1. Introduction

1.1. Need for Ultrafast Metrology

The development of mode-locked lasers in the mid-1960s gave rise to the problem of ultrashort pulse measurement, since optical pulses generated by this class of lasers were of significantly shorter duration than any photodetector response time. Despite the vastly increased capabilities of modern photodetectors in terms of both speed of response and sensitivity, the equally dramatic improvements in laser technology have sustained this disparity, and, indeed, with the emergence of attosecond pulse trains have extended it.

The need for metrology has increased along with the development of new sources and their application in a wide range of new fields. Of course, determining the pulse durations remains critical, both because this parameter is an important specification of the laser output needed for other applications and because it provides a diagnosis of the system operation.

Modern mode-locked lasers, for example, generate pulses with spectral bandwidths exceeding one octave, corresponding to pulses the brevity of which is well beyond anything that can be characterized by means of fast photodetectors. The operation of such lasers relies on a complex combination of linear pulse propagation, influenced by the chromatic dispersion of the laser material, the mirrors, and the intracavity dispersion-compensating devices, together with nonlinear effects, such as self-phase modulation of the pulse in the laser material or saturation of an intracavity absorption, such as in a semiconductor saturable absorber mirror (SESAM), as well as, in some cases, space–time coupling. The optimization of a mode-locked laser is made practicable by means of a diagnostic providing the electric field as a function of time or frequency, or at least providing some temporal information such as the second-order intensity autocorrelation [1, 2, 3, 4, 5, 6, 7]. Figure 1 displays the characterization results of the output pulse from a Ti:sapphire oscillator. One of the primary limits at present to the generation of few-cycle pulses directly from a laser is the dispersion of the intracavity mirrors and other optical elements. Historically, detailed measurements of laser output were able to identify this as a major obstacle to generating pulses of greater brevity.

Chirped pulse amplification (CPA) operates by lowering the peak power of the pulses in the amplifier gain medium, which would otherwise induce nonlinear phase distortion of the pulse or damage to the amplification medium [8, 9]. To achieve this, the pulses are stretched in time by means of a dispersive delay line, often based on angular dispersion from diffraction gratings or prisms. After amplification, the pulse is temporally recompressed by using an inverse dispersive delay line, or compressor, that compensates for the dispersion introduced by the stretcher and the propagation through the other amplifier elements. Obtaining peak performance from such a scheme requires a reliable and rapid method to characterize the output. Accurate characterization of the output pulses enables the optimization of the parameters of the system, such as the distance between the two gratings of the compressor and the angle of incidence of the input beam on the gratings. The usual optimization parameters in such an application are the duration of the recompressed pulses, since the peak power scales as the ratio of the energy per pulse to the duration, and the temporal contrast, since prepulses can hinder the control or observation of the physical processes of interest, for example the ionization of a target. Examples of optimization of CPA systems can be found in [10, 11]. Figure 2 presents an example of CPA optimization obtained with spectral phase interferometry for direct electric-field reconstruction (SPIDER) [11]. The spectral phase of the output pulse from a Ti:sapphire CPA system is plotted before and after optimization. The compressor optimization consisted in adjusting the angle of the diffraction gratings relative to the input beam and the relative distance between the two gratings. The large cubic spectral phase gives rise to significant prepulses, and the compressor optimization leads to a better pulse shape with a higher peak intensity.

The bandwidth of an optical pulse can be increased while maintaining a deterministic phase relation between different spectral components by means of various nonlinear optical processes such as self-phase modulation and harmonic generation. All of these require careful compensation of the spectral phase in order to lead to an output pulse with a shorter duration than the input. Further, these processes are dynamically complicated and sensitive to the details of the input pulse shape. Therefore, even characterizing the raw output pulse before recompression can be a difficult task [12, 13, 14, 15]. Figure 3 shows the characteristics of a filament pulse compressor that allows the generation of high-energy ultrashort optical pulses [16]. The output pulses have complicated spectral and temporal structures, and correlation between time and frequency can be visualized in the chronocyclic time–frequency space by calculating the spectrogram of the output electric field.

Shaped pulses, sometimes of a quite complex temporal structure, are now commonly used to both probe and manipulate fundamental processes in atoms and molecules [17, 18]. For instance, the study of primary processes in biologically relevant systems via ultrafast microscopy is now quite common. The details of the pulse shapes usually contain important information about the dynamical process under study, and this information, residing in both the temporal amplitude and the temporal phase of the field, can be extracted only by using modern techniques of metrology. For example, the important phenomenon of the self-action of intense optical pulses in nonlinear media gives rise to a complicated set of dynamics that has analogs in many branches of physics. The study of the changes in the shape of pulses propagating through such media provides access to these dynamics. Optical pulse shaping can also be used to generate trains of pulses useful in optical telecommunications or to generate shaped electrical waveforms after optical-to-electrical conversion by a photodetector. Figure 4 displays the intensity of a train of pulses generated by an optical pulse shaper based on a liquid crystal spatial light modulator placed in a zero-dispersion line.

There are also important technological applications of metrology. In optical telecommunication systems, such metrology is used to characterize modulators and the dispersion of fiber links. The propagation of light pulses carrying bits of information from transmitter to receiver demands long-distance transmission through various passive and active elements. Both linear processes (e.g., the frequency-dependent transmission and phase of the medium) and nonlinear processes (e.g., the intensity-dependent index and absorption) modify the electric field of the pulses, and these effects must be quantified in order to maximize the overall system performance. There is also a need to optimize the shape of the pulses that are used, which are typically carved from a continuous-wave (cw) quasi-monochromatic source by a modulator. Although telecommunication pulses typically have durations ranging from $1\mathrm{ps}$ to $1\mathrm{ns}$, the deleterious effects of propagation are significant because propagation distances of the order of $1000\mathrm{km}$ in a physical medium can be involved. Moreover, the pulses used in state-of-the-art commercial and research optical telecommunication systems are beyond the reach of all-electronic characterization. Further, temporal phase information is also needed. A review of high-speed diagnostics for optical telecommunication systems is presented in [19], and some examples of diagnostics used in the telecommunication environment can be found in [20, 21, 22, 23]. Figure 5 presents results of a pulse carver optimization using a real-time pulse characterization diagnostic based on linear spectrograms [24]. The electric field of the output of a Mach–Zehnder modulator driven by a $20\mathrm{GHz}$ sinusoidal RF drive depends on the phase difference between the two arms of the interferometer, which is controlled by a continuous voltage. The modulation format can be set to 33% return to zero ($8\mathrm{ps}$ pulses with identical phases) or 67% carrier-suppressed return to zero ($17\mathrm{ps}$ pulses with a π phase shift between adjacent pulses). Data-encoded optical signals require diagnostics that can acquire an invertible experimental trace in a single shot or can gather statistically significant samples of an optical waveform. Intensity sampling diagnostics use nonlinear cross-correlation schemes, while sampling systems based on homodyne detection are sensitive to the electric field of the waveform under test. These diagnostics are not detailed in this review, and relevant references can be found in [19].

Thus ultrafast metrology continues as an active field of research. In this review, we outline the basic approaches to pulse-field measurement and describe in detail several of the most popular and powerful methods. The aim is both to summarize the state of the art in this rapidly moving field and to provide sufficient analysis and design criteria that a researcher may begin to implement these methods in the laboratory.

1.2. Historical Developments

Considerable insight into the field can be gained from a look at the history of ultrafast metrology. Therefore we outline in a more or less chronological order the major advances over the past nearly four decades, since the invention of mode locking. To prefigure the structure of the review, the chronology is given in terms of several threads that have led to distinct techniques.

Of course, much has been written on the subject in recent years, and a number of excellent reviews of a few methods exist. A review of pulse measurement methods prior to 1974 can be found in the article by Bradley and New [25]. A review of concepts for shaping and analysis of short optical pulses can be found in a 1983 article by Froehly and coworkers [26], and a summary of methods available up to 1990 in the chapter by Laubereau in the book edited by Kaiser [27]. In more recent developments, a comprehensive description of frequency-resolved optical gating (FROG) is given in a book edited by Trebino [28], and a broader treatment of the field in the context of ultrafast optics is to be found in the book by Diels and Rudolph [29].

For pulses in the range of several picoseconds or longer, the temporal intensity can be measured by using a streak camera or a photodiode. Combined with a measurement of the spectrum of the pulse, this information can be used to provide a reasonable characterization. For pulses in the femtosecond and indeed attosecond range such methods are not possible, in part because detectors that can absorb across the spectral range of these pulses are not always available, but mostly because direct photodetection is not fast enough.

Thus a different approach is required, one that avoids the need for fast detectors. Nonetheless, it is clear that something with a response time as brief as the pulse itself is needed, and the initial work in the field made use of the most obvious short event to hand—the pulse itself. This was used to synthesize a rapidly responding material excitation by means of the nonlinear optical responses of several common processes and materials. This trend has continued, though it is now understood that measurements using linear systems may also provide sufficient information to measure a pulse field.

