Temporal imaging using a time pinhole

Zhao Wu; Jianji Dong; Jie Hou; Siqi Yan; Yu Yu; Xinliang Zhang

doi:10.1364/OE.22.008076

1. Introduction

The space-time duality transfers the idea of spatial phenomena into the temporal domain, leading to the concepts of temporal imaging. Temporal imaging is a waveform manipulation technique that enables compressing or expanding an input waveform and preserving its overall profile [1–5], thus improving the performance of signal processing and measurement [6–8]. The essential device in these applications is a time lens [6–11], which is a temporal quadratic phase modulation and widely implemented by the parametric mixing between the signal and a linearly chirped pump pulse, for its advantage of large phase shift [12–14]. Wherein, the chirped pump pulse is obtained by dispersing a short pulse through a section of fiber with pre-measured length. However, there is a very strict requirement of the pulse shape and dispersion for generating an ideal time lens [15], which is characterized by the pure quadratic phase shift and flatten amplitude in the operation window [1]. Moreover, for obtaining different scale ratio of the output waveform, the input and output dispersion should be chosen correctly with respect to the degree of the quadratic phase modulation [4].

The pinhole time camera has been proposed as an alternative to perform temporal imaging [16]. However, in practice, it is difficult to modulate a temporal short shutter with tunable duration. In this paper, we perform the temporal shutter by parametric mixing between the signal and a short pulse. The duration of the temporal shutter is adjusted by changing the spectral bandwidth of the pulse. Experimental results indicate that, the output waveform is the scaled and broadened profile of the input waveform. Moreover, the output waveforms under different shutter times are demonstrated, showing that, when the scale ratio is positive and close to 1, the output waveform is better with properly larger shutter time. Otherwise, the shutter time remains relatively small.

2. Theoretical analysis

The idea of time-pinhole based temporal imaging relies on the principle of the space-time duality, which indicates that the temporal dispersive propagation of a pulse is analogous to spatial diffractive propagation of a light beam, and a time-gate plays the role of a spatial pinhole, as illustrated in Fig. 1.

Fig. 1 (a) Schematic diagram of the conventional pinhole camera. An object I_i placed at a distance d_i before an aperture is projected to an image I_o at a distance d_o behind the aperture. (b) Temporal analog: an input waveform with pulse packet of “1101” propagates through a section of dispersion fiber, and then undergoes a short temporal shutter, which allows a small burst of the waveform to pass through. Finally, the waveform propagates in another dispersion fiber and forms the scaled profile of the input waveform.

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For a dispersive fiber with a flat amplitude and quadratic phase response over a certain bandwidth, its corresponding impulse response is given by $\sqrt{j / 2 π β_{2} L} \exp (- j t^{2} / 2 β_{2} L)$ in a traveling-wave coordinate system [12], where β₂ is the group velocity dispersion parameter and L is the length of the fiber. For simplification, neglecting the amplitude coefficient, $\exp (j a t^{2})$ and $\exp (j b t^{2})$ are set as the impulse response of the fiber with different dispersion on either side of the time-gate, where $a = - 1 / (2 β_{2, a} L_{a})$ , $b = - 1 / (2 β_{2, b} L_{b})$ . For a signal x(t) successively passing through an input dispersion $\exp (j a t^{2})$ , time gate g(t), and an output dispersion $\exp (j b t^{2})$ , the output field is expressed as

\begin{array}{l} y (t) = [x (t) \otimes \exp (j a t^{2}) \cdot g (t)] \otimes \exp (j b t^{2}) \\ = F_{b} {F_{a} {x (t)} \cdot \exp [j (a + b) t^{2}] g (t)} \cdot \exp (j b t^{2}) \end{array}

where ⊗ denotes the convolution operation, and F_a/b{•} denotes the integral operation, which is analogous to the Fourier transform (FT) integral but with a proper replacement of variable [1].

F_{a} {x (t)} = \int_{- \infty}^{+ \infty} x (τ) \exp (- j 2 a t τ) d τ = X (f = \frac{a}{π} t)

where X(f) is the FT of the input electric field x(t). Note that, Eq. (1) is obtained if the signal experiences a large dispersion [17–19], which is analogous to the spatial Fraunhofer diffraction.

