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Spatiotemporal compression of ultrashort pulses is one of the key issues of chirped pulse amplification (

The demonstration of the first working laser had opened up a door for photonics, a new technology that revitalized a number of fields of the physical, chemical and biological sciences. One of the early directions of its research was to reach short light pulses to make phenomena visible which are happening too fast even for high-speed photography to resolve. The technological development was so successful that the pulse durations overtook easily the response time of available electronic detectors at the time. Mode locking, as a method for the generation of short pulses, was suggested [

In the meantime, a new laser material was introduced and turned out to be even more advantageous for ultrashort pulse generation. The titanium-doped sapphire crystal [

Once ultrashort laser pulse generation had become a routine process, researchers turned toward reaching extreme high pulse intensities. Relatively early it became obvious that although direct amplification of laser pulses proved to be a successful method in the nanosecond pulse length regime, this technique is not applicable for ultrashort pulses, since self-phase modulation and related effects would arise during amplification. This issue was addressed by G. Mourou and D. Strickland when they suggested

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In experiments observing ultrafast phenomena, it is a fundamental requirement that amplified laser pulses reaching the target should be as short as the Fourier transform of their spectra allows for, thereby providing as high an intensity as possible. The most important parameter in the characterization of the temporal shape is the pulse duration. Since the devices for controlling the temporal shape are usually based on angularly dispersive elements (prisms, grisms, gratings), the residual angular dispersion is also of high interest.

Several diagnostic techniques have been developed so far for ultrashort pulse characterization. A major part of them can be classified as self-reference methods, since the pulse interferes with its own replica usually in a nonlinear crystal or detector. Such techniques are the interferometric autocorrelation [

Another group of diagnostics can be described as linear optical methods, as nonlinear processes are not involved in their schemes. They have the advantage of being able to measure weak pulses with high precision and being sensitive to small changes of the monitored parameters; although in several cases, they are restricted to having a relative measurement of pulse characteristics only. In this paper, we give a detailed overview of this latter category of diagnostic methods and discuss their advantages and disadvantages.

To follow the effect of the linear dispersion of the medium on the temporal shape of a pulse, a laser pulse with a Gaussian temporal envelope of electric field strength is considered, having a transform limited temporal duration (_{0} at the position _{0}, its carrier wave has a frequency of _{0} and—for the sake of simplicity—zero initial phase. In mathematical form, the pulse can be described as

Fourier transform provides a relationship between the spectral and the temporal representation of the laser pulse. In the spectral domain, the complex field strength of the pulse from Equation (1) can be written as

The function of the spectral phase describes the phase evolution of a light wave through any optical system. In the case of optical bulk material, its expression seems relatively simple (see Equation (4)), but in many cases, calculations with the refractive index functions can be rather complicated. The spectral phase expression of sophisticated optical elements or assemblies, like a compressor or stretcher, may become pretty cumbersome. In the experimental practice, moreover, the determination of the full spectral phase function is not possible. Hence, the usual way of handling spectral phase functions is to describe them through their Taylor series as

Once the coefficients of the series is determined to the required (or experimentally feasible) order, the spectral phase function can be constructed in a semi-empirical way, _{0}.

The derivatives of spectral phase, the so-called

The definitions above give

Phase derivatives considerably influence the temporal shape of the envelope. For example, let us consider a medium with first and second order dispersion. The evolution of the electric field will be given by

A pulse with an initial shape described by Equation (1) will have a form of

Schematic temporal effect of the first three phase derivatives.

For few-cycle pulses, the position of the carrier wave under the envelope plays a significant role since the outcome of many experiments could be drastically influenced if the electric field exceeds the ionization threshold by more or less one time during the course of a pulse relative to the previous ones. The quantity for the characterization of this property is the phase at

Definition of the carrier-envelope phase.

The different spectral components of an ultrafast laser pulse may propagate not only at different velocities but also into different directions. In this latter case, the propagation is affected by the

Angular dispersion can be defined in two ways. The most obvious one, based on geometrical optical considerations, is the wavelength dependence of the direction of propagation of the different spectral components, and is called _{PD}_{PF}

Comparison of the two types of angular dispersion in the case of plane wave and Gaussian beam approximation.

Illustration of spatial effects of the angular dispersion and pulse front tilt.