Historically, the lack of fast detectors led to the adoption of nonlinear optical processes for the purposes of pulse characterization. An early technique, and one that dominated the field for many years, was the measurement of the intensity autocorrelation of a pulse. This relies on the observation that the efficiency of a nonlinear process (such as second-harmonic generation) is higher for higher input intensities. Thus the second-harmonic signal when a pair of pulses is incident on a nonlinear crystal is greater when they arrive at the same time, as opposed to when they arrive separately. Therefore a measurement of the second-harmonic power as a function of the delay gives an estimate of the pulse duration. Although it is possible to determine something of the time dependence of the phase of the pulse from more sophisticated versions of the autocorrelation, it is not possible to get a complete map. Nonetheless, it was also understood that the temporal structure of the pulses was strongly dependent on the spectral phase, and various representations of the pulse were developed to help visualize this and to develop new methods of measurement. Among the most fruitful of these was the spectrogram, consisting of a time–frequency map that plotted what was called the instantaneous frequency of the pulse.

Many current methods of metrology borrow heavily from methods developed in other branches of optics, notably imaging and testing. The strong analogy between space and time in Maxwell’s equations was employed first in metrology in the concept of the “time lens.” This employs a temporal phase modulator (a time-domain analog to a spatial phase modulator—a lens) and a dispersive delay line (a frequency-domain analog to free-space propagation) to generate a time-stretched replica of the input pulse whose temporal intensity can be measured by using a relatively slow detector.

The ideas of time–frequency representations have proved to be one of the most lasting in metrology, through both spectrography and its cousin sonography. The former builds on the original notion by developing methods to measure the spectrum of sequential time slices of the test pulse. The latter, in contrast, measures the time dependence of adjacent spectral slices. The relationship of these time–frequency (or chronocyclic) representations to various pulse measurement schemes has been an important source of ideas in ultrafast metrology.

The most common form of spectrography relies on the idea that the nonlinear mechanism used to measure an autocorrelation effectively provides a time gate. Since the second-harmonic intensity is largest when the two replicas of the test pulse overlap in time, this mechanism can be thought of as the test pulse selecting a time slice of itself. Resolving the spectrum of the resulting second-harmonic radiation, rather than simply measuring the total second-harmonic power, then provides more information than the autocorrelation alone. This notion was developed in various ways, but a major breakthrough came when it was realized that the pulse field could be retrieved from spectrograms measured in this way by using methods from image processing. This idea forms the basis of one of the currently popular nonlinear methods—frequency-resolved optical gating. This form of spectrography has spawned numerous offspring, buoyed by the discovery of a powerful inversion algorithm based on the recognition that a matrix representation of the spectrogram should be of rank one. This allows powerful singular value decomposition methods to extract the most appropriate fields by iteration.

Sonography has also developed considerably since its first demonstration contemporaneously with frequency-resolved optical gating. Sonography requires a detector of reasonable speed, usually synthesized by using a nonlinear mechanism, and pulse reconstruction can be accomplished rapidly by using a deterministic algorithm applicable to a modified sonogram or an image-processing-related iterative inversion.

A second analogy from optics that has proved equally fruitful for pulse characterization is interferometry. This is a well-known and sensitive method for extracting phase information about an optical field and is commonly employed in precision metrology. The measurement of the time-dependent phase of an optical pulse was first demonstrated by interfering it with a reference pulse of known character. In this case, a source with a narrowband spectrum provides a usable reference. This is analogous to using a point source as a reference wave in optical testing. It is possible to make this into a self-referencing interferometer by selecting the narrow frequency band for the reference from the test pulse spectrum. This is analogous to the generation of a spatial reference wave in optical testing by selecting a single point from the input beam. The temporal interference pattern obtained by combining different frequencies from the test pulse spectrum, recorded for example by using a cross-correlation (synthesized by using the same nonlinear mechanisms as had been developed for the autocorrelator), enables the relative phases of the two spectral components to be determined.

A different approach to interferometry avoids the need for a fast detector and is instead based on measurements made in the frequency domain. In spectral interferometry, a test pulse is gauged by using a known reference, and the phase difference extracted by using a noniterative algorithm. This method, first applied to the measurement of pulse distortions through propagation, was shown to be an extremely sensitive tool for pulse characterization, capable of attaining the quantum limit for photodetection. The proposal of self-referencing spectral interferometry showed how the lack of a known reference pulse could be circumvented by interfering the test pulse with a frequency-shifted (or spectrally sheared) version of itself and measuring the resulting two-pulse spectrum. A nonlinear implementation of this idea—SPIDER—retains the direct and unique inversion characteristic of interferometry with the ability to acquire and process data on individual laser pulses at rates up to $1\mathrm{kHz}$. This opens the way to measurements of statistical properties of pulse trains.

The final optical imaging analogy that has proved useful in pulse measurement is chronocyclic tomography. In this approach, the pulse field is reconstructed from a set of spectra after phase modulation. The name comes from the idea that these spectra represent projections of the chronocyclic phase-space representation of the field in the form of a Wigner function. The phase modulation serves to rotate the phase space, thus giving a series of one-dimensional sections of this two-dimensional entity. Further analogs from optical imaging have been used to develop simplified versions of this method that require significantly fewer measurements at the cost of some assumptions about the character of the input pulses. One of the earliest attempts to reconstruct pulse fields this way was to use the dispersive properties of glass to temporally stretch the pulse, then to determine the time dependence of the stretched pulse intensity by using an intensity cross-correlation with the unstretched pulse. This idea was further extended by using phase retrieval algorithms. Several approaches to tomographic pulse reconstruction have also been made by using self- and cross-phase modulation to achieve the phase-space rotation, coupled with measurements of the spectrum of the modulated pulse. Because the nonlinear mechanisms involve an ancillary pulse that must have a known (if not precisely controlled) shape, the best approach to inverting such measurements also makes use of iterative image processing algorithms.

Although most of the development of metrology for ultrafast pulses has made use of nonlinear optical processes, this turns out to be an artifact of the time scales involved rather than a fundamental restriction. In fact, it has been shown that complete characterization can be achieved by using entirely linear optical filters, such as spectrometers and temporal modulators. The only requirements are that the apparatus consist of at least one filter with a time-stationary response (e.g., a spectrometer) and at least one with a time-nonstationary response (e.g., a phase modulator) [30]. All of the above-mentioned classes of measurement can be formulated in this way. It is only in recent years, however, that it has been possible to get active optical elements that have time and modulation scales appropriate for operating on subpicosecond pulses. Nonetheless, since modulation and photodetection with sub-$100\mathrm{ps}$ response times are required for a $40\mathrm{Gbit}∕\mathrm{s}$ optical telecommunications system, these elements are also available to build linear methods that have proved very useful in assessing the performance of systems and components in this application. Interferometric, spectrographic, and tomographic methods have been implemented by using linear temporal modulators and spectral measurements, with time-integrating or “slow” detectors (i.e., with electrical bandwidths much less than $40\mathrm{GHz}$).

If one goes beyond purely integrating detectors that measure only the energy of a pulse or the power in a pulse train, it is possible to achieve a different sort of measurement by using linear optics. In this case, using linear filters, one can, for example, select two different frequencies from the pulse spectrum and mix them on a fast photodiode. The phase of the resulting modulation of the photocurrent is closely related to the phase difference of the two optical components.

The subsequent sections of this review deal with each of these approaches in detail, providing both an analysis of the methods and a description of the current state of the art. To begin, a general analysis of pulse characterization is described in Section 2. This covers all known methods and indicates the necessary minimum conditions that all apparatuses must satisfy in order to operate successfully. The following sections describe each of the major approaches in turn: spectrography in Section 3, tomography in Section 4, and interferometry in Section 5. Some current areas of research are described in Section 6, together with the conclusions.

2. General Principles and Concepts of Pulse Characterization

2.1. Concepts and Protocols

What apparatus is required to characterize an optical pulse? Given the plethora of techniques purporting to achieve this aim, it is worth considering what are the necessary and sufficient conditions that must be satisfied by any method that provides a complete specification of an ultrashort pulse field. It is possible to formulate such conditions quite generally in terms of the theory of linear filters. The fact that this is possible already implies that apparatuses based entirely on linear optical elements are capable of pulse characterization, something that was not appreciated until relatively recently [31]. In practice, many of the popular methods make use of nonlinear optical processes, but this is because it has proved difficult to construct linear filters of the correct character or response time, rather than for any fundamental reason.

The inversion protocols for extracting the pulse shape from measured data are also made clear by working with linear transformations, which allows a categorization of different experimental methods and the development of a catalog of what is possible in principle. An important feature introduced by the use of nonlinear optics is that the inversion algorithms can become more complicated. In some cases they remain deterministic, but in others an iterative search for a solution satisfying the twin constraints of the signal form and the data must be implemented. Thus the two major considerations in pulse characterization are the physical arrangement of the linear and nonlinear components and the inversion procedure [32].

2.1a. Representation of Pulsed Fields

Before describing how pulse measurement methods operate, it will be well to set out some definitions and to delineate exactly what we mean by pulse characterization. An electromagnetic pulse may be specified by its electric field alone, at least below intensities that give rise to fields that will accelerate electrons to relativistic energies. Thus a useful notation is that of the analytic signal, whose amplitude and phase we seek to determine via measurement. The (real) electric field of the pulse is given in terms of the analytic signal by

A single pulse is said to be completely characterized if the function $E\left(t\right)$ is known on the domain $[-T,T]$. In practice one usually adopts the approximation that the pulse is also characterized by the function $\tilde{E}\left(\omega \right)$ on the domain $[{\omega }_0-\Omega ,{\omega }_0+\Omega ]$, where Ω is a frequency that is large compared with the bandwidth of the source (i.e., large compared with the inverse of the coherence time of the source). The sampling theorem prevents a function from having compact support in both domains, but it is usually a reasonable approximation to truncate the spectral function at large frequencies, where the spectral density falls below the noise level of the detector. With this approximation, all integrals are usually formally extended from $-\infty$ to $+\infty$. Figure 6 presents the temporal and spectral representations of a Gaussian pulse with flat spectral phase, a quadratic spectral phase ${\varphi }^{\left(2\right)}{\omega }^2∕2$, and a cubic spectral phase ${\varphi }^{\left(3\right)}{\omega }^3∕6$. The impact of these different phases on the temporal profile of the pulse can also be seen.

