Laser interferometry, a pivotal technology within the field of measurements, has found extensive applications in advanced manufacturing and cutting-edge scientific research.1 The continual advancement of laser interferometry over the past few decades focused on improving the key performances: measurement resolution, dynamic range, and update speed.2^–4 The single-frequency homodyne and heterodyne interferometry demonstrate the most precise measurement to date, making the detection of particle interactions5 and gravitational waves a reality,6 by referring to the frequency of the laser. However, the interference based on a single wavelength constrains the nonambiguity range (NAR) of an instantaneous phase measurement to half an optical period.7 To address this challenge, the dual-frequency and multiwavelength laser interferometry (MWI) have been introduced.7^–9 By employing the methods of synthetic wavelengths or beat frequencies, the dynamic range can be expanded to more than thousands of optical periods, which has been the mainstream alternative in lidar,10 optical coherence tomography (OCT),11 and fiber optic sensing.12 In recent years, MWI integrated with an optical frequency comb has led to the development of a dual-comb scheme,13^,14 enabling nanometer-resolved distance measurements over 30 km.15 However, the integral over time for detection is generally inevitable to achieve high precision, yielding the update rate up to milliseconds, which may not meet the requirements for observing rapidly changing events.16^,17

The dispersive spectral interferometry (DSI) technique can retrieve the phase from the optical spectral interferogram (IGM) of multiple wavelengths, holding the promise of simultaneously improving measurement resolution, range, and speed.18^–22 Using grating and charge-coupled device (CCD) for the acquisition of the IGMs, Joo and Kim19 achieved an NAR of ∼1.46mm with a resolution of 7 nm, and the NAR was further extended to 15 cm by improving the spectral resolution with virtual imaging phase array (VIPA).22 On the other hand, the developed time stretch dispersive Fourier transform (DFT) technique employing femtosecond laser can overcome the limitations of the working bandwidth of CCD, which has emerged as a powerful tool for capturing dynamic events,23^,24 ultrafast ranging,25 and real-time imaging.26^,27 Real-time DSI can be realized by integrating with DFT, thereby elevating the update rate of frequency-domain OCT to a megahertz level.27 Xia and Zhang28 and La Lone et al.16 extended its application to lidar, achieving submicrometer resolutions with a measurement time less than 1μs. Notably, the combination of DFT with dual-comb ranging alleviates the trade-off between measurement speed and precision and eliminates dead zones inherent in traditional methods, achieving high-precision measurements with a large dynamic range.29 Recent works in soliton molecular,30 gyroscopic,31 timing jitter,32 and vibration33 have sought to enhance the phase resolution to ∼10mrad by calculating the central phase based on the Fourier transform method.

The phase resolution limitation of DSI can be determined through a model involving the signal-to-noise ratio (SNR) and data size of the IGM, a methodology that has also been employed in wavelength-scanning interferometry34 and dual-comb ranging17 but has not been explored in real-time DSI. Although the theoretical resolution can potentially achieve submilliradian levels or even lower, the experimental results obtained thus far in DSI and real-time DSI remain considerably distant from this benchmark. This discrepancy strongly indicates that additional factors, such as laser-related noise and detector imperfections, which have not been rigorously analyzed, may degrade the phase results, and thereby obstruct further improvements in resolution.

In this work, we identify that the phase noise originating from the laser source and embedded in the spectral envelope has a significant impact on the resolution of real-time DSI. Subsequently, a model is developed to analyze this effect, and a real-time orthogonal polarization spectral interferometry (OPSI) is proposed to effectively mitigate the phase noise. The model quantifies the resolution degradation induced by phase noise from the perspectives of SNR and data volume, which aligns with existing research findings. It then explores the theoretical resolution of real-time OPSI in relation to detection bandwidth and measurement frame rate. The phase noise elimination is achieved by employing a pair of orthogonally polarized IGMs to remove the real-time spectral envelope. Experimental results show that real-time OPSI improves resolution to 1.1 mrad while maintaining a measurement frame rate exceeding 20 MHz even at low detection bandwidths. Our scheme has been validated using phase modulation signals of varying amplitudes and frequencies, confirming the measuring resolution and rate, and subsequently applied to measure the damped oscillation process of a piezoelectric stage. The measured oscillation curve is consistent with results from commercial interferometers and exhibits higher temporal resolution.

