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Abstract
We experimentally study the linear propagation of optical pulses affected by high-order dispersion. We use a programmable spectral pulse-shaper that applies a phase that equals that which would result from dispersive propagation. The temporal intensity profiles of the pulses are characterized using phase-resolved measurements. Our results are in very good agreement with previous numerical and theoretical results, confirming that for high dispersion orders m the central part of the pulses follow the same evolution, with m only determining the rate of evolution.
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1. Introduction
Chromatic dispersion is one of the most important phenomena in optics [1], with implications in a wide range of applications ranging from optical telecommunications to supercontinuum and ultrashort pulse generations [2–4]. For example, dispersion management is crucial to achieve high-speed data transmission over long distances [1]. In nonlinear cavities, dispersion plays an important part in the generation of optical pulses and determining their characteristics [5,6]. While in most of these applications, only low dispersion orders (second and possibly third) need to be considered, the effects of higher-order dispersion have been investigated in a variety of contexts. For example, the effects of high-order dispersion are important for broadband frequency generation [7–9] and four-wave mixing [10,11]. Dispersion up to the $10^{\rm th}$ order has been considered in the generation and modelling of octave spanning supercontinuum generation [3,12], while careful management of high-order dispersion have enabled the direct generation of pulses shorter than $10\,\rm {fs}$ [5,13–15]. Finally, high-order dispersion impacts the bit rate in telecommunication systems, as was confirmed in a number of studies including some reporting dispersion compensation schemes up to the fourth order [16–20].
There has been a growing interest in the generation and propagation of novel families of optical solitons, arising from the combination of Kerr nonlinearity and a single, negative high, even order of dispersion with the publication of several experimental and numerical results [21–29]. However, in most of this work, high-order dispersion effects are interacting with nonlinear effects such as Kerr or Raman, and only a few theoretical and numerical studies considered high-order dispersion acting alone [16,18,30–32]. We recently theoretically and numerically studied the linear propagation of optical pulses in the presence of a single, high-order of dispersion $m$ [33]. The aim of this work was to provide a conceptual understanding of the effects of high orders of dispersion. Our approach was based on the observation that the effect of the dispersion is small for the part of spectrum that is close to the central frequency while the remaining part of the spectrum is strongly affected, leading to the formation of a pedestal in the temporal domain [33]. Therefore, the evolution upon propagation is mainly determined by the central part of the pulse, which is well characterized by the evolution of its full width at half maximum (FWHM). While this approach was confirmed by numerical simulations, the evolution of optical pulses due to high-order dispersion is yet to be observed experimentally.
Here we report the experimental observation and characterization of the linear propagation of an optical pulse in the presence of a single, high-order dispersion. We use a programmable spectral pulse-shaper to apply a phase that equals the phase that results from propagation in the presence of high-order dispersion up to the $12^{\rm {th}}$ order. The temporal intensity profiles of the resulting pulses are characterized by a set of phase-resolved measurements [34]. Our experimental results are in very good agreement with previous theoretical and numerical results [16,33]. They confirm that the effect of high-order dispersion differs significantly from that of the conventional second order dispersion ($m=2$), and that for sufficiently large dispersion order $m$, the pulses follow the same evolution, but at a different rate which is determined by $m$.
The outline of this paper is as follows. In Sec. 2 we review the theory we developed earlier. In Sec. 3 we describe the experimental setup which we use to obtain the experimental results discussed in Sec. 4. Finally, in Sec. 5 we discuss our results and conclude.
