1. Introduction

Recently developments of LiNbO₃ modulators capable of yielding large modulation bandwidths and good operation performances in low driving voltages have initiated lots of dynamic research covering various areas such as multicarrier generations [1], optical pulse shapings [2], millimeter wave generations [3], and supercontinuum generations [4] for respective application on optical communication systems, radio-over-fiber systems, optical signal processing, wireless communications and so on. In all suggested schemes employed in references [1–4], the external modulations of continuous-wave (CW) laser signals were advantageously utilized. Compared with the conventional mode-locked lasers generally requiring the cavity length stabilization and polarization maintenance due to environmental perturbations, the external modulation technique should provide simple, robust, efficient and durable optical sources.

Among the externally modulated lasers, phase-modulated lasers, where CW laser signals are phase modulated, have been widely under study for various measurement applications such as optical frequency-domain reflectrometry (OFDR) [5], frequency-modulated continuous wave (FMCW) interferometry [6], and phase-lock interferometry [7]. Meanwhile various schemes for generating optical short pulses through phase modulation have been suggested [8–10].

Previous methods proposed for optical pulse generations based on phase-modulated laser signals can be generally categorized into three;

1. Conversion into amplitude-modulated optical signals by its passage through optical elements introducing different group delay to the phase-modulated laser signals [8]

2. Proper amplitude-only-filtering applied on extreme sidebands in a phase-modulated laser spectrum to obtain transform-limited optical pulses [9]

3. Optical filtering of the central part of a phase-modulated laser spectrum for repetition-rate-doubled pulse train generation [10].

Although the last categorized optical pulse was already reported in brief [10], pulse characteristics imposed on those pulses such as pulse durations and chirp information have never been verified and discussed in literatures. In this paper, we show that a fraction of a phase-modulated laser output can be turned into an optical pulse train with dual chirp states, where the frequency chirp is alternating periodically between an up-chirp state and a down-chirp state while pulse durations and pulse shapes are steadily maintained between two groups of optical pulse trains. To the best of our knowledge, dual chirped optical pulse generation has never been reported before. Up to now chirped pulses have been employed for several interesting research topics. Chirped optical pulses were proved to serve as very useful tools for the characterization of the intramolecular dynamics [11, 12] and the radiation of the terahertz signal [13–15]. In those experiments, chirp strengths and signs of optical pulses were commonly controlled by the adjustment of glass-based pulse compressors and pulse stretchers. In our laser configuration, the amount of frequency chirp and the pulse duration are decided by the modulation index and the modulation frequency. The modulation index is electrically decided depending on the available power of the radio frequency amplifier and the half wave voltage of the phase modulators. Furthermore optical pulse trains with the same amount of frequency chirp but opposite signs are simultaneously available and therefore we can improve measurement efficiencies regarding measurement time and measurement accuracy.

It is also found that the pulse duration and the linear frequency chirp parameter can be found in closed form expressions given as a function of the modulation index and the modulation frequency when the phase information of the dual-chirped pulse trains are well approximated to have the phase modulation function applied to the phase modulator even after spectral filtering. In experiments, a conventional frequency-modulated fiber laser was used to generate a phase-modulated laser signal in order to increase the modulation index. A 5 GHz repetition-rated optical pulse was generated with pulse duration 29.4 ps at the ~2.5 GHz modulation frequency when the output spectrum of the laser was filtered by an OBF with 0.3 nm FWHM and the center wavelength 1567.63 nm coinciding the carrier frequency of the frequency-modulated laser spectrum. Measured pulse duration was in good agreement with the analytical prediction value 28.2 ps and the numerical result 27.7 ps.

2. Principle of dual chirped pulse generation from a phase-modulated laser

The fundamental principle of a phase-modulated laser can be readily understood in timedomain by a simple model where a continuous-wave (CW) laser is frequency modulated in sinusoidal way by an external phase modulator [8]. When a CW laser of angular frequency ω_o is phase modulated by a sinusoidal signal of f_M frequency, single pass laser output can be expressed as

where E_o is input amplitude given as real, ϕ is the modulation index imposed on CW laser signal and Jn(ϕ) is the Bessel function of the first kind of integral order n.