2.1b. Correlation Functions and Chronocyclic Representations

The analytic signal describing a pulse field is not sufficient to specify the character of an ensemble of pulses. For example, each pulse from an amplifier system may be, indeed probably is, slightly different from its predecessor and successor, and thus each pulse represents a different realization of the ensemble. A complete specification of the ensemble is given by the probability distribution of the field at each point in time. However, it is usually sufficient to specify a set of correlation functions of the field, since experiments can be described in terms of a fairly small number of such functions.

The lowest order of these is the two-time correlation function $C(t,t^{\prime })=⟨E\left(t\right)E^{*}\left(t^{\prime }\right)⟩$, where the brackets indicate either a time average over the pulse train or an ensemble average over repeated experiments. Note that $C(t,t^{\prime })$ is not the same as the correlation function that is derived from the pulse spectral intensity ${\left|\tilde{E}\left(\omega \right)\right|}^2$ via the Wiener–Khintchine theorem. In that case, the Fourier transform yields the reduced correlation

A knowledge of the two-time correlation function allows one to determine whether the pulses in the ensemble are coherent, that is, to determine whether they have the same pulse field. A useful number characterizing the similarity of the pulses in the ensemble is the degree of temporal coherence, defined by [34]

Similarly, one can define a two-frequency correlation function $\stackrel{͌}{C}$, which is the double Fourier transform of its temporal counterpart, by

Beyond this approach, though, the correlation functions are difficult to measure. The reason is that the measured signals are functionals of the two-time correlation function and cannot always be simply inverted. This problem is usually ignored, and it is assumed from the beginning that the pulse train may be described in terms of a field. This makes possible more or less straightforward inversion algorithms.

Whether or not the pulse train is coherent, it is nevertheless useful to consider metrologic schemes in the two-dimensional space of the correlation function. The reason is that the output of all absorptive detectors is proportional to bilinear functional of the pulse field, and thus a linear functional of the two-time correlation function. However, it is frequently productive to work with a variation of the correlation function that uses the two-dimensional chronocyclic space $(t,\omega )$. The intuitive concept of time-dependent frequency can be most easily seen within this space.

One approach to defining a representation of the pulse in the chronocyclic phase space is a Fourier transformation of the correlation function with respect to the time difference of the two arguments:

Some examples of chronocyclic Wigner functions for common pulse shapes are shown in Fig. 7. The Wigner function of a Gaussian pulse with a flat spectral phase does not show a correlation between time and frequency. However, with a quadratic spectral phase (i.e. a linear group delay), the Wigner function acquires a slope indicating the correlation between time and frequency, and its contours provide some intuition about the pulse chirp via a graph of the time-dependent frequency. The Wigner function of a pair of phase-locked Gaussian pulses and a Gaussian pulse with cubic spectral phase take some negative values, although their marginals are, as expected, positive quantities.

2.2. General Strategies for Pulse Characterization

2.2a. Linear Systems Model and Photodetection

The oscillations of the electric field ɛ are too fast to be directly resolved by photodetection. Photodetectors are intrinsically square-law detectors, sensitive to the intensity of optical waves but not to their phase. Indirect approaches are therefore used to provide phase sensitivity with square-law photodetectors and to resolve the shape of short optical pulses. The basic elements required for the complete characterization of optical pulses are quite simple: at least one fast shutter or phase modulator, a spectrometer or an element to temporally stretch the pulse via dispersion, and one or two beam splitters. One can think of all elements except the beam splitters as two-port devices: a pulse enters at one port and exits at another. There may be ancillary ports for control signals, such as the timing signal for the shutter opening, for example, but these are essentially linear systems, in that the output pulse field scales linearly with the input pulse field. Thus the input–output relations for these devices are all of the kind

The beam splitter is a four-port device, having two input and two output ports. The input–output relations for this device are well known, and the main utility in pulse measurement applications is either in providing a means to generate a replica of a pulse (one input and two outputs) or to combine the unknown pulse with a reference pulse (two inputs and two outputs), or as elements of an interferometer in which phase-to-amplitude conversion takes place.

We take it that all detectors available have a response that is slow compared with the pulse itself. For pulses with temporal structure of duration less than $100\mathrm{fs}$ or so, this is usually the case. The measured signal from a square-law detector is related to the incident field, for our purposes, by

Linear filters may be separated into two classes: those with time-stationary response functions and those with time-nonstationary responses. For the former class, which includes the spectrometer and dispersive delay line, the shape of the output pulse does not depend on the arrival time of the pulse. For the latter class, which includes the phase modulator and the shutter, the output pulse shape clearly depends on the timing of the input pulse with respect to the shutter opening or the modulator drive signal.

Time-stationary filters are characterized by response functions of the form $H(t,t^{\prime })=S(t-t^{\prime })$, and a particularly useful class of time-nonstationary filters by $H(t,t^{\prime })=N\left(t\right)\delta (t-t^{\prime })$. Equivalently, in the frequency domain, these stationary filters take the general form $\stackrel{͌}{H}(\omega ,{\omega }^{\prime })=\tilde{S}\left(\omega \right)\delta (\omega -{\omega }^{\prime })$, and the nonstationary the form $\stackrel{͌}{H}(\omega ,{\omega }^{\prime })=\tilde{N}(\omega -{\omega }^{\prime })$, where the tilde represents a Fourier transform.

We may postulate a general linear filter function in the form of a temporal Fresnel kernel:

Some of these parameters become variables in the measured signal function (for example, the opening time of the gate or the center frequency of the spectrometer passband), while other parameters might be constant (for example the gate duration). The variable parameters are those on which the inversion is based. It is therefore important to ensure that the number and type of filters are adequate to the task.

2.2b. Measurement of the Marginals of the Wigner Function

Pulse energy spectrum. Possibly the simplest quantity that can be measured for an isolated pulse is its spectrum. It is therefore also one of the most important, since it can be used as a consistency check for all pulse characterization techniques: the reconstructed spectrum must match an independent direct measurement of the spectrum. The pulse spectrum is usually determined by the obvious expedient of sending the pulse into a spectrometer (usually a grating spectrometer is necessary to yield the necessary dispersion) and recording the output as a function of the setting of the passband of the instrument, Ω. Then the spectrometer output is

It is of particular importance that an equation equivalent to Eq. (2.23) can be written for all stationary filters; i.e., the output signal of a device built entirely with stationary amplitude and/or phase filters does not depend on the spectral phase of the pulse. The implication is that stationary-only filters are insufficient to gather information on the spectral phase of an optical pulse and can at best return information on the spectral intensity of the pulse.

In terms of the Wigner representation, the measurement of a pulse spectrum is written as

The important point is that all measurement techniques can be represented in terms of the overlap of the Wigner function of the test pulse (or pulse ensemble) and that of the apparatus. This provides an important insight into ways that the experimental data may be inverted to obtain the pulse field itself, as discussed in subsequent sections.

Measurement of the temporal intensity. The measurement of the temporal intensity of an optical pulse is in some sense the conjugate operation of the measurement of its optical spectrum. If the pulse under test is sent to a fast square-law detector (or equivalently, a fast shutter followed by a time-integrating detector), the measured output is

2.2c. Autocorrelations and Cross-Correlations

Intensity autocorrelation. The simplest technique for gathering at least moderate quantitative information about the temporal structure of an ultrashort pulse is the intensity autocorrelation. In a conventional autocorrelator, two pulse replicas are mixed in a nonlinear material, and the average power of the generated beam (measured with an integrating detector) is recorded as a function of the relative delay between the two replicas. By assuming a functional form for the temporal shape of the test pulse, one can estimate its duration from the autocorrelation trace. Because of its simplicity, autocorrelation is by far the most common method of measuring ultrashort optical pulses. However, the autocorrelation trace by itself provides little more than an estimate of the pulse duration.

A variety of schemes based on intensity correlation measurements were demonstrated during the late 1960s and early 1970s [40, 41, 42, 43, 44]. One particular form, the second-order intensity autocorrelation function (AC) became one of the standard techniques in the field for nearly two decades and is still in use today. This technique uses the lowest-order nonlinear process available, and therefore operates at the lowest power possible for a nonlinear process. This is important for making measurements of pulse trains from mode-locked laser oscillators, whose energy is in the picojoule to nanojoule range. The most common approach to extracting information from this AC data, however, involves fitting an AC calculated from a specific pulse shape.

