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Abstract
High speed surface defects detection of mirrors is of great significance, for detecting the quality of the mirrors on-site, and ultimately for monitoring the operating states of laser systems. The speeds of conventional proposals are relatively low as they utilize mechanically scanning methods or two-dimensional charge-coupled devices. Here, we propose a high speed surface detection method based on ultrafast single-pixel imaging, which consists of a spatial Fourier optical module for frequency-space mapping and a dispersive Fourier transform module for frequency-time mapping. An optical grating is utilized to map the wideband spectrum of dissipative soliton into the spatial domain under far-field diffraction, where the mirror is inspected. Dispersive Fourier transform is used to map the surface-defects-coded spectral information into the temporal domain, then recorded by a high speed single-pixel detector. The detection system permits continuous single-shot spectra measurement with a frame rate equivalent to the pulse repetition rate (8.4 MHz). We extract amplitude defects by demodulating light intensity, and obtain phase defects by demodulating the interference spectrum with a Mach-Zehnder interferometer structure. Experimental results show that the damaged mirror with a two-dimensional width of 10 × 13 mm can be obtained with a spatial resolution of 90 µm. The obtained phase accuracy after Hilbert transformation is 0.00217 rad, corresponding to a depth resolution of 51 nm. This scheme can find promising applications for surface defects detection of large aperture mirrors, and real-time monitoring of laser systems with high energy.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of information technology and space optics, precision optical elements, including mirrors, lens, prisms, and other elements with super-precision surface, are widely used in communication [1], laser [2,3], and other fields. Among them, as a key component in many optical systems, the quality of mirrors directly affects the performance of optical systems, such as microscopes [4–6], telescopes [7], high-energy laser systems, and so on [8,9]. Especially in high-energy laser systems, mirrors are widely used. While the optical performance of reflective optical elements limits the performance and service life of laser system to a great extent. For example, under the radiation from intense laser pulse, the optical mirrors will suffer high temperature and high pressure locally due to Joule heat absorption, resulting in thermal expansion [10]. The corresponding stress and deformation will cause defects and cracks in mirrors, thus affecting the downstream optical elements. In addition, pollutants caused by insufficient external cleanliness also have a great impact on the damage of reflective optical surfaces [11]. Surface scratches, surface contaminants and surface Talbot image effect may cause surface damage of optical elements. Obviously, damage of mirrors in a high-power laser line is a potentially catastrophic phenomenon, which not only affects its long-term stability, coating quality and surface shape accuracy, but also directly reduces the damage threshold of the optical system, and deteriorates its beam quality and service life [12]. Therefore, it is particularly important and urgent to monitor the performance of optical elements online, for diagnosing the operation state of high power laser system, and improving laser output efficiency and output quality [13,14].
Technically, the surface defects of mirrors can be divided into the amplitude defects and the phase defects [15]. The amplitude defects will cause energy scattering and affect the beam quality, while the phase defect causes energy converging into a small area and causing laser damage to mirrors or their film. In the past few decades, various amplitude defects measurement methods have been proposed to monitor the reflectivity of mirrors. For example, the cavity ring-down technique has been increasingly popular in implementation of accurate single point reflectance measurement [16]. Unfortunately, the realization of a large field of view can only be scanned mechanically, leading to limited spatial resolution and speed. Besides, the phase defects detection can be performed either with non-interference method or interference method. For non-interference methods, such as dark field method [17], the phase change is transformed into light intensity change to obtain the phase distribution in the measured part. However, the detection time is long. For interferometry methods, such as optical coherence tomography [18], they have high transverse resolution, but low longitudinal resolution, long scanning time and low detection efficiency. For some conventional defect detection systems, such as speckle interferometry or two wavelength interferometry, charge-coupled device and complementary metal-oxide-semiconductor are used to record the interference pattern, resulting in low speed. However, the damaging process of mirrors within high power laser system is a fast and non-repetitive event with a duration of microsecond or even to picosecond level. Therefore, the high frame rate requirement is beyond the capability of conventional measurement methods, which based on mechanical scanning or charge-coupled devices and complementary metal-oxide-semiconductors.
