Generation of tunable ultra-short pulse sequences in a quasi-discrete spectral supercontinuum by dark solitons

Saili Zhao; Saili Zhao; Hua Yang; Ying Huang; Yuzhe Xiao

doi:10.1364/OE.27.023539

1. Introduction

There have been numerous technological advances in various fields associated with the emergence of octave-spanning supercontinuum (SC) including frequency metrology, spectroscopy, medical imaging and optical communication, in which the coherence and stability properties are the key indications of the quality of the SC sources [1–6]. However, many applications like multiphoton microscopy and ultrafast spectroscopy require not only a high temporal resolution but also a high spectral resolution [7]. Unfortunately, there is a competing relationship between temporal resolution and spectral resolution for most pulse sources. On this occasion, the emergence of ultra-short pulse sequences in a high-coherent quasi-discrete spectral SC consisting of the isolated narrow-band sources provides a solution to this problem. Moreover, it would be attractive to achieve the tunability of temporal and spectral resolution by producing the ultra-short pulses of different temporal width in a quasi-discrete spectral SC with the narrow-band sources of different spectral width. Such tunable ultra-short pulse sequences in a quasi-discrete spectral SC is beneficial to the development for the detection of fast chemical processes [8], the investigation of molecular movement [9] and the ultrafast information transfer systems [10].

Previous studies mainly adopted a selective spectral filtering technique to obtain a useful multi-wavelength pulsed source, but the resulting pulse width is too long [11]. While there have also been several methods proposed to generate the ultra-short pulse sequences such as an array of birefringent crystals [12] and the superimposed Bragg fiber gratings [13], they are complicated and not easy to regulate. In this paper, we present a simple scheme to acquire the temporal ultra-short pulse sequences and spectral narrow-band sources in a quasi-discrete fiber SC generation. In time domain, the emergence of dark solitons leads to a series of ultra-short pulses yielded in the normal dispersion regions. In turn, spectral interference of these generated ultra-short pulses facilitates the formation of narrow-band sources in spectral domain, shaping a quasi-discrete spectral SC. These dark solitons are created by launching two identical pulses one of which is delayed relative to the other in the normal dispersion region of a fiber with two zero dispersion wavelengths (ZDWs) [14–16]. High-intensity few-cycle pulses are used as dual pumps so that no soliton fission occurs and pump energy is almost completely transferred into two normal dispersion regions, ensuring the generated SC with a high coherence and ultra-wide bandwidth [17,18].

Here, dual normal dispersion regions of fiber contribute to the formation of ultra-short pulses with different temporal width and narrow-band sources with different spectral width. Temporal width of ultra-short pulses and spectral width of narrow-band sources can be effectively regulated by tuning the pump power and initial time delay between two input pulses. In addition, the number of ultra-short pulses and narrow-band sources can be manipulated by the initial time delay and pump power. The above phenomena reveal that dark solitons open up a simple route for the generation of tunable ultra-short pulse sequences in a quasi-discrete spectral SC with narrow-band sources of different spectral width.

2. Propagation model and fiber properties

The propagation dynamics of ultra-short pulses in a fiber can be captured by employing the generalized nonlinear Schrödinger equation (GNLSE) [19,20]:

\frac{\partial A}{\partial z} = - \frac{a}{2} A + \sum_{k \geq 2} \frac{i^{k + 1} β_{k}}{k!} \frac{\partial^{k} A}{\partial t^{k}} + i γ (1 + i τ_{s h} \frac{\partial}{\partial t}) \times (A (z, t) {\int_{- \infty}^{\infty} R (t') | A (z, t - t') |}^{2} d t'),

where

A (z, t)

is the electric field envelope at propagation distance z in a retarded reference time frame

t = τ - β_{1} z

traveling at the envelope group velocity

β_{1}^{- 1} .

a

is linear loss constant,

γ

is the nonlinear coefficient, and

β_{k}

are the dispersion coefficients obtained by a Taylor series expansion of propagation constant around the pulse centre frequency

ω_{0} .

