Self-referenced spectral interferometry for ultra-short laser pulse characterization
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摘要:
对于超短激光脉冲，精确表征其时域特性（脉宽和相位）对其产生和应用都至关重要。自参考光谱干涉法，于2010年首次提出，该方法利用待测光自身通过三阶非线性光学过程产生测量所需的合适参考光，并借助傅里叶变换光谱干涉算法解析重建光谱相位，具有单发、准确、灵敏度高等优势。本文对该方法的实现光路和重建算法两条主线进行了综述。在光路层面，从最早提出的基于交叉偏振波产生的自参考光谱干涉法（XPW-SRSI）,到基于瞬态光栅效应的自参考光谱干涉法（TG-SRSI）全反射紧凑构型，对实现光路的灵敏度、波段覆盖与紧凑性三个维度进行了不断迭代升级。在算法层面，从针对于近傅里叶转换极限的脉冲重建方法，到针对时域展宽超过傅里叶变换极限两倍的大啁啾脉冲重建算法，进行了三条演进路线论述：光谱拼接方案，引入监督深度学习的重建方案，以及嵌入物理正向模型的无需训练神经网络；后者在无需任何预训练数据集的条件下，实现了大啁啾与高噪声场景下的单次准确重建。面对不断涌现的新型光束表征需求，超短激光脉冲的时域测量在未来仍需要备受重视。
Abstract:For ultrashort laser pulses, accurately characterizing their temporal characteristics (temporal width and phase) is crucial for their generation and application. Self-referenced spectral interferometry (SRSI), first proposed in 2010, utilizes the measured light itself to generate suitable reference light through third-order nonlinear optical processes, and employs Fourier transform spectral interferometry algorithms to reconstruct the input pulse. It has the advantages of single-shot, accuracy, and high sensitivity. This article provides an overview of the implementation of SRSI from two main aspects: the optical path and the reconstruction algorithm. On the optical path level, from the earliest proposed self-referenced spectral interferometry based on cross-polarization wave generation (XPW-SRSI) to the self-referenced spectral interferometry based on transient grating effect (TG-SRSI) with a compact total reflection configuration, the sensitivity, wavelength coverage, and compactness of the implementation path have been continuously iteratively upgraded. On the algorithm level, from pulse reconstruction methods targeting near Fourier transform limits to reconstruction algorithms for large chirp pulses with temporal broadening exceeding twice the Fourier transform limit, three evolutionary paths are discussed: spectral stitching schemes, reconstruction schemes incorporating supervised deep learning, and neural networks without training embedded in physical forward models. The latter achieves single-shot accurate reconstruction in large chirp and high noise scenarios without any pre-training dataset. In the face of emerging new beam characterization demands, the temporal measurement of ultrashort laser pulses still require significant attention in the future.
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Key words:
- ultrashort laser /
- pulse measurement /
- deep learning
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图 1 传统光谱干涉法(SI)与自参考光谱干涉法(SRSI)的光路对比示意图。(a)传统SI的光路设置。一束作为参考脉冲（需由外部独立光源提供），另一束为待测脉冲,经可调延迟线后与参考脉冲合束，进入光谱仪产生干涉条纹。(b)SRSI的光路设置。待测脉冲的一小部分通过三阶非线性过程产生自参考脉冲作为参考脉冲；其余部分经延迟后与参考脉冲合束进入光谱仪，形成携带相位信息的频谱干涉条纹。
Figure 1. Schematic comparison of traditional spectral interferometry (SI) and self-referenced spectral interferometry (SRSI). (a) Optical setup of conventional SI. One beam serves as the reference pulse (provided by an external independent source), and the other is the unknown pulse. After passing through an adjustable delay line, it is combined with the reference pulse and directed into a spectrometer to generate interference fringes. (b) Optical setup of SRSI. A small portion of the unknown pulse generates a self-referenced pulse via a third-order nonlinear process, which acts as the reference pulse. The remaining part is delayed, combined with the reference pulse, and sent into the spectrometer, forming spectral interference fringes that carry phase information.
图 2 (a)交叉偏振波产生原理，其中使用一个偏振片来分离产生的交叉偏振波。(b)自衍射过程，图中
$ {k}_{sd1} $ 和$ {k}_{sd2} $ 分别表示一阶和二阶自衍射信号。(c)瞬态光栅过程，图中$ {K}_{TG} $ 为产生的瞬态光栅信号。Figure 2. Principles of (a) cross-polarized wave generation (XPW), where a polarizer is used to separate the generated XPW. (b) Self-diffraction (SD) process, where
$ {k}_{sd1} $ and$ {k}_{sd2} $ are the first- and second-order SD signals, respectively. (c) Transient-grating (TG) process, where$ {K}_{TG} $ is the generated TG signal.
