Frequency-domain interferometer for femtosecond time-resolved phase spectroscopy

Dynamic and transient processes in warm dense matter

In this paper, we discuss some of the key challenges in the study of time-dependent processes and non-equilibrium behaviour in warm dense matter. We outline some of the basic physics concepts that have underpinned the definition of warm dense matter as a subject area in its own right and then cover, in a selective, non-comprehensive manner, some of the current challenges, pointing along the way to topics covered by the papers presented in this volume. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Spectrally Selective Time-Resolved Emission through Fourier-Filtering (STEF)

We demonstrate a method for separating and resolving the dynamics of multiple emitters without the use of conventional filters. By directing the photon emission through a fixed path-length imbalanced Mach-Zehnder interferometer, we interferometrically cancel (or enhance) certain spectral signatures corresponding to one emissive species. Our approach, Spectrally selective Time-resolved Emission through Fourier-filtering (STEF), leverages the detection and subtraction of both outputs of a tuned Mach-Zehnder interferometer, which can be combined with time-correlated single photon counting (TCSPC) or confocal imaging to demix multiple emitter signatures. We develop a procedure to calibrate out imperfections in Mach-Zehnder interferometry schemes. Additionally, we demonstrate the range and utility of STEF by performing the following procedures with one measurement: (1) filtering out laser scatter from a sample, (2) separating and measuring a fluorescence lifetime from a binary chromophore mixture with overlapped emission spectra, (3) confocally imaging and separately resolving the standard fluorescent stains in bovine pulmonary endothelial cells and nearly overlapping fluorescent stains on RAW 264.7 cells. This form of spectral balancing can allow for robust and tunable signal sorting.

Improving Spectral, Spatial, and Mechanistic Resolution Using Fourier Transform Nonlinear Optics: A Tutorial Review

Fourier transform nonlinear optics (FT-NLO) is a powerful experimental physical chemistry tool that provides insightful spectroscopic and imaging data. FT-NLO has revealed key steps in both intramolecular and intermolecular energy flow. Using phase-stabilized pulse sequences, FT-NLO is employed to resolve coherence dynamics in molecules and nanoparticle colloids. Recent advances in time-domain NLO interferometry using collinear beam geometries makes determination of molecular and material linear and nonlinear excitation spectra, homogeneous line width, and nonlinear excitation pathways straightforward. When combined with optical microscopy, rapid acquisition of hyperspectral images with the information content of FT-NLO spectroscopy is possible. With FT-NLO microscopy, molecules and nanoparticles colocated within the optical diffraction limit can be distinguished based on their excitation spectra. The suitability of certain nonlinear signals for statistical localization present exciting prospects for using FT-NLO to visualize energy flow on chemically relevant length scales. In this tutorial review, descriptions of FT-NLO experimental implementations are provided along with theoretical formalisms for obtaining spectral information from time-domain data. Select case studies that illustrate the use of FT-NLO are presented. Finally, strategies for extending super-resolution imaging capabilities based on polarization-selective spectroscopy are offered.

Asymmetric double-pulse interferometric FROG for visible-wavelength time-domain spectroscopy

To extend the detection range of time-domain spectroscopy into the challenging visible frequencies, we propose an interferometry-type frequency-resolved optical gating (FROG). Our numerical simulation shows that, when operating in a double-pulse scheme, a unique phase-locking mechanism can be activated and preserves both zero- and first-order phases (φ₀, φ₁)-indispensable for phase-sensitive spectroscopic study-that are otherwise inaccessible to standard FROG measurement. Followed by time-domain signal reconstruction and analysis protocol, we show that time-domain spectroscopy with sub-cycle temporal resolution is enabled and well suits the need of a ultrafast-compatible and ambiguity-free method for complex dielectric function measurement at visible wavelengths.

Cross-polarized common-path temporal interferometry for high-sensitivity strong-field ionization measurements

Absolute density measurements of low-ionization-degree or low-density plasmas ionized by lasers are very important for understanding strong-field physics, atmospheric propagation of intense laser pulses, Lidar etc. A cross-polarized common-path temporal interferometer using balanced detection was developed for measuring plasma density with a sensitivity of ∼0.6 mrad, equivalent to a plasma density-length product of ∼2.6 × 10¹³ cm^-2 if using an 800 nm probe laser. By using this interferometer, we have investigated strong-field ionization yield versus intensity for various noble gases (Ar, Kr, and Xe) using 800 nm, 55 fs laser pulses with both linear (LP) and circular (CP) polarization. The experimental results were compared to the theoretical models of Ammosov-Delone-Krainov (ADK) and Perelomov-Popov-Terent'ev (PPT). We find that the measured phase change induced by plasma formation can be explained by the ADK theory in the adiabatic tunneling ionization regime, while PPT model can be applied to all different regimes. We have also measured the photoionization and fractional photodissociation of molecular (MO) hydrogen. By comparing our experimental results with PPT and MO-PPT models, we have determined the likely ionization pathways when using three different pump laser wavelengths of 800 nm, 400 nm, and 267 nm.