Consider a material (say, a crystal) with second-order nonlinearity and two optical waves around the optical frequencies ${\omega }_1$ and ${\omega }_2$. The nonlinear susceptibility ${\chi }^{\left(2\right)}$ links the second-order contribution to the nonlinear polarization to the electric field of the two waves by

The AC described by Eq. (2.29) yields a direct measure of the root-mean-square (rms) pulse duration $\Delta t_I$ through the relation

Interferometric autocorrelation. The AC is often extended to its so-called fringe-resolved form [48] by using a collinear setup [Fig. 8(d)]. One advantage of the interferometric autocorrelation (IAC) is that it is sensitive to the phase of the electric field. Another advantage is that the quickly varying fringes lead to a natural calibration of the temporal axis, which is useful when characterizing few-cycle pulses. The upconverted signal is given by

N th-order intensity autocorrelation and cross-correlation. While autocorrelations provide a somewhat blurred version of the intensity of the pulse under test, a better picture can be obtained from a higher-order intensity correlation function, such as

Another area where cross-correlations have been used extensively is pulse shaping. Pulse shapers can transform an input pulse into a temporally shaped waveform with a temporal support much larger than the input pulse duration. In many cases, a description of the intensity of the output field is sufficient, and this can be obtained by cross correlating the output waveform with a replica of the input pulse (Fig. 4).

Finally, recovery of the intensity of a test pulse from triple cross-correlations of the intensity has been attempted [57]. The temporal intensity can in theory be reconstructed unambiguously from the two-dimensional triple correlation function measured as a function of the two relative delays between replicas of the test intensity.

2.2d. Classes of Pulse Characterization Devices

Measurement devices may be categorized by the arrangements of filters through which the test pulse is passed before being detected. A second classification involves the type of algorithm that is used to extract the pulsed field from the experimental data. The data is a function only of the filter parameters, and there should be a sufficient number of these, and of the right kind, that complete information about the test pulse is encoded in the data. The requirements that this places on the apparatus will be laid out in this subsection.

The filters are characterized by a set of parameters $\left\{p_i\right\}$. For example. the shutter transmits any portion of the pulse that falls within a time window of duration ${\tau }_g$ near the opening time τ. Likewise the spectrometer transmits any portion of the pulse that falls within a spectral window of width Γ near the passband center frequency Ω. The modulator adds a time-dependent phase onto the pulse, whose magnitude depends on the modulation index ${\psi }^{\left(2\right)}$, and the time of arrival of the pulse compared with the peak of the modulation at time τ. The dispersive line adds a spectrally dependent phase onto the pulse, whose magnitude depends on the second-order dispersion ${\varphi }^{\left(2\right)}$ and the position of the pulse spectrum with respect to the reference frequency ${\omega }_R$.

Although these are not the most general linear response functions possible, they are sufficient for our purposes. Moreover, the categories they represent are complete, in that any linear filter may be synthesized from a sequence of such filters. A pulse measurement apparatus therefore consists of a sequence of filters in series or in parallel, or both, followed by an integrating detector, as shown in Fig. 9.

Within this framework, the signal measured by a detector following a sequence of such filters is a function of the filter parameters. It may be written as the overlap of the Wigner function of the pulse with a chronocyclic window function $F(t,\omega ;\left\{p_i\right\})$ depending only on the properties of the arrangement of linear filters:

If arranged in series, there are four combinations. Two of these belong to the class of spectrographic measurements, and two to the class of tomographic measurements. If arranged in parallel, these elements give another four schemes. All of the latter are based on interferometry: two in the time domain, and two in the spectral domain. A final amplitude filter that is either a shutter (a time gate) or a spectrometer (a frequency gate) enables a slow detector to measure the interferogram. The full catalog of possible configurations is shown in Fig. 10: we describe each separately below.

It is instructive to revisit the autocorrelation in light of the chronocyclic representation. It consists of a delay [time-stationary filter ${\tilde{S}}_L^P(\omega ;\tau )=e^{i\omega \tau }$] followed by a shutter (time-nonstationary filter $N^A\left(t\right)$), so that the detected signal is

2.2e. In-Series Filtering Measurements

Spectrography. Spectrography refers to schemes in which a simultaneous measurement of the spectral and temporal intensity of the pulse is made. In particular, methods of this type are based on the measurement of the spectra of different temporal sections of the pulse, or on the measurement of the temporal intensity of different spectral sections (in which case it is known as a “sonogram”). For the former, one needs a fast shutter opening at time τ (with a speed comparable with, though not necessarily as fast as, the test pulse itself) followed by a high-resolution spectrometer with passband at frequency Ω [Fig. 10(a)].

For this arrangement, the Wigner function of the measurement apparatus is

The elements may also be used in the reverse order [Fig. 10(b)]. In this case, referred to as sonography, the test pulse first encounters a low-resolution spectrometer, then a very fast shutter. The Gabor sonogram is defined in an analogous manner to Eq. (2.37) by

The first method to provide complete information about an ultrashort optical pulse used a shutter based on upconversion of the spectrally filtered (and therefore temporally stretched) test pulse with the test pulse itself. The shutter speed is then equal to the duration of the test pulse, and something close to a sonogram of the test pulse can be measured [58].

In measuring either spectrograms or sonograms, it is important that the first filter encountered by the test pulse have low resolution in its appropriate domain (a nominally slow shutter for spectrography, and a low-resolution spectrometer for sonography), and that the second filter have high resolution (a high-resolution spectrometer for spectrography and a fast shutter for sonography). This makes the measured spectrogram or sonogram most similar to the true Gabor-type spectrogram or sonogram of the test pulse.

As discussed previously, nonlinear optics is not a necessity for pulse characterization. Its use in spectrographic techniques when characterizing sub-$100\mathrm{fs}$ pulses is required because there is no other way to build a shutter with a similar responses time. The test pulse itself is, ipso facto, the shortest-duration entity to which the experimenter has access, thereby setting a lower limit on the shutter speed. Because of this constraint, measurements of a sonogram of a femtosecond optical pulse always have lower resolution than the corresponding spectrogram.

When nonlinear optics is used, the measured spectrograms are nonlinear functionals of the test pulse Wigner function. A true spectrogram, such as the Gabor spectrogram, is a linear functional of the test pulse and the known shutter response Wigner function. Reconstruction of the pulse field from the Gabor spectrogram requires a deconvolution, but has in most cases a unique solution. Reconstruction of the pulse from nonlinear spectrograms requires an iterative nonlinear deconvolution where the convolution function depends on the unknown pulse. This problem might have multiple solutions. As a consequence, much of the effort devoted to these techniques has concentrated on devising robust iterative algorithms for extracting the field from the measured quantity, and this is discussed in Section 3.

An alternative approach to pulse reconstruction is to ensure that the apparatus operates with parameters that allow approximate direct inversion. This is possible with sonography [59], for example, and methods have been suggested for spectrography [60] as well as the spectrally and temporally resolved upconversion technique (STRUT) [61, 62].

Tomography. Tomographic pulse measurement is based on the notion of the time lens. This approach exploits the idea that the temporal intensity profile of the pulse can be transformed into the spectrum by suitable (linear) manipulations. The underlying principle can be illustrated by using the well-known dispersive properties of a grating pulse stretcher, as illustrated in Fig. 11(a). In this device, a pulse experiences a large group-velocity dispersion, since each wavelength traces a different path through the grating pair. Although the different wavelengths each occupy a different spatial position within the beam (an example of space–time coupling) after the second grating, a second pass through the apparatus undoes the space–time coupling while doubling the dispersion. Thus at the output of the stretcher the pulse duration is much longer than at the input, because each frequency in the pulse experiences a different transit time (or delay) through the grating pair. The Wigner functions of a pulse before and after quadratic spectral phase modulation ${\varphi }^{\left(2\right)}{\omega }^2∕2$ are related by

The inverse effect can also be made to happen. That is, the input pulse temporal shape can be made to appear in the output pulse spectrum. This requires a time-domain analog to a pair of gratings. Such a device turns out to be a temporal phase modulator [Fig. 11(c)], which shifts the frequency of different time slices of the pulse by different amounts, just as the grating stretcher shifts the time delay of different spectral slices of the pulse by different amounts. Clearly the response time of the modulator must be comparable with that of the pulse itself for this operation to provide a unique mapping, and for this reason it is only recently that such methods have begun to be practical in the picosecond and subpicosecond regimes. The Wigner functions of a pulse before and after quadratic temporal phase modulation ${\psi }^{\left(2\right)}{t}^2∕2$ are related by

The combination of the temporal modulator and dispersive stretcher allows one to perform an operation called “temporal imaging” by analogy to the operation performed by an optical imaging system in the spatial domain. Consider a standard optical imaging device, consisting of an object placed some distance before a lens, and an image plane (at which is placed a detector) some distance after the lens. The underlying physics of image formation is that light from the object undergoes diffraction in free space for a prescribed distance, then refraction by the lens, then further diffraction before being detected. For the appropriate adjustment of the distances and power of the lens, a magnified image of the object can be formed. The time–frequency analog is that the grating stretcher plays the role of diffraction and the temporal modulator the role of the lens. Using such a setup, a temporally magnified image of the input short pulse can be constructed, which is easy to measure by using detectors with response times much longer than the input pulse.

