## Abstract

We experimentally demonstrate tunable comb spacing of an original 10-GHz periodic frequency comb by spectral Talbot effect over an unprecedented range of even and odd comb spacing division factors, from 2 to 9. The implementation has been achieved by periodic electro-optic (EO) temporal phase modulation of the original comb (conventional mode-locked optical pulse train) with multilevel modulation functions, produced by an arbitrary waveform generator (AWG). These comb spacing division processes have been observed through the use of a high-resolution (20-MHz) optical spectrum analyzer. Comb spacing tuning is achieved without essentially affecting the spectral bandwidth and total energy of the original comb signal. Our results also confirm that the spectral Talbot method does not require carrier-envelope phase stabilization in the input frequency comb. Numerical studies on the impact of deviations in the applied phase modulation functions confirm the robustness of the technique, in agreement with the experimental results.

© 2013 OSA

## 1. Introduction

Generation of optical frequency combs and control of their key features, e.g. comb spacing and central frequency, are interesting for a wide range of applications, such as microwave photonics [1], wavelength-division multiplexing (WDM) transmitters [2], radio-frequency (RF) waveform generation and clock transmission [3], photonic signal processing [4,5], precision spectroscopy [6–8], and optical frequency metrology [9].

A promising and easy way to tune the comb spacing in periodic frequency comb lasers is based on the spectral self-imaging effect, i.e. spectral Talbot effect [10]. In this schematic, the comb spacing of a coherent frequency-comb signal (periodic optical pulse train) can be divided by a desired positive integer number (*m*) by simply applying a proper quadratic phase-only temporal modulation on the original pulse train. Implementations using cross-phase-modulation (XPM) [10] and electro-optic (EO) modulation [11] have been previously proposed and numerically demonstrated. The process can be applied over any given coherent frequency comb input, e.g. extending over arbitrarily large bandwidths, without affecting the comb spectral support (bandwidth) and the signal’s total energy. Additionally, no time-domain carrier-envelope phase stabilization is required in the input frequency comb to achieve the desired effect. These advantageous features should be contrasted with periodic frequency comb generation methods based on a combination of EO intensity and phase modulators [12]. These previous methods easily enable tuning the comb spacing of the generated comb in a continuous fashion. However, the attainable bandwidth with these methods is ultimately limited by the amount of phase modulation imposed in the system. For example, bandwidths larger than a few nanometers, several EO phase modulators need to be concatenated, leading to complex and inefficient setups.

The EO modulation implementation of spectral self-imaging is based on a relatively simple scheme using discrete (multilevel) temporal phase modulation of the original pulse train in a periodic fashion [11]. To achieve comb spacing division by a factor *m*, one should apply the same phase modulation function, generally consisting of *m* different phase levels (between 0 and 2π), over every *m* consecutive pulses. The necessary discrete phase modulation function is actually equivalent to the effect of a continuous quadratic phase modulation [10] at the discrete time pulse positions. In a recent experiment, this idea was exploited for discrete comb spacing tuning of a GHz-rate periodic frequency comb [13]. However, since the phase modulation was produced by a bit pattern generator, the employed time-domain phase modulation was restricted to be binary (two levels) only, drastically limiting the achievable frequency division factors. Comb spacing division only by two different even factors, i.e. 2 and 4, could be achieved (from 10 GHz to 5 GHz and 2.5 GHz, respectively).

We report here an experimental demonstration of comb spacing tuning of a periodic frequency comb by spectral Talbot effect over an unprecedented range of comb spacing division factors, from 2 to 9, including both even and odd division factors. This has been achieved by periodic EO temporal phase modulation of a 10-GHz optical pulse train with multilevel modulation functions, generated from an arbitrary waveform generator (AWG). Observation of all these comb spacing division processes has been possible through the use of a high-resolution (20-MHz) optical spectrum analyzer. We have also investigated in detail the influence of variations in the phase modulation functions on the comb spacing division processes. Our numerical studies evidence the robustness of this comb spacing-tuning mechanism, in agreement with the experimental results.

## 2. Operation principle

The sketch in Fig. 1
illustrates how to implement comb spacing-tuning by spectral Talbot effect using multilevel time-domain EO phase modulation. According to the spectral self-imaging theory [10,11], the discrete phase shift to be applied on the *n*-th temporal pulse of the original periodic pulse train so as to get comb spacing division by a factor of *m* (*m* = 2, 3, 4, …) should satisfy the following equation:

*m*and

*q*are mutually prime integer numbers and

*m*is the comb spacing division factor.

Essentially, the phase shift value in Eq. (1) can be assumed to be applied on the whole *n*-th temporal pulse period or, at least, along the whole duration of the *n*-th pulse. Furthermore, in practice, the actual phase values in Eq. (1) can be applied modulo 2π and it can be noticed that *ϕ _{n}* is periodic with period

*m*. Said another way, one needs to implement a periodic temporal phase modulation consisting of

*m*discrete phase steps per period. Thus, the modulation function period must be

*m*times the input pulse train period 1/

*f*, where

_{in}*f*is the input comb spacing. Notice that no wavelength shifting is expected in the output frequency comb with respect to the input one for the cases when (

_{in}*q·m*) is an even number. However, a wavelength shifting by half the output comb spacing should be obtained for the cases when (

*q·m*) is an odd number.

