## Abstract

Integer and fractional spectral self-imaging effects are induced on infinite-duration periodic frequency combs (probe signal) using cross-phase modulation (XPM) with a parabolic pulse train as pump signal. Free-spectral-range tuning (fractional effects) or wavelength-shifting (integer effects) of the frequency comb can be achieved by changing the parabolic pulse peak power or/and repetition rate without affecting the spectral envelope shape and bandwidth of the original comb. For design purposes, we derive the *complete family* of different pump signals that allow implementing a desired spectral self-imaging process. Numerical simulation results validate our theoretical analysis. We also investigate the detrimental influence of group-delay walk-off and deviations in the nominal temporal shape or power of the pump pulses on the generated output frequency combs.

© 2013 Optical Society of America

## 1. Introduction

The spectral self-imaging (SSI) effect, or spectral Talbot effect, is the frequency-domain counterpart of the well-known temporal Talbot effect [1]. SSI occurs when a coherent periodic frequency comb (periodic optical pulse sequence in the time domain) is temporally phase-modulated with a linear frequency chirp through either cross-phase modulation (XPM) [2] or electro-optic modulation (EOM) [3, 4] processes. Under specific phase modulation conditions, the free spectral range (FSR) of the frequency comb is divided by a desired integer number (*m*) while the comb spectral envelope remains unchanged (fractional SSI). Undistorted wavelength-shifting by half the FSR can also be induced on the original frequency comb (integer SSI). Besides the intrinsic physical interest of SSI phenomena, the capability to controlling key features (e.g. FSR or spectral line location) of periodic optical frequency combs can be potentially interesting for a wide range of applications, such as precision spectroscopy [5–7], optical frequency metrology [8], microwave photonics [9], wavelength-division multiplexing (WDM) transmitters [10], THz generation [11], coherent communications [12] and radio-frequency (RF) waveform generation and clock transmission [13].

The first proposal on SSI [2] exploits XPM in an optical fiber to linearly chirp the original periodic pulse sequence using the central (quadratic intensity profile) section of a long Gaussian pump pulse. A basic assumption in this work is that a ‘continuous’ (cumulative) quadratic phase modulation profile is needed to induce the desired SSI effects. Based on this assumption and the intrinsic finite extent of a Gaussian pulse, this original approach can be applied on periodic pulse sequences *with limited time duration only*, greatly limiting the practical use of the method. Recently, the SSI effect has been implemented by EOM using discrete (multi-level) temporal phase modulation of the original pulse train in a periodic fashion [3, 4], which can be shown to be equivalent to the effect of a ‘continuous’ quadratic phase modulation at discrete pulse positions. Despite the relative simplicity of this implementation, its capability is severely limited by the sampling rate of available electronic arbitrary waveform generation systems -needed for generation of the multi-level temporal modulation patterns- so that the input repetition rate is ultimately restricted to a few GHz.

In this paper, we numerically demonstrate that integer and fractional SSI phenomena can be induced on an *infinite-duration* coherent optical frequency comb using nonlinear XPM with a *periodic parabolic pulse train*, overcoming the speed limitations of EOM-based methods. Our proposal relies on the fact that the needed quadratic phase-modulation profile at the discrete input pulse locations is a temporally periodic function and as such, a cumulative phase chirp is not necessary for implementation of the effect [14]. Based on this observation, we derive, for the first time to our knowledge, the complete family of periodic parabolic pump pulse trains that can be used to induce a desired SSI effect on an incoming frequency comb. Free-spectral-range tuning (fractional effects) or undistorted wavelength-shifting (reversed integer effects) can be achieved by suitably changing the parabolic pulse peak power or/and repetition rate.

