1. Introduction

Multichannel fiber Bragg gratings (FBGs) have attracted great interest in the dense-wavelength-division-multiplexed (DWDM) systems. A single sampled FBG can show high performance in the operation for filtering and/or chromatic dispersion compensation simultaneously in multiple channels [1–3]. Such sampled multichannel FBGs can be implemented by modulating the seeding grating with either a periodic amplitude or phase sampling function or both [4–9]. In such structures, the free spectral range (FSR) of the comb filter is solely determined by the sampling period P. Recently, it has been shown both theoretically and experimentally that, in an amplitude sampled and chirped fiber grating, a spectral Talbot phenomenon appears if certain conditions between the grating period chirp and the sampling period are satisfied [10–15]. In Ref. [13], a phase shift between adjacent sampling periods was added in addition to the strong chirp to increase the design and fabrication flexibility of the amplitude sampled FBGs. With the Talbot effect, the FSR can be arbitrarily tuned by changing the grating chirp for a fixed sampling period. The spectral Talbot effect provides a great flexibility in fiber grating designs and advantages in energy efficiency enhancement compared to the conventional amplitude sampled uniform gratings. Comparing with the amplitude sampled FBGs, phase-only sampled FBGs (PSFBGs) [7–9, 16-19] have recently attracted great interests due to much higher energy efficiency than that of the amplitude sampled counterpart due to the limited duty cycle of the seeding grating over the sampling period in the amplitude sampled FBG. In a phase-only sampled FBG, the grating phase in each sampling period must be changed with either discrete or continuous mode. For a discrete phase-only sampling function, each period is divided into a number of discrete steps with either a fixedπ phase shift at optimized positions (Dammann type) or optimized phase shifts at a fixed position (multi-level type) by using the phase-shifted mask technique [18]. The phase-shifted phase mask approach has already been used to write multi-channel FBGs with five to nine channel with the help of high accurate e-beam fabricated phase mask, in which the typical number of phase transitions in each sampling period is ~10 or more [7–9]. With the increasing number of channels greater than 40, however, optimization for the discrete sampling function requires as many as 100 phase shifts within each sampling period of 1mm (the channel number is closely relevant to the number of phase transitions in each sampling period), which becomes too difficult to be realized. To address the issues of the too many phase transitions involved in the discrete phase-only FBGs, a continuous phase-only sampling approach was suggested, in which the continuous phase profile was encoded into every period of the seed grating by using the advanced stitch-error-free lithography mask [17, 18]. Optimization for 45- and 81-channel phase-only sampling FBGs has been carried out and demonstrated experimentally.

In this paper, we propose a novel method that utilizes the spectral Talbot effect in the PSFBGs to obtain the broadband high-count channels with possibly much less phase transitions than usual in the discrete phase-only sampled FBGs. Specifically, we first demonstrate the Talbot effect in the phase-only sampled FBGs, in which both integer or fractional Talbot effect were observed in both Dammann binary and multi-level type of PSFBGs. The optimized locations and/or the phase shifts were implemented by using simulated annealing (SA) algorithm [20] such that an array of spectral channels with equal and maximal reflectivity appears. The characteristics of the reflectivity and group delay of the PSFBGs with Talbot effect were then compared with that of PSFBGs with no Talbot effect. Finally, it is shown that the Talbot effect based phase-only sampled FBGs is capable of generating superior high-count DWDM channels in terms of channel bandwidth, channel number, channel uniformity and energy efficiency.

