Full-field ultrafast oscilloscope based on temporal imaging

Zhi Qiao; Xue Pan; Yudong Yao; Xiaochao Wang; Wei Fan; Xuechun Li

doi:10.1364/OE.27.007545

1. Introduction

As the development of ultrafast lasers has proceeded, various techniques for measuring femtosecond or even attosecond laser pulses have attracted considerable research interest [1–6]. However, most of the current techniques, such as frequency resolved optical gating (FROG) and spectral phase interferometry for direct electric field reconstruction (SPIDER), depend on a spectrum revealed after a nonlinear process using a charge-coupled device (CCD) camera, which has limited speed. Therefore, these methods are restricted to low repetition rate signals and make it difficult to realize high resolution and wide dynamic range simultaneously. [7, 8].

In order to achieve ultra-speed measurement for high-repetition rate ultrafast pulse and transient events, an ultrafast oscilloscope based on the temporal stretch or temporal imaging method was proposed for measuring the waveform of ultrafast pulses [9–12]. Similar to spatial imaging, the temporal imaging process can be realized using a time lens [13], which can be a sum or differential harmonics generation or of a four-wave mixing (FWM) effect [12, 14, 15]. Ultrafast oscilloscopes using temporal imaging are implemented to measure the temporal profile of an ultrafast pulse with high resolution, as well as over a wide dynamic range [16–18], but the temporal phase information is totally lost due to the assumption that the under-test laser pulse is transform-limited, which is only valid for some specific applications.

With the development of various spatial phase measurement methods [19–24], the iteration phase imaging method, including the Gerchberg-Saxton(GS) and ptychography algorithms, have been subjected to wide research due to their simplicity and high accuracy [25–28]. According to space-time duality, the methods for spatial phase measurement have equivalent temporal versions that are suitable for measuring temporal phase with the temporal imaging system. There is some research on the implementation of an iterative algorithm to measure laser pulses [29–35]. Recently, the commonly used GS phase retrieval method was used in a temporal imaging system to realize ultra-speed measurement of the building process of mode-locked lasers [36].

In this paper, we propose for the first time, a full-field ultrafast oscilloscope based on temporal phase methods, temporal annealing Gerchberg-Saxton (TAGS) algorithm, and temporal ptychography. These methods are analyzed numerically and verified experimentally. The all-fiber ultrafast oscilloscope is realized based on the FWM effect in highly nonlinear fiber. The results show that the temporal ptychography method can achieve accurate measurement of the temporal profile and phase for the signal and pump pulse. However, a scanning process during measurement is necessary, which means it is hard to achieve a single-shot real-time measurement. For some applications, real-time measurement of every single pulse is necessary. The TAGS method has single-shot measurement ability, but the method depends on the accuracy of the time lens. Considering the advantages and disadvantages of the two methods, we propose to use temporal ptychography to calibrate the pump pulse for the TAGS method. The experimental results show significant improvement compared with the normal TAGS method. Combined, the two methods shows good performance in the ultrafast oscilloscope. This achievement could solve the problem of full-field ultra-speed transient pulse measurement and should be helpful for the further development of a temporal phase retrieval method in temporal imaging systems.

Fig. 1 Schematic of four-wave mixing-effect temporal imaging.

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2. Temporal imaging system structure

Here, we take a temporal imaging system based on the FWM effect as an example. The schematic of a FWM temporal imaging structure is shown in Fig. 1, where $A_{i} n$ is the signal laser, u_s is the signal laser after dispersion of ϕ₁, and u_p is the pump laser after dispersion of ϕ_p. After the FWM process, the idler laser $u_{i} d l e r$ is generated and can be described as in Eq. (1) under a small signal assumption, where κ is the nonlinear coefficient.