2. Background
We now review our previous theory and associated results [33]. Dispersion causes different frequencies to have different propagation constants $\beta$. Different frequencies therefore acquire different phases $\varphi$ upon propagation over a distance $L$ of [1]
where $\beta _m\equiv \partial ^m\beta /\partial \omega ^m$ is the dispersion coefficient of order $m$, $\omega$ is the frequency, and $\omega _0$ is a reference frequency. In general the dispersion in a waveguide is a superposition of different dispersion orders. However, here we consider the effects of each dispersion order separately. For high orders of dispersion the phase upon propagation as a function of frequency typically looks like that as shown in Fig. 1(a): the phase is approximately constant for frequencies close to $\omega _0$, and it very rapidly varies for frequencies further away. Moreover, the phases do not differ much for different dispersion orders; see Fig. 1(a) which shows results for $m=10$ (red curve) and $m=18$ (blue curve). These observations suggest that the effect of high orders of dispersion can be approximated by separating the pulse spectrum into two parts. For $|\omega -\omega _0|<\Delta \omega$ the dispersion has almost no effect, and we refer to this as the coherent part of the spectrum. In contrast, for $|\omega -\omega _0|>\Delta \omega$ the dispersion causes the phase to be, in effect, random, since it varies very rapidly with frequency and with propagation distance. We refer to this as the incoherent part of the spectrum (see Fig. 1(b)). This implies that the pulse evolution is entirely captured by the value of $\Delta \omega$, irrespective of the order of dispersion, provided that it is sufficiently high.
Fig. 1. Illustration of the effect of high-order dispersion, taken to be even, on the field envelope. (a) Phase $\varphi (\omega )$ for $m=10$ (red) and $m=18$ (blue), versus normalized frequency $\ell \tau _0(\omega -\omega _0)$. Here $\beta _{10} = 3.63\times 10^{-2}\,\rm {ps^{10}/km}$, $\beta _{18} = 0.64\,\rm {ps^{18}/km}$, and $L = 10\,\rm {m}$. As the pulse propagates, its spectrum (b) separates into a coherent part of width $2\Delta \omega$, unaffected by the dispersion, and an incoherent part for which the phase is essentially random, and which, (c) corresponds to a pedestal in time. Red arrows indicate how the parts evolve with increasing propagation distance.
We choose $\Delta \omega$ to be the frequency where Eq. (1) takes the value of $1$ radian [33] and thus
Since $\Delta \omega$ monotonically decreases with $L$, the pulse evolution can be thought of as an increase of the fraction of the spectrum that is incoherent, and the concomitant decrease of the coherent part.In the time domain, the effect of the dispersion on the pulse shape is determined by the relative weights of the coherent and incoherent parts of the spectrum. Since the width of the spectrum is approximately $\tau _0^{-1}$, where $\tau _0$ is the full-width at half-maximum of the unchirped pulse, the effect of the dispersion is captured by the ratio $\Delta \omega /(\tau _0)^{-1}=\Delta \omega \tau _0$. This allows us to define the dimensionless length-like parameter $\ell$ through
While we illustrate our approach with a symmetric pulse, as seen in Fig. 1(b) and (c), this is merely for convenience and simplicity. In fact, our argument does not depend on the pulse shape and can be applied to any type of optical pulses as it only depends on the ratio between the coherent and incoherent spectral parts, as seen in Eqs. (2) and (3).
The argument developed above is based on the FWHM duration $\tau$ of the pulses rather than their root-mean square (RMS) width $\sigma$ [33]. While it is more convenient to base a rigorous mathematical description of the effect of dispersion on the RMS pulse width [16,18], it is dominated by the pedestal, the properties of which strongly depends on the dispersion order. In contrast, the FWHM directly relates to the width of that part of the pulse where the intensity is highest, revealing the insights of the pulse dynamics discussed earlier in this section. The FWHM pulse width is also less sensitive to experimental noise, which is most prominent in the pedestal [31,35].
3. Experimental setup
The intrinsic dispersion of conventional optical waveguides is usually dominated by the quadratic contribution ($m=2$) while the effects of higher-order dispersion are usually negligible. For example, a complex guided-wave structure is needed just to achieve dominant negative quartic ($m=4$) dispersion, as is required for the generation of pure-quartic solitons [21,36]. This strongly restricts the possibility of directly observing the linear propagation of optical pulses with high-order dispersion in waveguides. To overcome this limitation, we use a programmable spectral pulse-shaper in an experimental setup similar to that reported by Chang, et al. [37]. A schematic of the experimental setup is shown in Fig. 2(a). Optical pulses emitted by a mode-locked fiber laser and amplified by an Er-doped fiber amplifier are sent into a fiber-coupled spectral pulse-shaper [38]. This device, based on a programmable liquid crystal array, applies a phase function to the frequency components of the optical signal following Eq. (1) that is equal to that due to chromatic dispersion [38,39].
