Single Pulse Manipulations in Synthetic Time-frequency Space

Synthetic dimension in photonics is an emergent field for exploring physics in higher-dimensional space within lower-geometrical structure, which also points towards manipulating light by utilizing physical phenomena in the synthetic space Yuan et al. (2018a); Ozawa and Price (2019). Different degrees of freedom of photons can be used to construct synthetic dimensions, such as frequencies Yuan et al. (2016); Ozawa et al. (2016); Bell et al. (2017); Qin et al. (2018), orbital angular momenta Luo et al. (2015), pulse arrival times Regensburger et al. (2011, 2012), and others Lustig et al. (2019); Maczewsky et al. (2020); Wang et al. (2020). With synthetic dimensions, many potential applications have been proposed, including unidirectional frequency translation Yuan and Fan (2016), orbital angular momentum switch Luo et al. (2018), pulse narrowing Yuan et al. (2018b), and mode-locked topological laser Yang et al. (2020). Furthermore, it has also been shown that one can simultaneously build up two synthetic dimensions with different degrees of freedom of light and explore topological edge states Yuan et al. (2019); Dutt et al. (2020). This synthetic (2+1) dimensions require only one single cavity, which dramatically simplify the experimental requirements.

Group velocity dispersion (GVD) is a fundamental optical characteristic in a medium, and is of great importance in ultrashort pulse manipulations Reeves et al. (2003); Huang et al. (2015); Divitt et al. (2019), such as generation of optical solitons Stratmann et al. (2005); Lee et al. (2017), pulse compression Broderick et al. (1997); Colman et al. (2010); Tan et al. (2010), and group velocity control Gehring et al. (2006); Li et al. (2015); Qin et al. (2020), where the interplay between dispersive and nonlinear effects on optical pulses take place Agrawal (2010). Moreover, it has been found that, when a pulse propagates along a dispersive waveguide, one can consider the problem in a synthetic (1+1) dimensions, i.e., the optical field diffracts along a time dimension when it evolves along the spatial dimension Peschel et al. (2008); Plansinis et al. (2015, 2016).

In this work, we move a step further and show the possibility of multiple single pulse manipulations in a synthetic (2+1) dimensions including the temporal diffraction and frequency conversion in a synthetic time-frequency space while a pulse propagates along the spatial dimension. A dispersive waveguide incorporating segmented electrodes under travelling wave electro-optic modulation is considered [see Fig. 1(a)]. We show that one construct a two-dimensional synthetic space including the time and frequency dimensions, and pulse dynamics is studied when the spatial propagating dimension is treated as the synthetic time evolution. An effective gauge potential well is constructed in synthetic two dimensions with non-uniform distribution of modulation phases to confine light Lin and Fan (2014); Lumer et al. (2019); Cohen et al. (2020). By manipulating the effective well in multiple ways, we show rich physics of pulse manipulations, including confined pulse propagation, fast/slow light, and pulse compression. Fundamentally different from previous works Peschel et al. (2008); Plansinis et al. (2015, 2016), our results link to physics in a (2+1) dimensions, which points out exotic route towards manipulating pulse profile and frequency conversion process. Our work can find important applications of optical pulse engineering in various platforms, ranging from second/third-order dispersive waveguide-based systems to on-chip dispersive microresonator-based systems.

We begin with briefly illustrating synthetic (2+1) dimensions constructed in a waveguide shown in Fig. 1(a), modulated by a traveling wave with a sinusoidal radio frequency (RF) signal. The refractive index is governed by Gan and Yip (1997)

where $n_{0}$ is static refractive index and $\Delta n$ is the modulation amplitude. $\Omega$, $Q$, and $\phi$ are the frequency, wavevector, and modulation phase of the RF signal. For a pulse centered at the frequency $\omega_{0}$ propagating along the spatial dimension $z$, the applied modulation connects field components at discrete frequencies $\omega_{m}=\omega_{0}+m\Omega$, and forms the synthetic frequency dimension [see Fig. 1(b)] Qin et al. (2018). On the other hand, for a dispersive waveguide, the pulse experiences temporal diffraction, which brings up the concept of the synthetic dimension along the continuous retarded time frame ($\tau=t-z/v_{g}$, with $v_{g}$ being the group velocity at $\omega_{0}$) Peschel et al. (2008); Agrawal (2010). Hence, a synthetic continuous-discrete time-frequency space is constructed for the pulse travelling along $z$-direction inside the waveguide.