In DSI, IGMs affected by noise can be described as follows34: I1(ω)=2v02(ω)[1+cos(ωτ)]+nin(ω)=Iv0(ω)·Iideal(ω)+nin(ω),(1)where v02(ω) represents the spectral shape of the pulse and ω is the optical angular frequency. Iv0(ω)=2v02(ω) denotes the envelope and Iideal(ω)=1+cos(ωτ) represents the ideal cosine-shaped IGMs. τ corresponds to the optical path difference (OPD) introduced by the interferometer, which can be determined by calculating the phase slope of the IGM after processing with Fourier domain filtering. This result can be further refined by concentrating on the phase at the central frequency within a single optical cycle. These two approaches are commonly referred to as the Fourier transform (FT) method19^,35^–37 and the central frequency phase Fourier transform (CPFT) method.15^,31^,38nin(ω) reflects the intensity fluctuations introduced to the IGMs by the photodetector and analog-to-digital converter (ADC) during the detection process, commonly referred to as intensity noise. This noise imposes limitations on the accuracy of estimating τ using the FT and CPFT methods, which can be individually characterized by the Cramér–Rao bound39στlim=12σin2(Δω)2N(N2−1)≈23SNRin·N·δω,(2)στclim=σin2[Nωs2+N(N−1)ωs·Δω+16N(N−1)(2N−1)(Δω)2]≈1SNRin·N·ωc.(3)where σin2 represents the intensity noise level, with its reciprocal corresponding to the SNRin of the IGM. N denotes the data volume within a single IGM. The terms δω and Δω signify the spectral width and resolution, respectively, with Δω defined as Δω=δω/N. In addition, ωs and ωc represent the starting frequency and central frequency of the spectrum, respectively. From Eqs. (2) and (3), the resolution limit improvement of the CPFT method over the FT method can be expressed as στlim/στclim∝ωc/δω. The phase resolution limitation can be derived from Eq. (3) as σϕlim=στclim·ωc=1SNRin·N.(4)

According to Eq. (4), the evolution of resolution with respect to SNRin and data volume can be constructed, as illustrated by the lower surface in Fig. 1(b). When SNRin=30dB, a resolution of less than 5 mrad can be achieved with only 64 data points, and a resolution of 1 mrad is attainable when the data volume increases to 1024. Further enhancing the SNR can easily achieve submilliradian resolution, reaching ∼0.1mrad at 50 dB, which corresponds to picometer-level displacement resolution. However, the currently reported phase resolution results have not yet achieved the <5mrad level within the same data volume and SNR range,31^–33 which is still significantly distant from the theoretical predictions. This discrepancy suggests the presence of additional noise sources that affect the measurement results.

The initial pulse before the interferometer is influenced by the laser source and optical amplification,40 which can be represented as an addition of temporal noise np(t) to the temporal pulse v0(t), with the noise level being σp2. Consequently, the spectrum affected by the noise can be derived based on the FT properties of the pulse: v(ω)=v0(ω)+∑k=1Nvk(ω)·eiω·kΔT.(5)

The second term in Eq. (5) represents the spectral noise, where ΔT=2π/δω denotes the sampling interval in the Fourier domain, leading to distortions in the envelope after interference: $$\begin{gathered} Iv(ω)=|v(ω)|2=v02(ω)+2v0(ω)∑k=1Nvk(ω)·ei(ω·kΔT)+∑p=1N∑q=1Nvp(ω)vq(ω)·ei[ω·(p−q)ΔT] \\ =Iv0(ω)(1+δeiΔφ). \end{gathered}$$(6)

These distortions, represented by δeiΔφ [specifically, the second and third terms in Eq. (6)], introduce random phase noise into the interference phase ωτ, thereby degrading the estimation resolution of the central phase.