Fig. 2. (a) Schematic of the experimental setup. ML laser, mode-locked laser; EDFA, Er-doped fiber amplifier; SPS, spectral pulse-shaper; FREG, frequency-resolved electrical gating; MZM, Mach–Zehnder intensity modulator, ODL, optical delay line; PD, photodiode; OSA, optical spectrum analyzer. (b) ML laser retrieved temporal intensity (solid blue) and $\rm {sech}^2$ fit (red dashed) for similar FWHM duration. Inset: measured laser output spectrum (solid blue) and $\rm {sech}^2$ fit (red dashed).
The output pulses were characterized using a frequency-resolved electrical gating (FREG) setup [34]. After the spectral pulse-shaper, the optical pulses are split into two branches by a fiber coupler; the first part of the signal power is sent to a branch with a variable delay before being detected by a fast photodiode and transferred to the electrical domain. This electrical gate drives a Mach–Zehnder intensity modulator that gates the optical pulses from the other branch of the fiber coupler. Then, using an optical spectrum analyzer, the spectra are measured as a function of the delay, to generate a series of optical spectrograms [34]. Finally, the temporal intensity and phase of the pulses are retrieved from the measured spectrograms by using a numerical blind deconvolution algorithm ($256\times 256$ grid and retrieval error $< 0.003$) [40]. The validity of the retrieved pulses was assessed by taking their Fourier transform and checking that these matched the output spectrum measured with the optical spectrum analyzer. A similar setup was successfully used to characterize complex soliton dynamics [24,41,42].
We first characterized the experimental setup by measuring the temporal intensity profile of the pulses emitted by the mode-locked laser, when no phase mask is applied by the pulse-shaper. The results are shown in Fig. 2(b). The pulses exhibit a hyperbolic secant-shaped temporal intensity (solid blue line) with a duration of $\tau _0 = 1.57\,\rm {ps}$ at FWHM. The red-dashed line corresponds to a $\rm {sech}^2$ fit with the same FWHM duration. The associated measured output spectrum, shown in the inset (solid blue line), also displays a $\rm {sech}^2$ intensity profile (red dashed) with a $-3\,\rm {dB}$ bandwidth of $200\,\rm {GHz}$. This corresponds to a time-bandwidth product of $0.314$, indicating that the pulses are close to transform limited [1]. This means that the contribution of the dispersion of the single-mode fibers composing the experimental setup is negligible. This can be understood since the corresponding dispersion length $L_2 = T_0^2/|\beta _2| \approx 37\,\rm {m}$, where $T_0 = \tau _0/1.763$ and $\beta _2 = -21.4\,\rm {ps^2/km}$ [1] for our laser pulses. This is much longer than the length of approximately $10\,{\rm m}$ of the experimental setup.
4. Results
We used the setup described in Sec. 3 to measure how the shape of a hyperbolic secant-shaped pulse is affected by the application of an amount of dispersion of order $m$ corresponding to different dimensionless propagation distances $\ell$ (see Eq. (3)). For each discrete value of $\ell$ the spectral pulse-shaper applies a phase $\varphi$ following Eq. (1), and we measured the temporal intensity profiles of the output pulses using the FREG setup. The values of dispersion coefficient were chosen as $\beta _m = m!\tau _0^m/L_m$, where $L_m = 1\,\rm {m}$ and $\tau _0 = 1.57\,\rm {ps}$. The corresponding length $L$ can be calculated using Eq. (3). Examples of measured temporal intensity profiles (solid blue) for different orders of dispersion $m$ and propagation length parameters $\ell$ are shown in Fig. 3. All the recovered profiles are in excellent agreement with the corresponding calculated profiles (dashed red) for the same propagation parameters. For example, in Fig. 3(a) the measured output pulses exhibit temporal oscillations in the trailing edge, as expected for positive third order dispersion [1,37]. For $m = 4$ and $8$, the measured pulses exhibit symmetric intensity profiles but highly skewed spectrograms, as seen in the inset of Fig. 3(b) and (c), which show that the central frequency part remains essentially unaffected while low (high) frequencies propagate significantly faster (slower) than the central frequencies [33]. Finally, for $m=9$ and $\ell = 0.5$, shown in Fig. 3(d), the temporal profile displays small oscillations at the trailing edge, similar to the third and other odd dispersion order cases. These results confirm the ability of our setup to generate and capture the dynamics of an optical pulses under the effects of any dispersion order.