_{$τ_{s h} = {(ω_{0})}^{- 1}$ is associated with the effects such as self-steepening and optical shock formation. The response function $R (τ) = f_{R} h_{R} (τ) + (1 - f_{R}) δ (τ)$ includes both the delayed Raman and instantaneous electronic contributions, with $h_{R} (τ) =exp(- τ / τ_{2})sin(- τ / τ_{1}) (τ_{1}^{2} + τ_{2}^{2}) / τ_{1} τ_{2}^{2} .$
Two identical Gaussian pulses at $λ_{0} =835 nm$ with time delay $t_{d e l} = 0.2 ps$ as the initial envelope takes the form:
$A (z = 0, t) = \sqrt{P_{0}} {\exp (- \frac{t^{2}}{2 T_{0}^{2}}) + \exp (- \frac{{(t + t_{d e l})}^{2}}{2 T_{0}^{2}})},$ where peak power and initial width of each pulse are $P_{0} = 12 kW$ and $T_{0} = 10 fs,$ respectively. A 20-mm long silica photonic crystal fiber with two ZDWs located at 876 nm and 1174 nm serves as the transmission medium, which can be obtained by properly setting the geometry of an air-silica holey fiber according to the method proposed in Ref [21]. The specific values of $f_{R}$ and $τ_{1, 2}$ are 0.18, 12.2 fs and 32.0 fs in this silica fiber. Nonlinear coefficient and dispersion coefficients up to the sixth-order at the pump wavelength of 835 nm are as follow: $γ = 0.41 W^{- 1} m^{- 1}, β_{2} = 4 .9367 {ps}^{2} / km, β_{3} = 4 .85 \times 10^{- 2} {ps}^{3} / km, β_{4} = 3 .2023 \times 10^{- 5} {ps}^{4} / km, β_{5} =$ $- 2 .0394 \times 10^{- 7} {ps}^{5} / km, β_{6} = 8.0079 \times 10^{- 10} {ps}^{6} / km .$ The curve of dispersion and group delay corresponding to the pump is depicted in Fig. 1(a). Note that the self-steepening effect need be considered with the parameter $s = τ_{s h} / T_{0} =$ 0.2783 for the ultra-short pulse and the loss is neglected because of a short propagation length here.
Fig. 1 (a) The curve of dispersion (blue dashed) and group delay (red solid) corresponding to the pump as a function of wavelength in the fiber with two ZDWs. Two vertical dotted lines locate two ZDWs of fiber for 876 nm and 1174 nm. (b) Output pulse shape (top) and temporal evolution (bottom), as well as (c) output spectrum and the degree of coherence $| g_{12}^{(1)} |$ (top) as well as spectral evolution (bottom) of a single pump pulse in the fiber. The inset in (b) shows the temporal breakdown due to the shock wave formation stemming from self-steepening.
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Even though an all-normal dispersion fiber can be used as a medium for the generation of ultra-short pulse sequence, only a train of ultra-short pulses is formed and the resulting quasi-discrete SC spectral bandwidth is limited [22]. In a fiber with one ZDW, the interaction between dark solitons and bright solitons inhibits the formation of ultra-short pulse sequence, as shown in Ref [16]. Recently, I. Babushkin et al. have shown that the standard formation process of fundamental bright solitons by soliton fission is not available when a high-intensity few-cycle pulse as the pump propagates in a fiber with two ZDWs [17], which provides a new insight to acquire the ultra-short pulse sequences. In this paper, the injected pulse initial conditions and the input frequency within the normal dispersion region of fiber are set to initiate the interference pattern between the two dispersing pulses as well as subsequent generation of dark solitons and ultra-short pulse sequences.
3. The generation of quasi-discrete spectral SC in a fiber with two ZDWs
Let us first address the situation of only a single pump propagating in the fiber with two ZDWs. For the high-intensity few-cycle pump pulse, self-steepening as a critical effect creates optical shock formed at distance $z_{s} = 0.39 L_{N L} / s \approx 0.028 mm,$ where $L_{N l} = 1 / γ P_{0} \approx 0.22$ mm is the nonlinear length. In the initial stage of propagation, pump pulse undergoes a dramatically broadening in spectral domain accompanied by a catastrophic collapse in temporal domain. The broadening dynamics is governed by the combined action of self-phase modulation (SPM), shock formation due to self-steepening, as well as phase-matched Cherenkov radiation. Especially, the shock formation plays an important role in the significant spectral broadening. From the evolution of pulse shown in Fig. 1(b), one can observe that the scenario of temporal breakdown, which leads to a nearly-complete transfer of pump energy into two normal dispersion regions. Even though there are still a few optical waves with very low intensity remaining in the anomalous dispersion region, the ejection of fundamental bright solitons is not obtainable in the SC