This degradation can be quantitatively characterized by assessing the impact of phase noise on the SNR. First, np(t), being distributed vertically across each frequency component according to the FT, results in the phase noise effect as a noise accumulation of the entire spectrum. Second, the Fourier domain filtering process, employed in the FT or CPFT method, reduces the overall phase noise level to M/N, where M represents the width of the filtering window.41 It is crucial to select an appropriate value for M to avoid the excessive noise while preventing the truncation error of the valid signal, thereby optimizing phase resolution. Consequently, the SNR of DSI can be adjusted as follows: SNRDSI=(σin2+σp2·δωωr·MN)−1,(7)where ωr is the pulse repetition frequency and δω/ωr represents the number of frequencies contained within the spectrum. Phase resolution limitation can be modified as σDSIlim=1SNRDSI·N.(8)

Subject to the phase noise (at −70dB), the resolution deteriorates to above 10 mrad when the data volume falls below 100 points. Although the resolution improves with a data increment, it remains unaffected by a reduction in intensity noise [as illustrated by the upper evolution in Fig. 1(b)]. Ultimately, the resolution is restricted by phase noise of 4 to 5 mrad, a limitation that aligns with previously reported findings.

The key to improving resolution lies in correctly and completely removing the phase noise in an envelope. Several approaches, including synchronized storage of spectral information42 and the Hilbert transform,43^,44 have been proposed to mitigate the envelope’s impact on the phase retrieval. However, the former requires precise OPD design to ensure synchronicity between the origin spectrum and the IGM, complicating the system and increasing costs, whereas the latter, by assuming a fixed phase relationship when calculating the envelope using a “mathematical approach,” fails to accurately account for random phase noise.

We propose a “physical approach” to eliminate the phase noise utilizing an orthogonal interferometer. As depicted in Fig. 1(c), the polarization states for the target pulse and the reference pulse remain orthogonal after passing through the interferometer. By aligning the combined pulse at a 45 deg angle to the optical axis of the polarizing beam splitter (PBS), both pulses are decomposed into the fast and slow axes of the PBS. On the fast axis, the pulses share identical polarization directions, whereas on the slow axis, their polarization directions are opposite. This configuration produces a pair of IGMs with a π phase difference along the two axes, expressed as I1(ω)=Iv(ω)[1+cos(ωτ)] and I2(ω)=Iv(ω)[1−cos(ωτ)]. Thus, the envelope Iv(ω) can be extracted by summing them together, and the IGM is dynamically modified by Iv(ω) to yield the cosine-shaped interference spectral devoid of phase noise. The comparison of phase results obtained with OPSI and traditional de-envelope methods such as the Hilbert transform are discussed in the Supplementary Material.

After eliminating phase noise, phase resolution is primarily determined by the intensity noise introduced by the detection device and the data volume, as indicated in Eq. (4). Theoretically, increasing the data volume can continuously improve resolution, but achieving this in practice is challenging, as the measurement performance is governed by a multiparameter system space.45 In real-time DSI, with a detection bandwidth B and measurement frame rate F, the maximum achievable data volume is N=B/F (where F is equivalent to ωr/2π). Due to the limitations of photodetectors and ADC, B is inversely correlated with SNRin, which can be expressed as SNRin=k1/Bk2. The values of k1 and k2 vary depending on the detection equipment. Through fitting calculations on data from various manufacturers’ instruments, we found that there is always k2>1, likely because intensity noise includes not only shot noise, thermal noise, and amplifier noise, which have a linear relationship with B, but also potentially other electronic and environmental noise46 (details provided in the Supplementary Material). Hence, the limitation can be rewritten as σϕlim=(F·Bk2−1k1)1/2.(9)

This implies that opting for a higher bandwidth and more expensive device can paradoxically diminish phase resolution, rendering high-detection bandwidth a “burden.” Using the k1 and k2 of the devices used in our experiments, the theoretical resolution of real-time OPSI is depicted in Fig. 1(d). The results indicate that a 1 GHz bandwidth can achieve submilliradian level resolution at a 20 MHz frame rate and maintain a resolution better than 2 mrad at a 90 MHz rate. However, when the bandwidth increases to 50 GHz, the same resolution is only achievable at rates of 13 and 60 MHz. Thus, enhancing measurement speed inevitably sacrifices resolution, necessitating a careful balance between these two properties to suit the specific measurement scenario. It is worth noting that although high bandwidth may be a “burden” for improving resolution, it extends the dynamic range in real-time DSI, which is constrained by the Nyquist theorem rather than by noise.47