In this article, we propose a fast detection method of mirrors surface defects based on ultrafast single-pixel imaging. Over recent years, ultrafast single-pixel imaging brought us an high speed imaging processing method with high frame rate [19–22]. An ultrafast continuous imaging method, serial time-encoded imaging [23], attracts the attention of researchers with the advantage of high-speed data acquisition, which has been highly successful in fast transient dynamics observation [24], and applications that require fast continuous monitoring, including surface vibrometer [25] and surface defect detection [26]. In our demonstration, the detection system overcomes the limitations of low speed, and permits continuous single-shot measurements of rapidly evolving spectra at a frame rate equivalent to the pulse repetition rate of the laser (8.4MHz). Further, we add a reference channel to form a Mach-Zehnder interferometer (MZI) structure, so that the optical path change caused by thermal strain can be encoded in the phase information of the interference spectrum. As a proof-of-concept, we demonstrate an amplitude defects measurement scheme with a lateral and longitudinal resolution of 90 µm and 100 µm, respectively. As well as a phase defects measurement scheme with a resolution accuracy of 0.00217 rad (a depth resolution of 51 nm). This fast detection method of mirrors surface defects can also be extended to real-time measurement of large aperture optical elements, which enables monitor of operation state of high power laser system.
2. Main principle
In order to detect the surface defects with high speed, we follow the steps in Fig. 1(a). It consists of a coherent laser source, surface defects information mapping module, high speed detection and analysis. Primarily, the broad spectrum of the pulsed laser is mapped into special domain by the spatial Fourier optical module with frequency-space mapping, where the mirror under detection is located. Then, the spatial information is encoded so that each frequency component of the pulse corresponds to different spatial coordinates of the mirror, and the amplitude or phase variations are encoded into the spectrum. Next, the encoded spectrum of the pulse is time-stretched by large dispersion under the far-field approximation, that is, frequency-time mapping [27] is realized. Finally, the high speed optical signal is detected by optoeletronic system with an high bandwidth, and the imaged surface is reconstructed. In the following parts, we give a brief theoretical analysis of the two main mapping processes.
Fig. 1. (a) Processing of surface defects detection systems. (b) Measurement principle of mirrors surface defects based on far-field Fraunhofer diffraction. (c) Basic principle of time-stretch based on dispersion Fourier transform. PD: photodetector.
2.1 Frequency-space mapping
Optical linear invariant system can be described in either the time domain or the frequency domain [28]. In the time domain, h(t) is the impulse response function of the system. The output of the system y(t) is the convolution of the input function x(t) and h(t).
where, * denotes convolution. In the frequency domain, the frequency response H(ω) is the Fourier transform of the impulse response of the system, also known as the transfer function. It represents the effect of the system in the frequency domain, that is, it determines the change of each frequency component in the input spectrum when passing through the system. Similarly, the output spectrum Y(ω) is the convolution of the input spectrum X(ω) and H(ω). here, X(ω), Y(ω), and H(ω) are Fourier transform of x(t), y(t), h(t), respectively.The optical diagram of Fourier optical module is shown in Fig. 1(b). The frequency domain information Ein(ω,t) of ultrafast laser with wide spectral characteristics is mapped into the spatial domain through spatial disperser, in the case of far-field diffraction (Fraunhofer) [29].