We then consider a time-dependent and spatially non-uniform modulation phase $\phi(z,\tau)$, which can be achieved by controlling RF signals at each segmented electrode. Such phase distribution supports the effective gauge potential for light in the synthetic space Fang et al. (2012). In particular, we take the form of $\phi(z,\tau)=\phi_{0}$ as

i.e., at any $z$, the hopping phase along the frequency dimension is $\phi_{0}$ in a region with the center $\tau_{\texttt{c}}(z)$ and width $\Delta\tau_{\phi}(z)$, and equals to 0 at the remaining regions. Such phase distribution brings an effective gauge potential well in the synthetic space with the size dependent on $z$ [see Fig. 1(b)] Lin and Fan (2014); Lumer et al. (2019); Cohen et al. (2020), and can be used to manipulate light in different ways.

We now build detailed theoretical framework to study the system. For pulse propagating through the modulated waveguide, the electric field of the pulse can be expanded as $E(z,t)=\sum_{m}a_{m}(z,t)e^{i(\omega_{m}t-\beta_{m}z)}$, where $a_{m}(z,t)$ is the slowly-varying envelope for the frequency component at $\omega_{m}$ Agrawal (2010); Boyd (2020). The propagation constant $\beta_{m}$ is not equally spaced due to GVD [see Fig. 1(c)], which can be defined as $\beta_{m}=\beta_{0}+mQ+\Delta q_{m}$. Here, $\beta_{0}$ is the wavevector associated with $\omega_{0}$, and $\Delta q_{m}$ donates the wavevector mismatching. The pulse field follows the wave equation Boyd (2020)

where $\varepsilon_{0}$ and $\varepsilon_{r}(z,t)=n^{2}(z,t)$ are vacuum and relative permittivity, respectively. With the expansion of the field, we obtain the propagating equations for $m^{\texttt{th}}$ component in the retarded frame sup

Here, $c_{1}=k_{1}\Omega-Q$ is linear mismatching between light and RF signal, and $c_{2}=k_{2}\Omega^{2}/2$ is nonlinear mismatching caused by GVD, where $k_{1}$ and $k_{2}$ are Taylor expansion coefficiencies of wavevector $k(\omega)$ around $\omega_{m}$, representing the reciprocal of the group velocity and GVD, respectively. $g=\Delta n\omega_{0}/2c$ donates the coupling strength.

Equation (4) describes the dynamics of a pulse with multiple frequency components at $\omega_{m}$ propagating along $z$-axis in a reference frame moving at $v_{g}$. The first term on right-band side of Eq. (4) dominates the pulse dispersion behavior, which is a counterpart of wave diffusion in the spatial dimension, while the second term refers frequency conversions. Equation (4) is therefore the nonlinear Schrödinger equation within $(2+1)$ dimensions Agrawal (2010); Yuan et al. (2017), where the pulse experiences continuous temporal diffraction and discrete frequency conversion in two synthetic dimensions while it evolves at $z$-axis [see Fig. 1(b)]. Moreover, the hopping phase in Eq. (4) gives an effective nonlinear force $F=c_{1}+2c_{2}m+\partial\phi/\partial z$ pointing along the frequency dimension Yuan and Fan (2016). Different from previous studies, which have explored consequences of an effective linear force in the synthetic frequency dimension Yuan and Fan (2016); Qin et al. (2018); Li et al. (2021), here the effective force is nonlinear due to GVD. Yet, as we show in the following, a synthetic time-frequency space together with the nonlinear force give us alternative opportunity for manipulating the pulse propagation in the waveguide.

We simulate pulse propagations by Eq. (4) with excitations at one end of the waveguide at $z=0$. The input pulse has a profile as $f(\tau)=e^{-1.386[\tau/\Delta\tau]^{2}}$, where $\Delta\tau=30$ ps is the temporal full width at half maximum (FWHM). We assume that the input field contains only one frequency component at $\omega_{0}=1.2\times 10^{15}$ Hz (or 1550 nm). Note that Eq. (4) is valid when the condition $\Omega>\Delta\omega(z)$ is satisfied, i.e., field profiles at different frequency components in the spectral domain do not overlap, where $\Delta\omega(z)$ is the spectral FWHM for each frequency component at any $z$. For the input pulse, $\Delta\omega=2\pi\cdot 14.7$ GHz. The simulation performed with 21 modes ($m=-10,...,10$). Figure 2(a) shows the profile of the input pulse in the synthetic time-frequency space.

