1. Introduction

The letter presents a technique for Nth-order differentiation of periodic pulse train. In addition to the interest in optical computing and information systems, Nth-order differentiators are of immediate interest for generation of Nth-order Hermite-Gaussian (HG) temporal waveform from an input Gaussian pulse, which can be used to synthesize any temporal shape by superposition [1]. Several all-fiber schemes have been previously proposed based on long-period fiber gratings [2], phase-shifted fiber Bragg gratings (FBGs) [3,4], two-arm interferometer [5], and two oppositely chirped FBGs [6].

A schematic of the proposed general architecture is shown in Fig. 1. This approach exploits the well-known property of linearly-chirped FBGs, which apodization profile maps its spectral response [6–11]. The dispersion introduced by the FBG must meet two conditions: first, it must be high enough so the grating profile of the FBG maps the spectral response of the differentiator [8,9]. Second, it must meet the temporal Talbot condition [12], so dispersion does not affect to the waveform of the output pulses.

Besides the inherent advantages of FBGs (all-fiber approach, low insertion loss, and the potential for low cost), this scheme avoids the concatenation of N first order differentiator devices, which reduce energetic efficiency and increase the implementation complexity. Two different FBG based approaches have been previously demonstrated for all optical N-order time differentiation, namely multiple-phase-shifted FBG [4] and two oppositely chirped FBGs [6]. Concerning the first solution [4], it provides optical operation bandwidths in the tens-of- GHz, while our approach is specially suited for differentiating ultra-broadband optical waveforms (e.g. picosecond and sub-picosecond optical pulses). Regarding the second one [6], it requires two oppositely chirped FBGs, while this approach only requires one FBG. Furthermore, this approach can simultaneously multiply the input repetition rate. As a drawback, the system depends on the repetition rate of the input pulse train.

 

Fig. 1. Architecture of the system. Periodic pulse train is processed by an apodized linearly chirped FBG.

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2. Theory

The analytic expression of an Nth-order differentiator in temporal domain is f_out(t)=d^N_fin(t)/dtN, where f_in(t) and f_out(t) are the complex envelopes of the input and output of the system respectively, and t is the time variable. In frequency domain, F_out(ω)=(jω)^N F_in(ω), where F_in(ω) and F_out(ω) are the spectral functions of f_in(t) and f_out(t), respectively (ω is the base-band frequency, i.e., ω=ω_opt-ω₀, where ω_opt is the optical frequency, and ω₀ is the central optical frequency of the signals). Thus, an Nth-order differentiator is essentially a linear filtering device providing a spectral transfer function of the form H_N(ω)=F_out(ω)/F_in(ω)=(jω)^N. We are interested in obtaining an analytic expression of a feasible spectral response, so the ideal spectral response function must be windowed, H_N,w(ω)=H_N(ω)W(ω)=(jω)^NW(ω), where W(ω) is a window function.

The objective is to obtain a spectral response proportional to the differentiator spectral response. The chirped FBG introduces a dispersion term and we have H_r(ω)=(R(ω))^1/2 exp(jϕ_r(ω))∝|H_Nw(ω)|exp(jϕ_N(ω)+jω ² $\ddot{\varphi }$ _r/2) where H_r(ω), R(ω), and ϕ_r(ω) are the spectral response in reflection, reflectivity and phase of the FBG, ϕ_N(ω)=phase(H_N(ω))=phase(H_N,w(ω)) and $\ddot{\varphi }$ =∂2ϕ_r(ω)/∂ω ² is the first order dispersion coefficient of the FBG, which is a constant value for linearly chirped FBGs. Regarding the reflectivity, we have:

The refractive index of the FBG can be written as:

where n_av(z) represents the average refractive index of the propagation mode, Δn_max describes the maximum refractive index modulation, A(z) is the normalized apodization function, Λ₀ is the fundamental period of the grating, φ(z) describes the additional phase variation (chirp), and z∈[-L/2,L/2] is the spatial coordinate over the grating, with L the length of FBG. In the following we consider a constant average refractive index n_av=n_eff+(Δ_nmax/2), where n_eff is the effective refractive index of the propagation mode.

Notice that when N is odd, the differentiator spectral response presents a π-phase shift at ω=0. In our approach, this condition is attained by introducing a π-phase shift in the grating at z=0.

The chirp factor of the FBG, which is defined as C_K=∂²φ(z)/∂z², and the length of the grating L, can be calculated from [13]:

where c is the light vacuum speed, and Δω_g is the grating bandwidth.

The phase filtering of the FBG must be designed to cause a Talbot effect. In general, the pulses waveform of the periodic pulse train is affected by dispersion, but under Talbot condition [12] the pulses are reflected without undergoing distortion (self-image effect). This condition can be expressed as:

where s/m must be an irreducible rational fraction. Repetition rate multiplication can be achieved for m>1. As a result the reflected signal has a repetition rate m times that of the input signal.

Regarding the amplitude filtering, we apply an apodization profile to the grating that is accurately mapped on the spectral response under high dispersion condition [9], which can be expressed as:

where Δt_g can be calculated from the temporal length of ℑ^-1[H_N,w(ω)], and ℑ^-1 denotes inverse Fourier transform.

It is worth noting that in most cases T ²≫(Δt_g)², and it is probable that (5) not only satisfies (6), but greatly exceeds it, so from (4) we can deduce that it is necessary to use a longer FBG than strictly required for space-to-frequency mapping.

