A unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light

The analysis of optical pulse generation by phase modulation of narrowband continuous-wave light, and subsequent propagation through a group-delay-dispersion circuit, is usually performed in terms of the so-called bunching parameter. This heuristic approach does not provide theoretical support for the electrooptic flat-top-pulse generation reported recently. Here, we perform a waveform synthesis in terms of the Fresnel images of the periodically phase-modulated input light. In particular, we demonstrate flat-top-pulse generation with a duty ratio of 50% at a quarter of the Talbot condition for the sinusoidal phase modulation. Finally, we propose a binary modulation format to generate a well-defined square-wave-type optical bit pattern.


Introduction
Generation of ultrashort optical pulses at high repetition rates is a subject of increasing interest, which finds substantial application in ultrahigh-speed optical communications [1]. Ultrashort pulses obtained directly from passively mode-locked lasers suffer from the lack of electrical control of the pulse parameters, such as pulse width, pulse shape, and pulse position in a time slot. Moreover, it is not possible to tune the repetition rate for synchronization with other electrical signals. The above limitations can be overcome by the use of external modulators that permit ultrashort pulse generation from the continuous wave (CW) light emerging from a narrowband laser. Amplitude modulators suffer from large insertion losses and a low signal-to-noise ratio [2][3][4]. Alternatively, phase modulators have been widely employed for pulse pattern generation [5][6][7][8]. The quasi-velocity-matched guided-wave electrooptic modulator has allowed the design of compact, stable, and low-power ultrashort optical pulse generators [9]. In a different context, Sato has demonstrated optical pulse generation from a Fabry-Perot (FP) laser [10]. Here, no external modulation is employed. The physical mechanisms involved are the gain nonlinearities and the four-wave mixing process that originate the competition among the longitudinal modes supported by the laser cavity [11][12][13][14]. The CW light emerging from the FP laser is periodically phase-modulated with a frequency that is exactly the free spectral range (FSR) of the cavity.
The electrooptic method for optical pulse generation is based on the phase modulation with a sinusoidal signal of a CW beam from a narrowband laser diode. This produces harmonic sidebands (THz) around the optical carrier frequency so that the emerging waveform is strongly chirped. The optical field is launched through a group-delay-dispersion (GDD) circuit and compressed because the sweep rate acquired upon propagation partially compensates for the chirp. Among others, a single mode optical fiber (SMF) of adjusted length, a pair of diffraction gratings, an optical synthesizer, or a linearly chirped fiber Bragg [19][20][21]. Furthermore, we show that a remarkably simple formula describes the optical intensity at a quarter of the Talbot dispersion. On the framework of the space-time analogy [22], the above results constitute the temporal analogue of the field diffracted by a pure phase grating [23][24][25]. The parameters of the electrooptical modulator, the frequency of the driving signal and the modulation index, or alternatively the FSR in a FP laser, together with the GDD coefficient determine unambiguously the waveform achieved at the output. Specifically, we show flattop-pulse generation with a duty ratio of 50% for a modulation index of 4 π providing the sought theoretical support of the experimental results reported in [15,17]. The present description permits to identify a great variety of other pulse profiles. If we change continuously the dispersion amount in the GDD circuit, Fresnel patterns in intensity corresponding to a 1D sinusoidal phase-only grating appear, but now in the time domain, subsequently at the output of the arrangement. Of course, the same conclusion applies for other nonsinusoidal phase-only modulations.
This paper is structured as follows. In Section 2, the evolution of the optical field associated with a periodically phase-modulated input light through an arbitrary GDD circuit is provided in terms of the Talbot dispersion amount. We illustrate several examples concerning synthesis of different pulse waveforms at different dispersion amounts. In Section 3, the output pulse intensity is expressed in terms of a simple trigonometric formula when the output dispersion corresponds to a quarter of the Talbot dispersion. We identify an ultra-flat-toppulse pattern by binary phase modulation of CW light. Finally, in Section 4, the effect of the third order dispersion (TOD) of the SMF, or alternatively the spectral window of a LCFG, when used as a GDD circuit is discussed.