where, Eout(ω,t) is the output of the Fourier optical system. On the reflecting surface, two-dimensional planes M(x,t)=A(x,t)exp(jφ(x,t)) with different spatial information are loaded in the frequency spectrum of the laser, including amplitude information A(x,t) and phase information exp(jφ(x,t)). Here, Ein(ω,t) and Eout(ω,t) are Fourier transform pairs. According to the expansion of Hermite–Gaussian modes for the mask function M(x,t) [30], the frequency response can be described as:In this diffraction system, the spatial resolution [23] can be described as
2.2 Frequency-time mapping
The mapping from frequency to time is schematically shown in Fig. 1(c). When an optical pulse propagates through a dispersive medium with large group velocity dispersion (GVD), and only considering the second-order dispersion, its frequency domain information can be described as [32,33]:
Supposing that the optical bandwidth of the pulse, and the temporal pulse width are Δλ, and Δτ, respectively, the frequency to time mapping relation can be expressed as [35]:
where, D represents the GVD coefficient in ps⋅km−1⋅nm−1. The conventional dispersive mediums include single-mode fibers, gratings, chirped fiber Bragg gratings, and dispersion compensation fibers (DCFs) with a sufficiently large and linear GVD. Here, we use a section of DCF as dispersion element to perform a one-to-one mapping between wavelength and time. Then, the pulse spectrum is converted into time domain, recorded by oscilloscope and further processed by a computer.3. Experimental setup
The experimental apparatus of the proposed ultrafast mirrors surface defects detection scheme is illustrated in Fig. 2. The optical source is a home-made dissipative soliton (DS) laser source with a pulse repetition rate of 8.4 MHz, and the average power is appointment 0.3 mW [36]. As shown in Fig. 3(a), it provides a pulse source with a 13 nm rectangle-shaped optical spectrum centered at 1561 nm, with a duration of 29 ps. After compressed and amplified by a commercial high gain erbium-doped fiber amplifier, the average power is amplified up to 30 mW. The spectra amplified by EDFA and after reflection are also shown in Fig. 3(a) respectively. Then, it is injected into modules I and II, and the back ends of the two modules are connected with the spatial optical path to further extract the spatial information of the mirror. In the spatial light path, the broadband light source is collimated, and divided into one-dimensional (1-D) spectral shower by a reflective grating with 1200 lines/mm. Next, the 1-D linear array light is collimated by a plane convex lens with a focal length of 200 mm and irradiates on the target mirror. After reflection, the 1-D spatial information is encoded onto the back-reflected 1-D spectral shower, that is, the frequency-space mapping is accomplished. After that, the return light is separated into two paths. One for observing the average spectral by a conventional optical spectrum analyzer (OSA), and the other is stretched and converted into a temporal waveform through the DFT system, and then captured by a high-speed photodiode with a 50 GHz bandwidth, and digitized by a real-time oscilloscope with a 20 GHz bandwidth. In our case, the dispersion coefficient of DCF for time stretching is -215 ps⋅km−1⋅nm−1, and its length is 4 km and 2 km for the module I and module II, separately. Therefore, the corresponding spectral-to-temporal coefficient is 0.86 ns/nm and 0.43 ns/nm.
Fig. 2. Schematic of the proposed mirror surface defects detection system. DS: dissipative soliton; EDFA: erbium doped fiber amplifier; PD: photodetector; OC: optical coupler; OSA: optical spectrum analyzer; DCF: dispersion compensating fiber; OSC: oscilloscope.
Fig. 3. (a) Optical spectra of the DS, amplified by EDFA and the reflected light. (b) Single shot spectrum captured by the oscilloscope and the spectrum measured by OSA.
Here, we use two structures to detect the amplitude/phase defects, separately. The detection of amplitude defects corresponds to module I, which is connected though a circulator, and the reflectivity measurement data is obtained through spatial information coding. The detection of phase defects corresponds to module II, we use delay fiber as the reference channel in addition to a space channel to form a MZI structure, and the reference pulse is interfered with the reflected pulse. For laser-induced thermal damage, the resulting phase defects are mainly manifested by local refractive index changes and depth variation (surface bulges or depressions). Therefore, these two kinds of changes will be encoded in the reflected beam in the form of phase information in the MZI structure. Further, the refractive index or depth information variations can be obtained by demodulating the phase correspondence based on Hilbert transform. To verify the ability of the precise measurement for transient phase variation, we release instantaneous and rapidly changing water mist in front of the mirror, for simulating the rapid change of optical path caused by thermal strain in practical high energy laser system. It is noted that after the EDFA, the average power is amplified up to 30 mW. The amplification is needed because that there is a certain loss in the optical path, especially in the spatial channel with coupling efficiency is about 25%, so it is necessary to amplify the energy of DS source. In addition, the reference channel of MZI uses a tunable delay line to match the two channels. And the splitting ratio of two optical couplers are 90: 10 and 50: 50 respectively, so that powers in the two channels can be balanced.