u_{i d l e r} = κ u_{s}^{*} u_{p}^{2}

In the temporal imaging system, the pump laser is ordinarily a transform-limited pulse, which means that the pump laser after dispersion can be written as $u_{p} = exp (- j t^{2} / 2 ϕ_{p})$ (Here j is the same complex notation as with i). Therefore, the idler laser can be described by

u_{i d l e r} = κ {[F^{- 1} [F (A_{i n}) exp (j \frac{ω^{2} ϕ_{1}}{2})]]}^{*} \times [exp (- \frac{j t^{2}}{2 ϕ_{p}})]

After the output dispersion ϕ₂, the idler laser pulse $A_{o u t}$ can be described as

\begin{array}{l} A_{o u t} = F^{- 1} [κ \sqrt{- π j ϕ_{p}} exp (\frac{j ϕ_{2} ω^{2}}{2}) \\ [{\tilde{A}}_{i n}^{*} (- ω) exp (\frac{- j ϕ_{2} ω^{2}}{2}) \otimes exp (\frac{j ϕ_{p} ω^{2}}{4})]] \\ = κ \sqrt{\frac{ϕ_{p}}{ϕ_{p} + 2 ϕ_{2}}} exp (- j \frac{t^{2}}{ϕ_{p} + 2 ϕ_{2}}) \int_{- \infty}^{\infty} {\tilde{A}}_{i n}^{*} (- Ω) \times \\ e^{- j Ω^{2} [\frac{ϕ_{p}^{2}}{4 (ϕ_{p} + 2 ϕ_{2})} + \frac{2 ϕ_{1} - ϕ_{p}}{4}]} exp (- j \frac{ϕ_{p}}{ϕ_{p} + ϕ_{2}} Ω t) d Ω \end{array}

If the dispersion matches the imaging condition $2 / ϕ_{p} = 1 / ϕ_{1} - 1 / ϕ_{2}$ , the waveform of the idler laser will be the exact magnified waveform of the signal laser, which process is called temporal magnification [14, 15]. Moreover, the magnification factor is defined as $M = ϕ_{2} / ϕ_{1}$ , but the information of the temporal phase is "lost" or becomes indistinguishable from the waveform of the idler laser. As with the spatial image, the waveform of the idler pulse contains the information of the temporal phase when the temporal imaging is defocused. As shown in Eq. (3), the real image or a virtual image can be generated when the dispersion in the temporal imaging system fulfills the condition $2 / ϕ_{p} < 1 / ϕ_{1}$ or $2 / ϕ_{p} > 1 / ϕ_{1}$ separately. In order to recover the temporal phase, temporal imaging waveforms should be measured at the image plane and far-field. Therefore, dispersion of the temporal imaging system should make the real image plane around the minimum dispersion of ϕ₂. For example, when the dispersion ϕ₂

is between $Φ_{m i n}$ and $Φ_{m a x}$ , the dispersion for signal and pump laser should be $1 / ϕ_{1} - 2 / ϕ_{p} \approx Φ_{m i n}$ and $Φ_{m a x} ≫ ϕ_{1}$ .

Once the defocused images are measured around the real image plane, the complex amplitude of the signal pulses can be recovered by the TAGS algorithm and temporal ptychography [34, 37]. One advantage of the TAGS algorithm is that the laser under test and the timing error can be recovered simultaneously. Another advantage is the single-shot measurement ability. It is easy to implement a cascaded split structure or pulse reproduction structure to realize single-shot measurement of the multiple waveforms with different dispersion. Nevertheless, the TAGS algorithm has one drawback for application in a temporal imaging system, the transmission process or function must be accurate. In the complex temporal imaging system, there exists one or more time lenses that are used to temporally magnify the laser, but the time lens is usually generated by active phase modulation or a nonlinear phenomenon such as four-wave mixing effect, sum frequency, or differential frequency so that the time lens is not in perfect parabolic temporal phase. In order to obtain an accurate temporal phase retrieval measurement, temporal ptychography can be implemented as a calibration method for an ultrafast full field oscilloscope based on TAGS.