The amplitude of the electric field in the phase-modulated laser at ω=ω_o+n ω_M frequency becomes Jn(ϕ) at t=0. Figure 1(a) shows J_n(ϕ) for various sideband order n when the modulation index ϕ is 10π. From Fig. 1(a), it is clear that the red filled circles in the antistokes band on the right part of the figure have all positive amplitudes in other words they are in-phase. Consequently the sidebands in the anti-stokes band form an optical pulse train in time domain. On the contrary the phases of sidebands in the stokes band represented in blue filled circles changes their signs periodically and consequently destructive beatings takes place among sidebands in the stokes band. The phenomena can be explained more clearly by another Bessel function identity J-n(ϕ)=(-1)ⁿJ_n(ϕ). After a half period of modulation cycle is elapsed, or when ω_Mt is π, sidebands in stokes band becomes in-phase or added constructively to form an optical pulse train, whereas sidebands in anti-stokes band are added destructively at that moment. Owing to this iterative phase recovery for the stokes and anti-stokes bands, optical pulse can be formed by carving either of these extreme sidebands. Optical pulse trains are numerically obtained and plotted in Fig. 1(b) where the blue trace represents optical pulses consisting of anti-stokes bands only, and the red trace is for optical pulses consisting of stokes band only. By virtue of this phase correlation existing in these extreme sidebands, Mamyshev realized nearly transform limited optical pulse trains by selection of the extreme sidebands with an optical filter [9].

 

Fig. 1. Phase information of generated sidebands of a phase-modulated laser and optical pulse trains from spectral slicing of either of extreme sidebands. (a) Amplitude of the electric field component of a PM laser as a function of sideband order n when effective modulation index ϕ is 10π, and ω_Mt=0. (b) Optical pulse consisting of anti-stokes band in blue line and optical pulse consisting of stokes band in red line.

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Two extreme sidebands in a phase-modulated laser have been used to generate short optical pulse trains and the rest of sidebands distributed around angular carrier frequency ω_o were eliminated in previously suggested pulse generation schemes based on a phase-modulated laser [8, 9]. In Fig. 1(a), the sidebands located around the carrier frequency of a phase-modulated laser (the central portion of the figure) do not seem to represent any phase correlation among them. To verify phase evolution of each sideband around carrier frequency, central part of Fig. 1(a) is magnified in scale and separately plotted in Fig. 2(a). Each of two nearby sidebands is grouped in dashed line. As is clearly viewed, two nearby sidebands are inphase and the next two sidebands are also in-phase. However there is a relative phase difference of π between those two groups. There is no such periodicity observed in sidebands with higher sideband orders. Phase evolution of those central sidebands can be more simply described in a phasor diagram in Fig. 2(b) where all sidebands are drawn with equal amplitudes for a simple analysis.

 

Fig. 2. Phase relation between sidebands nearby the carrier frequency of a PM laser. (a) Phase correlation between a pair of sidebands. (b) Phasor diagram to show burst of optical pulse trains with repetition rate twice as fast as an imposed modulation frequency.

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If we choose a coordinate system where the carrier frequency f_o is stationary, sidebands in higher or lower frequencies at f=f_o±nf_M will rotate clockwise or counterclockwise respectively with an oscillating term of $e^{\pm i2\pi n{f}_{M}t}$. The instantaneous amplitude of the optical pulse is given by the vector sum of sidebands. When ω_Mt is zero, the vector sum of sidebands becomes zero, and complete destructive beating happens as shown in the top diagram of Fig. 2(b). After a quarter of the modulation period or when ω_Mt becomes π/2, the sum of phasors gives the largest possible amplitude corresponding to the second diagram of Fig. 2(b) from the top. However, as all phasors are not completely arranged in the same angle, constructive beatings take place partially. It shows that complete destructive and partially constructive beatings take place when ω_Mt becomes π and 3π/2. From this simple analysis, we can expect that the central spectral filtering of the phase-modulated laser signal must result in an optical pulse train with a repetition rate twice of the applied RF signal.