Optical pulses with parabolic temporal intensity profiles have attracted a great deal of attention for a wide range of applications from high-power ultrashort pulse generation to optical nonlinear processing of telecommunication signals [15–21]. In particular, it has been previously shown that quadratic phase modulation of an incoming optical pulse (probe) can be realized optically through XPM with a parabolic optical pulse (pump); this operation has proven especially useful for intensity profile restoration of optical pulses degraded by a variety of linear perturbations in communication links, including jitter, polarization-mode dispersion, higher order dispersion and time-varying dispersion [19–21]. Parabolic optical pulses have been experimentally generated with durations ranging from the sub-picosecond range to a few tens of picoseconds using either the intrinsic nonlinear reshaping undergone by an optical pulse propagating through an active/passive normally dispersive fiber [22–24] or linear pulse shaping techniques [25–27]. The latter has been practical implemented using superstructured fiber Bragg gratings [25], line-by-line pulse shaping in an arrayed waveguide grating [26] or a long-period fiber grating filter [27]. The approach proposed here for SSI requires the use of a periodic train of parabolic pulses each with a pulse time width filling the train temporal period. Such parabolic optical pulse trains, with pulse durations/periods up to the nanosecond range, could be potentially generated by use of spectral parabolic re-shaping of optical pulse trains with the adequate rate followed by dispersion-induced frequency-to-time mapping [28].

## 2. Operation principle

The temporal self-imaging effect (TSI), also referred to as temporal Talbot effect, is the
time-domain counterpart of the well-known spatial self-imaging phenomenon [1]. This space-time duality is due to the mathematical equivalence between the
equations describing the paraxial diffraction of beams in space and the first-order temporal
dispersion of optical pulses. The TSI occurs when a periodic train of optical pulses with
repetition period *T* ( = 1/*f _{0}*) propagates through a
dispersive medium in a first-order approximation; such, which dispersive medium exhibits a
linear all-pass response that is characterized by a quadratic phase variation in frequency,
namely $exp\left(i\text{\hspace{0.17em}}{\Phi}_{2}\text{\hspace{0.17em}}{\omega}^{2}/2\right)$, where${\Phi}_{2}$ is the first-order dispersion coefficient
and$\omega \text{\hspace{0.17em}}(=2\pi \text{\hspace{0.17em}}f)$denotes angular frequency. An appropriate amount of dispersion,
given by the so-called self-imaging condition [1], leads
either to an exact reproduction of the original pulse train (integer temporal Talbot effect) or
to repetition-rate multiplication, i.e. repetition period division, by an integer factor
(fractional temporal Talbot effect), as shown in Fig.
1(a).

In addition to the duality between space and time, a duality between time and frequency has been also identified [29, 30]: the function of the dispersion operator in the frequency domain is equivalent to the function of a time-lens operator in the time domain. The latter refers to the time-domain counterpart of a spatial thin lens and is essentially characterized by a quadratic phase temporal modulation of the form $exp\left(i\text{\hspace{0.17em}}\varphi \text{\hspace{0.17em}}{t}^{2}/2\right)$, where $\varphi $ is the linear chirp coefficient of the time-lens process. An important consequence of this duality is the frequency-domain counterpart of the TSI effect, the so-called SSI effect [2]. SSI occurs when a periodic sequence of optical pulses (periodic comb in the frequency domain) is linearly chirped by a time-lens process. Under specific temporal linear chirping conditions, the input periodic frequency comb is essentially unaffected by the time lens (integer SSI), whereas for other linear chirping conditions, the FSR of the frequency comb is divided by an integer factor (fractional SSI) without undergoing any further distortion in its spectral envelope, as shown in Fig. 1(b). The SSI condition on the linear chirp coefficient can be obtained from the frequency-domain analog of the well-established TSI theory [1] such that the needed temporal phase modulation function is:

*T*is the fundamental period of the input optical pulse train (i.e.

*f*= 1/

_{0}*T*is the FSR of the input frequency comb), the positive integer

*m*is the FSR division factor induced by the SSI effect under consideration (

*m*= 1 for integer effects and

*m*= 2, 3, … for fractional effects),

*s*is an arbitrary positive integer such than

*s*and

*m*are co-prime. We recall that when Eq. (1) is satisfied, the comb spectral envelope is unchanged after temporal phase modulation (time-lens) either (i) keeping the same FSR as the input (integer effects,

*m*= 1) or (ii) with a reduced FSR by a factor

*m*(fractional effects,

*m*= 2, 3, 4, …). Furthermore, similarly to its temporal counterpart [1], the spectral lines of the modulated comb match the positions of the lines in the input comb when the product (

*s*×

*m*) is an even number (direct effects); however, the lines are additionally frequency shifted by half the output FSR when the product (

*s*×

*m*) is an odd number (reversed effects).