generation. Similar phenomena have been confirmed in Ref [17], which indicates that the standard formation process of fundamental bright solitons by soliton fission is not scalable into the few-cycle regime. After nearly all energy is pushed into the normal dispersion regions, the pulse begins to broaden quickly in temporal domain due to the dispersion effect. It is well-known that propagation dynamics in the normal dispersion region exhibit a low sensitivity to input noise, and hence an important feature of this SC generation process is its inherent capacity of producing high-coherent spectra [19,20]. To demonstrate the coherence property of the generated SC, we calculated the spectrally resolved modulus of the complex degree of first-order coherence:
$| g_{12}^{(1)} (λ, t_{1} - t_{2} = 0) | = | 〈 E_{1}^{*} (λ, t_{1}) E_{2} (λ, t_{2}) 〉 | / \sqrt{〈 {| E_{1} (λ, t_{1}) |}^{2} 〉〈 {| E_{2} (λ, t_{2}) |}^{2} 〉},$ where the angular brackets denote an ensemble average on 100 pairs of the output SC spectra. These spectra are independently generated by using the input pulses with different random quantum noise. As observed in Fig. 1(c), the SC generation in two normal dispersion regions is high coherent when pumping high-intensity few-cycle pulse in the fiber. Note that the shock wave formation due to self-steepening leads to the spectral components at some wavelengths with extremely low intensity [17]. In this case, spectral fluctuation induced by random noise is obvious at the low intensity of output spectrum, which results in some dips in spectral coherence degree curve. However, ultra-short pulse sequence cannot be acquired with a single pump pulse, as shown in the output temporal profile of Fig. 1(b).
Subsequently, we study the propagating dynamics of dual pumps in the fiber, with temporal and spectral evolution shown in Fig. 2. In the few-cycle case, each pump immediately experiences self-steepening and then the corresponding shock front is formed. The shock front formation is accompanied by the rapid spectral broadening as shown in Fig. 2(b1). At the distance of z = 10 mm (Fig. 2(a2)), sinusoidally modulated intensity appears in the temporal overlap region with normal dispersion due to the interference between the leading edge of each pulse. The modulated intensity origins from a linear frequency sweep owing to SPM and group-velocity dispersion (GVD) [14–16]. With further propagation down the fiber (Fig. 2(a2)), the overlap region extends and sinusoidally modulation reshapes into an expanding train of increasingly isolated dark solitons. These dark pulses will either accelerate or decelerate on a variable background, and thus their darkness changes with propagation. From Figs. 2(a1) and 2(a2), one can observe that dark solitons are firstly generated on the leading edge and then on the trailing edge of temporal profile. These dark solitons on the trailing edge have a much narrower temporal width and spacing in contrast with those on the leading edge at the same propagation distance. The above phenomena can be attributed to two reasons. On the one hand, initial time delay $t_{d e l}$ plays a crucial role in the beat frequency between two pulses, and hence affects the temporal width and spacing of dark solitons [18]. On the other hand, the difference of group velocities in two normal dispersion regions results in the different temporal broadening, and therefore the frequency sweep rate is different. Figure 2(a) also shows the transition from the nearly sinusoidal modulation at higher amplitude to the wider and more separated dark solitons at lower amplitude in two normal dispersion regions. Such different temporal spacing between dark solitons leads to the formation of cycle-tunable ultra-short pulses in the fiber SC generation. Here, ultra-short pulse width is defined as the half-width at the 1/e-intensity point and obtained by measuring the ultra-short pulse with the highest peak intensity in each ultra-short pulse sequence. According to this definition, the output ultra-short pulse width in the short- and long-wavelength normal dispersion region is measured to be about 3.2 fs and 33 fs, respectively.
Fig. 2 (a1) Temporal and (b1) spectral evolution of two identical pump pulses with 0.2 ps time delay in the fiber. The related (a2) temporal and (b2) spectral profiles at different propagation distances as well as (b2) the degree of coherence $| g_{12}^{(1)} | .$