2. Theory of Talbot effect of PSFBGs

In a phase-only sampled FBG, the refractive index modulation Δn ₀ (z) of length L along the fiber axis z can be expressed as follows:

where Δn _DC (z) is the background or DC component of the effective refractive index modulation, Δn ₀ is the peak refractive-index modulation, G(z) describes the normalized apodization function, Λ(z) is the grating period at z with Λ₀ being the nominal grating period, and ϕ ₀ represents the additional chirp function that contains grating phase. The sampling function with a period of P can be written as:

where s _p(z) is the sampling function in one period. In amplitude sampled FBG, s _e(z) is usually a rectangular function while in phase-only sampled FBG, it is a function of the grating phase. δ(x) is the delta function and symbol “*” represents the operator of convolution, n is an integer (n=0, 1, 2…), and the period P is solely determined by the required channel spacing Δλ, i.e., FSR with the relationship:

where λ _B=2n ₀Λ₀ is the wavelength satisfying the Bragg condition, n ₀ represents the average refractive index of the propagating mode (we assume that the grating exhibits a constant average refractive index). The in-band amplitude and phase characteristics follow those of the seeding grating, and the amplitude and phase relations between the different spectral bands are determined by the Fourier spectrum (discrete Fourier coefficients) of the periodic sampling function. The sampling function s _p (z) of a discrete PSFBG for one period can be expressed as:

Where θ _m and z _m with m=0, 1, 2,…K denote the value and position of the phase transition, respectively. From Eq. (4), two types of phase sampling function can be derived, i.e., Dammann and multilevel sampling function. In Dammann sampling function, s _p (z) is a binary phase-only function that contains K alternative phase segments of 0 and π of varying segment lengths in each sampling period, i.e., θ _m=mπ at optimized position z _m. The Fourier coefficients of s _p (z) for a Dammann phase sampling can be written as:

In multi-level sampling function, arbitrary phase shift θ _m in the range (-2π, 2π) is arranged or optimized at equally divided position z _m within one sampling period. The Fourier transform of s _p (z) gives the coefficients of a multi-level sampling function as follows:

In both Dammann and multi-level phase-only sampled FBGs, the linear relation between coupling coefficient κ _n and Fourier coefficients S _n can be expressed as follows:

The maximum reflectivity of the nth channel in the reflection spectrum of a PSFBG can thus be determined using κ _n,

The desired WDM channel output in terms of the channel number, channel uniformity and maximization of the channel energy can be optimized by a minimization process that is set as follows:

where n=-N,…, 0,…, N, represents the channel order that comes from the corresponding Fourier order, the total channel is M=2 N+1. In Eq. (11), C is a cost function that will be minimized. W _n1 and W _n2 are the weight factors of reflectivity inside and outside the spectral band of interest, respectively, for the nth channel and may have values between 0 and 1, depending on the priority of the selected channel. r ₀ and r ₁ are the target reflectivity for the spectral band of interest and outside of the spectral band of interest, respectively. r0 is usually set to 1 for channels from -N to N, r1 is set to 0 for channels outside -N to N, and the weight factorsW _n1 andW _n2 are both set to 1 usually to maximize contributions equally from the inside and outside spectral band of interest to the cost function C.

For Dammann PSFBGs, solution is started by a trial set of z _m consisting of K elements, an optimization process is then used to determine a series of z _m that minimizes C. For multi-level PSFBGs, solution is to determine a series of θ _m at the pre-determined positions that minimizes C. The simulated annealing (SA) method was used to the multi-variable global optimization process. A set of initial trail z _m (0 ≤ z_m ≤ 1) (for Dammann) [or θ _m (0 ≤θ _m ≤ 2π) for multi-level] with m=1,…, K are chosen randomly. The cost function is then evaluated and minimized which results in a series of optimal solutions of z _m (or θ _m).

With the optimized solution of z _m (or θ _m), we can obtain the spectral output which contains 2N+1 uniform channels with possibly maximal magnitude. In the case of zero or small chirp introduced in the fiber grating, the channel spacing, i.e., FSR is solely determined by the sampling period P, Eq. (3). The total channel number obtainable is dependent on the number of the phase transition K adopted in one sampling period. When the grating chirp increases further such that the interference among the spectrally overlapped bands from different sampling periods re-exhibits a periodic set of nearly non-dispersive reflection peaks which are either a replica or fractional replica of the original ones, a spectral Talbot effect occurs. If the grating is in a linear chirp with:

where c _g is the chirp coefficient of the grating period, and L is the grating length. The Talbot condition among Λ₀, P and c _e can be expressed as follows [10, 11]:

where s and m are integers (m being positive), such that s/m is an irreducible rational number, the value of m decides if an integer or a fractional self-imaging Talbot effects occurs [10]. An integer spectral self-imaging Talbot effect occurs when m=1, in which a direct integer spectral Talbot effect occurs if s is an even integer, and an inverse integer spectral Talbot effect occurs if s is an odd integer. A fractional spectral self-imaging Talbot effect occurs when m>1, with (s/m) as a non-integer and irreducible rational number. The fractional Talbot effect results in a channel wavelength spacing m times smaller than the original one (i.e., the number of wavelength channels is effectively multiplied). This is valid for any value of m as long as the reduced wavelength spacing Δλ/m is larger than the bandwidth of the discrete bands. Similarly, a direct fractional or an inverse fractional spectral Talbot effect occurs when the product (sm) is an even or an odd integer. The spectral Talbot effect provides a new concept in phase-only sampled FBG design in that (1) the FSRs can be arbitrarily tuned by the chirp coefficient; (2) much wider spectral range than the conventional sampled FBG can be obtained due to the chirp coefficient introduced in the grating; and (3) much less phase transitions in the phase-only sampled FBGs will be needed than the conventional ones to achieve the same or more channel numbers. All the above mentioned features in the PSFBGs will be demonstrated in the following section.

3. Simulations and discussions

We will first examine the Talbot effect that occurs in the phase-only sampled FBGs. The simulation is based on the transfer-matrix method [21]. Two kinds of phase-only sampled FBGs, i.e., Dammann phase sampling function and multi-level phase sampling function are examined. Figures 1(a) and 1(b) gives the optimized phase sampling function on s _p (z) that contains K elements per sampling period for a Dammann PSFBG and a multi-level PSFBG, respectively. In Fig. 1(a), the number of phase transition points is six in each sampling period with the fixed phase values either 0 or π at optimized positions, while the number of phase transition in the multi-level PSFBG (Fig. 1(b)) is ten distributed at equally divided positions with optimized values in each sampling period. It is noted that the actual phase levels in Fig. 1(b) are only five because the symmetry principle was applied to reduce the variables and the complexity in the optimization process [22]. Figure 2 and Fig. 3 shows the multi-channel reflectivity spectra and the corresponding group delay of PSFBGs with the optimized phase transition shown in Fig. 1(a) (Dammann type) and 1(b) (multi-level type), respectively. In simulations of both Figs. 2 and 3, the following grating parameters are used: n ₀=1.485, Λ0=521.8855nm (corresponding Bragg wavelength λ _B=1550nm), Δn ₀=6.0 ×10^-4, and the sampling period P=1.0mm. Figure 2 shows the Talbot effect observed in a Dammann binary PSFBG with different Talbot conditions. Figures 2(a), 2(b), 2(c) and 2(d) show the spectral reflectivity while Figs. 2(e), 2(f), 2(g) and 2(h) show the corresponding group delay of Figs. 2(a), 2(b), 2(c) and 2(d), respectively. The total grating length is 6cm. In Fig. 2(a), s=0, m=1, c _g=0, i.e., an uniform PSFBG, the FSR is ~0.8nm (100GHz) determined by the sampling period P=1mm, as expected. In Fig. 2(b), s=1, m=1, c _g=5.2188×10^-4 1/mm, the FSR is still ~0.8nm (100GHz), an inverse integer spectral Talbot-effect, as expected. In Fig. 2(c), s=1, m=2, c _g=2.6094×10^-4 1/mm, it is seen that the FSR is reduced by a factor of 2 to 0.4nm (50GHz) with the same sampling period P=1mm. In Fig. 2(d), s=2, m=3, c _g=3.4792×10^-4 1/mm, it is a direct fractional Talbot effect, the FSR is reduced by a factor of 3 to 0.27 nm (33GHz).

 

Fig. 1. (a). Distribution of the phase transition at different positions with binary phase values (0 or π) of a Dammann PSFBG; (b). Distribution of the phase transition with optimized values at equally distributed positions (0.1mm) of a multi-level PSFBG.