Fig. 3. Examples of measured temporal intensity profiles for different dispersion order $m$ and propagation lengths $\ell$. (a) $m=3$ and $\ell = 0.95$, (b) $m=4$ and $\ell = 0.8$, (c) $m=8$ and $\ell = 0.65$, and (d) $m=9$ and $\ell = 0.5$. Dispersion coefficients are taken to be positive ($\beta _m>0$). The red dashed curves show the corresponding calculated temporal shapes. The insets show the corresponding measured spectrograms.
To assess the validity of the approach summarized in Sec. 2 and of our previous study [33], we measured the pulses’ FWHM for applied dispersion orders ranging from $m =2$ to $12$ as a function of $\ell$. Results of these measurements are shown in Fig. 4(a). The measured values of FWHM duration (dots) are in very good agreement with the numerically calculated evolution of the pulse duration, shown by the solid curves $C_m$ [33]. In particular, these results show that whereas for low dispersion orders, $m = 2,3$ and $4$, the FWHM follow different curves, for high dispersion orders ($m=8$ (purple), $m=9$ (green), $m=10$ (cyan) and $m=12$ (magenta)), the FWHM duration follow the same curve when considered as a function of $\ell$.
Fig. 4. (a) Numerically calculated (solid lines) and experimentally measured (dots) FWHM of an initially unchirped $\rm {sech}^2$ pulse centered around $\omega _0$ versus dimensionless parameter $\ell$ for different orders of dispersion. The vertical dashed-lines mark $\ell = 0.5$ and $\ell = 0.65$. (b) Same results but normalized by the FWHM evolution for $m=20$. Experimentally measured temporal intensity profiles at (c) $\ell = 0.5$ and (d) $\ell = 0,65$, for $m = 8$ (purple), $m = 9$ (green), $m = 10$ (cyan) and $m = 12$ (magenta).
This is highlighted in Fig. 4(b), which shows the same results but normalized by the numerically calculated FWHM evolution $C_{20}$ ($m=20$), which can be taken to be equal to that as $m\rightarrow \infty$. Plotted this way, it is clear that for large dispersion orders $m$ ($m\gtrapprox 8$), the FWHM duration lie on top of each other, consistent with Eqs. (2) and (3). This also confirms that our approach applies to any high even or odd order of dispersion $m$. The retrieved temporal intensity profiles for $m = 8$ (magenta), $m = 9$ (green), $m = 10$ (cyan) and $m = 12$ (magenta) at $\ell =0.5$, and $m = 8$, $10$ and $12$ at $\ell =0.65$ are shown in Fig. 4(c) and (d), respectively. This shows that for high dispersion order $m$, the central parts of the temporal pulses are almost identical at equivalent $\ell$. Note that our analysis only applies to this central part, where the intensity is highest; the pedestal of the pulses undergo an evolution that strongly depends on $m$.
Recall that according to the definition of $\ell$ in Eq. (3), the relationship between $L$ and $\ell$ is highly nonlinear [33]. For example, for the first and last experimental data points for $m=8$, $L = 1.72\times 10^{-2}\,\rm {m}$ and $L = 214.3\,\rm {m}$, for $\ell = 0.2$ and $0.65$, respectively. This corresponds to a factor of approximately $1.24\times 10^{4}$ in $L$ but only a factor $3.25$ in $\ell$. This indicates that for large $m$ the pulse evolution is characterized by a rapid initial increase of the pulse width, followed by a much slower rate of broadening in subsequent propagation.
