## Abstract

We introduce a new family of equivalent periodic phase-only filtering configurations that can be used for implementing the Talbot-based pulse rate multiplication technique. The introduced family of periodic Talbot filters allows one to design a desired pulse repetition rate multiplier with an unprecedented degree of freedom and flexibility. Moreover, these filters can be implemented using all-fiber technologies, and in particular (superimposed) linearly chirped fiber Bragg gratings. The design specifications and associated constraints of this new class of Talbot filters are discussed.

©2006 Optical Society of America

## 1. Introduction

Techniques for the generation and control of ultrahigh repetition-rate periodic optical pulse trains have become increasingly important for many applications including ultrahigh speed optical communications, ultrafast signal processing, and photonic sampling, among others [1]–[7]. An attractive method to generate optical pulse trains at repetition rates beyond those achievable by conventional mode-locking laser techniques is to multiply the repetition rate of a lower rate source outside the laser cavity. Two interesting linear filtering approaches for pulse rate multiplication have been recently demonstrated, namely the so-called temporal Talbot effect [3]–[5] and a solution based on the use of spectrally periodic phase-only filters [6], [7]. The most relevant features of these two methods is that (i) they are phase-only filtering techniques (in which the spectral amplitude of the original pulse train remains unaffected) and consequently, they offer a high energetic efficiency, and more importantly, (ii) they can be implemented using readily available fiber technologies. In particular, the Talbot effect approach only requires a linear propagation of the original pulse train through a suitable first-order dispersive medium, i.e. a phase filtering device providing a suitable quadratic phase spectral response, or equivalently, a suitable linear group delay, over the entire bandwidth of the pulses to be multiplied. In practice, this filtering process can be implemented using a simple and compact linearly-chirped fiber Bragg grating (LCFBG) [3], [4]. As an alternative solution, superimposed LCFBGs can be used to implement spectrally periodic phase-only filters for pulse repetition rate multiplication [6], [7]. In this approach, each of the superimposed LCFBGs implements a single frequency channel (single spectral period), in such a way that the phase variation in each period is also quadratic (linear group delay), while the spectral period is fixed by the desired output repetition rate.

It has been previously shown that temporal Talbot effects can be realized by reflecting the input periodic optical pulse train in a properly designed structure of concatenated FBGs, which acts like an equivalent dispersive delay line [8]. In this paper, we demonstrate that the two different approaches for pulse rate multiplication mentioned above, namely the temporal Talbot approach [3]–[5] and the quadratic-phase periodic filtering approach [6], [7], are formally equivalent. To be more concrete, we show that in the two cases the same filtering process (over an incoming periodic optical pulse train) is implemented. This analogy, which to the best of our knowledge has not been previously noticed, has two main consequences. First, the temporal Talbot effect can be re-interpreted as a simple spectrally periodic phase-only filtering process. Second and more importantly, based on this analogy, we determine a new complete set of possible periodic filtering configurations that can be used for implementing the Talbot-based pulse rate multiplication technique; we show that the conventional implementation using a single highly-dispersive medium (e.g. LCFBG) [3]–[5] and the quadratic-phase periodic filtering approach (e.g. based on SI-LCFBGs) [6], [7] represent only a sub-set among all the possible designs. For instance, a desired pulse repetition rate multiplication operation can be implemented using a periodic filter with a channel spacing different to (larger than) the desired output repetition rate. Our findings here provide an unprecedented degree of freedom for designing Talbot-based pulse repetition rate multipliers.