Broadband Optical Phase Modulation by Colloidal CdSe Quantum Wells

Two-dimensional (2D) semiconductors are primed to realize a variety of photonic devices that rely on the transient properties of photogenerated charges, yet little is known on the change of the refractive index. The associated optical phase changes can be beneficial or undesired depending on the application, but require proper quantification. Measuring optical phase modulation of dilute 2D materials is, however, not trivial with common methods. Here, we demonstrate that 2D colloidal CdSe quantum wells, a useful model system, can modulate the phase of light across a broad spectrum using a femtosecond interferometry method. Next, we develop a toolbox to calculate the time-dependent refractive index of colloidal 2D materials from widely available transient absorption experiments using a modified effective medium algorithm. Our results show that the excitonic features of 2D materials result in broadband, ultrafast, and sizable phase modulation, even extending to the near infrared because of intraband transitions.

Single 3.3 fs multiple plate compression light source in ultrafast transient absorption spectroscopy

Ultrafast transient absorption spectroscopy is a powerful tool to reveal excited state dynamics in various materials. Conventionally, probe pulses are generated via bulk supercontinuum generation or (noncollinear) optical parametric amplifiers whilst pump pulses are generated separately using (noncollinear) optical parametric amplifiers. These systems are limited by either their spectral density, stability, spectral range, and/or temporal compressibility. Recently, a new intense broadband light source is being developed, the multi-plate compression, which promises to overcome these limitations. In this paper, we analyze the supercontinuum generated by a single Multiple Plate Compression system to set a benchmark for its use in the field of ultrafast pump-probe spectroscopy. We have compressed the supercontinuum to 3.3 fs using chirp mirrors alone, making it an excellent candidate for pump-probe experiments requiring high temporal resolution. Furthermore, the single light source can be used to generate both probe and pump pulses due to its high spectral density (>14.5 nJ/nm) between 490 and 890 nm. The intensity has an average shot-to-shot relative standard deviation of 4.6 % over 490 to 890 nm, calculated over 2,000 sequential shots. By using only 1,000 shot pairs, a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta T/T$$\end{document}ΔT/T noise level of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.6\times 10^{-4}$$\end{document}2.6×10-4 RMS is achieved. Finally, as a proof of concept, the transient absorption spectrum of a methylammonium lead iodide perovskite film is taken, showing great signal to noise with only 1,000 shot pairs. These results show great potential for the employment of this technique in other spectroscopic techniques such as coherent multidimensional spectroscopy.

Spatially resolved spectral phase interferometry with an isolated attosecond pulse

We demonstrate spatially resolved supercontinuum spectral phase interferometry with an isolated attosecond pulse (IAP). The measured spatial-spectral interferogram over the broadband region indicates a high degree of IAP coherence in both spatial and spectral domains. In addition, the spectral-delay interferogram shows periodic temporal oscillations over the full IAP continuous spectrum, which indicates high temporal coherence. The supercontinuum spectral phase interferometry with broadband IAP will contribute to exploring spatiotemporal dispersive electronic dynamics through phase-based spectroscopy in the future.

Time-Resolved sub-Ångström Metrology by Temporal Phase Interferometry near X-Ray Resonances of Nuclei

We introduce an analytical phase-reconstruction principle that retrieves atomic scale motion via time-domain interferometry. The approach is based on a resonant interaction with high-frequency light and does not require temporal resolution on the time scale of the resonance period. It is thus applicable to hard x rays and γ rays for measurements of extremely small spatial displacements or relative-frequency changes. Here, it is applied to retrieve the temporal phase of a 14.4 keV emission line of an ^{57}Fe sample, which corresponds to a spatial translation of this sample. The small wavelength of this transition (λ=0.86  Å) allows for determining the motion of the emitter on sub-Ångström length and nanosecond timescales.