Finally, we performed similar measurements for two other input pulse shapes by adding an amplitude filter in our experimental setup. The results of these experiments for a Gaussian and a fourth-order super Gaussian (in the spectral domain) are shown in Fig. 5 (left column) and (right column), respectively. The temporal and spectral intensity pulses shapes when no phase is applied, are shown in Fig. 5(a) and (b). The FWHM input pulse duration of the Gaussian and super Gaussian pulses are measured to be $\tau _0 = 2.91\,\rm {ps}$ and $4.15\,\rm {ps}$, respectively. The retrieved temporal intensity profiles for $m = 8$, $10$ and $12$ at two propagation distances $\ell =0.5$ and $\ell = 0.65$ are shown in Fig. 5(c)-(d) and (e)-(f), respectively. These results are similar as for the $\rm {sech}^2$ pulse, confirming that the central parts of the temporal are almost identical for high dispersion order $m$ but at same $\ell$.
Fig. 5. Measured pulse shapes for Gaussian (left column) and super Gaussian (right column) input pulses. (a)-(b) Measured temporal intensity profile of the input (solid blue) and analytical fit (red dashed). Insets show the spectral intensity. Measured temporal intensity profiles at (c)-(d) $\ell = 0.5$ and, (e)-(f), $\ell = 0.65$ for $m = 8$ (purple), $m = 10$ (cyan) and $m = 12$ (magenta).
5. Discussion and conclusion
Thus far we exclusively considered media with a single order of dispersion. The $1/m$ power in Eq. (3) implies that for pulse propagation in the presence two different orders of dispersion, $m_1$ and $m_2$, say, where $m_1>m_2$, dispersion order $m_1$ initially dominates, whereas eventually $m_2$ does. There must therefore be a propagation length $L_e$ for which the effects of both orders are the same. This length is straightforwardly found to be
Our results do not provide a full description of the pulse’s temporal evolution. Previous studies, based on the RMS value of the pulse duration provided a mathematically more rigorous mathematical description [16,18]. However, the RMS pulse width is strongly affected by the pedestal and it thus depends strongly on the dispersion order and is also sensitive to experimental noise. In contrast, our results show that the FWHM is quite a robust measure of the pulse properties.
Optical waveguides with dominant high-order dispersion have been recently investigated [21,27,36]. The approach described in this work could be used to evaluate the relevant dispersion lengths in these devices, which is important for the generation and propagation of nonlinear pulses [33]. This could also be used to determined the dominant effects in waveguides with multiple high-order dispersion orders or in designing laser operating in novel regimes [43,44].
Finally, the range of dispersion orders and propagation distance achievable with our current setup is limited by the maximum delay range (or equivalently the maximum phase $\varphi$) that can be applied by the spectral pulse-shaper. In the spectral pulse-shaper, a phase change is created by steering the optical beam, inducing a temporal delay. As seen from Eq. (1), large dispersion orders $m$ lead to rapidly varying phase functions for frequencies away from $\omega _0$. This corresponds to a large beam steering causing some spectral frequencies not to be coupled back into the output fiber, effectively clipping the optical pulse spectrum [39,45]. For our experimental parameters this corresponds to a maximum dimensionless parameter $\ell \le 0.65$ ($\ell \le 0.5$) for the highest even (odd) dispersion order considered in this work. This also hinders the possibility to capture the dynamics of the incoherent pedestal. However, this limitation is not fundamental and could be overcome by using a pulse-shaper with a larger dynamic range [46,47]. The simple yet powerful setup used in this work could allow for future study in the case of linear systems with more complex dispersion profiles arising from more than one high-order of dispersion.
Funding
Australian Research Council (DE220100509, DP230102200).
Acknowledgments
The authors thank Mr. Patrick Blown from Finisar Australia for useful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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