## 2. Temporal Talbot effect as a periodic phase-only filtering

Let us assume a periodic optical pulse train *a*
_{1}(*t*) which is centered at the optical frequency *ω*
_{0} and has a repetition period *T*_{r}
. This pulse train is propagated through a first-order dispersive medium in the linear regime. A first-order dispersive medium is a phase-only filter characterized by a spectral transfer function of the form *H*(*ω*)∝exp[*j*Φ(*ω*)], where the spectral phase function can be expressed as Φ(*ω*)=Φ_{0}+${\mathrm{\Phi}}_{0}^{\left(1\right)}$(ω-ω_{0})+(${\mathrm{\Phi}}_{0}^{\left(2\right)}$/2)(ω-ω_{0})^{2}, with Φ_{0}=Φ(ω=ω_{0}) and

${{\Phi}_{0}}^{\left(k\right)}={[{\partial}^{k}\Phi \left(\omega \right)\u2044\partial {\omega}^{k}]}_{\omega ={\omega}_{0}}$

for *k*=1, 2. It is well known that the factors Φ_{0} and ${\mathrm{\Phi}}_{0}^{\left(1\right)}$(ω-ω_{0})(constant phase factor and average group delay introduced by the medium, respectively) are not responsible for the distortion experienced by the optical signal after propagation through the dispersive medium and can be consequently ignored in our following analysis. The only factor responsible for the optical signal distortion is the quadratic phase factor, characterized by the term ${\mathrm{\Phi}}_{0}^{\left(2\right)}$, which is usually referred to as the first-order dispersion coefficient of the medium. A first-order dispersive medium exhibits a linear group delay (as a function of frequency) with a slope given by the medium’s dispersion ${\mathrm{\Phi}}_{0}^{\left(2\right)}$. In what follows we will refer to Φ′(ω) as the quadratic term of the total phase function, i.e. Φ′(ω)=(${\mathrm{\Phi}}_{0}^{\left(2\right)}$/2)(ω-ω_{0})^{2}.

According to the previous studies on temporal Talbot effect, pulse repetition rate multiplication can be achieved if the first-order dispersion coefficient satisfies the following condition [4]

where *q*=1, 2, 3,…, and *m*=2, 3, 4, … such that (*q/m*) is a non-integer and irreducible rational number. This is the so-called fractional temporal Talbot condition. Under this condition, the original temporal sequence reappears at the output of the dispersive medium (the shape and duration of the individual pulses are not affected by the dispersive process) but with a repetition rate *m*-times that of the input one. This is true as long as the new repetition period is still longer than the time width of the individual pulses (to avoid interference among them). Assuming nearly transform-limited optical pulses, this additional condition can be mathematically stated as Δω_{in}>2Δω_{r,out}, where Δω_{in} is the total input pulse bandwidth and Δω_{r,out}=*mω*_{r}
is the multiplied repetition rate of the generated pulse train.

The input signal in our problem is a periodic function of time and as a result, it can be represented in the frequency domain as a series of ideal deltas (discrete Fourier components), spectrally spaced by the signal repetition frequency *ω*_{r}
=2π/*T*_{r}
. After propagation through the linear dispersive medium, each of the signal’s Fourier components is affected by an additional phase factor. Specifically, the quadratic phase distortion introduced by the dispersive medium on the signal’s discrete Fourier component of order *p* is Φ_{p}=Φ′(ω=ω_{0}+*pω*_{r}
)=(${\mathrm{\Phi}}_{0}^{\left(2\right)}$/2*p*
^{2}${\mathrm{\omega}}_{r}^{2}$, (*p*=0, ±1, ±2, …). If the medium’s dispersion coefficient satisfies the Talbot condition for repetition rate multiplication in Eq. (1), the dispersion-induced phase shifts fulfill the following equality:

It can be easily proved that the function exp(*j*|Φ_{p}|), with |Φp| given by Eq. (2), is a periodic function of *p* with a period given by the repetition rate multiplication factor *m*. In particular, |Φ_{p+m}|=(*p+m*)^{2}(*q/m*)π=|Φ_{p}|+(*q.m*)π. The function exp(*j*|Φ_{p}| is obviously periodic (with *m* period) when the product (*q.m*) is an even number. In the case when (*q.m*) is an odd number, the phase filter can be also made periodic by assuming a concatenated linear phase filtering operation of the form ${\left[\mathrm{exp}(j\left(\omega -{\omega}_{0}\right){T}_{r}\u20442m)\right]}_{\omega ={\omega}_{0}+p{\omega}_{r}}=\mathrm{exp}\left(j{\phi}_{p}\right)=\mathrm{exp}(jp\pi \u2044m)$. This filtering operation simply accounts for an additional time delay in the generated output pulse train by *Tr/2m*. The reader can easily prove that in this latter case, exp(*j*||Φ_{p+m}+φ_{p+m}|)=exp(*j*||Φ_{p}+φ_{p}|) for any p. Thus, in the Talbot approach, the dispersive medium distorts the phase of the input discrete spectrum in a periodic fashion, with a period given by the desired output repetition rate, i.e. (*mω*_{r}
).

Based on the argumentation above, the phase filtering process implemented by the dispersive medium over an incoming optical pulse train is equivalent to that implemented by a periodic quadratic-phase filter, of spectral period equal to the output repetition rate (*mωr*), in which the phase (or group delay) variation in each period is identical to that in the Talbot dispersive medium, i.e. linear group delay with a slope (dispersion coefficient) given by the Talbot condition in Eq. (1). A schematic of this equivalence is presented in Fig. 1, which shows the spectral phase and group delay responses of two equivalent filtering configurations over 5 spectral periods, assuming a repetition rate multiplication by *m*=3 and with the q parameter fixed to *q*=1.

It should be mentioned that the concept used in our work, i.e. reduction of a continuous phase distortion to the same phase distortion modulo 2π, is somewhat analogous to that exploited in the design of Fresnel lenses [9]; however, the typical aberrations introduced by a Fresnel lens are not present in our problem since the Talbot filters operate over a discrete spectrum.

An important parameter in characterizing a dispersive process is the total (maximum) group delay span. In general, the larger this total group delay span, the longer the filtering device must be. In particular, in a fiber phase filter (e.g. LCFBG), the delay span Δτ is directly proportional to the required fiber length, *L*≈(*c/2n*_{eff}
)Δτ, where c is the speed of light in vacuum and *n*_{eff}
is the effective refractive index of the mode propagating through the optical fiber. In the case of the conventional Talbot implementation based on a single dispersive medium (e.g. single LCFBG), the total group delay span Δ*τ*
_{0} must be calculated over the entire input pulse bandwidth (Δ*ω*_{in}
), see Fig. 1 (top plot); in particular,

where we have made use of the Talbot condition in Eq. (1) and *N*=Δ*ω*_{in}
/ω
_{r,out}
>2 is the inverse of the duty-cycle of the generated optical pulse train (assuming transform-limited pulses). In contrast, in the case of a periodic phase-only filtering device, the total group delay span Δ*τ*
_{1} must be calculated over a single spectral period, see Fig. 1 (bottom plot),

where we recall that ω_{r,out}=*mω*_{r}. A very relevant feature of a periodic phase-only filter (of period mωr) is that its total group-delay span is shorter than in the equivalent Talbot filter based on a single dispersive medium. This translates into shorter required devices. The difference is more significant for smaller output duty-cycles (i.e. for a larger *N*).

An important observation is that in the case of *q*=1 (case represented in Fig. 1), the total group delay span in the periodic phase-only filter is equal to the input repetition period, Δ*τ*
_{1}=*T*_{r}
. In this particular case, the design specifications of the periodic Talbot filter coincide exactly with those previously used for implementing pulse repetition rate multipliers based on SI-LCFBGs [6], [7]. The design of these filters has been conventionally justified on the basis of the following heuristic interpretation for the pulse rate multiplication process. Each optical pulse incident upon the filter is spread out in a sequence of multiple pulses with a repetition rate fixed by the filter’s spectral period and with a total sequence duration fixed by the total group delay span. A continuous train of optical pulses is obtained at the output if the duration of the sequence generated out of each incident pulse is fixed to coincide with the input repetition period (to fill the gap in between the individual pulses in the input sequence). Here, we have found out that this previously introduced periodic phase-only filtering method actually represents an alternative way of implementing a Talbot-based pulse repetition rate multiplication process.