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It is well know that a pair of pump pulses with a fixed phase relationship corresponds to a sinusoidally modulated power spectrum in frequency domain, which provides a mechanism for locking the relative phases in time domain. In particular, the phase between the dual pumps can be controlled by finely adjusting the time delay in order to maintain a peak or a null in the modulated power spectrum at some specific optical frequency. Therefore, dual pump pulses with suitable time delay $t_{d e l}$ result in the input spectral oscillation with period $1 / t_{d e l},$ as shown in the input spectrum in Fig. 2(b2). With the increase of propagation distance, new narrow-band sources emerge in the quasi-discrete spectral SC. At the same time, the energy of dual pumps is transferred into two normal dispersion regions almost entirely, which ensures the generated narrow-band sources contained in the quasi-discrete spectral SC with a high coherence, as shown in Fig. 2(b2). However, since the intensity in the dark parts of spectral interference fringes is very low, the corresponding signal-to-noise ratio is very small and spectral fluctuation induced by random noise is large, resulting in the decrease in spectral coherence at these parts. It is worth noting that the spectral width of these new generated narrow-band sources in the short-wavelength normal dispersion region is much narrower than in the long-wavelength normal dispersion region. Here, spectral width of narrow-band sources is defined as the full width at half maximum (FWHM) and obtained by measuring the narrow-band source with the highest peak intensity in each normal dispersion region. According to this definition, the output spectral width of narrow-band sources in the short- and long-wavelength normal dispersion region is measured to be about 4.4 nm and 26.3 nm, respectively. Such phenomena can be explained by the fact that each pulse in the temporal sequence of the interference pattern has a center frequency slightly different from the previous one [23]. The difference corresponds to the center frequency of spectral lines in the quasi-discrete structure, which further determines the spectral width of narrow-band sources.
We also plot the spectrograms at different distances in Fig. 3 to further illustrate the propagation dynamics of dual pumps. Since the dispersion length of each pulse $L_{D} = T_{0}^{2} / | β_{2} |$ $\approx 2 mm$ is much longer than $L_{N L}$ and $z_{s},$ SPM and self-steepening play the dominated role during the initial pulse evolution rather than GVD [20]. As shown in Fig. 3(a), two input pulses broaden considerably in spectral domain while only slightly in temporal domain. When the propagation distance exceeds 2 mm, two pulses rapidly broaden in temporal domain. Expectedly, temporal overlap occurs, as shown in Fig. 3(b). At the same time, the energy of each pulse has been pushed almost completely into two normal dispersion regions at z = 10mm. Due to the difference of group velocity in two normal dispersion regions, temporal overlap firstly occurs in the long-wavelength normal dispersion region (Fig. 3(b)) and then the short-wavelength normal dispersion region (Fig. 3(c)). Subsequently, owing to the linear frequency sweep from SPM and GVD in each normal dispersion region, intensity is sinusoidally modulated on temporal overlap region, generating a train of dark solitons separated by the ultra-short pulses. Along with propagation, temporal width of these ultra-short pulses is expanded because of the dispersion effect. However, dark solitons in the short-wavelength normal dispersion region cannot be clearly seen in Fig. 3(c). To get a closer look at this part, we plot the related temporal and spectral profiles in Figs. 3(d1) and 3(d2). Here, a series of ultra-short pulses separated by dark solitons are clearly observed in temporal domain. Accordingly, a quasi-discrete spectrum consisting of the narrow-band sources are shown in spectral domain. By comparing these ultra-short pulses in two normal dispersion regions, one can observe them with a much narrower temporal width on the short-wavelength side than on the long-wavelength side. Therefore, it is achievable to generate the tunable ultra-short pulse sequences in a quasi-discrete spectral SC including a series of narrow-band sources when pumping two identical pulses with a suitable time delay in the fiber with two ZDWs.
Fig. 3 The spectrogram at different propagation distances (a) z = 2 mm, (b) z = 10 mm, and (c) z = 20 mm for two identical pump pulses with 0.2 ps time delay propagating in the fiber. (d1) Temporal and (d2) spectral profiles of the short-wavelength normal dispersion region at z = 20 mm. (d1) is labeled in the rectangle box of (c).