4. Results and discussion
4.1 Amplitude defects measurement
This surface defects detection system can perform non-contact 2D imaging of target mirror, and its experimental layout of spatial optical path is shown in Fig. 4(a), corresponding to module I in Fig. 2. The 1-D spectral shower by the reflective grating is used as line-scanning array for horizontal scanning (x-axis). To perform a 2D sample surface detection, a motor-driven translation stage is used to move the sample for longitudinal scanning (y-axis). The inset shows the enlarged view of 2D moving parts. To test the performance of this system, we firstly measured the optical spectrum of pulses reflected by the sample as well as the temporal waveform detected by DFT. As can be seen in Fig. 3(b), a high consistency is shown between the temporal waveform captured by the oscilloscope and the spectrum measured by OSA. In our experiment, a defected mirror with a diameter of 30 mm is used as a sample.
Fig. 4. (a) Experimental setup of the mirror surface defects detection system. (b) Imaged area of the damaged mirror. (c) Reconstructed image of the damaged mirror. (d) The corresponding 1D profile of four typical defect locations. The corresponding scale bar is 1 mm.
Figure 4(b) shows a selected region of the sample with a reflectivity greater than 80% (non-damaged parts). The reconstructed target area with an imaging field of view (10×13 mm) is depicted in Fig. 4(c). Specifically, we enlarged a small defect and compared its actual image with the reconstructed image. The slightly distorted reconstructed image would result from variable coupling efficiencies during scanning processes, energy instability of the optical source, and nonlinear distortions during DFT. Therefore, the reconstructed image shows that the left side of the reconstructed image appears to have much lower intensity compared with that on the right side. In order to compare the intensity accuracy of the two sides, we calculate the intensity differences for the two selected times marked in Fig. 4(c). As shown in Fig. 5, the intensity difference probability distribution satisfies the Gaussian distribution with a standard deviation of 0.01538 (a. u.) and 0.06614 (a. u.), respectively. The corresponding detection accuracy are 0.00354 (a. u.) and 0.00215 (a. u.), respectively. The experimental data reveals that the distortions induced by the nonideal conditions result into certain errors, which can be further suppressed by optimizing the optical source, coupling efficiency, and nonlinear dispersion.
Fig. 5. (a) Intensity accuracy distribution of the left side of Fig. 4(c). (b) Intensity accuracy distribution of the right side of Fig. 4(c).
Here, we select four typical defect positions and plot the single-shot temporal signals obtained by DFT in Fig. 4(d) with a spectral-to-temporal coefficient of 0.86 ns/nm. The lateral resolution of the scan in x-direction is approximately 90 µm, which is limited by the spatial resolution of the spatial disperser, the spectral resolution imposed by DFT and the temporal resolution of the digitizer [20]. Besides, the longitudinal resolution in y-direction is 100 µm, depending on scanning interval. Obviously, the detection sensitivity can be further improved from 90 µm to 30 µm or even lower, according to the specific application scenario [17]. Based on the expression of the spatial resolution given by Eq. (7), and frequency to time mapping coefficient given by Eq. (1)0, the detection sensitivity of the system can be optimized by manipulating the spatial disperser and the DFT module. For instance, the decrement of grating period d and focal length f would result into better sensitivity. A dispersive medium with larger dispersion together with a photodetection system with a high bandwidth would also help to optimize the lateral resolution. For the longitudinal resolution, the minimum scanning interval can be reduced to several micrometers by adopting a servo motor with a fine step accuracy.