2.2f. In-Parallel Filtering Measurements

Interferometry refers to the situation where the phase of the test pulse is encoded into the intensity by means of mixing with a second pulse, which may be an ancillary reference pulse or the test pulse itself. These two categories are known as “test-plus-reference” and “self-referencing” interferometry, respectively. They are both direct techniques, in that it is possible to reconstruct the correlation function in either the time domain or the frequency domain directly (i.e., noniteratively) from the recorded intensity distributions. A general model of this category of measurement devices may be developed in terms of a sequence of in-parallel linear filters. In this model each pulse in the ensemble is split into two replicas at a beam splitter, and each replica is independently filtered before being recombined. The interference of the field from the parallel pathways introduces structure on the output intensity distribution, which then carries information about both the amplitude and the phase of the correlation function of the input field. If the ancillary port of the input beam splitter is empty, then the interferometer is said to be self-referencing. Alternatively, if the ancillary port is used to inject a characterized reference pulse, then it is possible to reconstruct the electric field of the test pulse in a rather straightforward manner [63, 64]. Of course, this approach requires one to first obtain a well-characterized reference pulse.

One significant advantage of direct techniques compared with phase-space techniques is that the entire space over which the phase-space or correlation functions are defined need not be explored if the pulse train is assumed to consist of identical pulses. Only a single section of one quadrature of the (complex) correlation function is necessary to obtain the electric field amplitude and phase, and these are precisely what is recorded by direct techniques.

Test-plus-reference interferometry. The most common form of test-plus-reference interferometry is Fourier-transform spectral interferometry (FTSI) [63, 64]. In this approach, the test and the reference pulse are delayed in time with respect to each other by τ before combining at the input beam splitter. The detected signal (interferogram) is then $S(\omega ;\tau )={|\tilde{E}\left(\omega \right)+{\tilde{E}}_R\left(\omega \right)e^{-i\omega \tau }|}^2$, where ${\tilde{E}}_R$ and $\tilde{E}$ are the spectral representations of the analytic signal of the reference and the test pulse. The spectral phase difference between test and reference pulses is encoded in the relative positions of the spectral fringes with respect to the nominal spacing of $2\pi ∕\tau$ and can be extracted by using a three-step algorithm involving a Fourier transform to the time domain, a filtering operation, and an inverse Fourier transform. The phase of the reference pulse must then be subtracted, leaving the spectral phase of the test pulse as required. A measurement of the test pulse spectrum then provides sufficient information to characterize the pulse. In common with all interferometric methods, the data set has one parameter, frequency, and may therefore be collected by using a one-dimensional detector array. This leaves the second dimension of a camera, for example, available for coding information about other degrees of freedom of the test pulse, such as the spatial phase. This method is therefore easily extended to full space–time characterization of the test field, again provided that a suitable (i.e., fully space–time characterized) reference pulse is available.

Self-referencing interferometry. It is possible to extract the phase of a field without a known reference pulse by gauging one spectral or temporal component of the field with another component. This is known as “self-referencing interferometry.” In this approach, the goal is to reconstruct the correlation function in either the time domain or the frequency domain directly (i.e., noniteratively) from one or several recorded intensity distributions. In fact, when the pulse train is coherent, it is necessary only to measure a section of the two-time or two-frequency correlation function in order to reconstruct the pulse field [34].

The in-parallel amplitude-only filters select either two frequency or two time slices of the pulse that beat together at the output of the interferometer. These are the time-domain analogs of Young’s double-slit interferometer [Figs. 10(e), 10(f)].

In the spectral domain [Fig. 10(e)], the center frequencies of the spectral filters are ${\omega }_{C1}$ and ${\omega }_{C2}$, and each has the same bandwidth Γ. The selected frequency components are recombined, giving rise to temporal fringes—or a time-dependent modulation of the intensity—at the output. These are resolved by using a time gate or a fast shutter. The signal recorded by the square-law detector is a function of the spectral filter center frequencies as well as the time of maximum transmission τ of the time gate,

A complementary form of interferometer consists of an in-parallel fast time-gate (time-nonstationary amplitude-only filters) followed by a spectral filter (time-stationary amplitude-only filter), as pictured in Fig. 10(f). The two replicas of the pulse are independently sampled with variable times,$\tau 1$ and $\tau 2$, before being recombined. The spectral beats, resulting from the overlap of the two time slices, are resolved by a spectrometer. The resulting signal, for the case of a very fast time gate and a very high-resolution spectrometer, written in terms of the center-time $(t=(\tau 1+\tau 2)∕2)$ and difference-time $(\Delta t=\tau 1-\tau 2)$ coordinates, is the temporal interferogram

A different class of interferometers makes use of a frequency shifter (a time-nonstationary linear phase filter) and a delay line (a time-stationary linear phase filter) arranged in parallel, followed by spectrometer after these signals are recombined [Fig. 10(g)]. The detected signal is a function of the delay ${\varphi }^{\left(1\right)}$, which acts as a fixed parameter, as well as the center frequency of the spectrometer passband, Ω,

An entirely analogous argument may be made for temporal shearing interferometers [Fig. 10(f)]. In this case, the delay line in one arm of the interferometer causes the pulses on recombining at the second beam splitter to exhibit temporal beats in their intensity that may be resolved by a fast time gate. This latter element is the amplitude-only filter that replaces the spectrometer required in the spectral shearing interferometer. In this arrangement, a temporal linear phase modulator may be used to provide a temporal carrier for the two-time correlation function in the interference term. This is accomplished by frequency shifting one of the pulses with respect to the other by a shear ${\psi }^{\left(1\right)}$ and by introducing a relative delay $\Delta t$ between the pulses. The signal detected as a function of τ, the delay of the time-nonstationary gate, which is assumed to be of infinitesimal duration, is then

2.2g. Joint Measurements

There are several modifications to the methods that have allowed some headway. The spectrum of the pulse helps in determining whether the pulse is close to the Fourier-transform limit and is an obvious second piece of data that is relatively easy to measure. Several iterative schemes have been developed to extract the pulse shape from a correlation and the spectrum [50, 65, 66, 67]. They provide varying degrees of success in extracting the pulse fields, but all share the same characteristic that they are very sensitive to noise in the data [46].

Attempts at retrieving the electric field of the pulse from a set of intensity autocorrelations measured after various amounts of second-order dispersion have been made [68, 69]. The use of the intensity autocorrelation in the temporally resolved optical gating (TROG) technique, instead of a direct intensity measurement, significantly increases the complexity of the retrieval compared with tomographic techniques.

Deterministic changes of the spectral phase of the pulse with a pulse shaper have also attracted some attention. For a given spectrum, the autocorrelation signal at $\tau =0$ is maximized by a flat spectral phase. Since this signal can be measured directly by doubling the pulse and measuring the energy of the converted pulse with a photodetector, an iterative algorithm can be used to modify the spectral phase and maximize the measured signal. For a given pulse, the spectral phase introduced by the pulse shaper when the maximum is reached corresponds to the opposite of the spectral phase of the input pulse, therefore leading to a measurement of the phase by adaptive pulse shaping [70]. In a multiphoton intrapulse interference phase scan (MIIPS), the spectrum of the upconverted signal is used as a feedback mechanism when a spectral phase is scanned across the spectral support of the pulse with a pulse shaper [71]. Iterations are required for accurately determining the spectral phase of the input pulse: the pulse shaper is also used to introduce a static spectral phase that attempts to compensate the spectral phase of the input pulse, and a specific multiphoton intrapulse interference phase scan trace is obtained when the shaper output pulse is Fourier-transform limited.

2.3. Conclusions

There are three general classes of measurement techniques for characterizing ultrashort optical pulses—spectrography, tomography, and interferometry—which lead to eight devices consisting of the smallest possible number of optical elements (two spectrographic, two tomographic, and four interferometric). All of these devices contain at least one time-stationary and one time-nonstationary filter, which may be linear in the input field.

The two spectrographic methods measure a smoothed version of the chronocyclic Wigner function by using sequential amplitude filters to make a simultaneous measurement of time and frequency. Tomographic methods measure projections of the chronocyclic Wigner function onto the frequency variable, following the application of a quadratic phase modulator to the input pulse. This serves to rotate the phase-space distribution of the pulse, so that a measurement of its modified spectrum reveals information about its initial orientation, and hence chirp. The data, consisting of a set of projections of the Wigner function for a range of phase-space rotations, can be deterministically inverted to retrieve the Wigner function itself.

Self-referencing interferometric techniques measure a point or a section of the two-frequency or two-time correlation function. A single section of either function is adequate for reconstructing the underlying electric field. Interferometers work by splitting each pulse in the ensemble into two replicas at a beam splitter, independently filtering the replicas, and then recombining them at a second beam splitter. The interference of the parallel pathways introduces structure on the output intensity distribution, which then carries information about both the amplitude and the phase of the correlation function. There are two interferometric devices that are analogs of Young’s double slits: either two spectral or two temporal slices are taken of the input pulse and fringes recorded in the temporal or spectral domains, respectively. Since these devices require time gates that are short compared with the input pulse duration, they are difficult to implement for femtosecond pulses. A more useful approach is via shearing interferometry. In this case, one of the pulses is shifted in frequency (or in time) with respect to the other, and an interference pattern recorded in the spectral (or temporal) domain. The simple and direct inversion algorithm gives a provably unique solution to the problem of pulsed field reconstruction.