Real-time OPSI necessitates a series of processing steps to extract the consecutive phase result from time series data, as illustrated in Fig. 2(a). Initially, time series data must be divided into individual single-shot time-domain IGMs. The pulse period provides a straightforward way to ensure uniform segmentation. However, the pulse period is hardly an integer multiple of the sampling interval of the oscilloscope, resulting in an inconsistency in length among frames. To address this, the raw data are interpolated thousands of times, reducing the relative coordinate error among adjacent IGMs to below 10 fs and inducing the phase error less than 10−5rad (further details in the Supplementary Material). Nevertheless, during the time stretch process, the fringe period of each time-domain IGM becomes chirped due to the high-order dispersion of the dispersion compensating fiber (DCF). This chirp can be corrected by mapping the IGM with the synchronized spectra obtained from the optical spectral analyzer, utilizing feature point extraction and polynomial fitting methods. Subsequently, the recovered optical spectra are de-enveloped using the IGMs with orthogonal polarization. A cosine fitting method is employed to determine variations in the central phase, as shown in the upper graph of panel V. The fitting function is expressed as Ifit(ω)=a+bcos(ωτ+φ), where a and b account for the influence of intensity noise on background amplitude and fringe modulation depth, whereas φ represents the phase ambiguity at zero frequency. By introducing a, b, and φ, the uncertainty of the fitting process is minimized, allowing for a more accurate estimation of the central phase ωcτ+φ. This cosine fitting method is fundamentally similar to the use of a linear function Φ=ωτ+φ for fitting the central phase in the CPFT method, as depicted in the lower graph of panel V.

Numerical simulations are performed to validate the model under varying levels of phase noise and intensity noise, as depicted in Figs. 2(b) and 2(c), respectively. The frame rate is set to 22.2 MHz, corresponding to the laser’s repetition rate, and a 4 GHz detection bandwidth is utilized. The phase results of real-time DSI indicated by green circles align well with the modified limitation σDSIlim (red curve). When the σin2 is fixed at −40dB, the resolution degrades nearly exponentially from 2 to 14 mrad as σp2 increases. By contrast, the resolution remains relatively stable at ∼8mrad as σin2 increases (with σp2 at −75dB), showing a flat evolution pattern similar to that observed in Fig. 1(b). This is because the impact of phase noise on SNR far outweighs that of intensity noise, which can be computed using Eq. (7). The resolution results of real-time OPSI, shown by purple circles, remain consistently ∼1mrad. The results significantly outperform those of real-time DSI, even at σp2 of −90dB, where the resolution improvement is more than double. Although the resolution deteriorates with increasing intensity noise, it remains at a submilliradian level for σin2<−40dB, consistently surpassing real-time DSI’s resolution. More surprisingly, the results even exceed σΦlim, represented by the orange line, as shown in the insets of Figs. 2(b) and 2(c). This improvement may be attributed to the effective reduction in intensity noise for each IGM during the phase noise elimination process.

The configuration of the experiment is illustrated in Fig. 3. The laser source is a homemade femtosecond fiber laser centered at ∼1566nm. The femtosecond pulses are first time-stretched by a 5 km DCF. The 1480 nm pump is introduced into the DCF through a wavelength division multiplexing (WDM) to amplify the stretched pulse via stimulated Raman effect, and the residual portion is then extracted by another WDM. Then, the laser is coupled into the polarization-maintaining fiber (PMF) propagating along the slow axis. A polarization-dependent isolator (ISO) can ensure the polarization direction, and the coupling efficiency is adjusted via the polarization controller (PC). A fiber-based Mach–Zehnder interferometer (MZI) is constructed by two 50/50 optical couplers (OC) to produce a time delay between two pulses. A phase modulator (iXblue) with a half-wave voltage of 5.9 V is installed on one arm to test the resolution. A circulator (Cir) and a collimator are installed on the other arm to measure the vibrations of the mirror introduced by the piezo stage (Micronix, California, USA, PPX-50), whereas a picometer interferometer (qutools, Munich, Germany, quDIS) is installed on the opposite side of the mirror to verify the measurement results. To construct the π-phase shifted IGMs in the fiber system, the polarization direction in PMF is changed to the fast axis by setting a 90 deg fiber splicing angle in one arm so that the interference signals distribute along two orthogonal polarization directions after MZI. By employing a PBS, two IGMs with a phase shift of π are formed. The time series signals are recorded by two PDs and a 4 GHz oscilloscope (Rohde & Schwarz, Munich, Germany), which are shown in the insets of Fig. 3.