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Fig. 2. Talbot effect observed in a Dammann binary PSFBG with different Talbot conditions. Figures 2(a), 2(b), 2(c) and (2d) show the spectral reflectivity and Figs. 2(e), 2(f), 2(g), 2(h) show the corresponding group delay of Figs. 2(a), 2(b), 2(c) and 2(d), respectively, in which two or three channels are zoomed with dashed lines for clarity purpose. (a) s=0, m=1(c _g=0, i.e., uniform PSFBG); (b) s=1, m=1 (c _g =5.2188×10^-4 1/mm); (c) s=1, m=2 (c _g=2.6094×10^-4 1/mm); and (d) s=2, m=3 (c _g =3.4792×10^-4 1/mm).

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Fig. 3. Talbot effect observed in a multi-level PSFBG with different Talbot conditions. Figures 3(a), 3(b), 3(c) and 3(d) show the spectral reflectivity and Figs. 3(e), 3(f), 3(g), 3(h) show the corresponding group delay of Figs. 3(a), 3(b), 3(c), 3(d) respectively, in which two channels are zoomed with dashed lines for clarity purpose. (a) s=0, m=1(cg=0, i.e., uniform PSFBG); (b) s=1, m=1 (c _g=5.2188×10^-4 1/mm); (c) s=1, m=2 (c _g=2.6094×10^-4 1/mm); and (d) s=2, m=3 (c _g =3.4792×10^-4 1/mm).

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It is also interested to see that the corresponding in-band group delay in Figs. 2(e), 2(f), 2(g) and 2(h) show symmetric shape that implies essentially zero dispersion in the center and small dispersion at the edges of the channel, which is similar to that of a uniform FBG although we are using a chirped seeding grating. This is an important feature of the spectral Talbot effect in which both reflection and the group delay show the self-imaging effect. Similarly, Fig. 3 shows the Talbot effect observed in a multi-level PSFBG with different Talbot conditions which is set exactly the same as that in Fig. 2, i.e., the total grating length is 6 cm, P=1mm. Not surprisingly, the various Talbot effects, i.e., inverse integer in Fig. 3(b) and direct fractional in Figs. 3(c), 3(d) respectively, can also be observed in the multi-level PSFBGs.

When the performance of the conventional PSFBGs and the Talbot effect based PSFBGs are compared, Fig. 4 shows the detailed comparison spectrum with and without Talbot effect for both Dammann (Figs. 4(a) and 4(b)) and multi-level (Figs. 4(c) and 4(d)) type of PSFBGs. In Figs. 4(a) and 4(b), the total grating length is 8cm, and the Dammann type phase transition is shown in Fig. 4(e), in which five transitions are adopted. Figure 4(a) shows the spectrum of a conventional PSFBG with no Talbot effect, i.e., c _g=0, in which the adjacent wavelength spacing (FSR)) is Δλ=0.4nm which is determined by the sampling period P=2mm. In order to obtain the same FSR (i..e, 0.4nm) as that with the conventional PSFBG, we employed a fractional Talbot effect, in which P=1mm, s=1, m=2, c _g=2.6094×10^-4 (1/mm), the same total grating length and the refractive index modulation as that without Talbot effect, i.e., 8cm and 6.0×10^-4, respectively, are used. The spectrum output is shown in Fig. 4(b). Similarly in Figs. 4(c) and 4(d), the spectrum output with (Fig. 4(d)) and without (Fig. 4(c)) Talbot effect for a multi-level type PSFBG is compared. The phase transition for the multi-level PSFBG is shown in Fig. 4(f), in which six phase levels are adopted. In both Figs. 4(c) and 4(d), the channel spacing is 0.4nm, the total grating length is 4cm and the refractive index modulation is 6.0×10^-4. It is seen clearly from both type of PSFBGs that much better channel uniformity, higher energy efficiency, much wider spectral range, and much higher channel isolation than those of the conventional PSFBGs can be obtained by using the Talbot effect. These features are inherit to the Talbot based PSFBGs, which are especially useful for generating broadband high-count-channel DWDM devices. With the Talbot effect in the discrete phase-only sampled FBGs, we can use very limited number of phase transitions in either Dammann or multi-level type of PSFBGs to generate high-count channels which is otherwise impossible to use the conventional discrete PSFBGs or have to resort to the complicated continuous phase profile.