First, we consider the normal case that pulse propagates in the waveguide under the modulation with $\phi(z,\tau)=0$. We choose modulation with $\Delta n=5\times 10^{-4}$ and $\Omega=2\pi\cdot 29.4$ GHz, which gives $g=10^{3}$ m^-1. For the waveguide with a length $L=2$ cm, we have $k_{2}=4\times 10^{-20}$ m^-1s², which lead to $c_{2}=682$ m^-1. $c_{1}$ can be approximated to be zero by tuning $Q=k_{1}\Omega$. All parameters are chosen with the experimental feasibility in the literature Luennemann et al. (2003); Kaushalram et al. (2019); Wang et al. (2018).

When the pulse evolves along the $z$ axis, it experiences frequency conversion and nonlinear oscillation near the $0^{\texttt{th}}$ mode in the frequency dimension under the effective nonlinear force [see Fig. 2(b)]. At $L=2$ cm, frequency components of pulse oscillate back to the single mode, which shifts to the $2^{\texttt{nd}}$ mode [see Fig. 2(c)]. Moreover, the pulse gets broadened due to GVD and has the temporal width $\Delta\tau(z=L)=79$ ps, as shown in Figs. 2(c)-(d). It agrees well with the theoretical calculated evolution of the pulse width for a Gaussian pulse [see Fig. S1(a)] sup .

We next consider modulations with non-uniform distributions of phases and explore the dynamics from Eq. (4) under effective gauge potential well. We choose parameters $\phi_{0}=\pi$, $\tau_{\texttt{c}}(z)=0$ and $\Delta\tau_{\phi}(z)=30$ ps which indicates a fixed well in the synthetic space. The simulation shows the confinement of light in middle region of the effective well (labelled by the dashed line) in the synthetic time dimension as shown in Figs. 2(e)-(f), while the frequency component is shifted to the $2^{\texttt{nd}}$ mode as previous case. The pulse width remains 30 ps while maintaining an approximate Gaussian waveform during the propagation [see Fig. S1(b)] sup . The result here shows an interesting combination between nonlinear oscillation along the frequency dimension and the light confinement due to the effective gauge potential well in the time dimension. As a comparison, we calculate the evolution of pulse with $c_{2}=0$, with other parameters unchanged, and find temporal confinement of light persists while the frequency conversion covers a broad range of modes sup . Not only the confined pulse propagation demonstrated here, the idea of utilizing effective gauge potentials in synthetic time-frequency dimensions provides more different ways to manipulate the pulse.

We further shift the center of the effective gauge potential well lineally dependent on $z$ with $\tau_{\texttt{c}}(z)=\eta_{1}z$, where $\eta_{1}$ is the shift parameter. This choice makes the modulation phase taking the form of $\phi(z,\tau)=\pi$ for $|\tau-\eta_{1}z|\leq 30$ ps and zero for other $\tau$. It provides a way to manipulate the group velocity of pulse controlled by $\eta_{1}$. In simulations, we use $\Omega=2\pi\cdot 29.4$ GHz, $\Delta n=10^{-3}$, and $k_{2}=6\times 10^{-20}$ m^-1s², which give $g=2\times 10^{3}$ m^-1, and $c_{2}=1023$ m^-1. With a positive $\eta_{1}=22$ ps/cm, we find slow light with group velocity delay of 57 ps and unchanged pulse width of 30 ps as shown in Figs. 3(a) and 3(c). On the other hand, one can see the generation of fast light in Figs. 3(b) and 3(d) with negative $\eta_{1}=-22$ ps/cm. Here large dispersion and strong modulation are chosen to efficiently manipulate the group velocity of the pulse. Figure 3(e) exhibits the output pulse profiles of slow and fast light at $z=L$, where the peak intensity decreases due to the dispersion loss. The corresponding nonlinear oscillation for the slow light is plotted in Fig. 3(f), while the nonlinear oscillation for the fast light is similar. At $z=L$, the output pulse has two major frequency components at $0^{\texttt{th}}$ and $2^{\texttt{nd}}$ modes [see Figs. 3(a)-(b)]. The single frequency conversion can be established by choosing a different propagation length. For example, one sees that oscillations shifts to a single mode at $m=2$ at $z=1.63$ cm. Further larger group velocity manipulation can be obtained by using larger $|\eta_{1}|$ and GVD at the cost of the intensity loss.