3. Spectrography

3.1. Introduction

Some of the earliest attempts at measuring the chirp of optical pulses were based on spectrographic concepts. Such ideas also underpinned the first attempts at precisely characterizing the electric field of pulses. The concept of “chirp” was developed for microwave pulses and refers to the existence of a time-dependent instantaneous frequency—or, equivalently, a frequency-dependent group delay—in which all of the frequencies of the pulse do not arrive at the observer simultaneously. In 1971, Treacy quantified this quantity for pulses from a mode-locked Nd:glass laser by measuring the time of arrival of spectral slices of the pulse [72]. This recording allowed the first evaluation of the optical chirp, which had been known for microwaves for some time. A description of early developments can be found in [26]. Different implementations of the same concept were then developed and are usually referred to as time-resolved spectroscopy [73, 74, 75]. Another precursor to the spectrographic techniques used nowadays is the measurement of optical spectra of the upconverted signal in an intensity autocorrelator [76]. Chilla and Martinez’s implementation of sonograms using nonlinear wave mixing in a crystal has inspired most setups for sonographic measurements of femtosecond pulses [58]. Spectrograms and sonograms are now widely used in ultrafast optics, and the development of phase retrieval algorithms enables full recovery of the amplitude and phase of the electric field without prior assumptions as to its functional form. The best-known example of this class of measurements is frequency-resolved optical gating (FROG) [28]. In this section, we describe the principles of spectrography, the apparatuses required for measuring spectrograms and sonograms, and the approaches available for extracting the pulse field from the experimental data. We also give some experimental implementations of these concepts adapted to ultrafast optics.

3.2. General Implementation of Spectrography

3.2a. Definitions

Spectrographic techniques aim at measuring simultaneously the arrival time and frequency of an optical wave, that is, a joint representation of the Fourier conjugate variables time and frequency. In the most general case, the measurement yields a time–frequency distribution that is uniquely related to the input pulse field. The usual approach to this measurement is to perform a sequential gating in the time and frequency domains by using a time-nonstationary and a time-stationary filter. The time-nonstationary filter can be delayed in time by a quantity τ with respect to the test pulse, and the transfer function of the time-stationary filter can be tuned along the optical frequency axis by an amount Ω. The measured quantity is therefore a function of these two variables, which have to be varied to cover completely the entire chronocyclic phase space occupied by the pulse. This permits a faithful estimation of the pulse time–frequency distribution. If the two filters are in sequence, and the second has very high resolution (i.e., is either a fast shutter or a narrowband spectral filter), this approach can be considered to make a simultaneous measurement of the time and frequency of the pulse.

Figure 12 shows two typical arrangements for spectrographic measurements consisting of two sequential filters. In Fig. 12(a) the first filter is a time-nonstationary device modulating the electric field with a gating function g, and the second filter is a time-stationary filter described by $\tilde{R}$. The gating function can be delayed in time by a delay τ relative to the pulse under test, and the stationary filter can be scanned in frequency, Ω describing a parameter relevant to this filter, for example the center of its passband. The signal measured by a time-integrating detector is

The order of the stationary and nonstationary filters can be inverted, so that the measurement is that of the temporal intensity of the pulse after spectral filtering [Fig. 12(b)]. The signal measured by a time-integrating detector is in this case

3.2b. Wigner Representation

The spectrogram of Eq. (3.2) and the sonogram of Eq. (3.4) can be written as a double convolution of the Wigner function of the test pulse $W_E$ with the Wigner function of the apparatus $W_g$:

3.2c. Chirp Representation

The first-order moments of the Wigner function can be linked to the group delay and instantaneous frequencies defined from the first derivatives of the spectral phase and the temporal phase of the electric field. The first-order moments of the spectrogram are by definition

3.3. Inversion Procedures for Spectrographic Techniques

The basic problem behind the inversion of the spectrogram is the determination of a relevant quantity describing the train of pulses under test (e.g., chirp, electric field, or Wigner function) from the measured time–frequency distribution. In some implementations of spectrographic techniques, the gate is unknown; for example it can be a function of the pulse under test itself in FROG, where the time-nonstationary filter is synthesized by a nonlinear interaction with a replica of the unknown pulse under test. The inversion approaches are classified here as chirp retrieval, Wigner deconvolution, and phase retrieval.

3.3a. Chirp Retrieval

A quantitative assessment of the chirp of the test pulse can be obtained from a spectrogram or sonogram by calculating its first-order moments by using Eqs. (3.10, 3.11), or simply by locating the delay at which the spectrogram has a maximum for each frequency or, equivalently, the frequency at which the spectrogram has a maximum for each delay (assuming that the pulse structure is simply enough that the maxima are unique). These properties were understood very early on and are at the basis of the works of Treacy [72] and Chilla and Martinez [58], who determined the group delay as a function of frequency for an optical pulse by spectrally filtering the pulse and determining the time of arrival of the wave packets centered at the corresponding frequencies. Precise estimation of the chirp is made difficult by the fact that the second-order moments of the time–frequency distribution along one axis increase significantly when tight filtering is performed along the conjugate variable [37]. For example, a spectrogram measured by using a short nonstationary filter leads to a large spread of the spectrogram along the frequency axis, which means that the practical determination of the instantaneous frequency requires a very high signal-to-noise ratio. Another limitation of this approach, in common with many methods, is that the chirp and the group delay are measures of the derivative of the phase with respect to time or frequency, respectively. To extract the full phase of the test pulse field, it is necessary to integrate the measured quantities, which can be done when the support of the field is continuous but is difficult otherwise. Such an approach would not perform well, for instance, in the characterization of pulses with disjoint spectral or temporal support or pulses with phase jumps (an example of the latter is a train of pulses used in telecommunication, such as carrier-suppressed return-to-zero pulses where adjacent pulses differ by a π phase shift [78]).

3.3b. Wigner Deconvolution

There is in principle a more direct way to extract the field from the spectrogram. When the gate response function is known, the corresponding apparatus Wigner function $W_g$ is known, and the Wigner function of the pulse under test can in principle be obtained by inverting the convolution of Eq. (3.5) [79]. The steps necessary to perform such an operation are the calculation of the double Fourier transform of the measured spectrogram or sonogram, the division of this quantity by the double Fourier transform of $W_g$, and the calculation of the inverse double Fourier transform of the obtained quantity, which leads to the Wigner function of the test pulse, followed by the calculation of the electric field of the test pulse from its Wigner function. However, direct deconvolution is highly sensitive to the precision with which the gate response function is known and to the signal-to-noise ratio and is prone to error at points in the phase space where the Fourier transform of the Wigner function of the gate takes zero values. Further, this approach does not take into account additional information such as the degree of coherence of the test pulse ensemble, nor can it easily include any assumptions about this. Therefore this approach is not widely used in practice.

3.3c. Phase Retrieval

The two-dimensional deconvolution method described in the previous subsection does not make use of the fact that the underlying quantity of interest is the one-dimensional electric field. In the common case where the test pulse ensemble is coherent, so that the test pulse field is well defined, the two-dimensional spectrogram or sonogram is a function of two one-dimensional functions describing the gate response and the test pulse field. This redundancy in the data may be put to good use. It is possible to make use of this fact to effect iterative deconvolution algorithms that work at lower signal-to-noise ratios than direct deconvolution.

The retrieval of E and g is equivalent to the retrieval of the phase of the short-term Fourier transform $\int dtE\left(t\right)g(t-\tau )\exp \left(i\Omega t\right)$. The spectrogram is by definition the modulus square of the latter quantity, and once the short-term Fourier transform is known, both E and g can be obtained by Fourier transformation. Spectrogram inversion therefore falls into the category of phase retrieval problems. These problems have been studied extensively in optics, owing to the fact that common square-law detectors, such as charge-coupled device arrays (CCDs) used in imaging, provide only intensity information. Various phase retrieval algorithms have been used for such inversion in the context of imaging, and phase retrieval for ultrafast optical metrology can be traced back to the spectrogram inversion by Kane and Trebino [80] and the later sonogram inversion by Wong and Walmsley [39]. The general approach to iterative inversion is to locate the intersections of two sets of two-dimensional functions corresponding to two constraints. The first constraint is that the modulus square of the short-term Fourier transform must match the experimentally measured spectrogram. The second constraint is that the experimental signal should be consistent with the functional form of spectrogram of a pulse gated by a gate; i.e., it can be written as Eq. (3.2) or (3.4). There can also be additional constraints, such as the functional dependence between the pulse and gate, or the spectral characteristics of the field or gate. Since the two sets of constraints are not convex, convergence is not guaranteed, but iterating by projecting the solution at each step onto each set has proved a robust way of inverting the spectrogram. Projection on the set of functions satisfying the modulus constraint is easily performed by replacement of the modulus with the square-root of the measured spectrogram. Projection on the set of functions satisfying the spectrogram mathematical form was initially performed by using an error minimization algorithm [81]. A more efficient algorithm to achieve this task is the principal component generalized projection algorithm (PCGPA) [82, 83] (Fig. 14).