Phase information along an optical pulse train starts to deviate from an ideal sinusoidal phase modulation function ϕsin(ω_Mt) found in Eq. (1) when the phase-modulated signal experiences the optical bandpass filtering. Figure 3 shows the numerical results of the generated optical pulse trains and the phase traces in time domain obtained by discretely changing the spectral width of the OBF. In these analyses, we assumed the frequency response of the OBF to be real (without frequency dependent group delay). Parameters used for a phase-modulated laser in the numerical analysis are the same as those used for describing Fig. 1(a) and E_o is set to be unity. For a simple and practical consideration, we have assumed a Gaussian shape for the transmission spectrum of an OBF. To model a Gaussian shaped OBF, we used exp(-0.5(ω-ω_c)²/(Δω _filter) ²) as an numerical expression where ω_c denote the center angular frequency and Δω _filter is given 2πΔf _filter where Δffilter represents the spectral half-width of the OBF. The number of sidebands N involved in dual-chirped pulse generations decided by the OBF can be approximated given as Δf _FWHM/fM where Δf _FWHM denotes the FWHM of the OBF given as $2\sqrt{\mathrm{\hspace{0.17em}ln}2}\Delta f_{\mathrm{filter}}$ Δf _filter. The filtered optical spectrum was obtained by taking the Fourier transform of Eq. (1) at first and then we product it with the frequency response function of the Gaussian shaped OBF. The consequent dual-chirped optical pulse was finally described in time domain by taking the inverse Fourier transform of the filtered spectrum in frequency domain. At the given modulation index 10π, the periodicity was not found any more in the higher order sidebands not included in the group of ~18 sidebands symmetrically located around the carrier frequency. For the further analysis we numerically found the number of sidebands N given as a function of modulation index under the condition that the periodicity should be kept. The result approximately shows the linear relation as below

Each trace in Fig. 3(a) described in red, green, or blue color was obtained when FWHM of the OBF was discretely chosen to be 14 f _M, 18 f _M, and 36 f _M, respectively to take into account of the relation suggested in Eq. (2) and show how dual chirped optical pulse trains evolve according to modulation indices and the number of sidebands. As expected from the discussion about Fig. 2, Fig. 3(a) shows that the repetition rate of the pulse trains becomes twice of the applied RF signal [10]. Pulse characteristics such as pulse duration and extinction ratio have inherent dependence on the spectral width of the OBF. The shortest pulse width was obtained under the condition of the largest expense of optical power as is represented in the red line. The broader the FWHM of an OBF became, the worse extinction ratio was imposed on the optical pulse trains. On the contrary, the broader the spectral width of an OBF became, the less perturbation on the ideal phase modulation function was kept as can be perceived intuitively. Phase information of each pulse train was retrieved and compared with the applied phase modulation function and all of them were plotted in Fig. 3(b). Blue and green traces were almost identical to each other and furthermore those two were also indistinguishable from the trace representing ideal phase modulation function 10πsin(ω_Mt). Noticeable phase change in the red curve reflects the effect of tight optical filtering which is responsible for the generation of an optical pulse train with the shortest pulse width and the highest extinction ratio in Fig. 3(a).

