Generation of an ultrashort pulse train through ultrafast parity-time symmetry switching

Wenhao Wang; Yanfang Zhang; Ben Li; Jing Wang; Jingui Ma; Peng Yuan; Dongfang Zhang; Hao Zhang; Hao Zhang; Liejia Qian; Liejia Qian

doi:10.1364/OE.492567

1. Introduction

Ultrashort laser pulse trains play an important role in ultrafast optics and have been widely used in optical frequency combs [1], high-speed optical communication [2], and refractive surgery in ophthalmology [3]. Traditionally, trains of laser pulses in the picosecond and femtosecond range are produced by mode locking technology. For either active or passive mode locking, the optical resonator that provides periodic gain or loss feedback is indispensable to selectively strengthen the optical modes with fixed phase differences while dissipating others. Hence, the pulse repetition rate is fundamentally limited by the cavity length, leading to a typical repetition range of 50-500 MHz. To obtain ultrashort pulses at higher repetition rate, several mechanisms have been proposed for the generation of optical pulse trains directly from continuous-wave (CW) light without the need for optical resonators. Related techniques can be classified into two categories. One category resorts to nonlinear optical effects, such as electro-optic modulation combs [4,5], cross-phase modulation [6–8], induced modulational instability [9], adiabatic Raman compression [10], parametric generation of phase-locked sidebands [11] and pulse compression via the balance between media dispersion and χ⁽³⁾ nonlinearity [12–14]. The other category explores linear optical techniques, such as beating multiple quasi-CW frequency lines [15,16] and interfering copies of chirped broadband pulses delayed with respect to each other [17].

Here, we propose an alternative scheme for the direct generation of ultrashort pulse trains by relying on the gain switching effect in the vicinity of parity-time (PT) symmetry threshold of non-Hermitian optical systems. In recent years, non-Hermitian optical systems respecting PT symmetry [18] have attracted considerable attention. Given that the complex optical refractive index plays the role of an optical potential, PT symmetry can be readily established based on optical structures with spatially distributed sections of loss and gain. One typical PT symmetry structure is a directional coupler of two waveguides with spatially distributed sections of gain and loss, as schematically depicted in Fig. 1(a). Light propagation in such coupled waveguides has been identified to be analogous to quantum wavepacket dynamics governed by a PT-symmetric Hamiltonian [19]. Below a certain gain/loss level, such systems support PT-symmetric optical modes, which exhibit the same average loss or gain [20,21]. However, when the gain or loss is increased, the PT symmetry of modes breaks, and a mode with the strongest gain (or smallest loss) dominates. The regions of PT symmetry and PT symmetry breaking are demarcated at the PT symmetry threshold, which is also known as the exceptional point [22]. It has been identified that the rich physics at PT-symmetric structure enable unprecedented capabilities of optical manipulation. Specifically, in the vicinity of the PT symmetry threshold, a small difference in the gain or loss leads to a dramatic difference in mode amplifications. Consequently, the strategic modulation of gain and loss in PT-symmetric optical structures enables a multitude of new capabilities, including single-mode lasing [23,24], unidirectional cloaking [25], asymmetric mode switching [26], exceptional-point-enhanced sensitivity [27,28], PT-symmetric compactons [29], PT-symmetric Thue-Morse aperiodic optical waveguide network [30], and optimization of Hermitian nonlinear wave mixing [31].

Fig. 1. (a) Scheme of the PT-symmetric coupler with a laser gain in one waveguide and linear absorption in the other waveguide. (b) Scheme of the PT-symmetric nonlinear coupler consisting of χ⁽²⁾ waveguides.

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The combination of the PT-symmetric optical structure with optical parametric amplification (OPA) further enables the ultrafast control of PT symmetry. As has been theoretically identified by Antonosyan et al. [32], parametric amplification in a directional coupler of waveguides with χ⁽²⁾ nonlinearity and loss [Fig. 1(b)] conforms to spectral PT anti-symmetry, under the assumptions of pump non-depletion, negligible media dispersion as well as degenerate signal and idler. Such device exhibits a dramatic difference in mode amplification below and above the PT symmetry threshold. It is further theoretically demonstrated that OPA in single waveguide or crystal can also exhibit dramatic gain transition related to PT symmetry breaking, once incoherent loss [33] or coherent couplings [31] to the external environment is appropriately introduced into the OPA. Ma et al. experimentally applies the ultrafast PT symmetry behavior to establish a χ⁽²⁾ nonlinear amplification with unprecedentedly-high conversion efficiency up to 56%, enabled by using the Sm³⁺-doped yttrium calcium oxyborate crystal with spectrally distributed loss and gain [34,35]. Here, we explore the PT symmetry switching effect to directly generate ultrashort pulse trains from CW light. To achieve this, a periodically amplitude-modulated pump laser is used to provide periodic and ultrafast control of PT symmetry. We further demonstrate that apodized switching that enables the production of clean ultrashort pulses without side lobes can also be achieved by engineering the PT symmetry threshold. Although the performance of PT-symmetric OPA waveguides has been theoretically identified, the work presented here is significantly different. First, a different pumping condition is studied. The introduction of an amplitude-modulated pump laser indicates that the dynamics of PT symmetry become time dependent. Second, previous studies on OPA in coupled χ⁽²⁾ waveguides are confined to the linear regime, wherein both the pump depletion and media dispersion are neglected. We identify the influence of pump depletion and media dispersion. Besides, we present a practical design of coupled OPA waveguides devoted for the generation of wavelength and duration tunable sub-picosecond pulse train and characterize the output performance.