Femtosecond two-photon absorption phase change spectroscopy of a ZnSe single crystal

We measured time- and frequency-resolved phase changes owing to two-photon absorption induced by ultrashort (10 fs) pulses transmitted through a transparent material (ZnSe crystal), using femtosecond two-photon phase change spectroscopy. The maximal phase change was 0.02 rad when a pulse with the energy density of 1.5 nJ was focused with the focus diameter of 100 μm. The maximal phase change occurred when a probe pulse was delayed with respect to a pump pulse by 64 fs. A calculation assuming two-photon absorption by noninteracting atomic gases was consistent with the experimental observations, which supported the observed phase change of the pulse that is due to the two-photon absorption.

Focus Tracking System for Femtosecond Laser Machining using Low Coherence Interferometry

We designed a real time, single-laser focus tracking system using low coherence properties of the machining femtosecond laser itself in order to monitor and correct the sample position relative to the focal plane. Using a Michelson Interferometer, the system collects data arising from part of the beam backscattered at the ablation spot. The data is analyzed by a custom software for position correction (employing an XYZ automated translation stage). With the focus tracking enabled we were able to etch channels with a stable cross-section profile on a bovine tooth with relief amplitude tens of times greater than the Rayleigh length of the system, keeping the sample inside the confocal parameter during most of the processing time. Moreover, the system is also capable of monitoring crater depth evolution during the ablation process, allowing for material removal assessment.

Interferometric Measurement of Transient Absorption and Refraction Spectra with Dual Frequency Comb

We demonstrate that a dual frequency comb-transient absorption (DFC-TA) technique can be combined with a time-domain interferometric detection to measure both the transient absorption and refraction spectra of molecules in solution. To do this, the pump-probe signal field of DFC-TA is allowed to interfere with a time-delayed local oscillator field in a time domain. We show that this DFC interferometric pump-probe spectroscopy (DFC-IPS) technique has a unique ability to extract the phase and amplitude information on the pump-probe signal using just a single-scan data, while conventional techniques require an independent signal measured without the pump field for the normalization of the pump-probe spectrum. As a proof-of-principle experiment, we here show that the DFC-IPS enables us to simultaneously measure the frequency-resolved (from 650 to 950 nm) transient absorption and refraction signals with an exceptionally broad dynamic range from femtosecond to nanosecond without using a mechanical translational stage for pump-probe time-scanning. We anticipate that our DFC-IPS technique with femtosecond time-resolution capability will be useful to investigate photoinduced chemical and biological reactions covering broad dynamic ranges.

High-precision determination of the cut angle of an electro-optic crystal by conoscopic interference

To precisely determine the cut angle of an electro-optic crystal, an improved conoscopic interference measurement method, developed to address the large influence of crystal alignment and melatope positioning errors on measurement results, is presented in this paper. Ray-tracing formulas are derived to calculate the cut angle and evaluate the measurement errors. A Twyman-Green interferometric partial system is employed to make auxiliary adjustments to the orientation of the crystal sample to ensure that the front surface of the crystal is completely perpendicular to the optical axis during the test. In this way, the alignment error can be controlled within 0.050 mrad. An interferogram processing method, by which the centers of gravity of each independent conoscopic speckle are connected, is put forward to position the crystal melatope. The average result is 3.734 mrad when the cut angle of a 6-mm-thick KDP electro-optic crystal is measured 12 times, with the measurement repeatability being approximately 5 times less than that of a sophisticated x-ray diffractometer. The experiment reveals that by using the method, the melatope positioning error can be reduced to less than 0.60 μm. The improved method avoids the shortcomings of traditional conoscopic measurement methods, with the measurement error being reduced from 1.750 to 0.100 mrad, and provides technical support for future applications of high-precision and low-cost crystal orientometers.

One-shot time-resolved holographic polarization microscopy for imaging laser-induced ultrafast phenomena

A time-resolved holographic polarization microscopy, based on angular multiplexing holographic technique, is proposed for imaging ultrafast phenomena in polarization-sensitive transparent materials. This method can retrieve and image the complex amplitude distributions of two orthogonal polarization components of two sequential vector wavefronts with ultrashort time interval by a single short recording. Some experimental results for imaging the pulse laser induced ultrafast events based on the method are given. It is demonstrated that this technique may provide a potential tool for characterizing ultrafast processes in polarization-sensitive materials, especially in the non-reproducible experiment conditions.