3. Numerical Simulation Analysis

3.1. Analysis of TAGS

For the TAGS method, the measurement structure is the same as in Fig. 1, where the idler laser is transmitted through several different dispersions to obtain the dispersed waveforms. The pump laser is transform-limited and much broader than the signal laser, therefore the dispersed pump laser in the four-wave mixing process is approximated as a parabolic temporal phase function, which can be written as $exp (- j t^{2} / 2 ϕ_{p})$ . ϕ_p is the dispersion for the pump laser and can be written as $ϕ_{p} = β_{2} L$ , where β₂ is the second-order dispersion within the fiber and L is the fiber length. A signal laser with a pulse width (FWHM) of 3 ps and a transform-limited pump laser are transmitted through a 250 m and 600 m fiber, respectively. The pump laser is obtained by spectrally filtering a mode-locked laser with a supergaussian function. The spectral width of pump laser is 3.3 nm. Therefore the dispersed pump laser after the pre-dispersion ϕ_p has a square pulseshape, which means that the effect of pump laser can be approximated as a parabolic phase for simplicity during the FWM process. Within the nonlinear fiber, the FWM process is numerically calculated by four-wave coupling equations. The pre-dispersion of signal and pump laser are chosen to make the temporal imaging system defocussing. According to the temporal imaging condition, the temporal focusing spot is the position of post-dispersion ϕ₂ of a 1500 m fiber. In the simulation, the temporally imaged waveforms are measured with dispersion of 2,4,6, and 8 km.

Fig. 2 Retrieval results of temporal waveform and phase. (a), (c), (e): Temporal profile for nonlinear phase of 0.2π, 2π and 4π; (b), (d), (f): Temporal phase for nonlinear phase of 0.2π, 2π and 4π (Black line: original profile and phase; Pink line: recovered profile and phase).

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First, the performance of the TAGS algorithm was verified for a signal laser with different temporal phases. Here we chose a typical nonlinear phenomenon, SPM, as an example to show the validity of the retrieval performance. In the simulation, the temporal nonlinear phase of signal laser is accumulated by SPM effect; then the dispersed waveforms of signal laser are used to recover the temporal phase. As shown in Fig. 2, the signal lasers with $0.2 π$ , $2 π$ , and $4 π$ rad nonlinear phase are recovered by the AGS algorithm. Here the nonlinear phase is the same as the B integral, which is often used in high-power laser systems. Figure 2 (a),(c),(e) are the retrieved intensity, and Fig. 2 (b),(d),(f) are the retrieved temporal phases. It can be seen that the retrieval performance for a small temporal nonlinear phase is better than for a large nonlinear phase. When the nonlinear phase is 4π, the retrieved waveform shows a relatively large error. However, for the retrieval of a temporal phase, the performance of a large nonlinear phase is better than for a small nonlinear phase. The reason for this difference is that the pulse shape is mainly influenced by the temporal phase; in other words, the dispersed pulse shape contains more information about the temporal phase, which makes the retrieval performance of a nonlinear phase better when it is large enough.

Fig. 3 Influence of temporal imaged waveform number on the retrieval results: (a) Temporal intensity; (b) Temporal phase; (c) Iteration error.(Black line, actual value; Blue square: N = 2; Red square: N = 4)

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For the normal GS method, only two dispersed waveforms are enough to recover the results. However, more dispersed waveforms means more constraints for the iteration algorithm, so the convergence will be faster and the accuracy will be higher. As shown in Fig. 3, the retrieval results for different number of dispersed waveforms are compared. The retrieved temporal profile and phase with only two dispersed waveforms show large error even though there is a unique solution for the retrieval theoretically. When the number of dispersed waveforms increases to four, as shown in Fig. 3, the retrieval performance is improved significantly. From the error of iteration in Fig. 3 (c), the iteration error for N = 4 is much lower than for the other. However, the increasing number of dispersed waveforms also induces more measurement error in the experiment because of the limited resolution of the measurement equipment. From our experience, four dispersed waveforms are enough for the AGS algorithm to obtain an accurate result, compared with two waveforms.

Fig. 4 Retrieval results with time shifting error: (a)Temporal intensity; (b) Temporal phase;(Black line, actual waveform; Pink square: without timing correction; Red square: with timing correction); (c) Iteration error (Black: with timing correction; Red: without timing correction).