图 3 XPW-SRSI原理图。CP为方解石平板，用于产生一个偏振方向与入射脉冲垂直的延迟待测脉冲；P为偏振片。
Figure 3. Principle of XPW-SRSI. CP is a calcite plate used to generate a delayed test pulse with polarization perpendicular to that of the incident pulse, and P is a polarizer.
图 4 傅里叶变换光谱干涉法从条纹中提取幅度与相位的流程。
Figure 4. Procedure for extracting amplitude and phase in fourier transform spectral interferometry.
图 5 SRSI中的迭代相位检索算法流程图。该算法通过反复迭代更新待测脉冲的频谱相位，直至收敛至稳定解。初始化相位设为通过傅里叶变换光谱干涉法测得的相位差
$ {\varphi }_{\text{diff}}\left(\omega \right) $ ，随后在每次迭代中利用非线性过程（如交叉偏振波产生）从当前待测脉冲$ {E}_{u}\left(t\right) $ 构造参考脉冲$ {E}_{r}\left(t\right) $ ，并提取其频谱相位$ {\varphi }_{r}\left(\omega \right) $ 。更新后的相位$ {\varphi }_{i+1}\left(\omega \right) $ 为$ {\varphi }_{\text{diff}}\left(\omega \right)+{\varphi }_{r}\left(\omega \right) $ ，用于下一轮迭代。Figure 5. Flowchart of the iterative phase retrieval algorithm in SRSI. The algorithm iteratively updates the spectral phase of the unknown pulse until convergence. The initial phase is set to the phase difference
$ {\varphi }_{\text{diff}}\left(\omega \right) $ obtained from Fourier transform spectral interferometry. In each iteration, a reference pulse$ {E}_{r}\left(t\right) $ is generated from the current unknown pulse$ {E}_{u}\left(t\right) $ via a nonlinear process (e.g., cross-polarized wave generation), and its spectral phase$ {\varphi }_{r}\left(\omega \right) $ is extracted. The updated phase$ {\varphi }_{i+1}\left(\omega \right) $ is$ {\varphi }_{\text{diff}}\left(\omega \right)+{\varphi }_{r}\left(\omega \right) $ for the next iteration.
图 6 用于 800 nm 脉冲测量的SD-SRSI实验装置图。VND，可变中性密度滤光片；BS1、BS2、BS3，厚度为 1 mm 的50/50分束镜；SM1，曲率半径R=−700 mm的球面反射镜；FS，厚度为0.1 mm的熔融石英玻璃片；ND，中性密度滤光片；SM2，曲率半R=−
1000 mm的球面反射镜；EPP-2000-HR，高分辨率光谱仪（分辨率0.15 nm）[14]。Figure 6. Experimental setup of SD-SRSI for pulse at 800 nm. VND, variable-neutral density filter; BS1, BS2, BS3, 50/50 1 mm thick beam splitters; SM1, spherical mirror with a radius of curvature R = −700 mm; FS, 0.1 mm thick fused silica glass plate; ND, neutral density filter; SM2, spherical mirror with a radius of curvature R = −
1000 mm; EPP-2000-HR, high-resolution spectrometer (0.15 nm resolution)[14].
图 7 (a)瞬态光栅自参考光谱干涉法原理图。(b)实验装置图；H1，黑板；P1、P2，熔融石英片；M1，四分之三区域镀银的平面反射镜；PM，抛物面反射镜；H2，光阑；L，透镜；M2，平面反射镜[15]。
Figure 7. (a) Schematic of TG-SRSI. (b) Experimental setup; H1, black plate; P1, P2, fused silica plate; M1, plane mirror with three quarters parts silver-coated; PM, parabolic reflector; H2, iris; L, lens; M2, plane mirror[15].