The time-gate g(t) with short duration is used to eliminate the effect of quadratic phase modulation $\exp [j (a + b) t^{2}]$ . In this case, little or no phase modulation occurs within the temporal shutter as illustrated in Fig. 2(a). Δφ is the maximum phase shift tolerable within the temporal shutter, and ΔT is the full-width at half-maximum (FWHM) of the time gate. Thus, the maximum width of the time-gate is given by

Δ T_{\max} = 2 \sqrt{| \frac{Δ φ}{a + b} |} = 2 \sqrt{| \frac{Δ φ}{\frac{1}{β_{2, a} L_{a}} + \frac{1}{β_{2, b} L_{b}}} |}

assuming ΔT≤T_max, the output waveform is then approximately expressed as

\begin{array}{l} {| y (t) |}^{2} \approx {| F_{b} {F_{a} {x (t)} \cdot g (t)} |}^{2} \\ = {| x (- \frac{β_{2, a} L_{a}}{β_{2, b} L_{b}} t) \otimes G (f = - \frac{t}{2 π β_{2, b} L_{b}}) |}^{2} \end{array}

where G(f) is the spectrum of the time-gate g(t). The output waveform is the convolution of the scaled waveform and the profile

G (f = - t / 2 π β_{2, b} L_{b})

. The scale of the output waveform depends on the output-to-input dispersion ratio

M = - β_{2, b} L_{b} / β_{2, a} L_{a}

. Specifically, the output waveform is reversed if the signs of dispersion on either side of the time-gate are identical, otherwise is non-reversed if the signs are opposite. On the other hand, the broadening effect of the output waveform is determined by the convolution operation with

G (f = - t / 2 π β_{2, b} L_{b})

, which relates directly to the resolution of the system. Therefore, the output waveform shows a scaled and broadened profile compared to the input waveform. Moreover, it can be seen from Eq. (4) that, for a fixed input dispersion and time-gate, the output waveform is scaled and broadened exactly by the same degree with the increase of the output dispersion.

Fig. 2 (a) Time-gate used to eliminate the effect of the quadratic phase term; (b) optimum width of the time-gate ΔT_opt versus the output-to-input scale ratio M for a fixed input dispersion.

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Ideally, the time-gate g(t) should be narrow enough for eliminating the impact of quadratic phase modulation. However, a too narrow profile of g(t) results in a broad profile of $G (f = - t / 2 π β_{2, b} L_{b})$ , thus inducing severe pulse broadening to the output waveform. Therefore, there exists a trade-off on the width of the time-gate. It can be found from Fig. 2(a) that, the optimum width of the time-gate is the tolerable maximum width, ΔT_opt = ΔT_max. The optimum width of the time-gate ΔT_opt versus the output-to-input ratio M is thus calculated to be

Δ T_{o p t} = 2 \sqrt{| Δ φ \cdot β_{2, a} L_{a} |} \cdot \sqrt{| \frac{M}{M - 1} |}

which is illustrated in Fig. 2(b), where the input dispersion is an opposite dispersion value of 5 km single mode fiber (SMF) (which is in consistent with the input dispersion in the following experiment), and Δφ is set as π/10.

In this paper, the proposed temporal shutter is experimentally realized by the parametric mixing between the signal and a short pulse. The generated idler signal is the logic AND-gate of the dispersed data signal and the pulse, and the signal within the duration of the pulse is extracted. As a result, the action of temporal shutter is achieved. It should be noted that, in our experiment, since the generated idler signal is conjugate with the input data signal due to the FWM effect [20], the data signal equivalently undergoes a dispersion with the opposite sign and same absolute value of actual dispersion that it experiences before the time-gate. Besides, the width of the pulse is adjusted by changing the bandwidth of its spectrum, for investigating the performance of the output waveform under different shutter times.

