The Talbot effect, an intriguing wave phenomenon manifested by periodic revivals of patterns, was first discovered by Henry Fox Talbot in 1836 [1] and explained by Lord Rayleigh in 1881 [2]. It is mostly known as the spatial Talbot effect—by illuminating a plane wave on an amplitude grating, the images of the grating are periodically reproduced at different distances from the grating plane [3]. Apart from the Talbot effect in space, the concept has also been developed in the time domain, given the space–time duality [4,5]. The temporal Talbot effect occurs when a periodic optical pulse train propagates through a dispersive element, for example a spool of optical fiber [6] or a linearly chirped fiber Bragg grating [7]. At the output, an identical or a repetition-rate multiplied pulse train can be created, which respectively corresponds to the condition of the integer or fractional temporal Talbot effect [4]. Specifically, the fractional self-imaging is of particular interest for the generation of ultrahigh repetition-rate optical pulse trains [6,8], and is hence appealing for such applications as microwave photonics [9] and optical communications [10].

However, to the best of our knowledge, all the temporal Talbot effects demonstrated so far are based on bright pulse trains, while their counterpart, the self-imaging effect of a dark pulse train, has remained unexplored. The dark pulse train here refers to a train of intensity dips buried in a continuous-wave (CW) background. Such temporal structures are naturally formed in laser cavities [11,12] and in microresonators [13], which are ubiquitous across many disciplines in optics [14]. Early works have also studied the propagation characteristics of dark pulses in optical fibers, mainly to understand dark soliton dynamics [15–17]. In terms of the linear propagation regime, no temporal self-imaging has been reported for dark pulse trains, which might exhibit distinct features from the conventional Talbot effect.

In this Letter, we present a detailed investigation of the Talbot phenomenon of an optical dark pulse train. The effect is theoretically analyzed and experimentally demonstrated by spectral Talbot phase shaping [18] and by single-mode fiber (SMF) propagation. Resembling its bright counterpart, a dark pulse train revives itself after a specific length of fiber propagation. An identical or a shifted copy of the original pulse train is recovered at one or one-half Talbot length, respectively. Besides, twofold temporal self-imaging is also observed at one-quarter of the Talbot length, where the dark pulse redistributes its waveform into two identical dark self-images but with shallower depths. However, at higher-order fractional self-imaging, a mixed pattern of bright and dark self-images is found to appear. These self-imaging patterns are explained through the complex interference between the Talbot pulse train and the CW background. The results of this work complement past Talbot studies on bright pulse trains, and may regenerate new interest in the linear and nonlinear effects based on dark pulse trains.

It is known that an optical bright pulse train corresponds to a frequency comb with an in-phase relation between comb lines. A number $q$ of bright self-images appears when each comb line of the bright pulse spectrum picks up an additional quadratic phase [19]:

Notably, the distinct phases of bright self-images described by Eq. (3) do not alter their intensity waveforms. By contrast, such a phase relation plays an important role for the dark pulse case. A dark pulse train can be comprehended as a bright pulse train being subtracted from a CW background by destructive interference. In the frequency domain, it corresponds to a bright pulse spectrum with a strong out-of-phase carrier located at the center mode of the comb. If the amplitude of the center comb mode is equivalent to the sum of amplitudes contributed from all other comb modes, a complete dip can be obtained in time. The temporal Talbot effect of the dark pulse train can be viewed as the interference of the unaffected CW component and the bright Talbot self-images with the phases defined in Eqs. (2) and (3).

Figure 1(a) illustrates a simulated dark Talbot carpet. For clarity, we show here only the first half of the full carpet, while the other half is simply a mirror image of the first half. To better visualize the patterns in the dark Talbot carpet, we plot the cross section of the carpet in Fig. 1(b) at specific Talbot planes $p/q=0, 1, 1/2, 1/3, 1/4$, where $T$ represents the period of the original dark pulse train. One can see that at the end of the current carpet ($p/q = 1$) a shifted copy with a half-period delay is obtained. Such a plane is actually located at half of the Talbot length. The Talbot length is defined where an exact replica of the input image is recovered ($p/q = 2$, not shown). At one-quarter of the Talbot length ($p/q = 1/2$), two identical, equally spaced dark pulses would appear in one period, leading to the repetition-rate multiplication. Since the Talbot operation is unitary, the depths of the two dark pulses are shallower than the original one by energy redistribution. In addition, the dark Talbot carpet exhibits unique features at higher-order fractional planes. For example, the threefold ($p/q = 1/3$) and fourfold ($p/q = 1/4$) self-images show mixed patterns of bright and dark structures.