 

Fig. 4. The detailed comparison spectrum with and without Talbot effect for both Dammann and multi-level type of PSFBGs. In the plots, the channel spacing(FSR) is all ~0.4nm, P is the sampling period, T is the total number of sampling period, s and m decide the chirp coefficient c _g, i.e., different Talbot conditions. (a) and (b): The total grating length is 8mm with (a) No Talbot effect, and (b) Talbot condition at s=1, m=2(c) and (d): The total grating length is 4mm with (c) no Talbot effect, and (d) Talbot condition at s=1, m=2; (e) and (f): The phase distribution of the Dammann and multi-level PSFBGs.

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It should be noted the difference in diffraction efficiency between the conventional (i.e., no Talbot effect) Dammann type phase-only and multi-level type phase-only sampled FBGs shown in Fig. 4(a) and Fig. 4(c), respectively. While Fig. 4(c) shows clearly 9 channels around 1550nm when using the optimized multi-level phase distribution shown in Fig. 4(f), Fig. 4(a) seems to have more channels covering a wider wavelength range than that in Fig. 4(c) although an optimized phase distribution of Fig. 4(e) is employed. This can be mainly attributed to the inherit difference in diffraction efficiency between these two different type of gratings. It is well known that the diffraction efficiency of a binary Dammann sampling function is typically 80%, implying nearly 20% of the power goes to the out-of-band channels which means high reflection peaks in out-of-band region [16]. This is especially true when the grating is long, e. g. 8cm shown in Fig. 4(a). With the increase of the grating length, both in-band and out-of-band reflection peaks increase which results in a high out-of-band peaks with the saturated central in-band peaks. On contrast, the multi-level type of phase-only sampled FBGs can show much lower suppressed out-of-band peaks than that of Dammann type of FBGs due to the high diffraction efficiency in the optimized multi-level phase distribution [16].

 

Fig. 5. High-count-channel generation using PSFBG based on the fractional Talbot effect. (a): no Talbot effect; (b) fractional Talbot effect, s=1, m=2, and the number of multi-levels is six, total 44 channels; (c) fractional Talbot effect, s=1, m=2, and the number of multi-levels is ten, total 80 channels; (d), (e) and (f) are the phase distribution of the multi-lever PSFBGs corresponding to (a), (b) and (c).

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Figure 5 shows the capability of using discrete phase-only sampled FBG to generate 44- and 80 channels with a FSR of 0.4nm. In all the plots, we assume the refractive index modulation is 8.0×10^-4. Figure 5(a) gives the spectral output of a conventional discrete multi-level phase-only sampled FBG with 40 phase transitions in one sampling period. The phase transition used for Fig. 5(a) is shown in Fig. 5(d). In Fig. 5(a), the following parameters are used: sampling period P=2mm, total number of sampling period is T=40, total grating length=8cm, s=0, i.e., c _g=0(no Talbot effect). From Fig. 5(a) it is seen that the effective channel number is ~36 with poor channel isolation. In Fig. 5(b), the spectral output from a Talbot effect based multi-level phase-only sampled FBG is shown, in which sampling period P=1mm, total number of sampling period is T=60, total grating length is 6cm, s=1, and m=2. The arrangement of the phase transition for Fig. 5(b) is shown in Fig. 5(e), in which total 6 phase transitions only are adopted. The number of output channel reaches 44 in Fig. 5(b). With the application of the Talbot effect, we show further the design of a 80-channel multi-level phase-only sampled FBG in which P=1mm, total grating length=10cm, s=1, and m=2. The total number of phase transition is 10, shown in Fig. 5(f). It is very clear from Fig. 5 that, based on the Talbot effect, the discrete phase-only sampled FBGs with very limited number of phase transitions (e.g. 6 or 10 in one period) can generate very high-count channels which could cover the whole C band, breaking the limitations with the conventional discrete phase-only sampled FBGs, in which only several channels(<10 channels) can be obtained with 10 or more phase transitions in one period.