According to the principle of data processing described above, the recorded data interval is subdivided 1000 times via interpolation, and then the time series data are divided into a series of consecutive frames with the roundtrip times of 45.045 ns. The time-to-frequency mapping is implemented by calibrating with the spectra measured simultaneously by the optical spectrum analyzer (Yokogawa, Tokyo, Japan), as shown in Fig. 4(a). Here, the dispersion coefficients up to fourth order are considered: ω=t/β2L−β3(t/β2L)2/2β2+β4(t/β2L)3/6β2, with β2=170.86ps2/km (D=131.15ps/km/nm), β3=0.2ps3/km, and β4=0.00735ps4/km. Figure 4(a) illustrates the resulting spectral evolution along with time. The de-enveloped results of interference spectra based on the π-phase shifting are depicted in Fig. 4(b), where the spectral trails with obvious errors are removed. The typical cosine-shaped curve can be seen in the single-shot IGM [left subfigure in Fig. 4(b)] that is fitted with φ and without φ, respectively. As predicted, the better fitting precision with the mean square error (MSE) of 1.16×10−3 can be achieved with the assistance of the zero-frequency phase.

Here, phase evolution results of real-time OPSI are retrieved at 1 ps time delay or 0.3 mm displacement. The resolution of ∼1mrad (<1 as in time) in Fig. 4(b) is nearly 6 times better than that retrieved by CPFT in Fig. 4(a), and this improvement aligns with the predictions from numerical simulations. CPFT enables the resolution of ∼6mrad (∼5 as) within the update rate of 20μs, which is in good agreement with the model [Fig. 2(b)]. It should be noted that the results also benefit from the fast measuring speed to eliminate most of the effects of thermal and vibration noise.

To verify the impact of sampling bandwidth on resolution, we conducted experiments under the sampling bandwidth of 4 GHz, 1 GHz, and 500 MHz. The de-enveloped data with the OPD of ∼3ps recorded under three different bandwidths are shown in Fig. 4(c), which displays a similar evolution. Notably, the data curves obtained under 1 GHz and 500 MHz exhibit smoother profiles compared with the 4 GHz case, as the reduced bandwidth filters out noise. The MSE of fitting for the three cases is 1.23×10−3, 1.10×10−3, and 1.11×10−3, respectively, with nearly identical fitting results. In Fig. 4(d), the standard deviations σΦ (στ) for all three cases are ∼1.1mrad (0.9 as) within a 20μs timeframe, demonstrating the capacity to achieve subattosecond resolution for single-shot spectra with limited bandwidth. Based on the experimental parameters, the dynamic ranges are calculated to be 3.3 ps (500 MHz), 6.6 ps (1 GHz), and 26.4 ps (4 GHz). It is worth noting that the dynamic range can be significantly extended by employing a longer DCF and a higher sampling bandwidth. For example, a 15 km DCF combined with a 33 GHz oscilloscope bandwidth can achieve a dynamic range of 660 ps, corresponding to a distance of ∼0.2m.

To validate the measurement resolution and speed, we introduce a periodical phase modulation via a phase modulator (PM). Figure 5(a) illustrates the spectral evolution under 200 kHz phase modulation with the driving voltage amplitude of 1 V, which clearly depicts the periodic variation. The corresponding phase evolutions of real-time DSI (retrieved by FT and CPFT, respectively) and real-time OPSI are presented in the upper part of Fig. 5(b), exhibiting the consistent evolution trajectories. When reducing the driving voltage to 5 mV, FT-derived results become inundated with noise, whereas the results obtained through CPFT and OPSI remain within a limited range [lower part of Fig. 5(b)]. Upon magnifying the vertical axis in Fig. 5(c), it is evident that under update rates of 22.2 and 2 MHz, OPSI consistently resolves the PM signal, but it becomes challenging to discern the modulation evolution from CPFT-based results. The measured phase modulation amplitude is 5.29 mrad, closely aligning with the theoretical response of 5.32 mrad of PM under 5 mV.

Real-time OPSI further showcases its capability to capture rapid changes on a submicrosecond scale. As illustrated in Fig. 5(d), when the modulation frequency is 2 MHz, it effectively captures the phase evolution, and the results closely match the theoretically calculated modulation signal (black curve). As the modulation frequency increases to 5 MHz, the signal remains clearly identifiable, although there are some distortions due to the limited repetition rate of the laser source. The use of a laser source with a higher repetition frequency can further boost the update speed.