A key observation for the work reported in this paper is that the phase function in Eq. (1) needs to be applied only on the specific
discrete times corresponding to the positions of the input pulses, and the resulting
discrete-time phase modulation is thus periodic with a fundamental period given by the product
*mT* [3]. An equivalent property was also
previously predicted for TSI [14]. In the following, we
show that the required time-lens process to induce SSI can then be obtained using XPM of the
original optical frequency comb (probe) with a *periodic* parabolic optical pulse
train (pump). Figure 2 explains schematically the principle
operation of SSI by XPM of an infinite-duration periodic pulse sequence (input periodic
frequency comb, probe signal in ${\lambda}_{Probe}$) by a periodic parabolic pulse train (pump signal in
${\lambda}_{Pump}$). Two different fractional SSI effects are illustrated, for
*m* = 2 and 4, which are respectively induced by two parabolic pump signals with
identical repetition periods and different peak powers. The probe and the pump signals are
assumed to be centered at different wavelengths such that their respective spectra do not
overlap. The pump signal is a periodic parabolic pulse train which exhibits a repetition period
of ${T}_{pu}$which, for the sake of simplicity, is here assumed to coincide with
the temporal width of each individual parabolic pump pulse, and a peak power of
${P}_{Pu\_peak}$.

In the proposed scheme, the probe signal co-propagates with the pump signal through a given length *L* of a highly nonlinear optical fiber (HNLF). Under ideal conditions (ignoring the dispersive and group-delay walk-off effects at first), the pump signal will induce a nonlinear phase shift of ${\varphi}_{NL}\left(t\right)=2\gamma L{P}_{Pu}\left(t\right)$ across the temporal profile of the probe signal [20, 21, 31], where ${P}_{Pu}\left(t\right)$is the temporal intensity profile of the pump signal at the fiber input, and $\gamma ={n}_{2}{k}_{0}/{A}_{Eff}$is the nonlinear coefficient per unit length of the fiber (*A _{Eff}* is the effective fiber core area,

*n*is nonlinear refractive index and ${k}_{0}=2\pi /{\lambda}_{Probe}$). A quadratic (parabolic) temporal phase modulation is thus applied across a number of consecutive probe pulses when a train of pulses, each with a parabolic intensity shape, is used as the pump. The maximum nonlinear phase shift, which occurs at the center of each individual parabolic pulse, is given by ${\varphi}_{NL,\text{\hspace{0.17em}}max}=2\gamma L\text{\hspace{0.17em}}{P}_{Pu\_peak}$.

_{2}On the other hand, by substituting $t={T}_{pu}/2$(where the maximum phase nonlinear shift occurs) in Eq. (1), the peak value of the pump-induced quadratic phase shift that is required to achieve a desired SSI process on a given input frequency comb is given by the following expression:

Using the above expression for maximum nonlinear phase shift, we can rewrite Eq. (2) as $2\gamma L\text{\hspace{0.17em}}{P}_{Pu\_peak}=\left(s/m\right)\pi {\left({T}_{pu}/2T\right)}^{2}$. Thus, the peak power of the required pump signal (${P}_{Pu\_peak}$) so as to achieve SSI by a factor*m*( = 1, 2, 3, …) can be derived to satisfy the following condition:

The original SSI theory [2] states that a ‘continuous’ quadratic phase modulation needs to be applied over the entire duration of the input periodic frequency comb; thus, the frequency comb was then assumed to have a finite duration. However, more recently, it has been shown that the required quadratic phase modulation is periodic with a fundamental period given by *mT* [3], a feature that was also previously anticipated for temporal self-imaging (Talbot) effects [14]. This property has been exploited to achieve SSI effects on infinite-duration periodic frequency combs through multi-level discrete EOM [3]. In our newly proposed implementation, we exploit this feature of self-imaging effects to extend the XPM-based SSI theory for application on infinite-duration frequency combs. For this purpose, the pump signal must be temporally periodic with a *fundamental period* given by ${T}_{pu}=mT$. Hence, the same SSI process can be achieved using a periodic quadratic pump signal with a temporal period given by *any* integer multiple of the fundamental period, i.e. with a temporal period given by