The principal component generalized projection algorithm works in the following way: assuming one has a pair of functions $(E_n,g_n)$ at iteration n, the outer product $E_n\left(t\right)g_n\left(t^{\prime }\right)$ is first calculated as a matrix (with row and column indices indicated by discretized values of t and $t^{\prime }$), and the short-term Fourier transform $\int dt{E}_n\left(t\right)g_n(t-\tau )\exp \left(i\Omega t\right)$ is then calculated by a sequence of row rotation and Fourier transformation. The modulus constraint is then applied by replacing the modulus of the calculated short-term Fourier transform with the square root of the measured spectrogram, retaining its phase. This new complex matrix undergoes the inverse of the operations applied to its predecessor. It is rendered into outer-product form, as required by the signal constraint, by means of a singular value decomposition (SVD) of the two-dimensional matrix. This decomposes the matrix into a sum of outer-product (i.e., rank one) matrices. The outer product corresponding to the largest eigenvalue is kept, and the corresponding eigenvectors are used as the set of solutions $(E_{n+1},g_{n+1})$ for the next iteration. In practice, this singular decomposition is slow, and an approximate decomposition is obtained by using matrix multiplications (power method). This algorithm does not use an explicit relation between the pulse and the gate, and is therefore referred to as “blind.” The algorithm yields in principle both the characteristics of the pulse and gate, and this is useful for implementations of spectrographic techniques where the pulse and the gate are not related, for example in cross-correlation FROG and linear spectrograms and sonograms. Some theoretical cases of ambiguity in this approach have been reported [84], though these can usually be removed using additional knowledge such as the optical spectrum of the test pulse or the relation between the pulse and the gate, as is often the case in FROG. The knowledge of the specific link between gate and field has been successfully inserted into the algorithm for polarization-gate FROG and second-harmonic generation (SHG) FROG [83, 85].

3.3d. Ambiguities, Accuracy, Precision, and Consistency

Ambiguities. An ambiguity in phase retrieval arises when more than one phase function can be assigned to the reconstructed field while satisfying all constraints of the inversion problem. Consider, for example, the spectrogram

Equation (3.12) demonstrates the important direction-of-time ambiguity of SHG-FROG, as explained below in Subsection 3.5b. In this case, the gate is derived from the test pulse so that $g=E$, and the inversion will yield either $E\left(t\right)$ or $E^{*}(-t)$. This implies that a single SHG-FROG measurement cannot determine the direction of time unless one has additional information about the test pulse structure. Various studies on ambiguities for specific implementations of FROG can be found in [28, 86, 87, 88, 89].

Accuracy. The accuracy of a diagnostic quantifies the similarity between the measured quantity and the physical quantity. This can be specified only if a well-known test pulse is available or by means of numerical simulations. There have been no extensive studies of these for modern ultrafast spectrographic methods, especially in the presence of noise or nonoptimal experimental conditions. Some simulations relevant to this issue can be found in [90].

Consistency. The consistency of the inversion of a spectrogram or sonogram specifies the degree to which the data reconstructed from the solution matches the experimental data; i.e., it tells how well the inversion algorithm for the problem worked. The two constraints that are used in inverting the data yield their own consistency criterion:

Precision. Evaluating the precision of a measurement device requires the ability to compare several retrievals of the same quantity by the same device. In the case of ultrafast pulse characterization, this may done in principle by characterizing the same test pulse several times, with the underlying assumption that the ensemble of test pulses is coherent, so that the electric field of each is the same. In practice, it is much more useful to be able to evaluate the precision of a given measurement from a single experimental trace. This obviously requires some redundancy. In the case of spectrographic techniques, such redundancy is likely to be present because of the size mismatch between the experimental trace and the measured quantities. Redundant data can be used to evaluate precision in conjunction with a statistical technique called bootstrapping, where multiple inversions of the data are performed after removal of some of the data points [92, 93].

3.4. Specific Implementation of Sonograms

3.4a. Treacy’s Sonogram

Among the earliest sonographic methods, Treacy’s sonogram [72] (also known as the “dynamic spectrogram”) was implemented by using the angular dispersion of a diffraction grating to spread the spectrum of the pulse in space. Temporal information about the arrival time of each spectral component was obtained by using two-photon fluorescence in a dye cell. In the original experiment, the dispersed pulse was correlated with a spatially inverted copy of itself, therefore comparing the time of arrival of optical frequencies symmetrically located on each side of a reference frequency. More recent implementations instead correlate the spatially dispersed pulse with a short pulse (for example, a replica of the pulse under test).

3.4b. Measurement of Sonograms with Nonlinear Crystals

Sonogram measurement of ultrashort pulses nowadays makes use of nonlinear crystals, which provide both reasonable signal amplitudes and appropriate temporal resolution [39, 58, 59, 94]. A typical setup is illustrated in Fig. 15(a). There, the test pulse is split into two replicas. One of the replicas is sent to the spectral filter that acts as the stationary filter, for example a slit in a zero-dispersion line. The output of this filter is cross correlated with the other (short) replica in a nonlinear crystal. The complete sonogram can be measured by scanning the frequency of the spectral filter and the delay between the filtered replica and the replica of the input pulse. One advantage of this implementation of the sonogram, as well as all subsequent implementations based on SHG, is that the experimental trace usually gives directly a good picture of the chirp. This is clearly seen in Fig. 15(b), which shows the correlation between time and frequency in the sonogram of a chirped pulse. As is shown below, implementations of spectrograms with SHG (SHG-FROG) do not benefit from such an intuitive structure. Martinez’s approach led to the measurement of the chirp of a colliding-pulse mode-locking laser by using the determination of the group delay as a function of the optical frequency.

Two-photon absorption has also been used to measure sonograms with high sensitivity [95]. The setup is similar to the setup shown schematically in Fig. 15, but it is necessary to remove the constant background arising from one-photon excitation that is present on the traces. Real-time implementations of the sonogram based on a scanning Fabry–Perot filter and a two-photon detector have also been demonstrated [96].

3.4c. Spectrally Resolved Cross-Correlation

A spectrally resolved cross-correlation approach has also been used to characterize femtosecond pulses. In this method, a narrowband reference pulse is generated from the test pulse by spectral filtering (for example, using a slit in a zero-dispersion line). The field resulting from the cross-correlation between the reference pulse and the unfiltered test pulse in a nonlinear crystal is then spectrally resolved by using a spectrometer. The resulting two-dimensional trace is

3.4d. Measurement of Sonograms with Fast Photodetection

Chirp measurements are important for optical telecommunications because of the detrimental effect of chromatic dispersion and self-phase modulation in optical fibers and the presence of time-varying phase modulation on the pulses generated by lasers and modulators. Telecommunication pulses have low peak power, and their polarization state can vary quickly, which makes diagnostics based on nonlinear optics difficult to implement. Since these pulses often have durations longer than a few picoseconds, time-resolved information can be obtained by using a streak camera, for example. A streak camera can display a two-dimensional image where one spatial direction corresponds to time (calibration of the space-to-time correspondence is, of course, required) and the other direction corresponds to a physical spatial coordinate. A sonogram can therefore be recorded by mapping the optical frequency onto a spatial coordinate at the Fourier plane of a monochromator [Fig. 16(a)]. The pulse under test goes into the monochromator (diffraction grating and imaging system) that maps the optical frequency onto the spatial coordinate x. The streak camera maps the temporal intensity onto spatial intensity along the y direction. The $(x,y)$ image therefore corresponds to the sonogram as a function of Ω and τ. Sonograms measured with fast photodetection have been used, for example, in the chirp evaluation of various externally modulated lasers [74, 97, 98], the measurement of the chromatic dispersion of optical fibers [73, 75], and the characterization of Raman radiation generated by propagation of optical pulses in a fiber [99].

A recent implementation of the sonogram for trains of pulses in the telecommunication environment is based on phase comparison in the RF domain [100]. These trains of pulses usually have repetition rates f of the order of $10\mathrm{GHz}$. As shown in Fig. 16(b), the train of pulses under test is spectrally filtered at the optical frequency Ω (upper part of the setup) and detected by a photodetector with bandwidth greater than f. This gives a RF signal whose phase is proportional to the group delay for the group of frequencies around Ω selected by the spectral filter. This phase can be measured by comparison with another RF signal at the same frequency, generated by sending the unfiltered train of pulses to an identical photodetector (lower part of the setup); the two RF signals are downconverted by mixing with an identical local oscillator running at a frequency close to f. The measurement of the RF phase as a function of the filtered optical frequency then yields the group delay in the pulse composing the pulse train. This implementation uses conventional telecommunication and RF components, is polarization insensitive, and generates its own temporal reference, all of which are significant advantages. Further, telecommunication signals can have large amounts of incoherent amplified spontaneous emission, and it has been suggested that the measurement process is insensitive to this noise background.

3.4e. Sonogram with Phase Retrieval

While the previous implementations of sonograms are based on the simplified retrieval of the chirp, it was pointed out by Wong and Walmsley that complete phase retrieval could be performed on a sonogram, provided that the complete trace of Eq. (3.4) is measured [39]. While the previous implementations typically use a narrowband spectral filter, this is not optimal for phase retrieval, and the stationary filter should in that case have a bandwidth comparable with that of the pulse under test. While spectrograms such as FROG are implemented with an unknown gate, the gate used in sonograms can be characterized in the spectral domain (e.g., by using a spectrometer to measure its transmission and using test-plus-reference spectral interferometry to measure the phase induced on the filtered pulse). The setup follows the general concept of the sonogram implementation, with a spectral filter such as a slit in a zero-dispersion line or a Fabry–Perot filter and fast photodetection provided by a nonlinear cross-correlation with the unfiltered pulse under test. Sonograms were also inverted by using the principal component generalized projection algorithm [77]. In these approaches, the time-nonstationary filter has finite duration and unknown shape. Deconvolution can be performed appropriately [101]. The phase retrieval can be used for all sonograms, provided that the entire experimental trace is measured and the transfer function of the stationary filter does not vary when its frequency is modified.

3.4f. Single-Shot Sonograms

Single-shot sonograms require the acquisition of the two-dimensional sonogram where the frequency and time variable are simultaneously scanned. Two experimental implementations have been demonstrated.