The paper is organized as follows. In Section 2, the coupled-mode equations that describe nonlinear wave-mixing in coupled OPA waveguides are presented, emphasizing the physical principle behind the ultrafast gain switching related to PT symmetry breaking. Pump non-depletion, near-degenerate signal and idler, neglected media dispersion have been assumed in this section to obtain the analytical solution. In Section 3, we demonstrate the direct generation of ultrashort pulse trains via the ultrafast gain switching effect. Moreover, the temporal and spectral characteristics of the output ultrashort pulses are discussed. In Section 4, we present a practical design of the coupled OPA waveguides, wherein the impacts of pump depletion and media dispersion are analyzed. Section 5 is the conclusion.

2. Ultrafast gain switching behavior of the coupled OPA waveguides

We consider the PT symmetric optical system as shown in Fig. 1(b). It consists of two χ⁽²⁾ waveguides. The pump laser is only injected into one waveguide, which is termed as gain waveguide. The other waveguide that exerts linear loss to the mixing waves is termed as loss waveguide. In practice, linear loss can be introduced by either metal deposition on the top of the waveguides [19,36] or by material absorption [34,35]. In particular, the pump laser is much more strongly localized than the signal and idler lasers [37] due to its high frequency. Therefore, the loss and coupling for the pump beams are neglected in the following studies. The evolution of the pump, signal and idler waves in such a system is subject to a comprehensive interplay between the nonlinear gain, linear loss and coupling. Under the slowly varying envelope approximation, the evolution of the pump, signal and idler waves in each waveguide can be described by the coupled-mode equations as follows [38]:

(1a)$$\frac{{\partial {A_{s1}}}}{{\partial z}} + \sum\limits_{n = 1}^\infty {\frac{{{{( - i)}^{n - 1}}}}{{n!}}} {k^{(n)}}\frac{{{\partial ^n}{A_{s1}}}}{{\partial {t^n}}} ={-} i\frac{{\chi _{eff}^{(2)}{\omega _s}}}{{2{n_s}{c_0}}}{A_{p1}}A_{i1}^\ast {e^{ - i\Delta k \cdot z}} - \frac{{{\mathrm{\gamma }_{s1}}}}{2}{A_{s1}} - i{C_s}{A_{s2}},$$

(1b)$$\frac{{\partial {A_{s2}}}}{{\partial z}} + \sum\limits_{n = 1}^\infty {\frac{{{{( - i)}^{n - 1}}}}{{n!}}} {k^{(n)}}\frac{{{\partial ^n}{A_{s2}}}}{{\partial {t^n}}} ={-} i\frac{{\chi _{eff}^{(2)}{\omega _s}}}{{2{n_s}{c_0}}}{A_{p2}}A_{i2}^\ast {e^{ - i\Delta k \cdot z}} - \frac{{{\mathrm{\gamma }_{s2}}}}{2}{A_{s2}} - i{C_s}{A_{s1}},$$

(1c)$$\frac{{\partial {A_{i1}}}}{{\partial z}} + \sum\limits_{n = 1}^\infty {\frac{{{{( - i)}^{n - 1}}}}{{n!}}} {k^{(n)}}\frac{{{\partial ^n}{A_{i1}}}}{{\partial {t^n}}} ={-} i\frac{{\chi _{eff}^{(2)}{\omega _i}}}{{2{n_i}{c_0}}}{A_{p1}}A_{s1}^\ast {e^{ - i\Delta k \cdot z}} - \frac{{{\mathrm{\gamma }_{i1}}}}{2}{A_{i1}} - i{C_i}{A_{i2}},$$