In

This can be converted easily into a stretch in pulse length based on Equation (12) [

Angular dispersion is practically always accompanied by some pulse front tilt and spatial chirp, both demonstrated in _{PD}_{0} per unit length along the cross sectional area [

One can see easily that spatiotemporal pulse distortions usually arise simultaneously; moreover, they are coupled with each other. A general mathematical analysis of these coupling effects is discussed in Reference [

The nature of ultrashort pulses makes it more convenient to examine their properties in the spectral domain instead of the time domain, simply because detectors that are fast enough do not exist yet. However, steady or slowly changing patterns can be created by the means of interferometry. By its nature, it is extremely sensitive to the phase properties of short pulses, which is also very important to know in the temporal reconstruction process. Therefore, the possibility to involve interference methods for ultrafast pulse characterization becomes evident. When interference patterns are recorded in the time domain, multiple shots and pulse scanning are usually required. On the contrary, however, in the spectral domain, it is feasible to have single-shot measurement without moving parts. The spectral distribution of the pulse can be easily measured by spectrographs, and can be transformed to the time domain to calculate the transform-limited pulse duration. Last but not least, some of the special propagation issues may require the detection of the intensity distribution of the pulse in space. The technique called spectrally and spatially resolved interferometry (

Methods using crossed-beam interferometry date back relatively early, well before the age of lasers. L. Puccianti published his first measurement results about the anomalous dispersion of oxy-hemoglobin [

At present, the

How these linear interferometric methods essentially work will be described in the next section, taking the general case of spectrally and spatially resolved interferometry.

The experimental implementation of

The arm of the interferometer containing the dispersive object is called the sample arm; the other one is the reference arm. The spatial and spectral intensity distribution of the laser pulses travelling through the corresponding arms are denoted by _{S}_{R}

Schematic experimental layout of the

In order to establish a spatial resolution, the angle between the crossed beams must also be adjusted. The outcoupling mirrors of the individual arms are tilted vertically so that the beams cross each other at an angle

The interference pattern on the two-dimensional detector surface of the imaging spectrograph is described by

Formation of interference in the case of crossed monochromatic beams.

In the formula above, _{S}_{R}_{0} denotes a reference point, where the phase difference arising from the tilted phase fronts is zero. At this height, the total group delay at the central frequency of _{0}_{0}.

The spectral phase shift in the arms of the interferometer can be approximated by Taylor expansion, similarly to Equation (5). If _{S}_{R}^{med}

A precondition of interference is that the components with identical frequency should coincide within the coherence time and the coherence length. Interference can emerge not only between such laser pulses, which were generated at the same time. Those interferometers which establish interference between consecutive laser pulses at their outputs are called asymmetric interferometers. In this case, the optical path difference of the arms must equal to the multiple of distance between subsequent pulses of the train. Based on Equation (10), pulses propagating in different pathways will overlap each other in time if the difference of the total group delays between the arms and the delay corresponding to _{coh}

In the frames of the spectral resolution of ultrashort laser pulses, the meaning of coherence time slightly differs from its common definition, since here the coherence of the individual modes of the frequency comb must be considered. This falls in the picosecond range, taking into consideration that the line width of a mode is approx. 150 Hz [

Simulated

_{0}, and the interference fringe corresponding to this height is set to be horizontal. In the interferogram, interference fringes diverge in a fan-shaped manner from left to right toward increasing frequencies, thereby increasing also the periodicity of the

Demonstration of separated effect of phase derivates on simulated interferograms. The values of the applied phase derivates are ^{2} (^{3} (

With increasing distance measured from the position _{0} along the spatial axis _{0}.

As it may become obvious from the preceding sections, _{F}_{RP}_{RP}_{RP}

Please note that this is basically a single-shot, linear method which describes the absolute spectrum and the relative spectral phase of the pulse to be characterized, so that the relative temporal shape can be calculated to great accuracy. Linear interferometric methods can be utilized in most of the practical applications like spectroscopy, linear and nonlinear dispersion measurements, time-resolved studies, as well as compressor alignment and beam monitoring along high intensity laser systems. On the one hand, these methods provide the experimentalists with more accurate information than the nonlinear measurements methods, which offer an absolute temporal shape, but with a factor of 2–5 higher error than the linear methods. On the other hand, the linear methods would never be able to provide the absolute pulse duration on their own, since the first-order auto- (and cross-) correlation function to be determined by linear interferometry offers coherence information of the interacting pulses only [

Although the

In the second step, the sample having a geometrical length of _{sample}_{sample}_{R}

After this adjustment, the equation _{S}_{R}^{(0)} and Δ

Depending on the nature of the sample, further measurements are also possible if the spectral phase shift of the sample can be tuned in such a way that neither the sample length nor the phase derivatives of the other components of the interferometer change (for example, changing the temperature of an optical element, the concentration of a solution, or the pressure of a gas). Besides such tuning of a state variable of the sample, equal group delays should be adjusted before each measurement if necessary, and the differences of the phase derivatives will finally be determined. If the dispersions of all the other elements within the system remain constant during the series of measurement, the correlation between the object’s variable quantity and the phase derivatives of the sample’s material can be represented by a fitted function.