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For the above simulation, the measurement timing error was ignored for simplicity. Normally the measurement timing error is related to the temporal resolution and timing jitter of the measurement equipment. For example, when the temporal resolution of the measurement equipment is 10 ps, the measurement timing error will randomly fluctuate around 10ps. Even when the timing error is very small, it still greatly affects the retrieval performance of the iteration algorithm.

As shown in Fig. 4, the retrieval result with timing error of 1, -3, -2, and 5 ps has obvious errors using the traditional modified GS algorithm. However, the retrieval result with TAGS shows significant improvement both in the temporal profile and phase. It should also be noted that the annealing process in the AGS algorithm should be started when the GS algorithm is converged. In the simulation, the annealing process begins after 100 iterations and the iteration error is steeply decreased after the annealing process starts. From the results of Fig. 4, it can be concluded that the TAGS algorithm can recover the signal and the timing error simultaneously, which is advantageous in real applications.

Fig. 5 Schematic of the temporal ptychography algorithm

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3.2. Analysis of temporal ptychography

The process of the temporal ptychography algorithm is shown in Fig. 5, where $u_{i n i} (t)$ and $ψ_{i n i} (t)$ are the initial guess for the initial complex amplitude of signal and pump. The signal pulse is temporally shifted $Δ t_{m}$ relative to the pump pulse, so the output of the interaction $v_{m} (t)$ is $u (t) ψ (t + Δ t_{m})$ for temporal shifting number m. Then the exit field $v_{m} (t)$ is transmitted through a dispersion of ϕ to get $w_{m} (t)$ , which can be updatedby the measured waveform $I_{m} (t)$ to get updated complex amplitude $w_{m}^{r} (t)$ as in Eq. (4). The updated complex amplitude $w_{m}^{r} (t)$ is inversely transmitted through a dispersion of ϕ and the obtained refreshed exit complex amplitude $v_{m}^{r} (t)$ is used to get updated signal $u^{r} (t)$ and pump $ϕ^{r} (t)$ using Eq. (5).

w_{m}^{r} (t) = \sqrt{I_{m} (t)} \frac{w_{m} (t)}{| w_{m} (t) |}

\begin{array}{l} u^{r} (t) = u (t) + \frac{ϕ^{*} (t + Δ t_{m})}{| ϕ (t + Δ t_{m}) |^{2}} (v_{m}^{r} (t) - v_{m} (t)) \\ ϕ^{r} (t + Δ t_{m}) = ϕ (t + Δ t_{m}) + \frac{u^{*} (t)}{| u (t) |^{2}} (v_{m}^{r} (t) - v_{m} (t)) \end{array}

For the temporal ptychography, the signal laser was temporally shifted with respect to the pump laser. The nonlinear phase of the signal laser is accumulated through the SPM effect, which is the same as with the TAGS algorithm. The signal laser and the pump laser were the same as for the analysis of the TAGS algorithm. First, the temporal shifting step $Δ t$ was 5 ps and the number of scanning steps was 21, which means that the scanning range was 105 ps. The dispersion for signal and pump laser were 250 m and 500 m. The dispersion after the time lens was 8000 m.

Fig. 6 Retrieval results with the temporal ptychography algorithm: (a) Dispersed waveforms for different time delay; (b) Spectrum for different time delay;(c),(d) Retrieved signal with 2π nonlinear phase and pump laser; (e),(f) Retrieved signal with 4π nonlinear phase and pump laser; (g),(h) Retrieved signal with 6π nonlinear phase and pump laser;(Black line: actual waveform; Green square: retrieved waveform; Red line: actual phase; Red square: retrieved phase.