## 3. Complete family of periodic Talbot filters

Based on the equivalence presented in the previous section, the same Talbot-based pulse repetition rate multiplication process can be realized using different filter designs, which are formally equivalent. In particular, let us assume that the input optical pulse train has a repetition rate ωr and that we want to design an optical filter for multiplying this repetition rate by *m*. In principle, this filtering operation can be realized using either of the following solutions: (i) a single dispersive medium (e.g. LCFBG) providing a first-order dispersion that satisfies the Talbot condition in Eq. (1) [3]–[5]; (ii) a periodic phase-only filtering device with a spectral period fixed by the output repetition rate, i.e., ω_{r,out}=*mω*_{r}
, and providing a linear group delay in each period with a slope (dispersion) given by Eq. (1). It should be noted that as compared with the conventional solution based on periodic phase filtering [6], [7], the solution defined in (ii) offers an additional degree of freedom, given by the parameter *q*, which can be fixed to be any arbitrary integer as long as (*q/m*) is a non-integer and irreducible rational number. The conventional periodic filtering solution is a particular case of the more general solution given here (for the case when *q*=1).

Moreover, the fact that a Talbot filter is a spectrally periodic filter with a fundamental period given by ω_{r,out}=*mω*_{r}
(when operating on the corresponding optical pulse train), implies that in general, the filter spectral response is periodic with a period given by *any* integer multiple of its fundamental period. Based on this generalization, the same rate multiplication process as above can be achieved using a periodic phase-only filter with a spectral period given by *any integer multiple of the desired output repetition rate*, i.e. with a spectral period given by *p*ω_{r,out}=*p*(*mωr*) where *p*=1, 2, 3, …, and where the spectral phase variation in each period is that required to induce the corresponding Talbot effect [in each period, the filter must exhibit a linear group delay with a slope fixed to satisfy the Talbot condition in Eq. (1)]. These general design specifications define a complete family of different periodic phase filters that allow implementing the same Talbot-based rate multiplication process.

The conventional previous solutions for pulse repetition rate multiplication are particular filter designs belonging to the more general family introduced here. Specifically, the previous periodic phase-only filtering design [6], [7] can be obtained by simply fixing *p*=1 in the general family of solutions. Similarly, the solution based on a single dispersive medium is the solution resulting from a sufficiently large value of *p* so that the spectral period *pω*_{r,out}
is larger than the input pulse bandwidth, Δω_{in}<*p*ω_{r,out} ; in this case, the general periodic filter is composed by a single spectral period [3]–[5].

Figure 2 shows an example of another filtering configuration equivalent to those shown in Fig. 1 (in particular, in the example shown in Fig. 2, we have fixed *p*=2 and *q*=1). It is important to note that the fact that a periodic optical pulse train can be obtained at the output of a spectrally periodic filter having a spectral period larger than the repetition rate of the output train (by an integer factor *p*) is a counter-intuitive fact, and this may explain in part why the complete family of periodic Talbot filters have not been found out before. The reader should be reminded that the introduced family of equivalences is only valid assuming a suitable periodic pulse train as the input signal to the filtering systems.

## 4. Design considerations

The maximum group delay span in a general periodic Talbot filter depends on the chosen parameter p and in particular, it is given by:

It can be easily proved that Eq. (5) reduces to Eq. (4) for the conventional periodic phase filtering solution (for which *p*=1). We also recall that the total group delay span in a Talbot filter based on a single dispersive medium is given by Eq. (3).

The number of required periods to cover a given input bandwidth Δ*ω*_{in}
using a general Talbot filter of parameter *p* is

where the operation *R*{•} rounds the argument to the nearest integer towards infinity.