Time-resolved quantitative-phase microscopy of laser-material interactions using a wavefront sensor

We report on a simple and efficient technique based on a wavefront sensor to obtain time-resolved amplitude and phase images of laser-material interactions. The main interest of the technique is to obtain quantitative self-calibrated phase measurements in one shot at the femtosecond time-scale, with high spatial resolution. The technique is used for direct observation and quantitative measurement of the Kerr effect in a fused silica substrate and free electron generation by photo-ionization processes in an optical coating.

In-line femtosecond common-path interferometer in reflection mode

An innovative method to perform femtosecond time-resolved interferometry in reflection mode is proposed. The experiment consists in the combined use of a pump-probe setup and of a fully passive in-line femtosecond common-path interferometer. The originality of this interferometer relies on the use of a single birefringent crystal first to generate a pair of phase-locked pulses and second to recombine them to interfere. As predicted by analytical modeling, this interferometer measures the temporal derivative of the ultrafast changes of the complex optical reflection coefficient of the sample. Working conditions are illustrated through picosecond opto-acoustic experiments on a thin film.

Laser-induced dispersion control

An intense laser pulse is used to control the spectral phase of a weak probe pulse as they overlap in fused silica. The laser-induced linear chirp is controlled by the delay time between pulses. Dependence from intensity and spectral phase of the pump pulse is also studied. Experimental data is validated by numerical simulation based on optical Kerr effect. Results show that laser-induced pulse shaping is possible and may be useful for intracavity pulse compression and shaping in enhancement cavities.

Extracting phase and amplitude modifications of laser-coupled Fano resonances

Fano line shapes observed in absorption spectra encode information on the amplitude and phase of the optical dipole response. A change in the Fano line shape, e.g., by interaction with short-pulsed laser fields, allows us to extract dynamical modifications of the amplitude and phase of the coupled excited quantum states. We introduce and apply this physical mechanism to near-resonantly coupled doubly excited states in helium. This general approach provides a physical understanding of the laser-induced spectral shift of absorption-line maxima on a sub-laser-cycle time scale as they are ubiquitously observed in attosecond transient-absorption measurements.

Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range

A new optical-fiber frequency domain interferometer (OFDI) device for accurate measurement of the absolute distance between two stationary objects, with centimeter measuring range and nanometer resolution, has been developed. Its working principle and on-line data processing method were elaborated. The new OFDI instrument was constructed all with currently available commercial communication products. It adopted the wide-spectrum amplified spontaneous emission light as the light source and optical-fiber tip as the test probe. Since this device consists of only fibers or fiber coupled components, it is very compact, convenient to operate, and easy to carry. By measuring the single-step length of a translation stage and the thickness of standard gauge blocks, its ability in implementing nanometer resolution and centimeter measuring range on-line measurements was validated.

Coherent time-stretch transformation for real-time capture of wideband signals

Time stretch transformation of wideband waveforms boosts the performance of analog-to-digital converters and digital signal processors by slowing down analog electrical signals before digitization. The transform is based on dispersive Fourier transformation implemented in the optical domain. A coherent receiver would be ideal for capturing the time-stretched optical signal. Coherent receivers offer improved sensitivity, allow for digital cancellation of dispersion-induced impairments and optical nonlinearities, and enable decoding of phase-modulated optical data formats. Because time-stretch uses a chirped broadband (>1 THz) optical carrier, a new coherent detection technique is required. In this paper, we introduce and demonstrate coherent time stretch transformation; a technique that combines dispersive Fourier transform with optically broadband coherent detection.

Single-shot ultrafast tomographic imaging by spectral multiplexing

Computed tomography has profoundly impacted science, medicine and technology by using projection measurements scanned over multiple angles to permit cross-sectional imaging of an object. The application of computed tomography to moving or dynamically varying objects, however, has been limited by the temporal resolution of the technique, which is set by the time required to complete the scan. For objects that vary on ultrafast timescales, traditional scanning methods are not an option. Here we present a non-scanning method capable of resolving structure on femtosecond timescales by using spectral multiplexing of a single laser beam to perform tomographic imaging over a continuous range of angles simultaneously. We use this technique to demonstrate the first single-shot ultrafast computed tomography reconstructions and obtain previously inaccessible structure and position information for laser-induced plasma filaments. This development enables real-time tomographic imaging for ultrafast science, and offers a potential solution to the challenging problem of imaging through scattering surfaces.