3. Experimental results and discussions

The experimental setup for the time-gate based temporal imaging is shown in Fig. 3. The pulse source is an erbium glass oscillating pulse-generating laser (ERGO-PGL) emitting pulses of 10 GHz repetition rate and centered at 1548.88 nm with FWHM of ~2 ps. The pulse sequence is amplified and spectrally broadened through self-phase modulation (SPM), generating a supercontinuum in a 1000 m dispersion-flattened high nonlinear fiber (HNLF) with nonlinear coefficient γ of 10 /W/km, from which both data and pump spectra are carved out. The pump pulses are spectrally filtered by a band pass filter (BPF) with different bandwidths, in order to achieve various pulsewidths. In the experiment, we use a BPF at 1548.88 nm with bandwidth of 1.6 nm. The 10 GHz pulse train for the data signal is obtained via a programmable spectral pulse shaper (Finisar WaveShaper) [21], which acts as a combined filter consisting of a BPF and a finite impulse response (FIR) filter. The BPF is at 1554 nm with 3 dB bandwidth of 2 nm, and the FIR filter with transfer function of $[1 + \exp (- j 2 π f τ) + \exp (- j 2 π f \cdot 3 τ)] / 3$ encodes the data “1101” on the pulse sequence, where τ is 10 ps. The encoded data signal is primarily dispersed in a standard SMF with premeasured length of 5 km, and the pulses in the packet are broadened and mixed together. The optical delay line (ODL) in the pump branch is used to aim the center of the gate pulse to the dispersed signal. Subsequently, the data signal and pump signal are amplified, and then undergo the following polarization control (PC). PC1 and PC2 are used to adjust the polarization states of the data signal and pump signal to get the optimal four wave mixing (FWM) efficiency in the following HNLF [20]. Then the data signal and pump signal are combined using a 3 dB coupler and injected into HNLF2. The powers of the data signal and pump pulse injected into the HNLF2 are 16.7 and 18.9 dBm, respectively. The HLNF2 is 135 m long with zero dispersion wavelength at 1549 nm, and nonlinear coefficient γ of 20 /W/km. The action of the time-gate is achieved using the FWM effect of the HNLF2 between the data signal and pump signal. At the output of the HNLF2, a BPF with bandwidth of 3.2 nm is used to extract the idler signal. Finally, the filtered signal propagates through another section of dispersion fiber with different length and then received by an Eye-1100C high-speed optical sampling oscilloscope (OSO), in which all-optical sampling technology is exploited.

Fig. 3 Schematic setup for the proposed method.

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The waveforms of the encoded data signal and the pump signal are illustrated in Figs. 4(a) and 4(b), respectively. The data signal is a repeating pulse train of “1101”, synthesized by a 10 GHz pulse sequence and its delayed replicas with delay times of 10 and 30 ps. The time gap between the pulse packet is 70 ps, and the FWHM of each pulse is 3.1 ps. The pump signal is a pulse sequence with repetition period of 100 ps and FWHM of 4.7 ps, acting as a temporal shutter. Since the pump pulse is filtered by a super-Gaussian shaped BPF with broad flat-top, the pump pulse shows a sinc-shape with small ripples at the bottom of the pulse. The asymmetry of the sinc-like pulse is due to its slightly asymmetric spectrum, originated from the chirped pulse source.

Fig. 4 Waveform of the (a) data and (b) pump pulse.

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The dispersed data signal and the pump signal combine and inject into the HNLF2. The spectrum at the output of HNLF2 is illustrated in Fig. 5. The spectrum of the idler signal appears beside the pump signal owing to the FWM effect of HNLF2, and is extracted by a 3.2 nm BPF. Note that the generated idler signal is conjugate with the input data signal due to the FWM effect [20], so the data “1101” equivalently experiences a section of dispersion compensation fiber (DCF) before the time gate, which is of the opposite sign and same value of 5 km SMF.

Fig. 5 Spectrum at the output of HNLF2.

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Then the extracted idler signal propagates through a section of DCF. The DCF is of premeasured length for compensating accurate dispersion of 0.5, 1, 2, and 4 km SMF. Hence, the corresponding scale ratios of the output-to-input waveform are −0.1, −0.2, −0.4, and −0.8, respectively. In this case, the signs of the dispersion on both sides of the temporal shutter are identical, which is in accordance with the spatial pinhole camera. According to the investigation on the optimum shutter size for the output waveform [16], in this situation, the optimum size of the temporal shutter almost remains unchanged for a fixed input dispersion (with the same absolute value and opposite sign of 5 km SMF). In the experiment, the pulse width is 4.7 ps. The output waveforms are illustrated in Figs. 6(a)-6(d), respectively. The output waveforms are reversed and stretched with the increase of the fiber length. On the other hand, the pulses in the packet are broadened and overlapped. The waveforms are fitted by the broadened Gaussian-shaped pulses sequence with “1011”. The time intervals of the data are 1, 2, 4, and 8 ps, respectively. Since the signal from the temporal shutter is a short segment of the dispersed data signal, the signal is weak and spectrally broadened. Thus, extra noise is introduced to the output waveform after amplification.

Fig. 6 The reversed waveforms from different length of DCF with opposite sign and same dispersion value of (a) 0.5 km; (b) 1 km; (c) 2 km; and (d) 4km SMF.

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We also investigate the output waveform in the situation that the signs of the dispersion on both sides of the temporal shutter are opposite. The pump signal is filtered by a 1.6 nm BPF. The output waveforms from SMFs with different length are illustrated in Fig. 7. The SMFs are of premeasured lengths of 0.5, 1, 2, and 4 km, thus the output-to-input scale ratios are 0.1, 0.2, 0.4, and 0.8, respectively. The output waveforms are non-reversed, and stretched with the increase of the fiber length. In the same way, the output waveforms are fitted by the broadened Gaussian-shaped pulse sequence of “1101”. The time intervals of the data in these figures are 1, 2, 4, and 8 ps, respectively.