(1d)$$\frac{{\partial {A_{i2}}}}{{\partial z}} + \sum\limits_{n = 1}^\infty {\frac{{{{( - i)}^{n - 1}}}}{{n!}}} {k^{(n)}}\frac{{{\partial ^n}{A_{i2}}}}{{\partial {t^n}}} ={-} i\frac{{\chi _{eff}^{(2)}{\omega _i}}}{{2{n_i}{c_0}}}{A_{p2}}A_{s2}^\ast {e^{ - i\Delta k \cdot z}} - \frac{{{\mathrm{\gamma }_{i2}}}}{2}{A_{i2}} - i{C_i}{A_{i1}},$$

(1e)$$\frac{{\partial {A_{p1}}}}{{\partial z}} + \sum\limits_{n = 1}^\infty {\frac{{{{( - i)}^{n - 1}}}}{{n!}}} {k^{(n)}}\frac{{{\partial ^n}{A_{p1}}}}{{\partial {t^n}}} ={-} i\frac{{\chi _{eff}^{(2)}{\omega _p}}}{{2{n_p}{c_0}}}{A_{s1}}{A_{i1}}{e^{i\Delta k \cdot z}},\textrm{ }$$

(1f)$$\frac{{\partial {A_{p2}}}}{{\partial z}} + \sum\limits_{n = 1}^\infty {\frac{{{{( - i)}^{n - 1}}}}{{n!}}} {k^{(n)}}\frac{{{\partial ^n}{A_{p2}}}}{{\partial {t^n}}} ={-} i\frac{{\chi _{eff}^{(2)}{\omega _p}}}{{2{n_p}{c_0}}}{A_{s2}}{A_{i2}}{e^{i\Delta k \cdot z}},\textrm{ }$$

where the subscripts represent the signal (“s”), idler (“i”), and pump (“p”) in the gain waveguide (“1”) and loss waveguide (“2”), respectively. A, ω, and n represent the complex amplitude, frequency, and refractive index of each interacting wave. There are five main components in these equations. The first term on the left-hand side describes the evolution of each wave along the z-direction of waveguides. The second term on the left-hand side describes the dispersion of the different fields inside the nonlinear waveguides, and the k⁽ⁿ⁾-terms are the nth-order dispersion coefficients of the medium. On the right-hand side of these equations, the first term describes the nonlinear wave-mixing in each waveguide, dominated by χ⁽²⁾ nonlinearity. χ⁽²⁾_eff is the effective nonlinear coefficient. Δk = k_p − k_s − k_i refers to the phase mismatch of the three interacting waves in each waveguide. c₀ is the speed of light. The second term on the right-hand side denotes the loss of each field along z, wherein γ are the loss coefficients. The third term corresponds to the coupling between the two waveguides, wherein C are the mode coupling coefficients.

Under the assumptions of pump non-depletion (i.e., ∂A_p₁/∂z = ∂A_p₂/∂z = 0), negligible dispersion and perfect phase matching (i.e., Δk = 0), Equation (1) reduces to four coupled-mode equations as:

(2)$$\begin{aligned} i\frac{{\partial {A_{s1}}}}{{\partial z}} &= {\Gamma _{s1}}A_{i1}^\ast{-} i\frac{{{\mathrm{\gamma }_{s1}}}}{2}{A_{s1}} + {C_s}{A_{s2}},\\ i\frac{{\partial {A_{s2}}}}{{\partial z}} &= {\Gamma _{s2}}A_{i2}^\ast{-} i\frac{{{\mathrm{\gamma }_{s2}}}}{2}{A_{s2}} + {C_s}{A_{s1}},\\ i\frac{{\partial A_{i1}^\ast }}{{\partial z}} &={-} \Gamma _{i1}^\ast {A_{s1}} - i\frac{{{\mathrm{\gamma }_{i1}}}}{2}A_{i1}^\ast{-} {C_i}A_{i2}^\ast ,\\ i\frac{{\partial A_{i2}^\ast }}{{\partial z}} &={-} \Gamma _{i2}^\ast {A_{s2}} - i\frac{{{\mathrm{\gamma }_{i2}}}}{2}A_{i2}^\ast{-} {C_i}A_{i1}^\ast , \end{aligned}$$

where Γ_s= χ⁽²⁾_effω_s/2n_sc₀·A_p and Γ_i = χ⁽²⁾_effω_i/2n_ic₀·A_p are known as the parametric gain coefficients for the signal and idler wave, respectively [39]. To conduct normalized analyses, χ⁽²⁾_effω_s,i/2 ns_,ic₀ = 1 is assumed. Equation (2) can be written in the Hamiltonian form as:

(3)$$i\frac{{\partial \boldsymbol{\alpha }}}{{\partial z}} = \mathrm{{\cal H}}\boldsymbol{\alpha },\textrm{ }$$

where $\boldsymbol{\alpha }$ = (A_s₁; A_s₂; A_i₁*; A_i₂*) and $\mathrm{{\cal H}}$ is a 4 × 4 Hamiltonian matrix. Furthermore, due to the assumption of a near-degenerate signal and idler (n_s = n_i, ω_s = ω_i), the parameters are almost the same at the signal and idler frequencies, i.e., γ_s₁ = γ_i₁ = γ₁, γ_s₂ = γ_i₂ = γ₂, C_s = C_i = C, Γ_s₁ = Γ_i₁ = Γ₁, Γ_s₂ = Γ_i₂ = Γ₂. The Hamiltonian can be further simplified into:

(4)$$\mathrm{{\cal H}} = \left( {\begin{array}{cccc} { - i{\mathrm{\gamma }_1}/2}&C&{{\mathrm{\Gamma }_1}}&0\\ C&{ - i{\mathrm{\gamma }_2}/2}&0&{{\mathrm{\Gamma }_2}}\\ { - \mathrm{\Gamma }_1^\ast }&0&{ - i{\mathrm{\gamma }_1}/2}&{ - C}\\ 0&{ - \mathrm{\Gamma }_2^\ast }&{ - C}&{ - i{\mathrm{\gamma }_2}/2} \end{array}} \right).$$

By applying a PT operator to Equation (3), we determine that the Hamiltonian $\mathrm{{\cal H}}$ given by Equation (4) possesses a spectral PT anti-symmetry [32]:

(5)$$\mathrm{{\cal P}{\cal T}{\cal H}} ={-} \mathrm{{\cal H}{\cal P}{\cal T}}\textrm{.}$$

Here, $\mathrm{{\cal T}}$ is a time-reversal operator that changes z → −z and performs a complex conjugation. The term ‘spectral’ is used because the parity operator $\mathrm{{\cal P}}$ operates in the spectral domain (rather than the spatial domain), interchanging the signal and idler waves as:

(6)$$\mathrm{{\cal P}} = \{{{A_{s1}} \leftrightarrow A_{i1}^\ast ,{A_{s2}} \leftrightarrow A_{i2}^\ast } \}\textrm{.}$$

The term anti-PT symmetry corresponds to underlining the negative sign on the right-hand side of Equation (5). The evolution of the signal and idler waves can be defined by the 4 eigenmodes of the 4 × 4 Hamiltonian matrix, which is given by Equation (4):

(7)$$\boldsymbol{\alpha }(z) = \tilde{\boldsymbol{\alpha }}(\sigma )\exp (i\sigma z),$$

where σ_j (j∈{1,2,3,4}) denotes the eigenvalue. The real part of the eigenvalue Re(σ_j) defines the phase velocity, whereas the imaginary part Im(σ_j) determines the gain/loss experienced by each eigenmode.

First, we characterize the gain characteristics of the coupled OPA waveguides with the initial conditions of C = 1, γ₁ = 0, Г₂ = 0, and γ₂ = 2. The gain characteristics are represented by −Im(σ_j), in which −Im(σ_j) > 0 and −Im(σ_j) < 0 indicate gain and loss, respectively. Fig. 2(a) shows −Im(σ_j) for each of the four eigenmodes calculated under varying Г₁. Obviously, the PT threshold in the evolution of the eigenvalues at Г₁ = 1 occurs based on the criteria |γ₁ − γ₂ − 2(Г₁ + Г₂)| = 4C that is identified for such devices [32]. The imaginary parts of eigenvalues split at the PT threshold (i.e., Г₁ = 1), indicating a fundamental change in the modal amplification. The parametric gain parameter Г₁ is proportional to the pump amplitude A_p₁. Therefore, Fig. 2(a) indicates that with regards to pump amplitudes below the threshold, the four eigenvalues manifest two sets of degenerate modes. Each set contains two eigenmodes with the same negative gain, indicating broken spectral PT symmetry of eigenmodes. Due to the beating between the two degenerate modes, the signal and idler periodically switch between the waveguides, as illustrated in Fig. 2(c). Regarding pump amplitudes above the threshold, one PT-symmetric mode (σ₁ in our case) experiences the largest gain. Accordingly, the signal exhibits exponential growth in the intensity versus the waveguide length z, as shown in Fig. 2(d). The amplification or loss of the signal depends on the largest mode gain. Fig. 2(b) presents the dependency of the consequent parametric gain G = |A_s₁(z)|²/|A_s₁(0)|² on the pump amplitude. The parametric gain fluctuates dramatically with the pump amplitude under a low pump power (Г₁< 1), whereas in the vicinity of the PT threshold (Г₁ = 1), the parametric gain significantly increases. A small perturbation in the pump amplitude leads to very large changes in the parametric gain. This gain transition is the fundamental basis for setting up ultrafast gain switching controlled by the pump beam. We term such a gain switching effect linking PT symmetry as ultrafast PT symmetry switching.