Equation (5) and Eq. (6) indicate that the broader the spectral period fixed by the designer (i.e. the larger the parameter *p*), the longer the required filtering device will be but at the same time, the smaller the number of spectral periods in the filter will be, thus resulting in a simpler solution. It should be emphasized that a larger number of spectral periods in the phase filter translates into a higher degree of complexity in regards to its practical realization. For instance, if the filter is implemented using SI-LCFBGs, each different spectral period is realized by a different LCFBG, in such a way that the number of spectral periods fixes the required number of LCFBGs to be superimposed on the same section of optical fiber [6], [7].

Among all the possible solutions of the general family of periodic Talbot filters, the one based on a single dispersive medium [3], [4] is the solution with the highest total group delay span [as given by Eq. (3)], or in other words, this is the solution that requires the longest filtering device for its practical implementation. In contrast, this is the simplest solution as the filter is composed by a single period (e.g. single LCFBG). In the other extreme, the conventional periodic phase filtering solution (*p*=1) [6], [7] is the one requiring the smallest group delay span [as given by Eq. (4)] or equivalently, the shortest filtering device (e.g. superimposed LCFBGs). However, this is the filtering design that requires the largest number of spectral periods, i.e. this is the most complex solution. In particular, in this case, the number of spectral periods is determined by the inverse of the duty cycle of the output sequence, *N*
_{1}=*R*{*N*}=*R*{Δω_{in}/ω_{r,out}}. In between the two mentioned extremes, we have now a complete collection of possible solutions where the specific tradeoff between device length and complexity of implementation can be chosen at the designer convenience.

## 5. Numerical example

As an example, we consider a 10GHz input pulse train (i.e. *T*_{r}
=100ps) composed by 25 transform-limited Gaussian pulses, each with a FWHM time width of 3 ps. The associated total input pulse bandwidth is Δ*ω*_{in}
/2π≈400 GHz. Three different but equivalent filtering configurations have been designed and numerically tested to multiply the input repetition rate by a factor of *m*=4 to obtain an output pulse repetition rate of 40 GHz.

As a first approach, we use a single linear dispersive medium with a first-order dispersion coefficient |${\mathrm{\Phi}}_{0}^{\left(2\right)}$| satisfying Eq. (1) for the given values of Tr and m (we also assume q=1). This filter has a quadratic phase profile along the entire spectral width of the input pulse train, introducing a total group delay span of *Δτ*
_{0}≈1 ns. If the dispersive filter is implemented using a LCFBG, a grating length *L*>10 cm is required.

Alternatively as a second approach, an equivalent filtering process is implemented using a periodic quadratic-phase filter with a spectral period equal to the output repetition rate, i.e. 40 GHz. In this case, 10 spectral periods are required to cover the entire input bandwidth, where the group delay profile in each spectral period is linear with the same slope (dispersion) as for the first filter [i.e. dispersion given by Eq. (1)]. In this case, the total group delay span is only *Δτ*
_{1}=*T*_{r}
=100ps. If the filter is implemented using SI-LCFBGs, a minimum of 10 LCFBGs should be superimposed on the same section of optical fiber; however, each one of these gratings (i.e. the grating structure) would be as short as *L*≈1 cm.

The same filtering operation is finally implemented by use of a different periodic filter where the spectral period is now fixed to 80 GHz, i.e. twice the desired output repetition rate (*p*=2). The new filter consists of 5 spectral periods and exhibits a linear group delay response in each period with the same slope (dispersion) as for the two previous filters [given by Eq. (1)]. This translates into a total group delay span of *Δτ*
_{2}=200ps. In practice, this filter can be implemented using only 5 superimposed LCFBGs of length L≈2 cm.

We simulated the three considered filtering operations assuming ideal phase-only filters with the spectral transfer functions described above. In each case, the output temporal profile was calculated by taking the inverse Fourier transform of the result of multiplying the spectrum of the input pulse train by the corresponding transfer function. The three simulated filtering configurations performed the expected pulse repetition rate multiplication process, delivering a 40-GHz output pulse train, where the individual output pulses were identical to those of the 10-GHz input train.