## 3. Experimental demonstration

We use the experimental setup illustrated in Fig. 2
. An actively mode-locked fiber laser (MLFL) produces the input temporal optical pulse train with a repetition rate of 10 GHz and an individual pulse time-width of ~2.5 ps. An AWG is employed to produce the required periodic step-like multilevel temporal phase modulation functions to be applied on the optical pulse train via an EO phase modulator (PM). In order to modulate each pulse period with the corresponding suitable phase shift, the MLFL and AWG need to be properly synchronized. For this purpose, the employed MLFL works in harmonic configuration, driven by the generated 10 GHz sinusoidal clock, which also acts as clock reference for the AWG sampling rate. Therefore, the AWG produces a modulation signal at 10 GS/s, each sample corresponding to a value of phase from Eq. (1). An optical delay line (ODL) is used to temporally overlap each pulse period with the corresponding modulation sample time slot. A polarization controller (PC) is used to maximize the modulation depth in the PM. A high-resolution (20 MHz) optical spectrum analyzer (Apex, AP2440A) is employed to monitor the obtained frequency comb features. The AWG has been programmed to produce periodic step-like waveforms so that to perform the needed phase shifts as derived from Eq. (1) with *q* = 1 and, respectively, even and odd values of *m*, ranging between 2 and 8 in the first case, and between 3 and 9 in the second one.

Figure 3
reports the experimentally measured optical spectra of the output periodic frequency comb. The predicted spectral Talbot effects are observed, leading to the anticipated comb spacing division processes by the corresponding factor *m*. In the bottom of Fig. 3 the periodic-comb spectrum of the original 10 GHz input optical pulse train is reported as well (output for *m* = 1). A few important comments should be made on the obtained results. The first consideration is that since the comb spacing tuning process is based on phase-only modulation, the total energy of the output frequency comb is essentially the same as that of the input signal (except for the expected loss introduced by the phase modulator of ~4dB); essentially, there is only a change in the way that the original amount of energy is redistributed along the frequency domain. In particular, reducing the factor m, the frequency density of spectral lines increases and, consequently, the mean peak power of the comb spectral lines is reduced. Considering that the absolute noise floor remains unchanged, the comb signal-to-noise ratio (SNR) decreases with increasing m. Moreover, the flatness of the spectrum envelope gets also worse as the division factor *m* is increased. This can be mainly attributed to the accuracy of the AWG output waveform shapes, which deteriorate as the number of discrete levels is increased. This effect can be noticed from Fig. 4
, which shows the ideal step-like phase functions uploaded in the AWG for each of the tested values of *m* (red curves) in comparison with the actual experimental waveforms coming out from the AWG (blue curves, measured through a 40-GHz electronic sampling scope).

The observed deviations between the ideal and experimentally implemented multilevel temporal modulation functions are due to the analog bandwidth limitation, and the associated pulse response and pulse response oscillations of the employed AWG, exhibiting a ~33 ps rising/falling time response, combined with the AWG amplitude quantization noise. Notice that at the output of the AWG, a modulation driver amplifier working in linear regime (not included in Fig. 2) was used to precisely set the peak-to-peak voltage (*V _{pp}*) in order to match the specifications of the PM. It should be also mentioned that the spectrum of the input optical pulse source used in our experiments exhibits a certain occasional wavelength shift. As a result, the predicted matching/shifting between the input and output comb spectra depending on the value of the product (

*q·m*) could not be properly attested.

## 4. Robustness against deviations in the phase modulation process

Finally, in order to investigate the effect of deviations in the values of the temporal phase modulation levels on the comb spacing division processes, we performed a comprehensive set of numerical simulations considering a 10-GHz pulse train consisting of 1200 5ps-width Gaussian optical pulses. An additive normal random noise is added to the whole temporal pulse train. Each pulse was modulated in phase according to the modulation function from Eq. (1) (for each tested value of *m* and *q* = 1), with a certain percentage of deviation (proportional to the corresponding phase level) added over each of these nominal phase levels *ϕ _{n}*, ranging from 0% to 20%, in steps of 4%. Each phase level is randomly increased or decreased by such a percentage and then the operation is repeated 20 times, finally getting and representing the average of each final result. We evaluate the quality of the output comb spectrum through two parameters, namely the average ratio between the maximum and minimum peaks of the obtained harmonics and the average ratio between the minimum harmonic peak and the maximum of the noise floor, both calculated versus the phase deviation percentage for each tested value of

*m*. The obtained curves (Fig. 5 ) are represented, as an example, for each even value of

*m*(

*q*= 1) as the phase variation percentage is increased (for odd values of

*m*, results are similar to those reported here for even values of

*m*). In Fig. 5(a) the trend is that the maximum-to-minimum harmonics peaks ratio worsens by increasing

*m*(and, for a given

*m*, by increasing the phase deviation amount); the same trend is observed for the minimum peak-to-noise floor ratio (see Fig. 5(b)). These trends are qualitatively in agreement with the obtained experimental results, as reported in Fig. 3.

## 5. Conclusions

In conclusion, spectral self-imaging has been exploited to realize a periodic frequency comb laser with a discretely tunable comb spacing. Frequency division factors ranging from 2 to 9 have been achieved by use of multilevel time-domain EO phase modulation of a 10-GHz optical pulse train, implemented through an electronic AWG. The process can be applied on any coherent periodic frequency comb input and the comb spacing tuning is achieved without affecting the spectral bandwidth and total energy of the original comb. The influence of deviations in the temporal phase modulation function on the output comb spectrum has been theoretically analyzed, confirming the experimentally observed deterioration in the comb spectrum features as the comb spacing-division factor is increased.

## Acknowledgments

Authors would like to thank Prof. Martin Rochette at McGill University in Montréal for the loan of the high-resolution OSA.

Dr. Antonio Malacarne is now with “National Laboratory of Photonic Networks - CNIT”, via G. Moruzzi 1, 56124, Pisa, Italy.

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