Fig. 2. (a) Negative imaginary parts of the four eigenvalues versus the pump amplitude Г₁ in the gain waveguide. (b) The parametric gain G = |A_s₁(z)|²/|A_s₁(0)|² versus the pump amplitude Г₁, calculated for the coupled OPA waveguides with a normalized length of z = 10. (c), (d) Evolution of the signal mode intensity along the two waveguides, calculated for a pump amplitude below the PT symmetry threshold (Г₁ = 0.95) and a pump amplitude above the PT symmetry threshold (Г₁ = 1.05). (e) Largest mode gain max(-Im{σ}) versus coupling coefficient C and pump amplitude Г₁. (f) The parametric gain versus the pump amplitude Г₁ and normalized length z. For all the plots, the parameters are set to γ₁ = 0, Г₂ = 0, and γ₂ = 2. Excluding (e), the coupling coefficient C = 1.

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Figure 2(e) presents the largest mode gain calculated for a varying pump amplitude under different coupling coefficients. The solid white line in this figure highlights the PT symmetry threshold. It is clear that this threshold is proportional to the coupling coefficient C. This result agrees well with the analytical solution of the PT threshold (|γ₁ − γ₂ − 2(Г₁ + Г₂)| = 4C). For Fig. 2(e), one can expect that with a larger coupling coefficient, the parametric gain becomes more sensitive to the pump amplitude, promising a shorter output duration. Fig. 2(f) presents the evolution of the parametric gain G along the waveguide length z calculated for a varying pump amplitude. When the pump amplitude is below the PT symmetry threshold, the signal exhibits gain fluctuation, and the fluctuation period decreases with the wave-mixing length z. When the pump amplitude is above the PT symmetry threshold, the signal no longer presents gain fluctuation.

3. Generation of an ultrashort pulse train via ultrafast PT symmetry switching

Next, we demonstrate the direct generation of ultrashort pulse train from CW light by pumping the coupled OPA waveguides with an amplitude-modulated laser. The dynamics of PT symmetry thus become time dependent. Analysis relying on the eigenvalues is no longer sufficient. Therefore, we characterize the evolution of the signal laser by numerically solving the coupled equations given by Equation (2). First, we assume a sinusoidal amplitude-modulated pump laser, as shown in Fig. 3(a), with a time-dependent envelope that is given by:

(8)$${A_{p1}}(t )= {A_1}\left( {\frac{{1 + m}}{2} + \frac{{1 - m}}{2}\cos ({2\pi \Omega t} )} \right),$$

where m denotes the modulation depth and Ω denotes the modulation frequency (Ω = 1 is assumed in this section). Such a pump laser contains one pair of sidebands in its spectrum [Fig. 3(b)]. Currently, sinusoidal amplitude-modulated lasers can achieve an extinction ratio of 10·log(I_max/I_min) = 30 dB in intensity, wherein I_max and I_min denote the intensity at the modulation peaks and valleys, respectively. We adopt such an extinction ratio in the following simulations, which corresponds to a modulation depth of m = 10^−1.5 in Equation (8). The seed signal adopts a CW laser light. Other parameters remain the same as those in Section 2.

Fig. 3. Temporal waveform (a) and frequency sidebands (b) of the input pump with sinusoidal amplitude modulation. The red circles represent the phases of the sidebands. Through the coupling waveguide with a length of z = 10, the CW signal is converted to an ultrashort pulse train in the temporal domain (c) and acquires more sidebands in the frequency domain (d). (e) The duration τ (black) and the temporal contrast (red) of the output signal pulse versus the propagation distance. (f) The evolution of signal waveform along the waveguide length z. For all the plots, the parameters are γ₁ = 0, Г₂ = 0, γ₂ = 2 and C = 1.

