition transform in the phase space. That is, an integral multiple of 2π (2Nπ) is equivalent to zero, where N is an integer. With this equivalence, the value of Eq. (1) keeps unchanged when we replace θ(z) = 2Nπ with θ(z) = 0 . Assuming that these transition points are located at sampling points

we can view these points as starting points of new super-periodic structures, where t is an integer. Such new super-periodic structures are reproductions of the original structure around z = 0 . In this way, such a SFBG comprises identical, repeating super-periodic structures, the so-called SFBG with super-periodic structures.

Here, we set the period of super-periodic structure as (s×m)P, where s is a positive integer. When only quadratic phase profile is considerer, Eq. (5) can be rewritten as

A simple analysis of Eq. (6) reveals that (s ² mN _k,2 ^́) must be an even number to satisfy Eq. (6). As long as any parameter in the set (s, m, N _k,2 ^́) is an even number, we can find suitable transition points. Note that Eq. (6) provides a general configuration for super-periodic structures. As reported in Ref [8], only the parity of N _k,2 ^́ has been taken into consideration. Here, besides N _k,2 ^́, m can be adjusted properly to obtain transition points. What is more, a new parameter s is introduced to get transition points for the first time, which allows a larger degree of freedom than before to obtain Talbot effect. When s > 1, the corresponding phase increment is greatly different from those investigated previously [6–9, 14] and we will give a detailed description next. In brief, the general configuration here accounts for a full-scale expansion of super-periodic structures.

Once super-periodic structures are adopted, “θ(k) is not a linear function any more, but a periodic function. The simplest example is that “θ(k) is equal to a constant, and this constant may be zero or mN _k,2 ^{́ π} according to Eq. (7) derived through the combination of Eqs. (2), (3) and (4).

On the other hand, “θ(k) can be a non-constant periodic function. There are probably a variety of such periodic functions and we will show a schematic diagram about such functions of certain case later in this work.

3.2 Concrete super-periodic structures and simulations

On the basis of the derivations and analysis above, we classify the general configuration of super-periodic structures into two cases.

 

Fig. 1. Phase profiles of (a) conventional chirped-SFBG, (b) proposed SFBG with and “θ(k) = 0, and (c) proposed SFBG with “θ(k) = mN _k,2 ^́ π; (d) chirp effect corresponding to phase profile (a).

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Fig. 2. Filtering characteristics of SFBGs with super-periodic structures (s =1, N _k,2 ^́ =1, m = 2)

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First of all, when s = 1 is offered, either m or N _k,2 ^́ is required to be an even number, so that spectral Talbot effect can be realized. In this case, “θ(k) is zero or mN _k,2 ^́ π, and detailed phase profiles are presented in Fig. 1. Fig. 1(a) illustrates the quadratic phase profile of conventional chirped-SFBG. For SFBGs with super-periodic structures, the phase profile for “θ(k) = 0 is shown in Fig. 1(b), while the phase profile for “θ(k) = mN _k,2 ^́ π is illustrated in Fig. 1(c). We can see that each period of phase profiles originates from the section around z = 0 . It is convenient to derive a corresponding chirp effect Λ(z) for Fig. 1(b), which is indicated in Fig. 1(d). However, it is not easy to get a resulting chirp effect for Fig. 1(c), and now we just introduce such a phase profile in theory.

To verify these analyses of spectral Talbot effect, we have to carry on some simulations through transfer matrix method [15]. The detailed parameters are specified as: “n = 5×10^-4, Λ₀ = 521.89nm, the effective index n_eff =1.485, the duty cycle p = 0.08, and the total number of sampling periods T = 24. The sampling period is set as P = 1.0 mm, implying a channel spacing of 0.8 nm for uniform SFBGs. If m is an even number (m = 2), the resulting filtering characteristics are shown in Fig. 2. Obviously, compared with uniform SFBGs, the number of channels over a given wavelength region has increased to m times, while group-delay characteristic is nearly identical in shape. Here, we skip analogous simulations about the parity of N _k,2 ^́ (such as N _k,2 ^́ = 2) as they have been discussed in Ref [8].

Secondly, there exists a more complicated case when s > 1. If s = 3, 5, 7, 9⋯, one between m and N _k,2 ^́ is required to be an even number, which is identical to the case s = 1. More importantly, if s = 2, 4, 6, 8⋯, the spectral Talbot phenomenon can always be realized, regardless of the parity of m or N _k,2 ^́.

As predicted previously, the phase increment here is a non-constant periodic function. In this case, the period of phase profile [Fig. 3(a)] is (s×m)P, not (mP). According to Fig. 3(a), we are capable of deriving a periodic function “θ(k) shown in Fig. 3(b), which might be the most remarkable difference between the two cases (s = 1 and s > 1).

 

Fig. 3. Schematic diagram of (a) phase profile and (b) phase condition (phase increment) for SFBGs with super-periodic structures when s > 1.

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Similarly, spectral Talbot phenomena are obtainable through numerical simulation. In Fig. 4, the channel spacing has reduced by m times and the group-delay characteristic is nearly identical with that of uniform SFBG. In addition, such derived parameter sets can be utilized to design temporal, periodic Talbot filters [14].

3.3 Discussions

By combining the result of Ref. [14] and ours, both temporal and spectral Talbot effect can be realized by super-periodic structures. The phase profile required by equivalent transition transform is the same and the fabrication process may be a bit complex. On the other hand, such SFBGs with super-periodic structures possess several outstanding features. Like conventional chirped-SFBGs, these SFBGs are with high energetic efficiency and tunable control of channel spacing (i.e., spacing is not being fixed solely by sampling period) [2, 4–7]. More importantly, compared with conventional chirped-SFBGs, these proposed SFBGs can provide a larger degree of freedom to design multi-channel filters. Three parameters in the set (s, m, N_k,2 ^́), rather than only m, can be adjusted properly. As far as we know, this obtained degree of freedom is the largest one to implement spectra Talbot effect. Furthermore, proposed SFBGs allow a larger tolerance on chirp coefficient because super-periodic structures can bring with a relatively lower deviation on phase profile [8].

 

Fig. 4. Filtering characteristics of proposed SFBGs with different parameters: (a) s=2, N _k,2 ^́ =1, m =3, (b) s = 3, N _k,2 ^́ =1, m = 2.

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4. Conclusion

We have introduced a general configuration of SFBGs with super-periodic structures for achieving spectral Talbot effect. As long as any parameter of the set (s, m, N_k,2 ^́) is an even number, a series of equivalent transition points can be determined, which corresponds to new super-periodic structures. On the basis of this principle, we succeed in achieving the spectral Talbot phenomena with a large degree of freedom. In the meantime, the phase profile and phase condition (phase increment) with different parameters are analyzed. The phase increment is zero or non-zero constant if s = 1, while the increment is a non-constant periodic function if s > 1.