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The dispersed waveforms and spectrum of an idler laser for different time delays are shown in Fig. 6 (a) and (b). It can be seen that the spectrum is spectrally shifted due to the time delay, and that the waveforms are also temporally shifted due to the GDD for different central wavelengths. Using the dispersed waveforms and the temporal ptychography mentioned above, the signal laser and pump laser are successfully retrieved as shown in Fig. 6 (c,d,e,f,g,h). The retrieval results for the pump laser show good performance even though the nonlinear phases of the signal laser are $2 π$ , $4 π$ , and $6 π$ . However, the retrieved waveform of the signal laser with large nonlinear phase shows large error due to the difficulty of recovering the temporal profile, while the retrieved temporal phase still shows good performance.

Fig. 7 Influence of time step on the retrieval error.

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From Fig. 6, it can be seen that the temporal ptychography shows good performance for the simultaneous retrieval of signal and pump laser information. Another key factor that influences the retrieval result is the temporal step in the scanning process. It is known that the temporal step should be around $1 / 3$ or $1 / 2$ of the signal pulse range. As shown in Fig. 7, the iteration error for different temporal steps indicates that the temporal steps of 5 ps and 7 ps are better for the retrieval results. Because the signal laser was a 3 ps (FWHM) Gaussian laser pulse, the overlap for temporal step of 10 ps was too large for retrieval. However, for the small temporal step, the nearby dispersed waveforms with different time delays have no obvious difference, so the retrieval performance is also imperfect. In our experience, the temporal step should be around 1-2 pulse duration of the signal laser.

In the simulation, the iterative loop can be random or fixed. We found that the random iteration provides better retrieval results because the random iteration has the ability to resist error of the dispersed waveforms and of the local optimum solution. From simulation with the temporal ptychography algorithm, it can be seen that the algorithm can accurately retrieve the signal and pump laser information simultaneously, which is a significant advantage in real applications.

Fig. 8 Experimental structure of temporal imaging system: (a) Experimental Structure; (b) Idler laser generated in the FWM process; (c) Temporal imaged waveforms after dispersion. (BPF: bandpass filter; FA: fiber amplifier; TDL: time delay line; HNLF: high nonlinear fiber; LPF: long pass filter)

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4. Results of Ultrafast Oscilloscope

The ultrafast oscilloscope is built based on the four wave mixing effect, which is shown in Fig. 8. A dissipative soliton (DS) mode-locked fiber laser (MLL) at 1053 nm was used as the laser source. The pulse from the MLL laser was a positive chirped laser with pulse duration of 10 ps and spectral width of 10 nm. The mode-locked laser had a repetition rate of 17 MHz. Then the mode-locked laser was split into two equal parts with an optical fiber coupler (OC) as shown in Fig. 8 (a). These two parts were filtered by two bandpass filters to generate the signal and pump laser. The central wavelength and spectral width (3 dB) of BPF $_{S}$ were 1047 nm and 1 nm, and those of BPF $_{P}$ were 1053 nm and 4 nm. A fiber amplifier (FA $_{1}$ ) was used to amplify the signal pulse and make the signal pulse accumulate nonlinear phase induced by SPM (self-phase modulation). Considering that the high peak power of the signal laser would induce extra nonlinear phase in the later transmission in the temporal imaging system, the fiber attenuator was used to decrease the peak power of the signal pulse [38, 39]. L $_{S}$ and L $_{P}$ were used to provide dispersion for the signal and pump pulses, which were 260 m and 562 m. A fiber time delay line (TDL) was used to synchronize the signal pulse and pump pulse, and to adjust their relative time delay. Before reaching the 15 m-high nonlinear fiber (HNLF, SC-5.0-1040), which was used to generate a time lens based on the four-wave mixing effect, the signal and pump pulses, respectively were amplified to obtain a high nonlinear efficiency.