The high resolution can also work in a larger temporal range with the recalibration of mapping relationships and compensation of thermal drift for the optical path. Figure 6(b) presents the normalized phase evolution over a temporal range of 500μs, and the standard variation for the calculated phase is 1.14 mrad with the update rate of 22.2 MHz. Moreover, the standard variation can be further reduced to 0.29 mrad (>0.3 as in time) by averaging, which limits the update rate to 40 kHz. Just as in the previous works, the integral over time could suppress the random noise effectively, as shown in Fig. 6(a). It is evident that the update rate must be maintained at a megahertz level when achieving a resolution below 0.5 mrad, equivalent to 1/12,000 of an optical cycle.

We utilized a 250 kHz frame rate to measure the rapid vibrations of a mirror induced by the damping oscillation of a piezo stage with nanometer precision. The distance measurement results are denoted as ΔD=Φ·c/2ωc, where c represents the speed of light, as illustrated by the red circles in Fig. 6(c). For comparison, displacement results are also obtained using a commercial interferometer with a maximum update rate of 25 kHz and a nominal resolution superior to 50 pm (at 100 Hz), shown by the blue squares. Within the 8 ms measurement range, real-time OPSI achieves results nearly identical to those of the interferometer, almost perfectly capturing an oscillation cycle of ∼125Hz with an amplitude of ∼150nm. As the picometer interferometer operates at its maximum frame rate of 25 kHz, its resolution proportionally decreases relative to its nominal resolution, and the two interferometric systems were subjected to different environmental disturbances. This results in an overall RMS value of 1.22 nm, with local deviations at the peaks and troughs exceeding 2 nm. The zoom-in views of regions (a) and (b) in Fig. 6(c) are depicted in Fig. 6(d), demonstrating that under varying displacement trends, real-time OPSI offers displacement resolution comparable with that of the picometer interferometer. At the same time, its temporal resolution is 10 times greater or even more, enabling its application in scenarios involving faster displacement changes.

This work demonstrates a real-time OPSI system that achieves submilliradian phase resolution while operating at a megahertz update rate. Initially, a detailed analysis of resolution limitation in DSI is conducted, focusing on the influence of intensity noise, phase noise, and detection bandwidth, and it is identified that the phase noise affecting the spectral envelope has hindered further resolution enhancement in high-resolution DSI experiments. To address this, a de-envelope method based on a set of IGMs with orthogonal polarization is employed, which effectively eliminates the phase noise in the envelope. In the data processing steps of real-time OPSI, interpolation, segmentation, and time–frequency mapping are performed to mitigate coordinate errors and the impact of higher order dispersion on IGMs. Then, a cosine fitting method is used after de-envelope to extract the central phase of the IGMs. Real-time OPSI achieved a single-shot resolution of 1.1 mrad (0.91 as in time) and further improved to 0.29 mrad (0.24 as in time) at a reduced update rate of 40 kHz, aligning with our theoretical resolution limitation and significantly enhancing measuring resolution compared with previous works in real-time DSI. In addition, the requirement for detection bandwidth can be reduced while maintaining the same resolution. Various amplitude and frequency modulation signals are employed to validate the high-speed, high-resolution measurement capabilities of our approach. Furthermore, our scheme is successfully applied to measure the rapid damped oscillations of a piezoelectric stage, producing results that closely match those of a commercial interferometer with an overall RMS of just 1.22 nm. Future work could explore the use of lower-noise detection equipment to further improve resolution and increase detection bandwidth to achieve high-precision measurements with a broader dynamic range.

Tianchang Lu is currently a PhD student at the College of Optical Science and Engineering, Zhejiang University. He received his BS degree in optical engineering from Nankai University. His research focuses on high-precision metrology based on femtosecond pulses and optical frequency combs.

Yudong Cui is currently an associate professor at the College of Optical Science and Engineering, Zhejiang University. He received his PhD from University of Chinese Academy of Sciences. His research interests include ultrafast laser generation, nonlinear ultrafast dynamics, and precision metrology and imaging for extreme optical measurements.

Xu Liu is currently a professor at the College of Optical Science and Engineering, Zhejiang University. He is a fellow of the Chinese Optical Society, Optica, and SPIE. His research focuses on super-resolution optical microscopy, optical thin films, and precision metrology and imaging for extreme optical conditions.

Biographies of the other authors are not available.