4.2 Phase defects measurement
The phase defects detection corresponds to module II in Fig. 2. To monitor the rapidly varying phase variation, we use a delay fiber as the reference channel to form a MZI structure, The location of the phase defects detection is marked in Fig. 4(b), and the spray of transparent water mist is shown in the Fig. 6(a). The spectrum measured by conventional OSA and the temporal spectrum captured by the DFT system are shown in Fig. 6(b), where high consistency validates a linear mapping through DFT with a spectral-to-temporal coefficient of 0.43 ns/nm.
Fig. 6. (a) The diagram of water mist. (b) Single-shot spectrum captured by the oscilloscope and the spectrum measured by OSA. (c) 300 groups of interference spectra. (d) Partially enlarged view of (c). (e) The comparison of DFT single frame spectra collected in the two states (with/without water mist). (f) Partially enlarged view of (e).
Primarily, we verify the stability of the interference without water mist by plotting 300 consecutive roundtrips interference spectra in Fig. 6(c), together with an averaged spectrum. The enlarged selected data is shown in the Fig. 6(d). The free spectral range (FSR) of the interference spectrum is about 80 ps, corresponding to an imaging distance of 110 µm. To extract the phase information, we perform Hilbert transformation of the interference spectrum [37]. For the phase accuracy, we calculate the phase difference between adjacent roundtrips at the same spectral position, and arrange the phase difference from small to large, eliminate all zero items, and finally depict the histogram distribution in Fig. 7. The phase difference probability distribution satisfies the Gaussian distribution with a standard deviation of 0.06998 rad, and the phase accuracy is 0.00217 rad. Such a phase accuracy is mainly determined by the MZI and the photo detection module, rather than the sample undertest. The more stable the MZI is, the more accurate the phase detection is. Besides, a photodiode with larger bandwidth and higher sensitivity would benefit the phase accuracy. Also, a large analogy-to-digital conversion bits in the oscilloscope will improve the phase accuracy of the detection system. For our system, the phase accuracy is accurate enough for charactering the typical phase defects in mirrors, and also may find potential applications in some other similar cases.
To verify the ability of the precise measurement for transient phase variation, we release water mist in front of the mirror, for simulating the optical path change caused by thermal strain. The comparison of DFT single frame spectra collected in a control experiment (with/without water mist) is shown in Fig. 6(e). Obvious interference spectrum change is observed in Fig. 6(f). Therefore, we can characterize the phase defects caused by thermal strain by demodulating the phase change of interference spectrum. Specifically, the pulse train is captured by the single-pixel photodiode and displayed on the real-time digitizer (oscilloscope).
After Hilbert transformation, the transient phase variation induced by water mist is shown in Fig. 8. Figure 8(a) shows the decoded instantaneous phase matrix. By arranging the collected data into a data matrix according to time (t) and roundtrips (n), the phase information at different positions can be clearly characterized over time. Figure 8(b) shows the decoded instantaneous phase in the single frame DFT spectrum. Specifically, we selected six typical spectra and plotted their phase changes in Fig. 8(c). The corresponding positions in the phase matrix of these six times are marked in Fig. 8(a). In these six typical moments, the phase less than 0.4 rad can be observed, which is considered to be a small phase change caused by pulse energy jitter or environmental perturbation. Besides, the large phase mutations less than 2π can be observed. For instance, the phase mutation (−0.95π, 0.97π) is regarded as the phase mutation caused by rapidly varying water mist during t = 0.74 ns, reflecting the rapid capture of phase defects, with a time resolution of 119 ns.
Fig. 8. (a) The decoded instantaneous phase matrix. (b) Instantaneous phase of single frame DFT spectrum. (c) Phase change diagrams of six locations within 2000 roundtrips, corresponding to a duration of 0.238 ms (2000×119 ns).