 

Fig. 1. (a) Simulated dark Talbot carpet (half a period is shown). (b) From left to right: the temporal waveforms of the original dark pulse train ($p/q=0$), the half-period shifted pulse train ($p/q=1$), and the twofold ($p/q=1/2$), threefold ($p/q=1/3$), and fourfold ($p/q=1/4$) fractional self-imaged pulse trains. (c) Phase distribution diagrams of the CW background (CW) and the bright Talbot self-images, displayed in accordance to (b).

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To understand the origin of such mixed structures, we introduce the phase distribution diagram, as illustrated in Fig. 1(c). It describes the phases of the CW background (CW) and the constituent bright Talbot pulses, which are calculated based on Eqs. (2) and (3) and marked in the circumference of a ring. Note that the radius of the ring here does not reflect the relative intensity. In fact, the intensity of the CW component should be stronger than the bright fractional self-images. At different fractional planes, the phase of the CW component remains unchanged, and the vector sums of the CW and the Talbot pulses determine whether the self-images will be bright or dark. For instance, in the scenario of $p/q=1/3$, two identical dark self-images are formed, owing to destructive interference between the pulses ($\pi /6$ phases) and the CW ($\pi$ phase), while a small bright self-image is built up, given the constructive interference of the pulse ($\pi /2$ phase) with the CW. The phase distribution diagram also explains that the two dark pulses at $p/q=1/2$ are of equal depths. Similar analysis can be applied to arbitrary fractional self-imaging of dark pulses. It is noted that the phase in Eq. (2) consists of a quadratic phase and a constant phase. While the quadratic part has been characterized experimentally [21,22], the constant phase term $c$ is generally neglected in the bright Talbot scenarios. In contrast, it cannot be overlooked in the dark cases, owing to the presence of a static CW background. From another point of view, the patterns in Fig. 1(b) indicate the existence of the constant phase in the bright Talbot self-images.

To experimentally confirm the aforementioned behaviors, we carry out our implementation using the setup illustrated in Fig. 2. A C-band CW laser is injected to a lithium niobate (LiNbO$_3$) electro-optical (EO) phase modulator to generate an EO comb. The modulator is driven by a radio frequency (RF) sinusoidal wave $V(t) = V_{\rm RF}\sin (\Delta \omega t)$ with a voltage $V_{\rm RF}$ and a modulation frequency $\Delta \omega =2\pi \times 30$ GHz. At the modulator output, the optical field becomes $E(t) = {E_0}\exp \left ( i{\omega _0}t + i\pi\frac{V_{\rm RF}}{V_{\pi}}\sin\Delta\omega t \right ),$ where $V_{\pi }$ is the half-wave voltage of the modulator and $E_0$ and $\omega _0$ are the optical field and the angular frequency of the laser carrier, respectively. Using the Jacobi–Anger expansion, the output field can be rewritten as $E(t) = E_0 \sum _{l=-\infty }^{+\infty } J_l(m)\exp (i\omega _0 t + il\Delta \omega t)$ with $m = \pi V_{\rm RF}/V_{\pi }$ being the modulation index and $J_{l}(m)$ being the Bessel function of the first kind. In the experiment, we operate at $m=4.6$. A series of comb lines is thus generated, and is directed to a programmable Fourier-transform pulse shaper (Finisar WaveShaper 4000S). After line-by-line shaping of the comb, the optical signal is then amplified, and its spectrum and temporal waveform are measured using an optical spectrum analyzer and a 500 GHz optical sampling oscilloscope (EXFO PSO-101), respectively. In the pulse shaping part, the phases of the comb lines are first corrected to be all in-phase; thereby, a bright pulse train is formed as shown in Fig. 3(a), with the corresponding spectrum in Fig. 3(b). Then we modify the phase of the center mode to be out-of-phase, and attenuate all the other comb lines so that a complete dip is obtained in time; meanwhile, we shape the spectrum into a Gaussian profile. Since the pulse shaper is working at its extinction ratio limit, we iteratively update the attenuation values to the pulse shaper by comparing the synthesized spectrum with the target one, until it eventually converges to a smooth Gaussian shape. A dark pulse train is thus synthesized, as shown in Fig. 3(c), and its waveform matches well with the simulation (dashed) reconstructed from its spectrum, as shown in Fig. 3(d). The subtle difference is probably due to small parasitic phase shifts in the amplitude shaping.