After the first experimental demonstration of pulse measurement by the ^{2} and 0.7 fs^{3} precisions in ^{2} and 2 fs^{3} can be considered typical values. The effect of bandwidth, fringe visibility and density, spherical phase fronts (based on Gaussian beam approximation), optical path fluctuations and intensity irregularities were also examined separately in Reference [

When the determination of specific phase derivatives are the aim of an experiment, the precision is affected by the measurement accuracy of the geometrical length of the object. Depending on the length of the object, e.g., interferometric length measurement methods (for under mm scale), micrometer screw gauge (for mm scale), vernier caliper (for mm and cm scale) or laser rangefinder (for cm and m scale) are to be used. Hence, the geometrical length can be determined with an accuracy higher than 0.1 percent, which is at least one order of magnitude better than the accuracy of the phase derivatives.

A modified version of crossed-beam interferometry-based techniques combined with

One of the most common methods used for the measurement of spectral phase shift is the so-called spectrally resolved interferometry,

One of the most widely used Fourier transform-based evaluation methods of _{0}_{0}

This method can be used with broadband laser pulses as well as with continuously operating white light sources, since the only precondition of the measurement is that the spectral bandwidth of the light source should be broad enough.

Steps of the spectral phase calculation by Fourier-transform

When overwhelming difference in dispersion of the sample and reference arms is present,

A typical example of spectrally and spatially resolved interferograms with the stationary phase point.

Let us suppose that the two beams are coaxial, and they reach the detector at the position _{0}. The stationary phase point is located at the central frequency _{0}

It is worth mentioning that the elliptical nature of the two-dimensional fringes (as illustrated in

A spectrograph equipped with two-dimensional detector for this measurement in order to ensure collinearity has been suggested. However, the position of the stationary phase point can be determined even if only a one-dimensional detector (e.g., a diode array) is available, positioned parallel to the wavelength axis. Stationary phase point method does not exploit the information about the dispersion of the sample encoded in the two-dimensional structure of the fringes.

The accuracy of measurement by

One of the simplest methods of interferogram evaluation is based on cosine function fitting. This method requires the intensity distribution of the individual arms. Substituting these values into Equation (16), a normalized intensity
^{th} column, corresponding to the angular frequency _{i}_{i}_{i}

Steps of the phase extraction from the interferograms.

From the aspect of the spectral phase, the values of _{i}_{i}_{i}

The influence of the

Similarly to the dispersion coefficients, the _{0}

Not only the relative

Schematic layout for the linear detection of the

Instead of using Michelson and Mach–Zehnder interferometers, a resonant ring layout was implemented to keep information of the pulse phase for several round trips (see

If the _{0}

The significance of this technique was emphasized by the

The characterization of the angular dispersion of ultrashort laser pulses is inevitable when complete spatiotemporal compression of the pulses is necessary. When the angular dispersion is not compensated carefully, it can lead to an unexpected decrement in the peak intensity. At the same time, spectral components may also be separated spatially, thereby inducing a so-called spatial chirp, which involves an asymmetry of the spectral content in the beam profile. A direct consequence of this spatial chirp is the tilt of the phase front, which raises some further unwanted increase in the pulse duration. Elimination of angular dispersion (

A simple and direct method to measure of propagation direction angular direction is based on spectrally and spatially recorded intensity distribution of the focused spot of the beam. The layout of the technique is shown in

Schematic diagram for measuring the propagation direction angular dispersion.

An achromatic lens or spherical mirror is used to focus the beam on the slit of the spectrograph. In case of angular dispersion, spectral components have different propagation directions, therefore, in the focal plane, also a different spatial position. This position

Unfortunately, this method can only detect the angular dispersion along the cross-section of the beam, which is defined by the slit of the spectrograph. In most cases, it is highly desired to ensure that the beam is free of angular dispersion in both horizontal and vertical direction. It can be executed only when the beam or the spectrograph is rotated by 90°. For this purpose, it is most convenient to use a mechanical beam rotator, which is practically a twistable periscope, but does not alternate the direction of the beam when it is passing through.