Fig. 7 The non-reversed waveforms from (a) 0.5 km; (b) 1km; (c) 2 km; and (d) 4 km SMF when the pump pulse is filtered by 1.6 nm BPF.

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For the positive output-to-input scale ratio M, the optimum size of the temporal shutter changes comparatively small when the scale ratios are 0.1, 0.2, and 0.4. However, the optimum shutter time increases rapidly as the scale ratio increases to 1 [16]. For the scale ratio of 0.4 and 0.8, we also investigate the output waveforms under different shutter times. The width of the pump pulse is changed by the BPF of different bandwidth. In the experiment, the pulse width is 8.7 ps from a 0.8 nm BPF, while it is 18.1 ps from a 0.32 nm BPF, respectively.

For the scale ratio of 0.4, the pump pulse is filtered by a 0.8 nm BPF centered at 1548.88 nm, and the output waveform is illustrated in Fig. 8(a). The waveform is broadened compared to that of Fig. 7(c), showing that larger shutter time degrades the output signal. In this situation, the residual quadratic phase modulation cannot be neglected within the wide temporal shutter, as indicated in Eq. (1). Hence, a temporal shutter that is too wide cannot achieve cascaded Fourier transformation as in Eq. (4), and then degrades the output waveform. For the scale ratio of 0.8, the pump pulse is filtered by the 0.8 and 0.32 nm BPF centered at 1548.88 nm, and the output waveforms are illustrated in Figs. 8(b) and 8(c), respectively. It can be seen that, the output waveform is improved with the increased temporal shutter size firstly, and then degrades with too wide temporal shutter. This is because that as the output dispersion increases, the residual phase is negligible within properly wider temporal shutter, which induces less pulse broadening, as illustrated in Fig. 8(b). However, the residual phase cannot be eliminated within too wide temporal shutter, which degrades the output waveform, as illustrated in Fig. 8(c). It can be concluded that, when the scale ratio is close to 1, the output waveform is better under properly wider temporal shutter. On the other hand, since the received signal becomes overlapped with the adjacent pulse packet through too large output dispersion, in the experiment, the length of the output SMF is less than 4 km.

Fig. 8 (a) Output waveform from 2 km SMF when the pump pulse is filtered by 0.8 nm BPF; (b) output waveform from 4 km SMF when the pump pulse is filtered by 0.8 nm BPF (c) output waveform from 4 km SMF when the pump pulse is filtered by 0.32 nm BPF.

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The degraded resolution of the result is due to the convolution operation of the scaled signal with $G (f = - t / 2 π β_{2, b} L_{b})$ , which broadens each pulse of the received waveform, as seen from Eq. (4). Fortunately, The resolution can be improved with a properly narrower profile of $G (f = - t / 2 π β_{2, b} L_{b})$ , i. e. tolerable wider temporal shutter g(t), as indicated from the comparison between Fig. 7(d) and 8(b). Thus, a better resolution is achieved when the scale ratio is around 1, as indicated in Fig. 2(b). In this case, the optimum size of the shutter is sufficiently large. Assuming an extreme situation that when the scale ratio is 1, the optimum size of the temporal shutter is infinite, thus recovering the input signal. This is the case of dispersion compensation [22]. On the other hand, a wider aperture improves the performance of the received signal, since a wider segment of the signal within the temporal shutter is extracted, leading to the improvement of the received optical power.

4. Conclusions

In this paper, we experimentally demonstrate a time-gate based temporal imaging system. The time-gate is realized by the parametric mixing between the data signal and a short pulse, and the duration of the time-gate is adjusted by changing the bandwidth of the pulse. Theoretical analysis combined with experimental result indicates that the received waveform is the scaled and broadened profile of the input waveform. Moreover, the impact of the shutter time on the output waveform is also investigated, showing that when the output-to-input scale ratio is positive and close to 1, the performance of the output waveform is better with properly larger shutter time. Otherwise, the output waveform is better with relatively smaller shutter time.

Acknowledgments

This work was supported by National Basic Research Program of China (Grant No. 2011CB301704), National Science Fund for Distinguished Young Scholars (No. 61125501), National Natural Science Foundation of China (NSFC) (Grant No. 61007042, and Grant No. 11174096), and New Century Excellent Talent Project in Ministry of Education of China (NCET-13-0240)

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Temporal imaging using a time pinhole

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (5)

Optics Express