Attosecond twin-pulse control by generalized kinetic heterodyne mixing

Attosecond double-pulse (twin-pulse) production in high-order harmonic generation is manipulated by a combination of two-color and carrier-envelope phase-control methods. As we show in numerical simulations, both relative amplitude and phase of the double pulse can be independently set by making use of multidimensional parameter control. Two technical implementation routes are discussed: kinetic heterodyning using second-harmonic generation and split-spectrum phase-step control.

Measurement of pressure dependent nonlinear refractive index of inert gases

The propagation of high intensity laser beams is excessively affected by optical nonlinear effects, thereby the knowledge of the nonlinear refractive indices of the beam guiding media is indispensable in the design of laser systems and experiments. Apart from undesired self-focusing, several areas of modern laser spectroscopy can utilize optical nonlinearity, from LiDAR measurements to filamentation. In this paper we report on a direct measurement of pressure dependent nonlinear refractive index of Ar, N2, Ne, Xe, and air between 0.05 mbar and 1 bar, based on the powerful technique called spectrally and spatially resolved interferometry. In this way the total value of nonlinear refractive index is measured, that is the sum of all elementary phenomena contributing to the intensity dependent refractivity of the gases.

Single-shot optical conductivity measurement of dense aluminum plasmas

The optical conductivity of a dense femtosecond laser-heated aluminum plasma heated to 0.1-1.5 eV was measured using frequency-domain interferometry with chirped pulses, permitting simultaneous observation of optical probe reflectivity and probe pulse phase shift. Coupled with published models of bound-electron contributions to the conductivity, these two independent experimental data yielded a direct measurement of both real and imaginary components of the plasma conductivity.

Resonant frequency-domain interferometry via third-harmonic generation

A method for measuring resonances using a combination of third-harmonic generation and frequency-domain interferometry is described and demonstrated in an index-matched dielectric material. The phase of the third-harmonic spectrum of a pulse generated from a resonant NdAlO(3) thin film and a temporally displaced sapphire substrate pulse was measured by analyzing the spectral interference pattern. The appropriate combination of substrate and film signals was obtained by translating the sample through the laser focus while observing the third-harmonic intensity.

Single-shot two-dimensional spectral interferometry for ultrafast laser-produced plasmas

Coherent white light was used as a light source for spectral interferometry of ultrafast laser-produced plasmas. Using a narrowband filter, two-dimensional images of field ionization in helium were obtained with a 14 fs time resolution.

Optical selectivity in optically dense media driven by optimized Gaussian-type ultrashort pulse pairs

We theoretically demonstrate that selective resonant excitation can be achieved in a dense collection of V-type three-level atoms by optimizing the pulse delay and peak intensity ratio of an applied phase-tailored ultrashort pulse pair. Near-dipole-dipole interaction plays an important role in the quantum control of selective excitations since it brings about an intrinsic frequency shift in the atomic resonance, which builds up various excitation pathways. As a consequence, we can control the quantum interference between various pathways by shaping the excitation pulse pair to steer the atomic excitation selectively toward a desired quantum state.

Two-dimensional femtosecond spectroscopy

The simplest two-dimensional (2D) spectra show how excitation with one (variable) frequency affects the spectrum at all other frequencies, thus revealing the molecular connections between transitions. Femtosecond 2D Fourier transform (2D FT) spectra are more flexible and share some of the remarkable properties of their conceptual parent, 2D FT nuclear magnetic resonance. When 2D FT spectra are experimentally separated into real absorptive and imaginary refractive parts, the time resolution and frequency resolution can both reach the uncertainty limit set for each resonance by the sample itself. Coherent four-level contributions to the signal provide new molecular phase information, such as relative signs of transition dipoles. The nonlinear response can be picked apart by selecting a single coherence pathway (e.g., specifying the relative signs of energy level difference frequencies during different time intervals as in the photon echo). Because molecules are frozen on the femtosecond timescale, femtosecond 2D FT experiments can separate a distribution of instantaneous molecular environments and intramolecular geometries as inhomogeneous broadening. This review provides an introduction to two-dimensional Fourier transform experiments exploiting second- and third-order vibrational and electronic nonlinearities.

Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse

The superluminal group velocity of an ultrashort optical Bessel beam pulse is measured over its entire depth of field, corresponding to approximately 2x10(4) optical wavelengths. The method used is to measure the traveling ionization front induced by the pulse.

Single-shot spectral interferometry with chirped pulses

We present a method for obtaining time-resolved measurements of the amplitude modulation and the phase shift of a chirped probe pulse interacting with a femtosecond-laser-produced plasma. Based on spectral interferometry, the technique allows for single-shot measurements and keeps the temporal resolution associated with the full bandwidth of the probe pulse. We demonstrate the efficiency of this technique by probing femtosecond-laser breakdown of plastic targets.