Since this method is limited to measure along one spatial dimension only, at least two subsequent measurements required for complete beam profile characterization. This leads to a lengthy and iterative alignment of the stretcher-compressor stages of the

When a broadband light beam is affected by angular dispersion, its spectral components will propagate in various directions. If the beam is focused directly on the sensor chip of a two-dimensional detector by an achromatic imaging element, a relatively elongated spot will appear in the image compared to a beam without the effect of dispersion. The intensity distribution of this elongated spot is in direct correlation with the spectral angular deviation. However, the spot itself is not suitable for accurate measurements. On one hand, chromatic aberration of the spot must be separated from optical aberrations of different nature. On the other hand, this elongated spot contains no information on the spectral calibration required for the measurement,

A method for single-shot, two-dimensional measurement of propagation direction angular dispersion of broadband light sources (e.g., ultrashort laser beams) was introduced recently. In this case, spectral calibration is achieved by spectrally filtering the beam in order to create well-separated peaks in the spectrum. Since these components are still overlapping spatially, we use an achromatic lens to image them onto a two-dimensional detector. In this way, the spectrally separated components of an angularly dispersed beam will appear as dissociated spots on the surface of the detector according to the orientation of the angular dispersion.

The role of the spectral filter can be filled with various, either passive or active solutions. For example, multiple bandpass interference filters or wavelength-selective reflectors can be used for this purpose. Another very simple idea is to block parts of the spectrally resolved beam in the stretcher; although it might be not suitable, if the stretcher itself is the object of examination. Acousto-optical programmable modulators can be used also to create an arbitrarily modified spectrum. A less expensive solution to create separated spectral peaks is the use of a Fabry–Perot interferometer (

Typical layout for simple, two-dimensional detection of propagation dispersion angular dispersion (

The scheme of the technique can be seen in

When the beam has weak angular dispersion only, it is possible that the spectral spots cannot be separated effectively with the base length modulation of the

Experimental verification with different prisms showed that the precision does not decrease significantly compared to the one-dimensional method, as standard deviation from the expected values was 0.15 μrad/nm [

The _{PF}

Experimental set-up of the interferometric detection of phase front angular dispersion with an inverted Mach–Zehnder interferometer.

The angle between the phase fronts will then be measured by spectrally and spatially resolved interferometry, and will depend on the wavelength due to the angular dispersion. If a small angle is adjusted between the beams in the direction of polarization, then this (wavelength-independent) angle will also occur between the phase fronts, and thereby the interference pattern will be modulated in the direction of polarization. The frequency of this modulation is directly proportional to the angle and to the frequency of the light. The manually adjusted angle between the beams can be measured as the function of the wavelength by a two-dimensional imaging spectrograph having a slit oriented in the direction of polarization. The derivative of the measured angle with respect to the wavelength results in a value twice the phase front angular dispersion occurring in the direction of polarization.

The angle between the phase fronts as defined by Equation (17) will depend on the wavelength. Using the parameters _{i}

An inappropriate adjustment of the optical elements in a

Accuracy of the phase front angular dispersion depends on the detector noise similarly as it was shown in the case of

Spatial invertion of the beam profile can be effectively used in nonlinear autocorrelators as well. Since no spectral dimension is needed during the evaluation, both spatial directions can be inverted, hence pulse front tilt can be detected in 2D in a single-shot measurement [

Experiments with intense ultrashort pulses require precise characterization of both spatial and temporal properties. For this purpose, the class of diagnostic tools based on linear schemes has been found to be simple, fast and reliable. In this review article, we demonstrated their capabilities and some of their potential applications. The combination of interferometry with spectrally resolved detection provides a flexible, multifunctional technique, which can be displayed best by spectrally and spatially resolved interferometry (

The authors are grateful for Mihaly Gorbe and Peter Jojart for the helpful discussions. Financial support from the European Union, the European Social Fund through grant no. TAMOP 4.2.2/B-10/1-2010-0012, the Hungarian Scientific Research Found (OTKA) under grant no. K75149, and the EU FP7-Infrastructures-2011-1 program (contract 284464, Laserlab-Europe––The Integrated Initiative of European Laser Research Infrastructures III) are acknowledged.

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