*p*= 1, 2, 3, …. Additionally, as discussed above, the individual pulses in the optical pump train must exhibit a parabolic temporal intensity shape, with duration equal to

*T*, Eq. (4), and with a peak power satisfying Eq. (3). This set of design specifications defines a

_{pu}*complete family*of different pump signals that allow implementing a desired SSI process by simply suitably changing the peak power or/and the repetition rate of the pump pulse train.

## 3. Simulation results

We have performed several numerical simulations to validate the introduced theory on SSI by
parabolic XPM. We assume an input probe signal characterized by a FSR of
${f}_{0}=40\text{GHz}$ (*T* = 25 ps), and numerically simulate a periodic
train of 1,500 Gaussian pulses, each with a 1.6-ps FWHM, corresponding to a Gaussian spectral
envelope with a 275.8-GHz FWHM. In all simulated cases, we assume a typical HNLF with
*L* = 1km, $\gamma $ = 25 W^{−1}km^{−1}, zero-dispersion
wavelength at λ_{0} = 1551nm and a flat dispersion profile with a slope of 0.017
ps/nm^{2} ·km [2, 21, 32]. The dispersion characteristics
of the assumed HNLF are shown in Fig. 3. Therefore, first, we set the probe and pump wavelengths so that they are symmetric with
respect to λ_{0} to avoid walk-off between the two pulses and second, we place
them close enough to λ_{0} to reduce dispersion-induced pulse broadening of the
propagating pulses, provided also that the respective spectra do not overlap
(λ* _{Probe}* = 1560nm and
λ

*= 1542nm). Thus, in our simulations, the probe pulses are not affected by dispersion during propagation through the HNLF considering the GVD specifications of the fiber and bandwidth of the considered probe signal. For simplicity, the polarization effects are ignored assuming that the pump and probe signals propagate in the fiber with the same polarization state [31]. Moreover, the probe peak power should be low enough to neglect the corresponding self-phase modulation (SPM) contribution. The probe peak power of ${P}_{Pr\_peak}$ = 5mW guarantees this assumption since $L<<{L}_{Probe\_NL}=1/(\gamma {P}_{Pr\_peak})=8\text{km}$ where ${L}_{Probe\_NL}$is the distance for which SPM plays a significant role [31]. The simulations are based on the nonlinear Schrödinger equation which is numerically solved by the split-step Fourier method [31].*

_{Pump}We first consider that the probe signal launched at the input of the HNLF is synchronized
with a pump signal whose period is set to ${T}_{pu}=2T$, considering *m* = 2 and *p* = 1, as
illustrated in Fig. 4(a) , top. The pump peak power is
then fixed to satisfy the SSI condition in Eq. (3)
with *s* = 1 and *m* = 2, i.e., ${P}_{Pu\_peak}$ = 31.4 mW. As shown in Fig.
4(a), bottom, the resultant spectrum at the output of the HNLF is a replica of the input
spectrum (4(a), middle) but exhibits a FSR that is exactly half of that of the input probe
signal; i.e., we observe a spectral line density (SLD) multiplication by *m* = 2.
For comparison, the spectral Gaussian envelope (GE) of the input frequency comb is also
illustrated (dotted dark green lines) in all the spectral simulation results to show the
fidelity of the Gaussian spectral envelope shape of the output signals with respect to the input
one.

We further observe that by changing the period of the pump pulse train to be equal to three times the probe signal period (${T}_{pu}=3T$) and adjusting its peak power to satisfy Eq. (3) with *s* = 1 and *m* = 3, such that${P}_{Pu\_peak}$ = 47.1 mW, the spectrum of the probe signal at the output of the HNLF is again a self-imaged version of the input signal’s spectrum but with a FSR now reduced by a factor 3, as presented in Fig. 4(b).