Absorption of femtosecond laser pulses in interaction with solid targets

We have studied the effects of the plasma density scale length on the absorption mechanism of the femtosecond (fs) laser pulses interacting with solid targets. Experiments and particle-in-cell (PIC) simulations demonstrate that the vacuum heating is the main absorption in the plasma in the interaction of fs laser pulses with solid targets when no prepulses are applied. The energy spectrum of hot electrons ejected out of or injected into the plasma show a bitemperature distribution. While the first temperature of the two groups of hot electrons can be attributed to the "pull-and-push" exertion of the laser field, the second temperature refers to the electrons accelerated by the static part (in front of the target) and the oscillating part (in the plasma layer) of the laser-induced electric field, respectively. PIC simulations also show that with an appropriate density scale length, the femtosecond laser energy can be absorbed locally through different mechanisms.

Observation of spatial asymmetry of THz oscillating electron plasma wave in a laser wakefield

The asymmetric spatial distribution of electron density perturbation is observed by using a frequency-domain interferometry technique. The wake amplitude of the outside bump is enhanced by the elliptical distribution of the pump laser pulse. This asymmetry can be explained with a two-dimensional analytical model expanded from cylindrically symmetric linear theory.

Measurement of shock wave rise times in metal thin films

We have measured the rise time of laser-generated shock waves in vapor plated metal thin films using frequency-domain interferometry with subpicosecond time resolution. 10%- 90% rise times of <6.25 ps were found in targets ranging from 0.25 to 2.0 microm in thickness. Particle and average shock velocities were simultaneously determined. Shock velocities of approximately 5 nm/ps were inferred from the measured free surface velocity, corresponding to pressures of 30-50 kbar. Thus, the shock front extends only a few tens of lattice spacings.

Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air

A novel interferometry technique is presented by which, in one shot, one can measure phase changes with a resolution of tens of femtoseconds while extending the measurement over picoseconds or even longer. The method is based on spectral (frequency-domain) interferometry with a pair of linearly chirped pules as probes. With this technique we obtained single-shot measurements of the rapid phase changes induced by optical field ionization of air. This allowed us to calculate the time profile of the electron density created by an intense short laser pulse.

Direct measurement of self-phase shift due to fiber nonlinearity

We directly measure the time-independent nonlinear self-phase shift of a pulse after fiber propagation by spectral interferometry. Both the soliton and the nonsoliton case were studied. We verify numerically that the measured phase shift approximates this time-independent phase well.

Spectral interferometry as an alternative to time-domain heterodyning

A novel method of time resolving the amplitude and phase of a transient nonlinear-optical polarization, based on Fourier transformation of a spectral interference pattern, is discussed. We show that with this technique the delay at which the polarization is generated can be retrieved. The method is demonstrated in a transient four-wave-mixing experiment on a dye solution.

Determination of the optical-axis orientation of a uniaxial crystal by frequency-domain interferometry

Frequency-domain interferometry has been used to find the optical axis of a uniaxial, birefringent crystal. The dependence of the interference fringe spacing on the incident angle and the direction of the optical axis were determined in a 4.6-cm-thick KDP crystal.

Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry

We demonstrate that dual-channel spectral interferometry in conjunction with a well-characterized reference pulse can be used to time resolve the polarization state of extremely weak ultrashort coherent signals from linear-and nonlinear-optical experiments by measuring the intensity and the phase of two orthogonal polarization components. In this way the signal is completely characterized.

Femtosecond Sagnac interferometer for phase spectroscopy

We developed a novel femtosecond Sagnac interferometer that enables us to take single-shot measurements of both the difference transmission and the difference phase spectra. Owing to the excellent stability of the Sagnac interferometer, a difference phase spectrum was obtained with a high phase change sensitivity of ~lambda/100 after a 1000-shot accumulation. The spectral range, wavelength, and time resolution were 530-650 nm, 1 nm, and 200 fs, respectively.

Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma

A frequency-domain interferometer for probing the variations of the dielectric constant of a plasma with sub-100- fs temporal resolution and lambda/2000 phase resolution is described. Imaging the plasma on the entrance slit of a spectrograph provides spatial resolution along a diameter of the focal spot. The technique is used to map out the expansion of the critical density surface of a femtosecond laser-produced plasma with subnanometer spatial resolution along the laser axis.