During the HNLF, the idler laser was generated through the FWM effect, as shown in Fig. 8 (b). The central wavelength of the idler laser was 1058 nm, which can be adjusted by tuning the time delay between the signal and pump laser. After the HNLF, the idler laser was filtered out by a long pass filter (LPF) with a cutting wavelength of 1055 nm. Then the idler laser was transmitted through different lengths of fiber. As shown in Fig. 8 (c), the temporally imaged waveforms have different pulse shape and duration with dispersions. In the experiment, all the fiber was 1060XP with a second order dispersion of 22.94 $p s^{2} / k m$ . Considering that the duration of temporal imaged waveforms were all hundreds of picoseconds, a high bandwidth (45 GHz) photoelectric detector (Newport, 1014) and a high speed (30 GHz) oscilloscope (Agilent, DSO93004L) were used to acquire the waveforms. The calculated temporal magnification factor was 7.7 for the dispersion of a 2 km fiber, which means that this ultrafast oscilloscope had a bandwidth of 230 GHz, at least. The bandwidth can be further increased if fiber of longer length is used.

Fig. 9 Retrieved results of signal laser with TAGS algorithm: (a) Retrieved temporal profile; (b) Retrieved temporal phase; (c) Iteration error

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4.1. Temporal imaging with TAGS algorithm

Based on the experimental structure in Fig. 8, the signal laser was first retrieved using the TAGS method. During the experiment, the nonlinear phase was accumulated when the signal pulse was transmitted in the fiber amplifier. The temporally imaged pulse was temporally magnified by dispersion fibers with different lengths, including 2, 4, 6, 8, and 10 km. Using the measured temporally imaged waveforms, the signal pulse could be retrieved by the TAGS algorithm, which is shown in Fig. 9. As shown in Fig. 9 (a) and (b), it can be seen that the signal pulse has a pulse duration of 5 ps and the peak-to-valley nonlinear phase is almost 1.5π rad.

The error evolution during the iteration is shown in Fig. 9 (c). As shown in the Fig. 9 (c), the total iterations of TAGS algorithm is 1000 and the annealing process was started when the iteration exceeded 200. When the annealing process started, the error dropped quickly, which indicates the efficiency of the time error correcting. The iterative order was totally random, which is more effective than fixed order for overcoming the local optimum solution. Therefore, itcan be seen that the error fluctuates randomly, but still decreases gradually. In order to eliminate the influence of noise, a numerical temporal window filter was used to restrict the retrieval range in the calculation. This made the retrieved temporal phase in Fig. 9 (b) zero outside the window. In order to obtain a better and faster iteration, the initial guess for the signal pulse is a Gaussian pulse with 10 ps pulse duration before the time lens. It was found that an accurate initial guess is unnecessary and does not affect the final result much. However, the pulse-shape initial guess still has good influence on the iteration, compared with total random and noise-like guesses.

Fig. 10 Retrieved spectrum and waveforms of signal pulse: (a) Retrieved spectrum (red) and measured spectrum (black); (b) Retrieved waveforms (square) and measured waveforms

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In order to investigate the performance of the retrieved result, the recovered spectrum and temporally imaged waveforms were compared with the actually measured results, which is shown in Fig. 10. The spectrum was measured with a high resolution spectrometer (YOKOGAWA, AQ6370B). Compared with the measured spectrum, the recovered spectrum has the same spectral width but the spectral shape shows a large error. This means that the information about the fine structure was lost during the retrieval process. The recovered temporally imaged pulses match well with the actually measured waveforms in Fig. 10.

Fig. 11 Retrieved results for different nonlinear phases: (a, b) Retrieved waveform, phase, and spectrum for π nonlinear phase;(c, d) Retrieved waveform, phase, and spectrum for 2π nonlinear phase; (e, f) Retrieved waveform, phase, and spectrum for 3π nonlinear phase; (Black line: measured spectrum; Red square: retrieved spectrum)

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Furthermore, the retrieved results for signal pulses with different nonlinear phases are compared in Fig. 11. It can be seen that the result for low nonlinear phase is obviously better than the others. It also seems that the retrieved spectrum is clipped for high nonlinear phase. Considering the assumption in the theoretical analysis, the pump pulse is regarded as a pulse with uniform amplitude and a perfect parabolic temporal phase. However, in the experiment, the pump pulse came from the dispersed and filtered mode-locked laser pulse, which means that the pump pulse has a unique waveform and that the time lens deviates from the parabolic temporal phase.