图 8 (a)当待测脉冲中分别引入0 fs2（居中，黑色）、+500 fs2（上方，品红色）和-500 fs2（下方，蓝色）的二阶色散时，测量得到的TG信号光谱（绿色点线）、待测脉冲光谱（红色点划线）、反演得到的待测脉冲光谱（黑色实线），以及由 TG-SRSI（短点线）、WIZZLER（虚线）和理论计算（点划线）得到的频谱相位。(b)三种不同条件下反演得到的待测脉冲时域波形：0 fs2（粗细黑色实线）、+500 fs2（品红色点划线）、-500 fs2（蓝色短点线）。红色点线和黑色点划线分别表示待测脉冲的傅里叶变换极限脉冲时域波形和TG信号的时域波形。(c)测量得到的TG信号光谱（蓝色点划线）、待测脉冲光谱（黑色实线）以及反演得到的待测脉冲光谱（红色短点线）。分别是有（黑色短虚线）和无（品红色双点划线）置于布儒斯特角位置的1 mm厚熔融石英片时，由TG-SRSI反演得到的频谱相位。青色虚线为无玻璃片时的理论计算频谱相位。(d) 有（粗细黑色实线）和无（品红色双点划线）1 mm厚玻璃片时反演得到的待测脉冲时域波形，其脉宽分别为10.6 fs和15.1 fs。蓝色虚线和红色短点线分别表示待测脉冲和TG信号的傅里叶变换极限脉冲时域波形，其脉宽分别为9.6 fs和6.8 fs[15]。
Figure 8. (a) Measured spectra of the TG signal (green dot) and the testing-pulse (red dash-dot-dot), the retrieved spectrum of the testing-pulse (solid black), the spectral phase obtained by TG-SRSI (short-dot), WIZZLER (dash) and the calculation (dash-dot) when introduced at 0 fs2 (center and black), +500 fs2 (up and magenta), -500 fs2 (down and blue) in the testing-pulse, respectively. (b) Retrieved temporal profile of the testing-pulse at 0 fs2 (thick and thin black solid), +500 fs2 (magenta dash-dot), -500 fs2 (blue short-dot); three different conditions. The red dot and the black dash-dot-dot lines are the FTL temporal profile of the testing-pulse and the TG signal, respectively. (c) Measured spectra of the TG signal (blue dash-dot), the testing-pulse (black solid), and the retrieved spectrum of testing-pulse (red short-dot). Retrieved spectral phase by TG-SRSI with (black short-dash) and without (magenta dash-dot-dot) the Brewster-angle located 1 mm thick fused silica plate. Cyan dashed line is the calculated spectral phase without the glass plate. (d) Retrieved temporal profile of the testing-pulse with (thick and thin black solid) and without (magenta dash-dot-dot) the 1 mm thick glass plate, which are 10.6 and 15.1 fs, respectively. Blue dashed and red short-dot lines are the FTL temporal profile of the testing-pulse and the TG signal, which are 9.6 and 6.8 fs, respectively[15].
图 9 (a)基于反射式显微物镜的瞬态光栅自参考光谱干涉法光路图。P1，镀铝熔融石英片；P2，非线性材料；A，黑色挡板；L，透镜；M，平面反射镜；S，光谱仪；a和b为箭头所指部分的截面图，c为黑板。所有图示均为从右侧观察的视图。(b)BOX-CARS 相位匹配几何示意图[22]。
Figure 9. (a) The optical setup of the RMO based TG-SRSI. P1, aluminum-coated, fused silica plate; P2, nonlinear material; A, black plate; L, lens; M, reflective plane mirror; S, spectrometer; a and b are cross sections of the portions indicated by the arrows and c is the black plate. All are seen from the right side. (b) BOX-CARS phase-matching geometry[22].
图 10 (a)由光谱仪直接测量得到的测试光束（红色曲线）、TG信号（蓝色曲线）及其干涉图（黑色曲线）的光谱强度。(b)使用TG-SRSI反演得到的光谱（黑色实线）和频谱相位（黑色点线）。使用SHG-FROG反演得到的光谱（蓝色实线）和频谱相位（蓝色点线）。红色曲线为光谱仪直接测得的测试光束光谱。(c)使用TG-SRSI（黑色曲线）和SHG-FROG（蓝色曲线）反演得到的时域波形（实线）和相位（虚线）。(d)光谱仪直接测得的三个TG信号光谱。自下而上的光谱分别对应输入脉冲能量为65 nJ、75 nJ 和85 nJ时的情况[22]。
Figure 10. (a) Spectral intensity of the test beam (red curve), the TG signal (blue curve), and the interference between them (black curve) measured directly with the spectrometer. (b) The spectrum retrieved (black solid curve) and the spectral phase (black dotted curve) by using TG-SRSI. The spectrum retrieved (blue solid curve) and the spectral phase retrieved (blue dotted curve) by using SHG-FROG. The red curve is the spectrum of the test beam measured directly by the spectrometer. (c) The temporal profiles (solid curves) and phases (dashed curves) retrieved by using TG-SRSI (black curves) and SHG-FROG (blue curves). (d) Spectra of three TG signals measured directly by the spectrometer. Spectra from the bottom to the top correspond to when the input pulse energies are 65, 75, and 85 nJ, respectively[22].