It should be useful and helpful to discuss the physical origin behind the much improved channel capability and quality using Talbot based PSFBSs. The physical mechanism behind these features are the re-distribution and subsequently the interference of the shifted central wavelength from each single sampling period due to the strong chirp introduced to the seed gratings. As discussed in Ref. [11], the spectral Talbot phenomenon is based on the spectral interference among many cascaded gratings spatially separated by the sampling period P. When the grating period is uniform (i.e., no chirp), the spectral contribution from each of the sampling period is the replication of the first period, i.e., the spectral reflection peaks (complex amplitude) from different sampling periods are overlapped with each other in the exact same wavelength region. In the chirped PSFBGs, however, the central wavelength of the reflection peak from each sampling period shifts a δλ determined by the grating chirp coefficient although the shape and magnitude of the reflection peak from each sampling period are exactly the same as those in the uniform grating. Figure 6 shows the reflection amplitude from each individual sampling period of a PSFBG. In Fig. 6, the reflection amplitude from the first, second, third, the 15^th, and 60^th sampling period is shown, respectively. The final output of the PSFBG is the coherent interference of all these complex amplitude from each individual sampling period. The parameters used in Fig. 6 are: total grating length 6cm, P=1mm, T=60(total number of sampling period), and c _g=2.6094×10^-4 (1/mm). As can be seen, the total spectral bandwidth is extended to ~24nm for the 6cm-long grating, compared with the total ~5nm bandwidth (determined by the bandwidth of each individual reflection peak) in the case of no Talbot effect. Besides the significant extension of the total bandwidth, the great improvement in the channel uniformity can easily be explained from the interference among the wavelength shifted reflection peaks (Fig. 6). The shifting of the wavelength from each individual sampling period results in a uniformly distributed interference spectrum over the total wavelength range in contrast to the sampled uniform FBG (no Talbot effect). It should be noted that higher values of m (fractional Talbot, Eq. (13)) can result in smaller channel FSR (i.e., higher channel density, see Fig. 2(d) and Fig. 3(d)) with the same number of phase transitions in one sampling period as those with lower m (Fig. 2(c) and Fig. 3(c)), the s value in Eq. (13), however, should be increased accordingly in order to maintain the same total effective spectral range.

 

Fig. 6. The amplitude of reflection coefficient from a sequence of decomposed individual gratings (the first three, the 15th, and 60th shown in the graph with chirp coefficient s=1, m=2) of a PSFBG. The grating parameters are the same as those given in the Fig. 2.

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The fabrication tolerances of grating parameters, especially the grating chirp coefficient c _g have been estimated using numerical simulations. It is found that if c _g deviates from the nominal values by 1.5%, the channel isolation and uniformity is practically unaffected. The deviation translates into distortions in the spectral channels, which are slightly more irregular and broader. For a 6cm-long PSFBG with a chirp coefficient of s=1, m=2, the average 3-dB channel bandwidth is 0.13497 nm with a maximum fluctuation of 13.90%, which is on the same level as that seen in the counterpart of the amplitude sampled FBGs [12], which implies that no additional difficulties will be imposed to the fabrication of phase-only sampled FBGs considering the successful implementation with less than 1.4% control error in the grating chirp in amplitude sampled FBGs [14].

4. Conclusions

In this paper, we obtained the high-count channels with very limited phase transition segments based on the Talbot effect in discrete phase-only sampled FBGs. Various integer or fractional Talbot effects are examined for both Dammann and multi-level type of PSFBGs using different Talbot conditions. It is shown that with the total number of 10 phase transitions only in one sampling period, 80 channels that cover the whole C band can be obtained. The Talbot effect based discrete PSFBG (T-DPSFBG) provides a novel method for generating high-channel-count FBG filters with great uniformity and energy efficiency that is otherwise difficult or impossible using conventional discrete phase-only sampled FBGs.