Spectral interferometry

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Spectral interferometry (SI) or frequency-domain interferometry is a linear technique used to measure optical pulses, with the condition that a reference pulse that was previously characterized is available. This technique provides information about the intensity and phase of the pulses. SI was first proposed by Claude Froehly and coworkers in the 1970s. [1] [2]

Contents

A known (acting as the reference) and an unknown pulse arrive at a spectrometer, with a time delay between them, in order to create spectral fringes. A spectrum is produced by the sum of these two pulses and, by measuring said fringes, one can retrieve the unknown pulse. If and are the electric fields of the unknown and reference pulse respectively, the time delay can be expressed as a phase factor for the unknown pulses. [3] Then, the combined field is:

The average spacing between fringes is inversely proportional to the time delay . [4] Thus, the SI signal is given by:

where is the oscillation phase.

Furthermore, the spectral fringes width can provide information on the spectral phase difference between the two pulses ; narrowly spaced fringes indicate rapid phase changes with frequency. [5]

Comparison with the Time Domain

Compared to time-domain interferometry, SI presents some interesting advantages. Firstly, by using a CCD detector or a simple camera, the whole interferogram can be recorded simultaneously. Furthermore, the interferogram is not nullified by small fluctuations of the optical path, but reduction in the fringe contrast should be expected in cases of exposure time being bigger than the fluctuation time scale. [6] However, SI produces phase measurements through its cosine only, meaning that results arise for phase differences in multiples of which can lead to solutions that degrade the signal-to-noise ratio.

There have been efforts to measure pulse intensity and phase in both the time and the frequency domain by combining the autocorrelation and the spectrum. This technique is called Temporal Information Via Intensity (TIVI) [7] and it involves an iterative algorithm to find an intensity consistent with the autocorrelation, followed by another iterative algorithm to find the temporal and spectral phases consistent with the intensity and spectrum, but the results are inconclusive.

Applications

Spectral Interferometry has gained momentum in recent years. It is frequently used for measuring the linear response of materials, such as the thickness and refractive index of normal dispersive materials, [8] the amplitude and phase of the electric field in semiconductor nanostructures [9] and the group delay on laser mirrors. [10]

In the realm of femtosecond spectroscopy, SI is the technique on which SPIDER is based, thus it is used for four-wave mixing experiments [11] [12] [13] and various phase-resolved pump-probe experiments. [4] [14]

Experimental Difficulties

This technique is not commonly used since it relies on a number of factors in order to obtain strong fringes during experimental processes. Some of them include: [15]

Spectral Shearing Interferometry

In cases of relatively long pulses, one can opt for Spectral Shearing Interferometry. For this method, the reference pulse is obtained by sending its mirror image through a sinusoidal phase modulation. Hence, a spectral shift of magnitude can be correlated to the produced linear temporal phase modulation and the spectrum of the combined pulses then has a modulation phase of:

where the approximate relation is appropriate for small enough . Thus, the spectral derivative of the phase of the signal pulse which corresponds to the frequency-dependent group delay can be obtained. [16] [17] [18]

Spectral Phase Interferometry for Direct Electric-field Reconstruction

SPIDER set-up Spectral phase interferometry for direct electric-field reconstruction.png
SPIDER set-up

Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) is a nonlinear self-referencing technique based on spectral shearing interferometry. For this method, the reference pulse should produce a mirror image of itself with a spectral shift, in order to provide the spectral intensity and phase of the probe pulse via a direct Fast Fourier Transform (FFT) filtering routine. However, unlike SI, in order to produce the probe pulse phase, it requires phase integration extracted from the interferogram.

Self-Referenced Spectral Interferometry

Self-Referenced Spectral Interferometry (SRSI) is a technique where the reference pulse is self created from the unknown pulse being. The self referencing is possible due to pulse shaping optimization and non-linear temporal filtering. [19] [20] It provides all the benefits associated with SI (high sensitivity, precision and resolution, dynamic and large temporal range) but, unlike the SPIDER technique, neither shear nor harmonic generation are necessary in order to be implemented.

For SRSI, the generation of a weak mirror image of the unknown pulse is required. That image is perpendicularly polarized and delayed with respect to the input pulse. Then, in order to filter the reference pulse in the time domain, the main portion of the pulse is used for cross-polarized wave generation (XPW) in a nonlinear crystal. [21] The interference between the reference pulse and the mirror image is recorded and analyzed via Fourier transform spectral interferometry (FTSI). [6] Known applications of the SRSI technique include the characterization of pulses below 15 fs. [22]

Frequency-Resolved Optical Gating

Frequency Resolved Optical Gating (FROG) is a technique that determines the intensity and phase of a pulse by measuring the spectrum of a particular temporal component of said pulse. [23] This results in an intensity trace, related to the spectrogram of the pulse , versus frequency and delay:

where is a variable-delay gate pulse. FROG is commonly combined with Second Harmonic Generation (SHG) process (SHG-FROG).