As expected, for a non-dispersive XPM operation, the temporal waveform of the probe signal is not affected by XPM; i.e., the upper row in Fig. 4 represents the probe temporal waveforms at both the input and the output of the fiber. It is also worth noting that the SLD multiplication by *m* ideally yields the same order of reduction in the power spectral density (PSD) amplitude.

In the second set of simulations, we fix the period of the pump signal to
${T}_{pu}=4T$and we observe that by modifying the pump peak power
*only*, different SSI effects can be induced, namely integer (*m*
= 1) and fractional (*m* = 2, 4) effects, as shown in Figs. 5(a)–5(e). In particular,
in Fig. 5(c) the pump peak power is fixed to satisfy the
SSI condition in Eq. (3), with *s*
= 1 and *m* = 1, such that ${P}_{Pu\_peak}$ = 251 mW. As predicted, the resultant output spectrum is an exact
replica of the input spectrum shifted by half of the original FSR (*reversed
integer* SSI condition).

Subsequently, just by reducing the pump peak power to ${P}_{Pu\_peak}$ = 125.5 mW and${P}_{Pu\_peak}$ = 62.75 mW, SLD multiplications by *m* = 2 and *m* = 4 are also achieved, as shown in Figs. 5(d) and 5(e), respectively. One can readily notice that as anticipated, the same SSI process for *m* = 2 has been obtained using different pump signals: one with period of ${T}_{pu}=2T$ (with *p* = 1) and peak power of ${P}_{Pu\_peak}$ = 31.4 mW (Fig. 4(a)) and the other with period${T}_{pu}=4T$ (with *p* = 2) and peak power of ${P}_{Pu\_peak}$ = 125.5 mW (Fig. 5(d)).

Similar to TSI, the observation of SSI effects depends only on the magnitude of the linear chirp coefficient and not its sign, as explicitly indicated in Eq. (1). The parabolic pulses we have used so far in our simulations, typically referred to as “bright” parabolic pulses, exhibit positive chirp. Nonetheless, there is another class of optical parabolic pulses, namely “dark” parabolic pulses, which can provide the same linear chirp variation but with negative sign [21, 26]. Likewise, we anticipate that application of dark parabolic pulses as the pump signal will also yield the same SSI results. Notice that a dark parabolic pulse has zero amplitude at its center and reach its peak power at the pulse edges. We replicated the second set of simulations with a dark parabolic pump signal with period of ${T}_{pu}=4T$and the corresponding pump peak powers used for SSI with *m* = 1, 2 and 4. Figures 5 (f)–5(j) show the induced SSI results which are nearly identical to the corresponding results for bright parabolic pump pulses, depicted in Figs. 5(a)–5(e). It should be, however, noted that a dark pulse requires half of the average power of a bright pulse to produce the same magnitude of chirp, since the average power of bright and dark pulses is given by $(2/3){P}_{Pu\_peak}$ and $(1/3){P}_{Pu\_peak}$, respectively [26].

The evolution of the probe signal’s energy spectrum as a function of parabolic pump peak power can be visualized using the SSI carpet shown in Fig. 6(a), an illustration that is inspired from the well-known TSI carpet [1]. In Fig. 6(a), the period of the parabolic pump signal is fixed to ${T}_{pu}=4T$. In weaving this SSI carpet, we omitted the empty portions of the probe spectrum between the discrete frequency comb lines, and enlarged the frequency comb lines to be clearly visible. We also peak-normalized the probe spectrum for each pump peak power value.

As can be directly inferred from the carpet, the hopping power from one mode (corresponding to an specific SLD multiplication factor) to the neighbor mode is equal to ${P}_{MTM}=\pi /2\gamma L=62.75\text{mW}$, in the reported example.

For any given specific mode, the predicted SSI effect is observed with no distortion on the comb spectral envelope only if Eq. (3) is exactly satisfied; a small pump peak power deviation from the exact SSI condition (Eq. (3) will induce distortions in the comb spectral shape. To determine the acceptable power deviation from the SSI condition, we define a maximum acceptable extinction ratio of 3dB between the amplitudes of the highest and lowest in-mode frequency comb lines, over a 80-GHz frequency range around the comb central frequency, and a minimum acceptable extinction ratio of 15 dB between the amplitudes of the highest in-mode and out-mode frequency comb lines, as shown in Fig. 6(b).