The apodization profile can be obtained from the expression [8]:

3. Examples and results

In this section, examples of 1st and 4th order differentiators are designed and numerically simulated. We assume a carrier frequency (ω₀/2π) of 193 THz, an effective refractive index n_eff=1.45, a band of interest (Δω/2π) of 5 THz centred at ω ₀(ω ₀-Δω/2≤ω _opt≤ω ₀+Δω/2), a grating bandwidth Δω_g=Δω, a maximum reflectivity of 50 %, and a pulse train period T=40 ps.

For the first example (1st order differentiator) the corresponding ideal spectral response is H₁(ω)=jω. We choose a function based on a hyperbolic tangent as window, W(ω)=W_th(ω)=(1/2)[1+tanh(4-|16ω/Δω_g|)], and we have H_1,w(ω)=H₁(ω)W_th(ω). The desired reflectivity is obtained from (1):

where C_R=2.1238 is a normalization constant to get a maximum reflectivity of 50 %.

From the temporal length of ℑ^-1[H_1,w(ω)] we obtain Δt_g≈2 ps. Using expressions (5)and (6) we have $\mid {\ddot{\varphi }}_r\mid =\frac{s}{m}2.5464\times {10}^{-22}{s}^2⁄\mathrm{rad}$ and |$\ddot{\varphi }$ _r|≫1.5915×10^-25 s ²/rad.

We choose $\ddot{\varphi }$ _r=-1.2732×10^-22 s ²/rad, where s=1 and m=2 have been selected. This implies that the input repetition rate is multiplied by two, so we have an output period of repetition T_out=T_in/2=20 ps. The odd order of 1st differentiator implies that π-phase shift must be introduced in the grating at z=0.

Using (7), we obtain Δn_max=2.8160×10^-4, n_av=1.45014. Additionally, using (3) and (4), we obtain C_K=-7.3508×10⁵ rad/m² and L=41.346 cm. The fundamental period of the grating can be obtained from Λ ₀=πc/(n_avω₀)=535.574 nm. Finally, using (7) we obtain the apodization profile function:

where C_A=1.2011 is a normalization constant selected to get a normalized apodization profile function 0≤A(z)≤1, and N=1.

As a second example we design a 4th-order differentiator using the same methodology. We obtain again Δt_g≈2 ps, and we design the same technological parameters as in the first example (so repetition rate is also doubled). The apodization profile is given by (9), where C_R=181.02, and N=4 (same L and C_A as for first example).

Figures 2(a), 2(b), and 2(c) show the results from our numerical simulations corresponding to the first example, and Fig. 2(d), 2(e), and 2(f) show the corresponding to second example. The spectral responses of the designed FBG and the ideal differentiator are showed in Fig. 2(a) and 2(d) for first and second example, respectively. The output pulse corresponding to an input gaussian pulse described by f_in,1(t)∝exp(-t ²/(2σ ²)) with σ=800 fs, are showed in Fig. 2(b), and 2(e) for first and second example, respectively. The output pulse corresponding to an antisymmetric HG pulse described by f_in,2(t)∝∂ f_in,1(t)/∂t∝t·exp(-t ²/(2σ ²)) are showed in Fig. 2(c), and 2(f) for first and second example, respectively.

 

Fig. 2. Plots (a) and (d) show the amplitude of the spectral response corresponding to the FBG (solid), and to an ideal differentiator (dashed) for first and second examples, respectively. The temporal waveforms are showed in plots (b) and (c) for first example, and in plots (e) and (f) for second. Plots (b) and (e) correspond to a Gaussian pulse as input, and plots(c) and (f) correspond to an antisymmetric Hermite-Gaussian pulse as input. In plots (b), (c), (e) and (f) we show the input pulse in dashed line, the ouput pulse for the designed system in solid line, and the output pulse for the ideal differentiator in dotted line (indistinguishable from solid line in (b) and (c), and hardly distinguishable from solid line in (e) and (f)).

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From (4) and (6) we obtain that the length of the grating requires L≫0.051 cm for space-to- frequency, which can be satisfied from L>5 cm. The length of the grating designed (41.346 cm) is much longer, but is within the accuracy of currently available fabrication techniques, as it can be seen in [11], where Talbot effect was achieved in a 96 cm linearly chirped FBG.

Considerations about the effect of group delay ripple can be found in [14], where it has been associated to the amplitude jitter of the rate-multiplied pulse train.

4. Conclusion

In this letter, an apodized linearly chirped FBG is designed and numerically simulated to simultaneously perform amplitude filtering to obtain the spectral response of the Nth-order differentiator, and phase filtering to cause temporal Talbot effect. The amplitude filtering exploits the fact that apodization profile of linearly chirped FBG maps its spectral response. This idea was used in a previous article published recently [6]. The main difference is that second chirped FBG for compensation of dispersion introducing by first FBG proposed in [6] is not required. This method reduces the number of required FBG to one and in addition the use of the Talbot effect allows repetition rate multiplication. These advantages must be weighed against the disadvantage that the system depends on the input repetition rate, and that the FBG length required for temporal Talbot effect can be much longer than the strictly required for space-to-frequency mapping.

It is worth noting that, unlike other pulse shaping techniques based on chirped FBGs [7–11], where the objective is shaping a specific output pulse from a known input pulse waveform, this system performs an operation (Nth-order temporal differentiation) than can be applied over different input waveforms to get the respective output waveforms.