In our demonstration, the accuracy of phase defects detection is 0.00217 rad, and the corresponding depth accuracy of phase defects is 51 nm. The detection time accuracy is 119 ns depending on the frame rate (8.4 MHz). Moreover, our detection system can reflect the phase change between (−π, +π), according to the phase unwrapping method of Hilbert transform. During the thermal damage process, the absorption of laser energy by the optical elements leads to the rise of local temperature. Due to thermal expansion, it will be subjected to thermal stress, resulting in melting, ablation or rupture. Those events finished within picoseconds or even to microseconds. Generally, the depth of initial laser-induced damage by high-power laser are ranging between 0.5 µm and 50 µm, corresponding to a phase variation less than 0.66π [38,39]. Therefore, the performance of our detection system can meet most of the phase defects detection requirements.
In this detection system, the frame rate can be optimized. The horizontal (x-axis) scanning speed depends on the repetition rate of laser pulses (8.4 MHz). The longitudinal (y-axis) scanning speed determined by the speed of mechanical scanning servo motor. In our demonstration, longitudinal scanning can be completed within 650 ms. Further, a virtually imaged phased array can be used to extend the high speed 1-D detection into 2-D detection, so the effective frame rate can be improved into orders of MHz. Besides, the application of this technology is not limited to specific bands for laser source, whether near infrared or near ultraviolet. It should be noted that the requirements of light source for high performance and detection efficiency need to be guaranteed: highly coherent, low noise, high flatness and high stability. In our spatial optical path, Fraunhofer far-field diffraction is completed by a collimator, a reflective grating and a convex lens. Therefore, any mirror-like surface detection with certain reflectivity can be applied. It is possible to detect curved surfaces, such as parabolic and hyperbolic mirrors, once the corresponding surface compensators are utilized so that the parallel beam can be collected effectively after reflected by mirrors with different shapes. While it may fail when applying for some setups with irregular geometries, where the beam intensity can not be collected uniformly.
5. Conclusion
We have proposed a high speed surface defects detection system of mirrors based on ultrafast single-pixel imaging technology. The spatial Fourier optical module and the DFT mapping module are employed to achieve frequency-space and frequency-time mapping separately, and then surface-defects-encoded optical signal is obtained by a high-speed photodiode. In addition to the amplitude defects detection, we build a MZI structure to perform phase defects detection. The two kinds of defects have been experimentally tested on a damaged mirror. The obtained lateral resolution of amplitude defects and phase accuracy are 90 µm and 0.00217 rad, respectively. For such a 1-D imaging scheme, a detecting time of 119 ns is needed. The frame rate to obtain a full 2-D image just relies on the mechanically sweeping speed. As a proof of concept, we detected both the amplitude and phase defects of a damaged mirror surface with a two-dimensional width of 10 × 13 mm. It is anticipated that by adopting high coherence wide light source or beam shrinking system, this fast surface defects detection method of mirrors is promising to be extended to the real-time measurement of large aperture optical elements, and monitor of operating states for high power laser systems.
Funding
National Natural Science Foundation of China (62075021); China National Funds for Distinguished Young Scientists (61825501); Graduate research and innovation foundation of Chongqing, China (CYS21057).
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.