Theoretical analysis
After phase modulation, the optical field of the narrowband CW light is expressed as Here o E is the constant amplitude, o ω denotes the carrier optical frequency, and ( ) is the phase modulation function. For our purposes, we assume that ( ) is a periodic function with period T. Note that the perfect sinusoid is enclosed as a particular case. As a result of the periodicity of the phase ( ) t V , we can rewrite Eq. 1 in terms of a Fourier series expansion, namely, The periodic optical input intensity is where Of course, from Eq. 1, This implies that Kronecker delta function.
Aside from an irrelevant constant factor, the phase delay of an ideal GDD circuit is with 1 Φ and 2 Φ denoting the group delay and the GDD coefficient, respectively. Note that we assume no losses neither in the coupling of the input into the dispersive circuit or in the propagation. If we consider that the GDD circuit is implemented using a SMF, z , with z the propagation distance. The parameters 1 β and 2 β are the inverse of the group velocity and the group velocity dispersion (GVD) parameter of the fiber, respectively.
Note that we neglect higher-order dispersion terms and nonlinear interactions. Roughly speaking, both assumptions are satisfied when the bandwidth of the input light is less than 3 2 3 β β , with 3 β the TOD parameter of the fiber, and the power carried by individual pulses is not enough to excite nonlinear mechanisms in the fiber [26]. In section 4 we will further consider the narrowband assumption.
After propagation inside the GDD circuit (see Fig.1) the output field becomes From now on, the description of the signal is given in a reference framework moving at the group velocity of the wave packet, i.e., . From Eq. 6, the output intensity can be written as with ( ) Two findings are clear from the above equations. First, Eq. 7 indicates that is a periodic function of τ . Its period is, in principle, equal to the modulation period T . Second, from Eqs. 7 and 8 it is clear that the output optical intensity changes periodically with the dispersion coefficient 2 Φ . The period is just the Talbot dispersion, In this way, we obtain . This means that a change in the dispersion by 2 2 T Φ is equivalent to a temporal shift of half a period at the output intensity. We explore further implications of the above facts.
Next, we consider, as an example, the case of perfect sinusoidal modulation, Here, θ ∆ is the modulation index in radians. Of course For this case, the Fourier coefficients are expressed by the Bessel functions of the first kind, To illustrate waveform formation, we consider a realistic example concerning sinusoidal modulation where 2 Φ and f are set to 16 and 40 GHz, respectively. Different new pulse waveforms not yet reported are obtained by changing θ ∆ . In particular, we mention short pulse generation for 4 π θ = ∆ . Here, a duty cycle (DC) of 33% is achieved. Note that in this case the signal is free of annoying wings and tails but a high dc-floor level is present, as shown in Fig. 2(a). In Fig. 2(b) the numerical simulation shows a short pulse with a DC of approximately 18%. Although part of the energy lies outside the main pulse, the remaining dcfloor level is low. Note that, aside for a temporal shift of half a period, the same profiles are achieved when the dispersion is set to where q is an arbitrary integer. We also claim that the above shapes can be achieved with normal GDD as well as with anomalous one.