图 11 实验装置图。(a) M1，特制3/4镀膜反射镜；P1，0.5 mm厚熔融石英片；PM1、PM2，90°镀铝离轴抛物面反射镜，焦距f=25.4 mm；P2，0.15 mm厚YAG晶体片；A，光阑；L，透镜，焦距f=200 mm；M2，反射镜。插图a为M1的正面视图；插图b为光阑A的形状。(b)装置光路实物照片[24]。
Figure 11. Experimental setup. (a) M1, special 3/4 coated reflective mirror; P1, fused silica plate, 0.5 mm thick; PM1, PM2, 90° aluminum coating off-axis parabolic mirror, f = 25.4 mm; P2, YAG plate, 0.15 mm thick; A, aperture; L, lens, f = 200 mm; M2, reflective mirror. Inset a, the front side of M1; inset b, the shape of A. (b) Picture of the optical setup of the device[24].
图 12 1 kHz重复频率下飞秒脉冲的表征结果。(a)反演得到和直接测量的未知输入脉冲及TG参考脉冲的光谱，其中“R”表示“反演光谱”，“D”表示“直接测量光谱”。(b)反演得到的未知输入脉冲的时域波形（黑色实线）、傅里叶变换极限脉冲时域波形（红色点划线）及相位。(c)输入能量分别为5 nJ、4 nJ 和 3 nJ时自产生的 TG 参考脉冲的光谱强度。(d)不同输入脉冲能量下表征过程中的脉冲宽度波动[24]。
Figure 12. Characterization results of femtosecond pulses at 1 kHz repetition rates. (a) Retrieved and directly measured spectra of the unknown input pulse and TG reference, “R” for “retrieved spectra” and “D” for “directly measured spectra.” (b) Retrieved temporal profile (black solid line), the Fourier transform limit pulse temporal profile (red dashed-dotted line), and the phase of the unknown input pulse. (c) Spectra intensity of self-created TG reference pulses with input energy of 5 nJ, 4 nJ, and 3 nJ, respectively. (d) Pulse duration fluctuation in the characterization with different input pulse energies[24].
图 13 84 MHz重复频率下飞秒脉冲的表征结果。(a)反演得到和直接测量的未知输入脉冲及TG参考脉冲的光谱，其中“R”表示“反演光谱”，“D”表示“直接测量光谱”。(b)反演得到的未知输入脉冲的时域波形（黑色实线）、傅里叶变换极限脉冲时域波形（红色点划线）及相位。(c)对数坐标下反演得到的时域强度波形（黑色线）和傅里叶变换极限脉冲时域波形（红色线）。(d)不同输入脉冲能量下表征过程中的脉冲宽度波动曲线[24]。
Figure 13. Characterization results of femtosecond pulses at 84 MHz repetition rates. (a) Retrieved and directly measured spectra of the unknown input pulse and TG reference pulse, “R” for “retrieved spectra” and “D” for “directly measured spectra.” (b) Retrieved temporal profile (black solid line), Fourier transform limit pulse temporal profile (red dashed-dotted line), and phase of the unknown input pulse. (c) Retrieved temporal intensity profile (black line) and Fourier transform limit pulse temporal profile (red line) on a logarithmic scale. (d) Pulse duration fluctuation curves in the characterization with different input pulse energies[24].
图 14 实验装置示意图。FH1、FH2：四孔小孔光阑；M1：四分之三镀膜反射镜；M2：四分之三镀膜反射镜；SM：中心开孔平面反射镜；CM：凹面反射镜；KP：薄熔融石英片；L1、L2：透镜；S：单孔光阑；SP：光谱仪[25]。
Figure 14. Schematic of the experimental setup. FH1, FH2: Four-Hole Pinhole. M1: three-quarter coated mirror. M2: Three-quarter coated mirror. SM: Plane mirror with hole in the center. CM: Concave mirror. KP: Thin fused-silica plate. L1, L2: Lens. S: Single-hole aperture. SP: spectrometer[25].