References
1. T. K. N. Sweet, M. H. Rolley, W. Li, M. C. Paul, A. Johnson, J. I. Davies, R. Tuley, K. Simpson, F. M. Almonacid, E. F. Fernández, and A. R. Knox, “Design and characterization of hybrid III–V concentrator photovoltaic-thermoelectric receivers under primary and secondary optical elements,” Appl. Energy 226, 772–783 (2018). [CrossRef]
2. B. Dromey, S. Kar, M. Zepf, and P. Foster, “The plasma mirror–a subpicosecond optical switch for ultrahigh power lasers,” Rev. Sci. Instrum. 75(3), 645–649 (2004). [CrossRef]
3. R. Cubeddu, R. Polloni, C. Sacchi, and O. Svelto, “Picosecond pulses, TEM00mode, mode-locked ruby laser,” IEEE J. Quantum Electron. 5(9), 470–471 (1969). [CrossRef]
4. S. Matsuyama, H. Mimura, H. Yumoto, Y. Sano, and K. Yamauchi, “Development of scanning x-ray fluorescence microscope with spatial resolution of 30 nm using Kirkpatrick-Baez mirror optics,” Rev. Sci. Instrum. 77(10), 103102 (2006). [CrossRef]
5. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008). [CrossRef]
6. Y. Wu, P. Wawrzusin, J. Senseney, R. S. Fischer, R. Christensen, A. Santella, A. G. York, P. W. Winter, C. M. Waterman, and Z. Bao, “Spatially isotropic four-dimensional imaging with dual-view plane illumination microscopy,” Nat. Biotechnol. 31(11), 1032–1038 (2013). [CrossRef]
7. Z. He, K. Yin, and S. T. Wu, “Miniature planar telescopes for efficient, wide-angle, high-precision beam steering,” Light Sci. Appl. 10(8), 10 (2021). [CrossRef]
8. M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. 24(3), 160–162 (1999). [CrossRef]
9. X. Zeng, K. Zhou, Y. Zuo, Q. Zhu, J. Su, W. Xiao, X. Wang, X. Huang, X. Jiang, and D. Jiang, “Multi-petawatt laser facility fully based on optical parametric chirped-pulse amplification,” Opt. Lett. 42(10), 2014–2017 (2017). [CrossRef]
10. C. C. Widmayer, M. R. Nickels, and D. Milam, “Nonlinear holographic imaging of phase errors,” Appl. Opt. 37(21), 4801–4805 (1998). [CrossRef]
11. M. D. Feit and A. M. Rubenchik, “Influence of subsurface cracks on laser-induced surface damage,” SPIE. (2004).
12. W. Shuang-Chun and F. Dian-Yuan, “Small-scale self-focusing of intense laser beams in the presence of vector effect,” Chinese Phys. Lett. 17(10), 731–733 (2000). [CrossRef]
13. F. Y. Genin, M. D. Feit, M. R. Kozlowski, A. M. Rubenchik, and J. Yoshiyama, “Rear-surface laser damage on 355-nm silica optics owing to Fresnel diffraction on front-surface contamination particles,” Appl. Opt. 39(21), 3654–3663 (2000). [CrossRef]
14. S. Palmier, I. Tovena, R. Courchinoux, M. A. Josse, J. Luc Rullier, B. Bertussi, J. Y. Natoli, L. Servant, and D. Talaga, “Laser damage to optical components induced by surface chromium particles,” SPIE. 5647, 156–164 (2005). [CrossRef]
15. H. Bercegol, P. R. Bouchut, L. Lamaignere, B. L. Garrec, and G. Raze, “The impact of laser damage on the lifetime of optical components in fusion lasers,” SPIE. 5273, 312–324 (2004). [CrossRef]
16. Y. Gong, B. Li, and Y. Han, “Optical feedback cavity ring-down technique for accurate measurement of ultra-high reflectivity,” Appl. Phys. B 93(2-3), 355–360 (2008). [CrossRef]
17. A. Conder, J. Chang, L. Kegelmeyer, M. Spaeth, and P. Whitman, “Final optics damage inspection (FODI) for the national ignition facility,” SPIE. 7797, 77970P (2010). [CrossRef]
18. M. T. Tsai, F. Y. Chang, Y. J. Lee, J. D. Lee, H. C. Wang, and C. K. Lee, “Defect detection and property evaluation of indium tin oxide conducting glass using optical coherence tomography,” Opt. Express 19(8), 7559–7566 (2011). [CrossRef]