Fig. 12 Measurement results of temporal ptychography: (a) Experimental measured waveforms with different relative time delay; (b) Retrieved temporal profile and phase of signal laser after pre-dispersion; (c) Retrieved temporal profile and phase of signal laser before pre-dispersion; (d) Retrieved temporal profile and phase of the pump laser after pre-dispersion; Retrieved waveform.

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4.2. Temporal imaging with temporal ptychography

As shown above, the uncertainty of the pump pulse induces large error for the TAGS algorithm. Therefore, another method is required to retrieve the pump pulse as a calibration technique. From the theoretical analysis and simulation, the temporal ptychography method was found suitable for this task considering that it provides accuracy, high data redundancy, and can retrieve the signal and pump pulse simultaneously with a temporal scanning process.

For the temporal ptychography, the experimental structure was the same as for the TAGS method shown in Fig. 8. A fiber delay line (TDL) was used to change the delay between the signal and pump pulse and for each delay, the generated idler pulse was transmitted into a length of fiber to get the dispersed waveform. After measuring the dispersed waveform with different time delay, the signal and pump pulse could be recovered using temporal ptychography, which is shown in Fig. 5.

The measured waveforms for different time delays are shown in Fig. 12 (a). In the experiment, it was easy to change the relative time delay using the fiber TDL due to the all-fibered temporal imaging structure. It can be seen that with different time delays, the temporal imaged waveform is shifted along time and that the temporal profile varies simultaneously. This is reasonable considering the adding to the temporal phase of the signal pulse by the pump pulse. The variable temporal imaging waveform is induced by temporal aberration, which is similar with the spatial aberration of a lens, such as off-axis aberration. Theoretically, the signal pulse should scan for the whole duration of a pump pulse. However, only half of the pump pulse was scanned in the experiment because the idler pulse could not be accurately measured without distortion of residual pump pulse. This is due to insufficient extinction rate of the transmission of the LPF $_{1}$ . In the experiment, the idler pulse was separated from the residual pump pulse by the GDD of the fiber. Half of the idler pulse could be totally separated and accurately measured, while the other part was still mixed with the pump pulse. Therefore, the measured idler pulses after dispersion with different scanning time delays were unsymmetrical, as shown in Fig. 12 (a). For the same reason, the retrieved pump pulse after pre-dispersion was successfully retrieved within the time window, as shown in Fig. 12 (d). The pump pulse outside the time window randomly fluctuated and was meaningless. The recovered temporal profile and phase of the signal pulse before and after the pre-dispersion are shown in Fig. 12 (b) and (c), respectively. It can be seen that the recovered signal pulse has a duration of 3.5 ps and that the nonlinear phase is approximately π. Because temporal ptychography is an iterative method and a number of waveforms are necessary to recover the results, the computation time was around 10 min with a dual core laptop. This could be optimized using a high performance computer.

Fig. 13 Spectrum retrieved by temporal ptychography: (a) Retrieved temporal profile and phase of signal pulse; (b) Retrieved spectrum (red) and measured spectrum (black) of pump pulse;(c) Error evolution during iteration.

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In order to verify the accuracy of the retrieval results, the retrieved spectra of the signal and pump pulses was compared with the experimentally measured spectrum, as shown in Fig. 13 (a) and (b). It can be seen that the spectrum of the signal pulse matches well with the actually measured spectrum, while the spectrum of the pump pulse shows large error. As mentioned above, only part of the pump pulse is scanned by the signal pulse and the pump pulse outside the time window is noise. Therefore, the retrieved spectrum is the long wavelength part, shown in Fig. 13 (b) and the short wavelength part is noise. Figure 13 (c) is the evolution error of the iteration process by temporal ptychography, and even when the timing error correction starts after 50 iterations, which is similar to that with the TAGS algorithm, the iteration error is still very stable. The comparison of the retrieved spectrum and measured spectrum indicates that temporal ptychography is a good way to retrieve the signal and pump pulse information in a temporal imaging system. However, due to the necessary scanning process, temporal ptychography could not realize single-shot measurement, which restricts the application for high speed transient measurement of ultrafast signals.