 

Fig. 2. Experimental setup. EDFA: erbium-doped fiber amplifier; OSA: optical spectrum analyzer; OSO: optical sampling oscilloscope; SMF: single-mode fiber. The length of SMF varies according to the required fractional self-imaging, and is not connected when the Talbot phase is given by the programmable pulse shaper.

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Fig. 3. Bright and dark pulse trains. (a) Waveforms and (b) spectra of a bright pulse train. (c) Waveforms and (d) spectra of a dark pulse train. Solid: experiment; dashed: simulation.

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We first demonstrate the fractional self-imaging properties of the dark pulse train via the same pulse shaper used in the setup. Specifically, we assign additional Talbot phase sequences, described in Eq. (1), on top of the applied amplitude and phase functions, in order to emulate the dispersive element [18,23]. Figure 4(a) shows the temporal waveform of the pulse train when its spectrum is impressed by an alternating Talbot phase of $[0,\pi /2]$ ($p/q=1/2$). Compared with Fig. 3(c), the repetition rate of the dark pulse train in Fig. 4(a) is indeed doubled, while its spectral shape in Fig. 4(b) is almost the same as that in Fig. 3(d). Further, if a periodic Talbot phase of $[0,\pi /3,4\pi /3]$ ($p/q=1/3$) is applied across the original spectrum, two dark self-images and one bright self-image are observed within one time period, as shown in Fig. 4(c). The spectrum after the phase shaping in Fig. 4(d) is again similar to Fig. 3(d), while the slight amplitude deviations are attributed to the imperfect amplitude and phase controls of the pulse shaper. All the experimental results here are in good agreement with the simulation, and confirm the theoretical analysis presented in Fig. 1.

 

Fig. 4. Temporal Talbot effect of a dark pulse train implemented by spectral Talbot phase shaping. (a) Waveforms and (b) spectra of twofold self-imaging. (c) Waveforms and (d) spectra of threefold self-imaging. Solid: experiment; dashed: simulation.

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In addition, we show the temporal Talbot effect of the dark pulse train by fiber propagation. To satisfy the Talbot condition of $p/q$, the required fiber length can be calculated from $|\beta _2|\Delta \omega ^2 L/2 = \pi p/q$ [19], where $\beta _2$ is the group velocity dispersion of the fiber. In the experiment, we use SMF with $\beta _2 = -21.6$ ps$^2$/km, and the fiber length is measured using an optical time-domain reflectometer. Three spools of SMF of 8.18 km, 4.08 km, and 2.72 km are prepared in order to demonstrate the Talbot images at $p/q=1, 1/2, 1/3$, respectively. Figures 5(a)–5(c) show the corresponding waveforms of the pulse train after propagating through these fibers. Specifically, a dark pulse train is reproduced after 8.18 km of SMF, which verifies the integer Talbot effect ($p/q=1$) in the dark pulse regime. The fractional self-imaged pulse trains here are also consistent with the results obtained in Fig. 4, where the pulse asymmetry in Fig. 5(b) is due to the slight fiber length mismatch from the perfect Talbot condition. Figure 5(d) shows the spectrum of the waveform in Fig. 5(a) (longest SMF propagation) as an example, confirming that the spectral shape is well maintained and the pulse propagation is in the linear regime.

 

Fig. 5. Temporal Talbot effect of a dark pulse train observed after SMF propagation of different lengths. Waveforms of (a) onefold, (b) twofold, and (c) threefold self-imaging. (d) Corresponding spectrum of the waveform in (a). Solid: experiment; dashed: simulation.

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In conclusion, we study in this work the temporal Talbot phenomena of optical dark pulse trains. The experimental results from direct Talbot phase shaping and through SMF propagation agree with the theoretical predictions. Resembling the conventional temporal Talbot effect, the self-imaging of a dark pulse train also redistributes the waveform into self-images. These self-images can be either bright or dark, and are experimentally observed and theoretically explained. The equivalent dark Talbot carpet should also be observed in the reflection direction of a periodic amplitude grating, and may appear in other mode bases as well, for example the orbital angular momentum modes [24,25]. Besides, this work may also trigger interest in investigating the Talbot phenomena of dark space–time non-separable fields that revive synchronously both in space and time [26,27].