Figure 3 analyzes the output pulse train obtained by use of any of these three equivalent Talbot filters. In Fig. 3, the bottom plot (solid curve) represents the temporal variation (optical intensity) of the pulse train generated at the filter output, the plot at the left represents the corresponding energy spectral density and the larger two-dimensional image at the center of the plot represents the joint time-frequency (TF) energy distribution of the generated pulse train. In this image, different intensities are represented with different colors according to the colormap shown at the right of Fig. 3. The joint TF representation of a given signal provides information on the temporal location of the signal’s spectral components [10] and depends essentially on the temporal/spectral phase structure of this signal. The TF energy distributions have been computed using conventional Spectrograms with a Gaussian time window of duration ~31.25ps. For comparison, the temporal intensity distribution of the input pulse train is also represented in Fig. 3 (bottom plot, dashed curve).

The fact that the output pulse train exhibits an identical joint TF distribution for any of the three considered filters confirms that the exact same filtering process is implemented in all these cases. In other words, the output pulse trains generated in the three considered cases are identical both in amplitude and in phase. Moreover, the obtained TF pattern in the three cases is expected for a Talbot-based repetition rate multiplication process, according to a numerical study we have conducted recently [11]. Notice that the energy spectrum is not affected by the filtering process since ideal phase-only filters are assumed. As a result, the output energy spectrum consists of a periodic set of discrete frequency components with frequency spacing identical to that at the input, i.e. fixed by the original repetition rate (10 GHz). Thus, as expected for a phase-only filtering process, only the repetition rate of the intensity temporal profile is multiplied but the rate of the complex electric field corresponding to the generated pulse train actually remains unaffected (this is why the different output pulses in one period of the input train exhibit different mean wavelengths). This limitation of the temporal fractional Talbot effect has been previously recognized and studied and different solutions for field-to-intensity conversion have been demonstrated to achieve real pulse repetition rate multiplication (in amplitude and phase) [5].

For comparison purposes, a fourth filtering process is finally simulated for implementing the same pulse repetition rate multiplication operation (by *m*=4) on the same input pulse train as in the previous cases. This last filter simply added a π phase shift over one spectral mode every 4 modes of the original signal. The results corresponding to this repetition rate multiplication process are presented in Fig. 4, with the same definitions as for Fig. 3. Though the pulse repetition rate was increased (in intensity) in the same way as in the previous cases, the joint TF distribution reveals a different phase structure for the generated pulse train to that obtained with the Talbot filters. Thus, as expected, a different filtering process to that of the Talbot technique is now implemented.

## 6. Conclusions

We have introduced a new complete set of equivalent periodic phase-only filtering configurations that can be used for implementing the Talbot-based pulse rate multiplication technique. We have shown that the conventional implementation using a single highlydispersive medium (e.g. LCFBG) [3]–[5] and the previously introduced quadratic-phase periodic filtering approach for pulse rate multiplication (e.g. based on SI-LCFBGs) [6], [7] actually represent a sub-set of solutions among all the possible designs in the introduced family.

Specifically, we have demonstrated that the repetition rate of an incoming optical pulse train of repetition period *T*_{r}
can be multiplied (by a factor *m*) using a periodic phase-only filter with a spectral period given by any integer multiple of the desired output repetition rate, i.e. with a spectral period given by *p*ω_{r,out}=*p*(*m*ω_{r}) where *p*=1, 2, 3, …, and where the spectral phase variation in each period is that required to induce the corresponding Talbot effect, i.e. in each period, the filter must exhibit a linear group delay with a slope fixed to satisfy the fractional Talbot condition in Eq. (1).

The introduced family of periodic Talbot filters allows one to design a desired pulse repetition rate multiplier with an unprecedented degree of freedom and flexibility. Moreover, these filters can be implemented using all-fiber technologies, and in particular (superimposed) LCFBGs.

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