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To minimize the gain window, the PT symmetry threshold is set just at the peak amplitude of the modulated pump (i.e., A₁ = 1 in Equation (8)), indicating that the gain occurs only at the pump modulation peaks. After a normalized propagation distance of z = 10, the CW seed signal laser translates into an ultrashort pulse train, as shown in Fig. 3(c), with a repetition rate identical to the pump modulation frequency. Regarding the individual pulse, the temporal duration τ defined as the full width at half magnitude (FWHM) is τ = 0.07, which is ∼15 times shorter than the modulation period (T = 1/Ω = 1) of the pump laser. In frequency domain, the CW signal laser translates into a broadband one with tens of equally spaced frequency lines [Fig. 3(d)]. These frequency lines exhibit a constant phase distribution, indicating that the output pulses are transform-limited. The spectral amplitude profile is in super-Gaussian type, with a bandwidth Δv (the FWHM of the spectral intensity profile) of ∼9. The time-bandwidth product is τ·Δv = 0.63. The duration of the output pulses is determined by the gain slope near the PT threshold, i.e., ∂G/∂A_p₁. According to Equation (7), the gain slope also increases with the interaction length z. Therefore, under a fixed pump amplitude modulation, a longer z leads to a larger change in the parametric gain G in the vicinity of the PT threshold and consequently a shorter output pulse, as illustrated in Fig. 3(e).

The temporal intensity profile of the output signal pulses [Fig. 3(c)] is sinc-type, wherein each pulse is accompanied by a pair of weak side lobes. The production of these side lobes can be interpreted from the gain characteristics in the vicinity of the PT threshold. As illustrated in Fig. 2(b), the parametric gain fluctuates with the pump amplitude under a low pump power (below the PT threshold). Correspondingly, amplitude oscillations occur at both the front and back edge of the output signal. Nevertheless, such amplitude oscillations are barely noticeable in the intensity profile of the output pulse (in linear intensity coordinate), because the absolute value of the oscillating gain is several orders of magnitude smaller than that gain experienced by the signal peaks. As shown in Fig. 3(c), only the first pair of side lobes is observed before and after the main pulse. The intensity ratio between the main pulse and the side lobes, which is defined as the temporal contrast, is approximately 100:1. The temporal contrast varies with the waveguide length z, as illustrated in Fig. 3(e). Evolution of the signal temporal waveform along the waveguide is presented in Fig. 3(f). At the very beginning of pulse train formation, the signal peaks experience high parametric gain (up the PT threshold) while the side lobes have not occurred yet [Fig. 3(f)], so the temporal contrast sharply increases.Then side lobes appear, whose relative intensity (against the intensity of the signal peaks) first increases and then gradually stabilizes. After a normalized propagation distance of z > 8, the output contrast gradually stabilizes at ∼20 dB.

We next demonstrate that the side lobes related to the gain fluctuating behavior can be entirely suppressed by engineering the PT threshold. This scheme can be termed as apodized PT switching. For apodization, we set the PT threshold at the valley amplitude of the modulated pump [Fig. 4(a)], rather than at the peak amplitude. Other parameters change to C = 2 and γ₂ = 4C = 8 accordingly. The dependency of the eigenvalues on the pump amplitude in this case is depicted in Fig. 4(b). It clearly shows that at when the Г₁ > 0, only one supermode with strongest gain dominates, while all the other three modes get suppressed. Besides, the supermode exhibits a monotonic increasing, rather than gain fluctuating. It is for this reason that the side lobes will not be produce. A sharp gain slope occurs in the vicinity of the PT threshold (Г₁ = 0) and leads to signal fast dissipation at the modulation valley. Therefore, the output pulse train can exhibit a high extinction ratio caused by both parametric gain and linear loss. The ultrafast gain switching effect is still valid at the peak of pump (Г₁ = 1) and the CW signal laser translates into a train of ultrashort pulses with FWHM duration τ = 0.10. Such an output duration is just slightly wider than the case illustrated in Fig. 3 (τ = 0.07). To highlight the performance of apodization, the temporal intensity profile of the output signal pulse is also plotted in semilogarithmic coordinates in Fig. 4(d). Being very different from output pulses that are accompanied by several side lobes (blue line), all of the side lobes are suppressed in this apodization scheme (red line). Such an apodized PT symmetry switching enables the production of pulse trains with an ultimate-high extinction ratio. As in our simulated case, the output pulse train (red solid line in Fig. 4(d)) exhibits an extinction ratio as high as 140 dB.

Fig. 4. (a) Temporal waveform of the input pump and the PT threshold is calculated for the condition γ₂ = 4C = 8. (b) Negative imaginary parts of the four mode eigenvalues of the Hamiltonian versus the pump amplitude Г₁. Under linear (c) and semilogarithmic (d) coordinates, the output pulse with apodization design (γ₂ = 4C = 8) and without apodization design (γ₂ = 2C = 2) after a normalized propagation distance of z = 10. For all the plots, γ₁ = 0 and Г₂ = 0.