 

Fig.3 . umerically obtained optical intensity (a) and phase (b) for a laser output in time domain depending on the spectral width of an OBF. Red, green, and blue curves are when FWHMs of the OBFs are 14 f _M, 18 f _M, and 36 f _M, respectively. (c) 3D plot of dual-chirped optical pulse trains with varying the number of sidebands at the modulation index 10π

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It is also qualitatively predictable that the relation of the pulse duration with the spectral width is different from the case found in the transform limited optical pulse if phase states in central sidebands are kept in minds as discussed in Fig. 2. In Fig. 3(c) 3D plot of the dual-chirped optical pulse trains expected upon varying the number of sidebands at the modulation index 10π is also shown to clearly represent the general characteristics of dual chirped pulse trains. It is notable that optical pulse trains with high extinction ratios and short pulse durations have large tolerance over the number of sidebands. As the number of sidebands increased from the value ~10 to the value ~25, the pulse duration gradually increases still maintaining high extinction ratios whereas the phase modulation function of the dual chirped pulse remained almost identical with the applied phase modulation signal when the selected number of sidebands was kept above 18. To extend our analysis further more about chirp characteristics, we only take care of a dual chirped pulse train which is generated under the condition of the selection of the sidebands holding periodicity as described in Eq. (2).

We can extract analytic phase information about a pulse across the pulse duration in closed form from Eq. (1) when the pulse width is relatively short compared to the period of a pulse train and the phase distortion is negligibly small. As pulse duration is noticeably short compared to the half of the phase modulation period, it is reasonable to take a Taylor expansion for the phase function near the each peak position of a pulse train. There are two pulse groups in the pulse train depending on the frequency chirp of a pulse in Fig. 3. The center positions for the first group are found when ω_Mt becomes (2k+0.5)π, while those for another group are found when ω_Mt becomes (2k-0.5)π and k is an integer in both expressions. From Eq. (1), the frequency chirp of output pulses becomes

The frequency chirp around the peak position can be approximately given by expanding a cosine term in Eq. (3) around a pulse position. Then, we have a very simple expression for the frequency chirp, which is linear with respect to time in small range,

This indicates the optical pulse is linearly chirped and the chirp state keeps switching; pulses in the first pulse group are chirped negatively whereas those in the second pulse group are positively chirped. Positively chirped pulses are conventionally called up-chirped pulses and negatively chirped pulses are called down-chirped pulses.

In the last paragraph, dual chirp characteristics of the dual chirped optical pulse are roughly explored by driving the frequency chirp expression. However it is demanding to combine all the parameters such as the spectral half-width Δf, the number of sidebands N, the modulation index ϕ and the modulation frequency f _M for extracting analytical expressions for the pulse duration and the frequency chirp. For the analysis we begin with the conventional Gaussian-shaped pulse expression including the information of frequency chirp in time domain like

where C means the linear frequency chirp parameter and T _o is the pulse half-width. The peak amplitude term was omitted, as it is not required for finding the pulse duration and the frequency chirp parameter. From Eq. (5), another frequency chirp expression can be found by taking the first time derivative of the phase information and it is given as

Compared with Eq. (4), frequency chirp is given as a function of linear frequency chirp parameter and pulse duration. Fourier transform of Eq. (5) serves as another equation necessary for extracting two parameters (C and T _o). In the Fourier transform expression we took only the spectral half-width expression Δf,

As the spectral half-width of the spectrum in dual chirped optical pulse trains is identical to the spectral half-width of the employed OBF Δf _filter, the value of spectral half-width of the spectrum can be substituted with $~\frac{N}{2\sqrt{\ln 2}}{f}_M.$. Finally if we combine Eq. (4), Eq. (6) and Eq. (7), the analytic expression of linear frequency chirp parameter can be found

where ${\frac{1}{\varphi }\left(\frac{\Delta f}{f_M}\right)}^2$ is defined as ζ and it can also be described as $\frac{1}{\varphi }{\left(\frac{N}{2\sqrt{\ln 2}}\right)}^2$. The pulse duration is also obtained by inserting the known value of linear frequency chirp parameter into the equation generated by using Eq. (4) and Eq. (6) and the result is

The linear frequency chirp approximation is generally accepted in actively mode locked laser and it also holds in our case as long as pulse duration is comparably short. It is notable that similar pulse characteristics are typically observed in conventional FM mod-locked lasers. However competition between up-chirped pulses and down-chirped pulses usually degrades stability inside an optical cavity, and random oscillations including sudden switching between up-chirped and down-chirped pulses states take place [16, 17].