Using this criterion, we estimate that the acceptable range of pump peak power deviation from the SSI condition so that SSI is observed without significant mode-hopping is about 20mW.

In the next set of simulations, we employ a sinusoidally modulated optical signal, which is much
easier to generate than the parabolic pulses, as the pump signal in order to illustrate the
influence of the use of a simpler but non-ideal pump signal to induce the desired SSI effects.
Figure 7 presents the simulation results
illustrating SSI with SLD multiplication factors of *m =* 2, 3, 4 and 5. For
comparison, the parabolic pulse train is also depicted (dotted red lines) along the time domain
to show the difference between the ideal parabolic pulse train and the simulated sinusoidal pump
signal. First, we assume a sinusoidal pump signal with a period set to ${T}_{pu}=2T$ and a peak-to-peak power that satisfies the SSI condition in Eq. (3). The resultant probe spectrum at the output of
the HNLF (Fig. 7(a), bottom) shows that the sinusoidal
pump signal induces SSI phenomena almost without distorting the original comb spectral envelope.
In this case, the sinusoidal pump resembles the parabolic pulse train in all the input probe
pulse positions. However, the sinusoidal pump signal fails to provide the required quadratic
phase modulation as *m* is increased. In the case of *m* = 3,
Fig. 7(b), only one out of each three consecutive probe
pulses receives the correct phase shift and two of them receive a phase shift that deviates
~0.33 rad from the nominal one, significantly affecting the fidelity of the Gaussian envelope
shape in the output frequency comb. Figure 7(c) shows
that for an SLD factor of *m* = 4, half of each four consecutive pulses are phase
modulated correctly whereas phase deviations of ~0.78 rad are induced on the rest of the pulses.
distortions on the output frequency comb envelope are more significant as the FSR division
factor *m* is increased, as observed for instance for the case of
*m* = 5 in Fig. 7(d).

Finally, in order to analyze the influence of group-delay walk-off effects on the observation of SSI phenomena, we change the wavelength of pump signal to 1525nm to impose a walk-off parameter equal to around $\delta =5ps/km$. As illustrated in Fig. 8(a), group-delay walk-off effects will affect the initial synchronization of the pump and probe signals as they propagate along the fiber so that at the fiber output, the two signals become desynchronized by ${\tau}_{WO}=L\delta $ (in this simulation ${\tau}_{WO}=5\text{\hspace{0.17em}}ps$). This effect essentially affects the SSI phenomena by distorting the original comb spectral envelope. However, this detrimental effect can be reduced if we deliberately set-up an initial pump-probe desynchronization so as the pump signal leads the probe by${\tau}_{WO}/2$, as shown in Fig. 8(b), or, as assumed in our previous set of numerical simulations, by using a HNLF whose zero-dispersion wavelength lies between the pump and probe signal central wavelengths such that the two signals undergo the same group velocity [21, 31, 32].

## 4. Conclusions

In summary, we have demonstrated that integer and fractional SSI effects can be induced on an infinite-duration coherent optical frequency comb using XPM with a periodic parabolic pump pulse train. We have derived the complete family of parabolic pulse profiles that can be used to induce a desired SSI effect on an incoming frequency comb. Our numerical simulations confirm that FSR tuning or wavelength-shifting can be achieved on the comb, without distorting the comb spectral envelope, by suitably changing the parabolic pulse peak power or/and repetition rate. We have also investigated the effect of deviations in the nominal temporal shape or power of the pump pulses on the generated output frequency combs.

## Acknowledgments

The authors are very grateful to the referees of the original submission, whose comments and suggestions have been very useful to improve the quality and extent of the reported work. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chair in “Ultrafast Photonic Signal Processing”. R.M acknowledges financial support from the Ministère de l'Éducation, du Loisir et du Sport (MELS) du Québec through the Merit Scholarship Program for Foreign Students.

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