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In addition to the direct generation of ultrashort pulse trains, ultrafast PT symmetry switching can also be used as an external pulse compressor, which can further shorten the pulse duration of an ultrashort pulse train. To demonstrate this, we recalculate the coupled OPA waveguides in Fig. 3 by replacing the sinusoidal amplitude-modulated pump with a pump pulse train, as depicted by the black solid line in Fig. 5(a). The FWHM duration of the individual pump pulse is τ_p = 0.2. The other parameters remain the same as those in Fig. 3(c). Such a pump pulse train leads to the production of a train of signal pulses of τ = 0.039 in duration, indicating an external pulse compression ∼5 times compared to the pump pulse duration. This comparison highlights that the output pulse duration of ultrafast PT switching is determined collaboratively by the gain slope near the PT threshold and the steepness of the pump laser amplitude. Fig. 5(b) shows the output signal spectrum, wherein the number of induced frequency lines is almost twice that in Fig. 3(d). The time-bandwidth product is τ·Δv = 0.66.

Fig. 5. (a) Temporal waveform of the input pump pulse train (black) and output pulse with shortened duration (blue). (b) The amplitude (vertical lines) and phase (red circles) of each sideband for the output pulse.

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4. Practical considerations

In the above sections, all the parameters, such as the pump amplitude A₁ and the coupling coefficient C, have been normalized. In addition, the analyses are carried out under the assumptions of a nondepleted pump, near-degenerate signal and idler, perfect phase-matching and negligible media dispersion. Only with these assumptions can the equations for signal and idler waves be described by a spectral PT anti-symmetric Hamiltonian, and the gain characteristics can be represented by the eigenvalues. In this section, we identify the generation of ultrashort pulse trains based on coupled OPA waveguides with practical parameters. The influences of pump depletion and media dispersion (up to the third-order dispersion) are discussed. The analytical solutions based on the Hamiltonian given by Equation (4) as well as the simplified coupled-wave equations given by Equation (2) no longer apply. Characterization of the time-dependent evolution of the signal laser in this case necessitates numerically solving the six coupled-wave equations given by Equations (1a)-(1f).

Concretely, the pump laser adopts a sinusoidal amplitude-modulated laser with a central wavelength of 527 nm, a modulation frequency of 10 GHz (corresponding to a modulation period of T = 100 ps), a peak intensity of I_pin = 10 MW/cm² and an extinction ratio of 30 dB. The seed signal adopts a quasi-CW laser at 1030 nm. The wavelength of the idler is 1079 nm. Therefore, the signal and idler wave are no longer degenerate. The PT symmetric optical structure adopts two periodically poled lithium niobate (PPLN) OPA waveguides of 10 µm² in aperture. The effective nonlinear coefficient χ⁽²⁾_eff is 28.2 pm/V [40]. The seed signal injected into the gain waveguide has an initial intensity of I_sin = 100 W/cm². The pump laser is injected only into the gain waveguide, corresponding to an injection ratio of I_sin/I_pin = 10⁻⁵. At such a pump intensity, the gain waveguide provides a parametric gain coefficient of Г_s₁ = 234 m^-1 and Г_i₁ = 224 m^-1. The group velocity mismatch (GVM) between the signal and pump is 1/v_gp − 1/v_gs = 803 fs/mm, with v_gp and v_gs as the group velocities at the pump and signal frequencies, respectively. The GVM between the signal and idler is 1/v_gs − 1/v_gi = 20 fs/mm. The coupling coefficient between the gain waveguide and lossy waveguide is C = 0.9$\sqrt {{\Gamma _{s1}}{\Gamma _{i1}}}$= 206 m^-1. The loss coefficient of the loss waveguide is assumed to be γ₂ = 2C = 412 m^-1.

Figure 6(a) shows the temporal profile of the output ultrashort pulse train after propagating through 43.7 mm-long PPLN waveguides (corresponding to a normalized length of z = 10). In this case, the pump depletion at the output end is approximately 0.17%. The peak intensity of the output ultrashort pulse train is 18.0 W/cm². It is worth pointing out that under the laser intensity level and PPLN length in our scheme, χ⁽³⁾ nonlinearity as well as higher order optical nonlinearities are negligible.

Fig. 6. Practical simulations considering pump depletion and media dispersion. The normalized temporal waveform of the pump (black) and signal (blue) with an injection ratio of I_sin/I_pin equal to (a) 10⁻⁵ and (c) 2 × 10⁻³. (b), (d) The amplitude (vertical lines) and phase (red circles) of each sideband for the output signal corresponding to plots (a) and (c), respectively. (e) The pulse duration of the output signal (black), (f) the intensity of the strongest side lobe (blue) and pump depletion (red) versus the injection ratio I_sin/I_pin.