图 15 SRSI算法的局限性及改进。(a)在大GDD情况下，原始SRSI算法能够提取相位的范围。(b)不同延迟下参考脉冲的光谱及其各自的相位差。(c)扣除拟合的一阶和零阶相位后的相位差。
Figure 15. Limitations and improvements of the SRSI algorithm. (a) Range over which the vanilla SRSI can extract phase in cases with large GDD. (b) Spectra of reference pulses with varied delays and their respective phase differences. (c) Phase difference after subtracting the fitted first and zeroth-order phases.
图 16 原始SRSI与改进型SRSI对具有高阶色散的大啁啾高斯脉冲的反演相位对比。(a)脉冲的模拟光谱轮廓。(b)由共轭脉冲在正延迟和零延迟下产生的参考脉冲的光谱相位曲线。所有光谱相位均已扣除一阶相位项，并在中心波长处设为零。(c)改进型SRSI反演得到的频谱相位。(d)SRSI反演得到的频谱相位[25]。
Figure 16. Comparison of the retrieved phase of vanilla SRSI and improved SRSI for large chirped Gaussian pulse with high order dispersion. (a) Simulated spectral profile the pulse. (b) Spectral profiles of the reference pulses generated by conjugate pulse with positive delay and zero delay. All spectral phases have subtracted the first-order phase term and set to zero at the center wavelength. (c) Retrieved spectral phase by improved SRSI. (d) Retrieved spectral phase by SRSI[25].
图 17 原始SRSI与FROG的测量结果对比。(a)测量得到的归一化光谱：干涉图、未知脉冲以及零延迟下产生的参考脉冲。(b)第28、29和30次迭代的频谱相位。(c)FROG测量和反演得到的光谱。原始SRSI和FROG反演得到的频谱相位。(d)原始SRSI和FROG反演得到的未知脉冲时域波形。(e，f)测量和反演得到的FROG轨迹图[25]。
Figure 17. Measurement results of vanilla SRSI and FROG. (a) Measured normalized spectra of interferogram, unknown pulse, and reference pulse generated with zero delay. (b) Spectral phases of 28th, 29th, and 30th iterations. (c) Measured and retrieved spectra by FROG. Spectral phases retrieved by vanilla SRSI and FROG. (d) Retrieved temporal profiles of unknown pulse by vanilla SRSI and FROG. (e, f) Measured and retrieved FROG traces[25].
图 18 改进型SRSI的测量结果。(a)测量得到的归一化光谱：干涉图、未知脉冲和参考脉冲。(b)从负延迟和正延迟干涉图中提取的频谱相位。(c)扣除一阶和零阶分量后的频谱相位。(d)第4次（红线）、第5次（红色虚线）和第6次（红色点线）迭代的频谱相位。(e)第4次与第5次迭代（红线）以及第5次与第6次迭代（红色虚线）的频谱相位差。(f)FROG测量和反演得到的光谱。改进型SRSI和FROG反演得到的频谱相位。(g)改进型SRSI和FROG反演得到的时域波形[25]。
Figure 18. Measurement results by improved SRSI. (a) Measured normalized spectra of interferograms, unknown pulse, and reference pulses. (b) Spectral phases extracted from interferograms with negative and positive delay (c) Spectral phases after subtracted the first-order and zero-order component. (d) Spectral phases of 4th (red line), 5th (red dashed line), and 6th (red dotted line) iterations. (e) Spectral phase differences of the 4th and 5th iterations (red line), and the 5th and 6th iterations (red dashed line). (f) Measured and retrieved spectra by FROG. Spectral phases retrieved by improved SRSI and FROG. (g) Retrieved temporal profiles by improved SRSI and FROG[25].
图 19 Dense-1D-U-Net神经网络。(a)神经网络整体结构；(b)稠密连接块结构
Figure 19. Dense-1D-U-Net neural network. (a) Overall network structure; (b) structure of dense block.
图 21 经典深度学习架构和无监督神经网络先验。(a)经典的监督学习包括独立的训练（在模拟数据上）和预测阶段。(b)提出的UNNP框架，将训练和预测整合为一步优化过程，直接使用测量的干涉图。
Figure 21. Architectures of classic deep learning and the Untrained Neural Network Prior. (a) Classic supervised learning involving distinct training (on
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