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4. Impact of pulse parameters on the quasi-discrete spectral SC
To study the impact of input pulse parameters on the generation dynamics of a quasi-discrete spectral SC with ultra-short pulse sequences, we performed a series of simulations only with different time delay $t_{d e l}$ and plot the results in Fig. 4. From Figs. 4(a) and 4(b), one can observe that the output profiles are the same with an identical $| t_{d e l} |,$ but a little horizontal movement occurs in temporal domain between $+ | t_{d e l} |$ and $- | t_{d e l} | .$ This is because even though $| t_{d e l} |$ determines the beat frequency and further output profiles, temporal positions of input pulses ( $+ | t_{d e l} |$ or $- | t_{d e l} |$ ) have a non-negligible effect on the temporal profile, causing a little shift in x axis.
Fig. 4 Output (a1) pulse shapes and (b1) spectra versus time delay $t_{d e l}$ varying from −0.50 ps to 0.50 ps for an input of two identical pulses in the fiber. Output (a2) temporal and (b2) spectral profiles with 5 different $t_{d e l}$ values. (c1-c4) Output spectrograms with $t_{d e l} =$ 0 ps, 0.05 ps, 0.15 ps, 0.50 ps.
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Since the formation of ultra-short pulse sequences is attributed to the generation of dark solitons, it is important to study whether dark soliton can be produced in the normal dispersion regions. With $t_{d e l} = 0,$ two co-propagating input pulses are completely overlaid, effectively a single pump with twice peak power. In this case, input pulses broaden considerably in temporal and spectral domains, but without the formation of dark solitons, as shown in Fig. 4(c1). With the increase of $| t_{d e l} |,$ temporal and spectral broadening is weakened due to the effect of spectral interference rather than energy superposition. Indeed, dark soliton still cannot be formed under the condition with a small $| t_{d e l} |,$ as shown in $t_{d e l} = 0 .01 ps .$ For asmall $| t_{d e l} |,$ one can view the temporal components of two pulses located at the same normal dispersion region as a long single pulse with a small dip in temporal domain before GVD comes into play. In this view, SPM leads to a long positive frequency sweep with a small negative glitch in the center due to the small dip. Subsequently, GVD compresses the small negative sweep, which enhances the central intensity and converts the dip into a bump. Finally, a positive frequency sweep emerges in the center of the long single pulse due to SPM acting on the bump, so that no oscillations form in the pulse center, resulting in the absence of dark solitons [24]. When $| t_{d e l} |$ exceeds a certain threshold $(\approx 0 .03 ps),$ dark solitons begin to emerge on the leading edge and trailing edge of output temporal profile, as shown in $t_{d e l} =0 .05 ps .$ With further increase of $| t_{d e l} |,$ temporal width of dark solitons and ultra-short pulses narrows down and their number increases, as shown from $t_{d e l} =0 .05 ps$ to $t_{d e l} =0 .15 ps .$ This is because the larger $| t_{d e l} |,$ the larger the frequency difference between two output pulses at the same temporal position, which leads to the spatial interference with a quicker oscillation frequency. When $| t_{d e l} |$ is large enough, dark solitons will firstly disappear on the trailing edge, like the case with $t_{d e l} =0 .50 ps$ and then on the leading edge. Such phenomena result from whether dispersion induced temporal broadening within a limited fiber length can compensate the initial time delay between the parts of two pulses that locate at the same normal dispersion region, which affects the formation and disappearance of ultra-short pulses separated by dark solitons.
When looking at the output spectrum with a small $| t_{d e l} |$ (from 0 to 0.03 ps), one only sees a smooth spectrum in two normal dispersion regions. With $| t_{d e l} | > 0.03 ps,$ a quasi-discrete spectral SC begins to show up, as shown in Fig. 4(b2) with $t_{d e l} = 0.05 ps .$ With further increase of $| t_{d e l} |,$ the number of narrow-band sources increases and their spectral width narrows down. This is because a larger $| t_{d e l} |$ leads to a larger temporal difference between the same spectral components of two output pulses, which makes the spectral interference with a faster oscillation rate and the generated narrow-band sources with a larger number as well as a narrower spectral width [24,25]. From these results shown in Fig. 4, the number of ultra-short pulses and narrow-band sources as well as the temporal width of ultra-short pulses and spectral width of narrow-band sources can be manipulated in the quasi-discrete spectral SC generation by choosing an appropriate $| t_{d e l} | .$