3. Experiment and discussion

Dual chirped pulses are made possible after central parts selection of a whole phase-modulated laser spectrum. Figure 3 and numerical expressions driven in the last paragraph show that the FWHM of the OBF should be chosen under the full consideration of the phase-modulated signal characteristics such as the modulation index and the modulation frequency to achieve dual chirped pulses with high extinction ratio and short pulse duration. In the current experiments, the maximum spectral bandwidth was still far less than the FWHM of the OBF (0.3 nm) which was only available during experiments due to the maximum value of the modulation frequency (2.5 GHz) and the maximum modulation index (~π). Both values were limited by the bandwidth of the used RF devices. Consequently we could not generate a phase-modulated optical signal for dual chirped optical pulse generation. Therefore instead of the conventional phase-modulated laser we applied a so-called frequency-modulated (FM) laser, where frequency modulation takes place inside of an optical cavity, to the dual chirped pulse configuration to increase the modulation index [18]. In that case, the modulation index is technically multiplied by f _ax/(2πf _d) where f _ax is the fundamental mode spacing of a laser cavity, and f _d is the amount of frequency detuning between f _M and m times of f _ax [18]. m is the harmonic order of an FM mode-locking. As a result, the effective modulation index Φ is simply given as

and we can reach an expression for an FM laser signal like Eq. (1),

We should note that even though the FM laser was temporally employed instead of the phase-modulated laser according to the narrow spectral bandwidth achievable from a phase-modulated laser in the current experimental conditions but it should not be considered as any restriction or drawback to the dual-chirped pulse generation because the large amount of modulation index is believed to be commercially available in near future considering the development speed of the low V_π LiNbO₃ modulators found in literatures [19–21]. The repetition rate and the chirp sign can also be flexibly selected for specific measurement conditions by use of an additional modulator following our dual-chirped pulse setup.

Figure 4(a) shows the FM erbium-doped fiber ring laser and measurement configurations used for the dual chirped optical pulse generations and their characterizations A 10 m long erbium-doped fiber, which was pumped by a 980 nm pump laser diode, was used as a gain medium. Frequency modulation was achieved by using a LiNbO₃ phase modulator. The insertion loss and the V_π, which is the voltage necessary for π radians phase shift, of the LiNbO3 phase modulator were 3 dB and 5.5 V, respectively. The total cavity length was approximately 21.6 m, and the corresponding fundamental mode spacing f _ax was given as 9.27 MHz. 30% of circulating energy was extracted and used for pulse characterization by an optical spectrum analyzer (OSA), an intensity auto-correlator and a fast sampling scope. FWHM of an OBF was 0.3 nm in wavelength domain or 37.5 GHz in frequency domain, and its transmission spectrum was well approximated to have a Gaussian shape. Due to high insertion loss of an OBF of ~6 dB used in current experiment, the optical signal was amplified by a following EDFA with 10 dBm saturation output power for further analysis.

Figure 4-(b) shows the optical output spectrum of an FM laser before (in the blue trace) and after (in the red trace) the optical band-pass filter. This experimental result was used for numerical analysis in this paper. Optical spectrum in blue line is an FM laser spectrum and red line shows an optical spectrum after central portion of FM spectrum were selected by the OBF. Carrier frequency of an FM laser was 1567.63 nm and transmission peak of an OBF was tuned to closely match that wavelength. Laser output power just after the 30 % optical coupler was 3.34 dBm. Modulation frequency f _M was 2494.21 MHz, and it was slightly detuned from the 269^th harmonic frequency of f _ax by +150 kHz where f _ax is the fundamental mode spacing of a laser cavity. Effective modulation index was calculated by using Eq. (10) and it was ~9.8 π radian. Expected spectral width of an FM laser given as ~2Φf _M was ~157 GHz and it was quite close to measured spectral width ~150 GHz.