4.3. Pre-calibrated temporal phase measurement

As mentioned above, the TAGS algorithm is suitable for single-shot measurement, but an accurate pump pulse is necessary for successful retrieval results. Temporal ptychography can recover the signal and pump pulse simultaneously and the retrieval results are more accurate than with the TAGS algorithm. However, the temporal ptychography method requires a time scanning process, which restricts its application for single-shot measurement. Based on the advantages and disadvantages of the two methods, we propose using the temporal ptychography method to calibrate the pump pulse for the TAGS algorithm.

The recovered pump pulse that occurs after pre-dispersion in Fig. 12 (d) was used as the real pump pulse and then it is numerically interpolated to match the TAGS method. It should be mentioned that the time delay between signal and pump pulse is very important for successful retrieval. Here, the time delay was calibrated first in the temporal ptychography measurement and then the time delay was kept the same for the TAGS method.

The results of the temporal ptychography method used in the above section was implemented as the pump pulse for the TAGS algorithm. The condition of the signal pulse was the same as in Fig. 9. Then the signal pulse was retrieved with the calibrated pump pulse and the results are shown in Fig. 14. It should be noted that the relative time delay between the signal and pump pulse is important for the retrieval results, so that we use the same time delay (TDL) for the temporal ptychography.

Fig. 14 Results retrieved by the TAGS method with pre-calibration through temporal ptychography: (a) Retrieved spectrum (red) and measured spectrum (black) of signal pulse; (b) Retrieved temporal profile and phase of signal pulse; (c) Recovered dispersed waveforms (solid line) and measured waveforms (square).

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Compared with the retrieved spectrum in Fig. 10 (a), the recovered spectrum of a signal pulse has almost same shape as the measured spectrum except that there exists error in the relative intensity, which is influenced by the noise in the measured waveforms. The recovered signal pulse has a duration of 2.3 ps, which is coincident with the autocorrelation of the signal pulse. For the experimental results, the computation time is around 3 min with a dual core laptop, which is shorter than for temporal ptychography.

The retrieval result of the pre-calibrated TAGS algorithm shows better measurement results than with the normal TAGS algorithm. The results indicate that combining the two methods, TAGS and temporal ptychography, makes them suitable for phase imaging in a temporal imaging system.

5. Conclusions

For an ultrafast oscilloscope, two methods, TAGS and temporal ptychography, were used to realize full-field measurement. The TAGS and temporal ptychography methods were numerically analyzed in the simulation and the results show that the two methods can both provide accurate retrieval results. The ultrafast oscilloscope based on the FWM effect in an all-fiber structure was experimentally investigated to verify the performance of the full-field measurement. In order to achieve single-shot measurement and to improve the performance of the TAGS algorithm, the temporal ptychography method was implemented to pre-calibrate the pump pulse in the ultrafast oscilloscope for the TAGS algorithm. The calibrated TAGS method showed significant improvement for the retrieval of signal pulse information. The results from simulation and experiment indicate that a full-field ultrafast oscilloscope with at least 230 GHz bandwidth is suitable for the measurement of ultra-speed transient and dynamic processes. This result promises further application in the measurement of complex ultra-speed transient processes and will promote the development of temporal phase imaging methods and applications.

Funding

National Natural Science Foundation of China (NSFC) 11604350.

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Full-field ultrafast oscilloscope based on temporal imaging

Abstract

1. Introduction

2. Temporal imaging system structure

3. Numerical Simulation Analysis

3.1. Analysis of TAGS

3.2. Analysis of temporal ptychography

4. Results of Ultrafast Oscilloscope

4.1. Temporal imaging with TAGS algorithm

4.2. Temporal imaging with temporal ptychography

4.3. Pre-calibrated temporal phase measurement

5. Conclusions

Funding

References

Cited By

Figures (14)

Equations (5)

Optics Express