19. Z. Lou, K. Han, R. Song, B. Yan, and Z. Liu, “Damage characteristics of the optical element in resonant cavity irradiated by high-power continuous-wave laser,” Proc. SPIE 10192, 101920O (2017). [CrossRef]
20. L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014). [CrossRef]
21. K. Nakagawa, A. Iwasaki, and Y. Oishi, “Sequentially timed all-optical mapping photography (STAMP),” Opt. Express 8(9), 695–700 (2014). [CrossRef]
22. H. Mikami, L. Gao, and K. Goda, “Ultrafast optical imaging technology: principles and applications of emerging methods,” Nanophotonics 5(4), 497–509 (2016). [CrossRef]
23. K. Tsia, K. Goda, D. Capewell, and B. Jalali, “Performance of serial time-encoded amplified microscope,” Opt. Express 18(10), 10016 (2010). [CrossRef]
24. H. Ihee, V. A. Lobastov, U. M. Gomez, B. M. Goodson, R. Srinivasan, C. Y. Ruan, and A. H. Zewail, “Direct imaging of transient molecular structures with ultrafast diffraction,” Science 291(5503), 458–462 (2001). [CrossRef]
25. H. Chen, C. Wang, A. Yazaki, C. Kim, K. Goda, and B. Jalali, “Ultrafast web inspection with hybrid dispersion laser scanner,” Appl. Opt. 52(17), 4072–4076 (2013). [CrossRef]
26. A. Yazaki, C. Kim, J. Chan, A. Mahjoubfar, K. Goda, M. Watanabe, and B. Jalali, “Ultrafast dark-field surface inspection with hybrid-dispersion laser scanning,” Appl. Phys. Lett. 104(25), 251106 (2014). [CrossRef]
27. C. Ropers, G. Herink, F. Kurtz, B. Jalali, and D. Solli, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017). [CrossRef]
28. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19(3), 161–237 (1995). [CrossRef]
29. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]
30. R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, “Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor,” IEEE J. Quantum Electron. 22(5), 682–696 (1986). [CrossRef]
31. G. J. Tearney, M. Shishkov, and B. E. Bouma, “Spectrally encoded miniature endoscopy,” Opt. Lett. 27(6), 412–414 (2002). [CrossRef]
32. K. Goda, D. R. Solli, K. K. Tsia, and B. Jalali, “Theory of amplified dispersive Fourier transformation,” Phys. Rev. A: At., Mol., Opt. Phys. 80(4), 043821 (2009). [CrossRef]
33. K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013). [CrossRef]
34. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolve optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997). [CrossRef]
35. F. Xing, H. Chen, S. Xie, and J. Yao, “Ultrafast three-dimensional surface imaging based on short-time Fourier transform,” IEEE Photonics Technol. Lett. 27(21), 2264–2267 (2015). [CrossRef]
36. L. Gao, Y. Cao, S. Wabnitz, H. Ran, L. Kong, and Y. Li, “Polarization evolution dynamics of dissipative soliton fiber lasers,” Photonics Res. 7(11), 1331–1339 (2019). [CrossRef]
37. Y. Watanabe and M. Sato, “Quasi-single shot axial-lateral parallel time domain optical coherence tomography with Hilbert transformation,” Opt. Express 16(2), 524–534 (2008). [CrossRef]
38. W. Zhu, L. Chen, Y. Liu, Y. Ma, D. Zheng, Z. Han, and J. Li, “Defect detection based on a lensless reflective point diffraction interferometer,” Appl. Opt. 56(26), 7435–7441 (2017). [CrossRef]
39. W. Dai, X. Xiang, Y. Jiang, H. J. Wang, X. B. Li, X. D. Yuan, W. G. Zheng, H. B. Lv, and X. T. Zua, “Surface evolution and laser damage resistance of CO2 laser irradiated area of fused silica,” Opt. Lasers Eng. 49(2), 273–280 (2011). [CrossRef]