The width of effective gauge potential well can also be used to control the pulse width. We consider a varying well along $z$ with $\Delta\tau_{\phi}(z)=30-\eta_{2}z$ and $\tau_{\texttt{c}}(z)=0$, where $\eta_{2}$ is the width-varying parameter. Thus the configuration of modulation phase becomes $\phi(z,\tau)=\pi$ for $|\tau|\leq 30-\eta_{2}z$ ps and zero for other $\tau$. We take parameters $\eta_{2}=12.5$ ps/cm, $k_{2}=6\times 10^{-21}$ m^-1s², $\Delta n=10^{-3}$ and $\Omega=2\pi\cdot 88.2$ GHz, which lead to $g=2\times 10^{3}$ m^-1 and $c_{2}=920$ m^-1 in the simulation and show results in Fig. 4. Figure 4(a) show the pulse profile in the synthetic time-frequency space at $z=L$. Although the pulse converts to multiple modes, the temporal width of the pulse is largely compressed. As shown in Fig. 4(b), the narrowing of the effective gauge potential well forces pulse compression gradually while it propagates along the $z$ axis. Figure 4(c) plots the pulse width versus $z$, showing a trend of overall decrease. The output pulse profile has width of 5 ps, with the enhanced peak intensity. Simultaneously, the frequency conversion follows nonlinear oscillation as shown in Fig. 4(d). The output with multiple frequency components can be tuned by changing the length of the waveguide. For example, at $z=1.64$ cm, the field exhibits one major frequency component at the $2^{\texttt{nd}}$ mode. Moreover, one can control the distribution of frequency components in the output pulse by using an input pulse with a Gaussian distribution of multiple modes. Unidirectional and bidirectional frequency transports together with temporal pulse manipulations can be achieved sup . As the last note, the pulse compression can not lead to an output field with an infinitely small temporal width (broad frequency bandwidth). The condition $\Omega>\Delta\omega(z)$ at any $z$ shall be satisfied.

We propose a modulated waveguide system with parameters based on lithium-niobate waveguide system. The numbers used in simulations require waveguide with large dispersion and fast electro-optic modulation, which are experimentally achievable in the second-order nonlinear waveguide with the state-of-art technology. Dispersion near $10^{-22}$ m^-1s² has been reported Kaushalram et al. (2019), which can be further enlarged by structure engineering or operating at higher dispersion wavelength. $\Delta n=10^{-3}$ corresponds to $7$ V/$\mu$m voltage amplitude of an applied external electric field. If larger modulation is needed, high voltage up to 65 V/$\mu$m has been demonstrated Luennemann et al. (2003). Shorter pulse manipulation is possible with smaller dispersion but larger modulation frequency, such as 100 GHz, which is commercially available Wang et al. (2018). Moreover, recent advances of integrated waveguide platform brings opportunity to achieve synthetic time-frequency dimensions in integrated photonics, on which modulators with frequency $\sim$200 GHz have been demonstrated Liu et al. (2020). The construction of a synthetic time-frequency space can be further extended beyond by adding other degrees of freedom such as orbital angular momentum Luo et al. (2015) or pseudospin Dutt et al. (2020). Besides, our analysis shall be valid for other systems if one scales parameters consistently. Our model shows promise for studying pulse propagating in dispersive resonators with synthetic dimensions Shan et al. (2020), and in third-order dispersive waveguide or microresonator based systems Hsieh et al. (2006); Mia et al. (2019); Xue et al. (2015); Yang et al. (2016).

In summary, we propose a synthetic (2+1) dimensions for manipulating pulse propagation in a dispersive waveguide under dynamic modulations. With the effective gauge potential well for photons and nonlinear oscillation in the synthetic space, multiple pulse propagation behaviors including confined pulse propagation, fast/slow light, and pulse compression have been shown. Our work provides an alternative platform for actively manipulating single pulse in different ways, which is highly re-configurable, and hence shows promising potentials for pulse engineering in integrated photonics and optical communications.