The parameter N is introduced by using $N^{2} = L_{D} / L_{N L} = γ P_{0} T_{0}^{2} / | β_{2} |$ related to pump power and governs the relative importance of SPM and GVD effects on pulse evolution along the fiber. Since the SPM and GVD is the main mechanism to drive the formation of dark solitons, it is meaningful to study the dependence of dark solitons on N. We performed a series of simulations by changing the injected pump power to regulate N, and plot the output pulse shapes and spectra in Figs. 5(a) and 5(b). The increase of pump power makes the nonlinearity take effect earlier and contributes to the broadening in the spectral and temporal domains [19,20]. Even though two input pulses only broaden slightly in spectral and temporal domains within a limited fiber when N is small ( $\leq 4$ ), a quasi-discrete spectral SC consisting of the narrow-band sources still can be created due to spectral interference. However, temporal overlap does not occur between two output pulses, resulting in the absence of dark solitons in the normal dispersion regions. When N is larger than 4, input pulses significantly broaden in spectral domain, accompanied by the pump energy transferred into two normal dispersion regions almost completely. Such a high-efficient frequency conversion is conductive to the generation of an ultra-wideband SC spectrum. The temporal broadening is also extended as N grows, resulting in the formation of dark solitons and ultra-short pulses due to temporal overlap in two normal dispersion regions. More importantly, there are a growing number of ultra-short pulses and narrow-band sources emerging in the quasi-discrete spectral SC generation as N increases. The larger N, the larger the tunable range in the temporal width of ultra-short pulses and spectral width of narrow-band sources. In addition, note that the integer part of parameter N is equal to the calculated fundamental soliton order of each pump pulse. According to the simulation results shown in Fig. 2 and Fig. 4, the number of output dark solitons is affected by the value of fiber length and time delay. However, even in a long enough fiber, the generated dark soliton order does not necessarily match the calculated fundamental soliton order since these dark solitons produced by pumping two identical pulses with a suitable time delay are not fundamental dark solitons in the strict sense [14–16,22].
Fig. 5 Output (a) temporal pulse shapes and (b) spectra versus N ( $N = \sqrt{γ P_{0} T_{0}^{2} / β_{2}}$ ) related to pump power for an input of two identical pulses with a 0.2 ps time delay in the fiber.
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5. Conclusion
In this work, we showed an interesting mechanism to acquire the tunable ultra-short pulse sequences in a quasi-discrete spectral SC generation via pumping two identical pulses with a suitable time delay in a fiber with two ZDWs. Temporal width of these generated ultra-short pulses is different in two normal dispersion regions due to the combined effect of group-velocity dispersion and initial time delay. In turn, spectral width of these narrow-band sources contained in the quasi-discrete spectral SC is also different in two normal dispersion regions. Moreover, temporal width of ultra-short pulses and spectral width of narrow-band sources can be further regulated by varying the pump power and initial time delay. We believe these results presented here will provide a new insight to explore how to achieve the tunability of temporal and spectral resolution.
Funding
National Natural Science Foundation of China (61275137); Natural Science Foundation of Hunan Province (2018JJ2061); China Scholarship Council.
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Generation of tunable ultra-short pulse sequences in a quasi-discrete spectral supercontinuum by dark solitons

Abstract

1. Introduction

2. Propagation model and fiber properties

3. The generation of quasi-discrete spectral SC in a fiber with two ZDWs

4. Impact of pulse parameters on the quasi-discrete spectral SC

5. Conclusion

Funding

References

Cited By

Figures (5)

Equations (3)

Optics Express