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Compared to the results calculated under neglected dispersion [Fig. 3(c)], the output signal pulses exhibit obvious temporal slippage (Δt = 17.8 ps) with respect to the pump peaks. Such temporal slippage is caused by the GVM between the pump and signal. It is worth noting that such a temporal slippage is less than the temporal walk-off calculated from linear dispersion, as (1/v_gp − 1/v_gs)×L = 800 fs/mm × 43.7 mm = 34.9 ps. This indicates that the effective interaction length between the ultrashort pulse train and the pump laser is less than the physical length of the waveguide. The FWHM duration of the output pulse train (τ = 7.4 ps) is ∼13.5 times shorter than the pump modulation period (100 ps). The pulse slightly broadens compared to the result [Fig. 3(c)] simulated under the assumption of negligible media dispersion. These results indicate that in the context of producing a picosecond pulse train, the media dispersion only leads to a temporal slippage between the pump and signal laser, whereas its impact on the output pulse duration is trivial. In the frequency domain, the media dispersion leads to the presence of 1^st-order spectral phase, as illustrated in Fig. 6(b). High-order dispersion does not manifest; thus, the output signal pulses are still transform-limited with a time-bandwidth product of τ·Δv = 0.64. These results also reveal another advantage of our scheme in the direct generation of ultrashort pulse train, compared to those methods based on the balance of media dispersion and χ⁽³⁾ nonlinearity [12–14]. Since the ultrashort pulse train is produced from χ⁽²⁾-based ultrafast PT symmetry switching, the gain switching effect is only sensitive the pump amplitude, and not sensitive to the media dispersion.

To identify the impact of pump depletion, we increase the injection ratio I_sin/I_pin from 10⁻⁵ to 2 × 10⁻³ while maintaining the values of the other parameters. Fig. 6(c) shows the normalized intensity profiles of the output pump and signal lasers. The pump depletion reaches 16.7%. In this case, the duration of the output signal pulses (∼6.4 ps) is still almost the same as that obtained under weak injection (where the pump depletion is as low as 0.17%). However, the relative intensity of the side lobes significantly increases. Correspondingly, the signal spectrum has a deep depression around the central wavelength [Fig. 6(d)], and the spectral phase exhibits high-order dispersion, indicating a degradation of the pulse quality. Figs. 6(e) and 6(f) summarize the impacts of pump depletion on the temporal duration and the side lobe intensity of the output pulse train, respectively. These results indicate that a pump depletion lower than 1% has a negligible impact on the output pulse train.

5. Conclusion

In conclusion, we have proposed a new mechanism for producing ultrashort pulse trains in OPA waveguide systems. Our scheme is based on the nonlinear gain switching near the PT symmetry threshold of the coupled OPA waveguides system. For pump powers below the threshold, the modes form pairs with broken PT symmetry and the same gain/loss, whereas above the threshold, one supermode experiences the largest gain. The combination of ultrafast all-optical gain switching and a periodically amplitude-modulated pump enables time-domain selective mode amplification, as the signal laser experiences large parametric gain only at the peaks of the modulated pump. Such a gain characterize further enables the direction generation of an ultrashort pulse train without the need for optical resonators.

We have identified the duration and temporal contrast of the output ultrashort pulse under different conditions. We have also tested the ultrafast PT symmetry switching performance in the presence of pump depletion and media dispersion. We believe this work not only opens a new avenue for producing ultrashort pulse trains via photonic waveguiding structures, but also fosters other new applications that require ultrafast all-optical gain switching. Besides, by engineering other strategic modulation of gain/loss, parametric amplification in PT symmetric systems can also be designed to provide spectrally selective mode amplification or spectrally selective mode amplification. In addition to implementing OPA in PT-symmetric structures, the concept of ultrafast PT symmetry switching will be extended to different physical mechanisms, including Kerr-type optical nonlinearity.

Funding

Fundamental Research Funds for the Central Universities; National Natural Science Foundation of China (62122049); Shanghai Rising-Star Program (21QA1404600); The Science and Technology Commission of Shanghai Municipality (22JC1401900).

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (62122049), Shanghai Rising-Star Program (21QA1404600), and The Science and Technology Commission of Shanghai Municipality (Grant No. 22JC1401900).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Generation of an ultrashort pulse train through ultrafast parity-time symmetry switching

Abstract

1. Introduction

2. Ultrafast gain switching behavior of the coupled OPA waveguides

3. Generation of an ultrashort pulse train via ultrafast PT symmetry switching

4. Practical considerations

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (13)

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