But the same principle can be applied exploiting different physical process, like polarization-gated FROG (PG-FROG) or transient-grating FROG (TG-FROG). [24]

Other Linear Techniques

There is a variety of linear techniques that are based on the main principles of spectral interferometry. Some of them are listed below.

Dual-Quadrature Spectral Interferometry
The acquisition of the two quadratures of the interference signal resolves the issue generated by the phase differences being expressed in multiples of . The acquisition should happen simultaneously via polarization multiplexing, with the reference beam under circular polarization. [6]
Fourier-Transform Spectral Interferometry
It is a technique created for direct determination of , mainly used for femtosecond pump-probe experiments in materials with long dephasing times. [25] [26] It is based on the inverse Fourier transform of the signal:

Related Research Articles

In signal processing, group delay and phase delay are two related ways of describing how a signal's frequency components are delayed in time when passing through a linear time-invariant (LTI) system. Phase delay describes the time shift of a sinusoidal component. Group delay describes the time shift of the envelope of a wave packet, a "pack" or "group" of oscillations centered around one frequency that travel together, formed for instance by multiplying a sine wave by an envelope.

<span class="mw-page-title-main">Chirp</span> Frequency swept signal

A chirp is a signal in which the frequency increases (up-chirp) or decreases (down-chirp) with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly applied to sonar, radar, and laser systems, and to other applications, such as in spread-spectrum communications. This signal type is biologically inspired and occurs as a phenomenon due to dispersion. It is usually compensated for by using a matched filter, which can be part of the propagation channel. Depending on the specific performance measure, however, there are better techniques both for radar and communication. Since it was used in radar and space, it has been adopted also for communication standards. For automotive radar applications, it is usually called linear frequency modulated waveform (LFMW).

<span class="mw-page-title-main">Transmittance</span> Effectiveness of a material in transmitting radiant energy

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<span class="mw-page-title-main">Optical coherence tomography</span> Imaging technique

Optical coherence tomography (OCT) is an imaging technique that uses interferometry with short-coherence-length light to obtain micrometer-level depth resolution and uses transverse scanning of the light beam to form two- and three-dimensional images from light reflected from within biological tissue or other scattering media. Short-coherence-length light can be obtained using a superluminescent diode (SLD) with a broad spectral bandwidth or a broadly tunable laser with narrow linewidth. The first demonstration of OCT imaging was published by a team from MIT and Harvard Medical School in a 1991 article in the journal Science. The article introduced the term "OCT" to credit its derivation from optical coherence-domain reflectometry, in which the axial resolution is based on temporal coherence. The first demonstrations of in vivo OCT imaging quickly followed.

In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

In physics, the Hanbury Brown and Twiss (HBT) effect is any of a variety of correlation and anti-correlation effects in the intensities received by two detectors from a beam of particles. HBT effects can generally be attributed to the wave–particle duality of the beam, and the results of a given experiment depend on whether the beam is composed of fermions or bosons. Devices which use the effect are commonly called intensity interferometers and were originally used in astronomy, although they are also heavily used in the field of quantum optics.

<span class="mw-page-title-main">Chirped pulse amplification</span>

Chirped pulse amplification (CPA) is a technique for amplifying an ultrashort laser pulse up to the petawatt level, with the laser pulse being stretched out temporally and spectrally, then amplified, and then compressed again. The stretching and compression uses devices that ensure that the different color components of the pulse travel different distances.

Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

Frequency-resolved optical gating (FROG) is a general method for measuring the spectral phase of ultrashort laser pulses, which range from subfemtosecond to about a nanosecond in length. Invented in 1991 by Rick Trebino and Daniel J. Kane, FROG was the first technique to solve this problem, which is difficult because, ordinarily, to measure an event in time, a shorter event is required with which to measure it. For example, to measure a soap bubble popping requires a strobe light with a shorter duration to freeze the action. Because ultrashort laser pulses are the shortest events ever created, before FROG, it was thought by many that their complete measurement in time was not possible. FROG, however, solved the problem by measuring an "auto-spectrogram" of the pulse, in which the pulse gates itself in a nonlinear-optical medium and the resulting gated piece of the pulse is then spectrally resolved as a function of the delay between the two pulses. Retrieval of the pulse from its FROG trace is accomplished by using a two-dimensional phase-retrieval algorithm.

<span class="mw-page-title-main">Spectral phase interferometry for direct electric-field reconstruction</span>

In ultrafast optics, spectral phase interferometry for direct electric-field reconstruction (SPIDER) is an ultrashort pulse measurement technique originally developed by Chris Iaconis and Ian Walmsley.