Flat-top-pulse generation
In this section we particularize the above key equations when dispersion is set to a quarter of the Talbot dispersion. From Eqs. 7 and 8, for 4 Equation 10, which is one of the main results of this paper, provides theoretical support for electrooptic flat-top-pulse generation, as will be shown next. At this point it is worth mentioning that the spatial analogue of the above formula was derived in [24,25], in the context of Fourier optics, to describe the properties of the irradiance distribution corresponding to the Fresnel diffraction patterns of a one-dimensional phase grating. Therefore, one should anticipate the above result within the framework of the celebrated space-time analogy. We note that Eq. 10 is valid for a general periodic phase function ( ) τ V . If we consider the sinusoidal modulation From Eq. 11 we observe that when 4 π θ = ∆ , the argument within the exterior trigonometric function ranges from 2 . The analytical curve shown in Eq. 11 is plotted in Fig. 3 for 4 π θ = ∆ and an input frequency of 40 GHz. The temporal width of the individual pulses is 12.5 ps. In this way, a nearly flat-top-pulse with a DC of 50% is achieved. Equation 11 provides an analytical formula for the waveform that was experimentally obtained in references [15] and [17]. Furthermore, due to the periodic nature of the optical field, the same result is achieved for a GDD dispersion , with q an arbitrary integer. The existence of multiple GDD amounts was pointed out in [15].
Next, we seek a different phase modulation format that allows ultra-flat-top-pulse generation. With this aim, we consider the periodic binary phase-only modulation of the carrier frequency given by with T being the period. The modulation ( ) τ V is plotted in Fig. 4(a). For this case, the argument inside the trigonometric function in Eq. 10 has two values, 2 π − and 2 π , respectively. Consequently, the output intensity shows a binary shape at 4 In order to clarify our description, we have calculated numerically and plotted in Fig.  4 for the phase modulation in Eq. 12 and 2 Φ ranging the whole first Talbot period. As expected, for dispersions , and , the irradiance presents a constant value. Whereas for 4 , according to Eq. 13 an ultra-flat-top optical pulse train is obtained, see Fig. 4(c). This kind of pulse could be employed for RZ modulation formats in optical signal transmission and, in particular, for differential phaseshift-keyed transmission.

GDD circuit analysis
A. Standard SMF It is usual to perform the FM to AM conversion process by means of a SMF. The strongly chirped light emerging from the electrooptical modulator is temporally distorted and compressed due to the propagation inside the fiber. A rigorous analysis of the quadratic approximation in Eq. 5 must be carried out to test the performance of the setup. Generally speaking, the spectral bandwidth of the incoming signal, ω ∆ , should be limited to . To obtain a rough estimation for the optical bandwidth of the phase-modulated signal, we plot in . From this plot we can assume that the main contribution to the output intensity comes from the Bessel functions with an order lower than 10. Thus, the condition for the validity of the parabolic approximation reads . So, the above inequality is widely satisfied even for the fastest commercially available electrooptical modulator, which works in the GHz range.

B. LCFG
The response of a LCFG operating in reflection is assumed to be a phase quadratic function only over a limited bandwidth ∆Ω . As a result two conditions must be fulfilled for the use of a LCFG as dispersive element. First, the carrier frequency should match the central frequency of the reflected spectral band of the grating. Second, the full spectral bandwidth of the phasemodulated light, ω ∆ , must be lower than ∆Ω . The spectral bandwidth of the element is related to the length through the expression eff n c L 2 2 ∆Ω Φ = [27], with eff n the effective refractive index. We assume that the LCFG is designed to match the condition

Conclusions
The evolution of an input field, consisted in a periodically phase-modulated CW light, into an arbitrary GDD circuit has been carried out in terms of the Talbot dispersion. The periodicity of the phase function has allowed us to derive an analytical formula for the FM to AM conversion process at one quarter of the Talbot dispersion. Furthermore, we have provided theoretical support for the recently experimentally demonstrated generation of a flat-top-pulse train using a phase modulator and a LCFG. We have also considered the generation of ultraflat-top light pulses by phase modulation with a square-wave-type signal, which could be employed for RZ modulation format. We numerically show that a SMF or LCFG can be used as an efficient GDD device in terms of the spectral bandwidth of the modulated signal. We would like to mention that the mathematical framework developed in this work is also applicable to the case of a multimode laser source, such as a FP laser, working in the frequency-modulated supermode regime. In this case, the frequency of the equivalent modulator will be given by the free spectral range of the longitudinal modes supported by the laser cavity. In the framework of the space-time analogy, the above results constitute the temporal analogue of the Fresnel diffraction field diffracted by a pure phase grating.

Appendix
We note that the Fourier series expansion of (A1) From the above equation, we have Let us now particularize Eq. 8 for 4 . We obtain Substitution of Eqs. A6 and A7 into Eq. A5 after some simple algebra leads to which is Eq. 10 in the text.