 

Fig. 4. (a) Experimental configuration for dual chirped optical pulse generation from an FM erbium doped fiber ring laser and measurement setups for laser output. (b) Optical spectra before and after an optical band-pass filter with a FWHM of 0.3 nm or 37.5 GHz.

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Optical pulse trains corresponding to optically filtered spectrum in Fig. 4(b) were measured using a high speed sampling scope and shown in Fig. 5(a). Repetition rate of the generated pulse train was ~5 GHz which is twice of the applied modulation frequency. Measured intensity profiles for neighboring pulses look identical in this measured pulse train. Periodic distortions at the pulse tailing edges result from the insufficient electrical bandwidth of the photo-detector and the sampling scope. Optical pulse was also measured with an intensity auto-correlator. For the pulse characterization, we assumed Gaussian optical pulse shape deduced from the Gaussian spectral shape and autocorrelation function given in red trace in Fig. 5(b) showing the best correspondence with the experimental result in blue trace in Fig. 5(b) was found numerically. FWHM of the pulse duration retrieved after deconvolution was 29.4 ps and by use of this value the linear frequency chirp parameter ±2.25 was acquired using Eq. (7). General disagreement observed between two spectra was generated by the non-perfectly symmetric FM laser spectrum within the OBF transmission function. At the modulation index 9.8 π radian, the optimum number of sidebands symmetrically found around the carrier frequency corresponding to 1567.5 nm in wavelength is calculated to be ~17 and it matches ~40 GHz spectral width. If we consider the FWHM of the OBF 37 GHz, it is only 7 % larger value and therefore we can apply Eq. (8) and Eq. (9) for analytically finding of the linear frequency chirp parameter and pulse duration. Obtained linear frequency chirp parameters and the FWHM of the pulse duration are ±2.18 and 28.2 ps. Two values predicted from the analytic solutions show close correspondences with the measurement results and thus it proves the effectiveness and usefulness of the analytic expressions for pulse duration and linear frequency chirp parameter.

 

Fig. 5. Optical pulse trains corresponding to optically filtered spectrum in Fig. 4(b). (a) Measurement result of an optical pulse train using a high speed sampling scope. (b) Intensity autocorrelation of dual chirped optical pulse obtained in experiments (blue trace) and in numerical expectation (red trace).

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4. Conclusion

We have experimentally demonstrated and theoretically proved a generation scheme of a dual chirped optical pulse train from a phase-modulated laser signal for the first time to our knowledge. More specifically a fraction of a phase-modulated laser output was turned into an optical pulse train with dual chirp states, where the frequency chirp is alternating periodically between an up-chirp state and a down-chirp state. At every half of a modulation cycle, chirp states change periodically, while pulse duration and pulse shape are steadily maintained. Linear frequency chirp parameter and the pulse duration were analytically derived from the phase-modulated laser model in time-domain combined with the conventional Gaussian-shaped pulse expression embedded with frequency chirp in time domain. Experimentally to increase the modulation index because of the limited bandwidth and the lack of RF modulation power, a phase-modulated spectrum was generated through a conventional fiber-based FM laser. And then a number of sidebands located around the carrier frequency of an FM laser were selected through an OBF. Optical pulses can be grouped into two depending on their chirped states. Neighboring pulses have opposite chirped states and the frequency chirp of each pulse was well approximated to be linear along the pulse duration. In experiments, 5 GHz repetition rated optical pulses were generated with pulse duration as short as 29.4 ps when FM erbium doped fiber ring laser modulated at ~2.5 GHz was followed by a Gaussian shaped OBF. The FWHM of the OBF was 0.3 nm in wavelength or 37.5 GHz in frequency, and the center wavelength corresponding to the carrier frequency of the FM laser signal was at 1567.63 nm. Although the optical pulse obtained in this experiment had rather wide pulse duration, we believe that ultrashort pulse trains with the femosecond pulse for the applications of chirped pulse is surely feasible when the high modulation frequency and the high modulation index are available in phase-modulated laser configuration based on our scheme.