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In optics, femtosecond pulse shaping refers to manipulations with temporal profile of an ultrashort laser pulse. Pulse shaping can be used to shorten/elongate the duration of optical pulse, or to generate complex pulses.

Multiphoton intrapulse interference phase scan (MIIPS) is a method used in ultrashort laser technology that simultaneously measures, and compensates femtosecond laser pulses using an adaptive pulse shaper. When an ultrashort laser pulse reaches a duration of less than a few hundred femtosecond, it becomes critical to characterize its duration, its temporal intensity curve, or its electric field as a function of time. Classical photodetectors measuring the intensity of light are still too slow to allow for a direct measurement, even with the fastest photodiodes or streak cameras.

The method of reassignment is a technique for sharpening a time-frequency representation by mapping the data to time-frequency coordinates that are nearer to the true region of support of the analyzed signal. The method has been independently introduced by several parties under various names, including method of reassignment, remapping, time-frequency reassignment, and modified moving-window method. In the case of the spectrogram or the short-time Fourier transform, the method of reassignment sharpens blurry time-frequency data by relocating the data according to local estimates of instantaneous frequency and group delay. This mapping to reassigned time-frequency coordinates is very precise for signals that are separable in time and frequency with respect to the analysis window.

Ultrasound-modulated optical tomography (UOT), also known as Acousto-Optic Tomography (AOT), is a hybrid imaging modality that combines light and sound; it is a form of tomography involving ultrasound. It is used in imaging of biological soft tissues and has potential applications for early cancer detection. As a hybrid modality which uses both light and sound, UOT provides some of the best features of both: the use of light provides strong contrast and sensitivity ; these two features are derived from the optical component of UOT. The use of ultrasound allows for high resolution, as well as a high imaging depth. However, the difficulty of tackling the two fundamental problems with UOT have caused UOT to evolve relatively slowly; most work in the field is limited to theoretical simulations or phantom / sample studies.

Metal-mesh optical filters are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

Laser linewidth is the spectral linewidth of a laser beam.

An anamorphic stretch transform (AST) also referred to as warped stretch transform is a physics-inspired signal transform that emerged from time stretch dispersive Fourier transform. The transform can be applied to analog temporal signals such as communication signals, or to digital spatial data such as images. The transform reshapes the data in such a way that its output has properties conducive for data compression and analytics. The reshaping consists of warped stretching in the Fourier domain. The name "Anamorphic" is used because of the metaphoric analogy between the warped stretch operation and warping of images in anamorphosis and surrealist artworks.

<span class="mw-page-title-main">Nano-FTIR</span> Infrared microscopy technique

Nano-FTIR is a scanning probe technique that utilizes as a combination of two techniques: Fourier transform infrared spectroscopy (FTIR) and scattering-type scanning near-field optical microscopy (s-SNOM). As s-SNOM, nano-FTIR is based on atomic-force microscopy (AFM), where a sharp tip is illuminated by an external light source and the tip-scattered light is detected as a function of tip position. A typical nano-FTIR setup thus consists of an atomic force microscope, a broadband infrared light source used for tip illumination, and a Michelson interferometer acting as Fourier-transform spectrometer. In nano-FTIR, the sample stage is placed in one of the interferometer arms, which allows for recording both amplitude and phase of the detected light. Scanning the tip allows for performing hyperspectral imaging with nanoscale spatial resolution determined by the tip apex size. The use of broadband infrared sources enables the acquisition of continuous spectra, which is a distinctive feature of nano-FTIR compared to s-SNOM. Nano-FTIR is capable of performing infrared (IR) spectroscopy of materials in ultrasmall quantities and with nanoscale spatial resolution. The detection of a single molecular complex and the sensitivity to a single monolayer has been shown. Recording infrared spectra as a function of position can be used for nanoscale mapping of the sample chemical composition, performing a local ultrafast IR spectroscopy and analyzing the nanoscale intermolecular coupling, among others. A spatial resolution of 10 nm to 20 nm is routinely achieved.

Quantum optical coherence tomography (Q-OCT) is an imaging technique that uses nonclassical (quantum) light sources to generate high-resolution images based on the Hong-Ou-Mandel effect (HOM). Q-OCT is similar to conventional OCT but uses a fourth-order interferometer that incorporates two photodetectors rather than a second-order interferometer with a single photodetector. The primary advantage of Q-OCT over OCT is insensitivity to even-order dispersion in multi-layered and scattering media.

References

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  19. ↑ Oksenhendler, T.; Coudreau, S.; Forget, N.; Crozatier, V.; Grabielle, S.; Herzog, R.; Gobert, O.; Kaplan, D. (April 2010). "Self-referenced spectral interferometry". Applied Physics B. 99 (1–2): 7–12. Bibcode:2010ApPhB..99....7O. doi:10.1007/s00340-010-3916-y. S2CID   124727290.
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