The thick nonlinear crystal approach [102] follows lines similar to those developed in the poor man’s FROG and the GRENOUILLE devices, which are presented in Subsection 3.5 [103, 104]. Phase-matching conditions in a nonlinear crystal can be used to provide strong spectral filtering, therefore enabling, for example, angular dispersion, while the nonlinearity itself provides the gating mechanism. This was implemented in a type II crystal, following Fig. 17(a). The combination of a cylindrical and a spherical lens magnifies the input beam in the horizontal transverse direction and focuses it in the vertical direction. A Wollaston prism is then used to split the incoming pulse into two orthogonally polarized beams that propagate at an angle. This therefore encodes the relative delay between the two pulses on the horizontal spatial axis. The tight vertical focusing in a type II KDP crystal ($\mathrm{K}{\mathrm{H}}_{\mathrm{2}}\mathrm{P}{\mathrm{O}}_{\mathrm{4}}$) leads to a SHG process that is mostly narrowband along the extraordinary axis and broadband along the ordinary axis, with a one-to-one relation between the output angle and the wavelength of the extraordinary wave being phase matched. In other words, the crystal essentially performs the narrowband spectral filtering along the extraordinary axis and gates the broadband pulse under test with the filtered pulse. Another combination of cylindrical and spherical lenses is used to map the vertical angle and the horizontal position on a two-dimensional detector to allow the single-shot measurement of the sonogram as a function of optical frequency and time.

Single-shot sonograms have also been measured by using the encoding of the optical frequency on a spatial coordinate, using a zero-dispersion line and spatially dependent time gating by correlation with the short pulse under test in a noncollinear geometry on a two-photon detector [105, 106]. Following Fig. 17(b), the pulse under test is spatially dispersed by using a diffraction grating and a cylindrical lens so that the optical frequency is encoded on the horizontal variable. A short reference pulse (in practice, a replica of the pulse under test) is incident at the Fourier plane of the zero-dispersion line and produces a cross-correlation signal on a two-photon CCD array, where the relative delay between the two interacting pulses is encoded onto the vertical direction.

3.5. Specific Implementations of Spectrograms

Spectrogram measurement apparatuses can be classified by their implementation of the time-nonstationary filter and the specific measurement geometry.

3.5a. Early Attempts

One of the earliest attempt to measure a complete spectrogram used SHG as the time-nonstationary filter [76, 107, 108]. The setup is an intensity autocorrelator followed by a spectrometer that measures the upconverted pulse spectrum, an experimental combination that is now referred to as SHG-FROG. In this case, the gate is the pulse itself, and the experimental trace is related to the test pulse field by

3.5b. Frequency-Resolved Optical Gating

A complete measurement of the upconverted spectra, together with a means for inverting the data to extract the test pulse field, is generally referred to as FROG (frequency-resolved optical gating). In this method, the time-nonstationary filter is obtained via a nonlinear interaction in a medium with a nearly instantaneous response. The spectrum of the output of the nonlinear mixing process is measured for all delays between the input test pulse replicas, and the pulse is recovered from the measured spectrogram by means of a phase retrieval algorithm. The reader is referred to [28, 109] for extensive descriptions of this technique, together with details of various implementations and experimental results. The most popular experimental implementations are described in the following subsections, with examples of an experimental trace in the case of a Gaussian pulse with second- and third-order dispersion.

Second-harmonic generation frequency-resolved optical gating. In SHG-FROG, the pulse is mixed with a delayed replica in a nonlinear crystal with large spectral acceptance, as indicated in Fig. 18 [60, 110, 111]. The experimental spectrogram is

SHG-FROG has a major drawback that is derived from the fact that the gate is the electric field of the test pulse itself. This leads to a spectrogram that is rather unintuitive, and the sign of a chirp is, for example, not visible on a SHG-FROG trace. Inversion of a SHG-FROG trace is ambiguous in the direction of time, as explained previously. The spectrograms of a Gaussian pulse with second- and third-order dispersion are shown in Fig. 18. Determination of the chirp from first-order moments of the SHG-FROG trace is not possible, and the traces are both symmetric with respect to the relative delay τ. Issues due to the direction-of-time ambiguity can be alleviated in various ways. Some prior knowledge of the electric field of the pulse under test (e.g., knowing that the pulse is positively chirped) or the introduction of a recognizable feature (e.g., a trailing pulse using multiple reflections in a piece of glass) can break this ambiguity. Another approach is to perform an additional measurement of the SHG-FROG trace after addition of chromatic dispersion of a known sign, but this is rather impractical in applications requiring single-shot operation. The deleterious effects of limited spectral acceptance of the nonlinear crystal can be compensated by numerical correction of the experimental trace or by crystal dithering [118]. While SHG-FROG is usually implemented in a noncollinear geometry, some collinear SHG-FROG setups, useful when the experimental setup has some spatial constraints, have been investigated [119, 120, 121]. Collinearity of the two replicas of the pulse in the SHG crystal leads to an interferometric modulation of the experimental trace, which can be used independently of the SHG-FROG trace for phase retrieval [121, 122, 123]. A recent example of the application of SHG-FROG to the determination of pulse formation in a nonlinear optical process is shown in Fig. 19 [124].

Polarization-gate frequency-resolved optical gating. Polarization-gate FROG (PG-FROG) uses the principle of the Kerr shutter [125]. The basic physical principle is that cross-phase modulation is polarization dependent, i.e., the phase shift induced by a pump pulse on a probe pulse depends on the relative polarization state of the two pulses. Consider the interaction of a low-energy probe pulse with a high-energy pump pulse. The phase shift induced on the probe is ${\varphi }_{\mathrm{CO}}\left(t\right)=(4\pi n_{2}L∕\lambda )I_{\text{PUMP}}\left(t\right)$ if the pulses are copolarized, and ${\varphi }_{\mathrm{ORTHO}}\left(t\right)=(4\pi n_{2}L∕3\lambda )I_{\text{PUMP}}\left(t\right)$ if the pulses are orthogonally polarized. If the probe pulse is polarized along $\widehat{x}$ and the pump pulse is polarized along $\widehat{x}+\widehat{y}$, there is an induced time-dependent birefringence, and the probe after interaction is proportional to $E\left(t\right)\{(\widehat{x}+\widehat{y})\exp \left[i{\varphi }_{\mathrm{CO}}\left(t\right)\right]+(\widehat{x}-\widehat{y})\exp \left[i{\varphi }_{\mathrm{ORTHO}}\left(t\right)\right]\}$. The field transmitted through an analyzing polarizer set along $\widehat{y}$ is then $E\left(t\right)\{\exp \left[i{\varphi }_{\mathrm{CO}}\left(t\right)\right]-\exp \left[i{\varphi }_{\mathrm{ORTHO}}\left(t\right)\right]\}$, which is proportional to $E\left(t\right)I_{\text{PUMP}}\left(t\right)$ for small birefringence. A gate proportional to $I_{\text{PUMP}}\left(t\right)$ can therefore be implemented in such a geometry. Spectrally resolving the transmitted pulse as a function of the delay between the probe pulse and the pump pulse yields the PG-FROG trace

Self-diffraction frequency-resolved optical gating. Self-diffraction FROG (SD-FROG) uses the diffracting properties of an index grating written in a material by the interference of two replicas of the pulse under test via the Kerr effect [80]. The efficiency of diffraction of each replica on the grating is proportional to the interference term between the electric field of the two replicas. The SD-FROG trace [Fig. 21(a)] is obtained by delaying one replica with respect to the other and spectrally resolving one of the replicas and can be written as

Third-harmonic generation frequency-resolved optical gating. Third-harmonic generation FROG (THG-FROG) has been implemented by using surface harmonic generation in a glass plate [128] and more recently in organic films [129]. Two replicas of the pulse under test are mixed noncollinearly in a medium, so that they overlap spatially at one of the surfaces of the medium [Fig. 21(b)]. This is actually the most sensitive FROG setup based on third-order nonlinear effects for femtosecond pulses, and it has the advantage of a large phase-matching bandwidth. The traces of THG-FROG can be difficult to interpret; for example, the THG-FROG trace of a Gaussian pulse with second-order dispersion does not show the familiar correlation between time and frequency of linear chirp. However, the THG-FROG traces usually have some asymmetry; for example, the THG-FROG trace of a Gaussian pulse with third-order dispersion has a more familiar shape. Since there is no third-harmonic generation from a beam with circular polarization, a collinear fringe-free THG-FROG setup can be built by using two beams with opposite circular polarizations [130].

Transient grating frequency-resolved optical gating. Transient gradient FROG (TG-FROG) is based on a three-beam geometry similar to what is called BOXCARS in nonlinear spectroscopy [131]. As in Figs. 21(c), 21(d), the input test pulse $E\left(t\right)$ is split into three replicas $E_1\left(t\right)$, $E_2\left(t\right)$, and $E_3\left(t\right)$. The field generated by four-wave mixing is proportional to $E_1\left(t\right)\cdot E_2\left(t\right)\cdot E_3^{*}\left(t\right)$. Depending on the choice of the delayed pulse (either $E_2\left(t\right)$ or $E_3\left(t\right)$), the TG-FROG trace obtained by frequency resolving the generated field is, respectively

Four-wave mixing frequency-resolved optical gating. The strong wave-mixing effects in semiconductor optical amplifiers (SOAs) have been used to characterize pulses in the telecommunication environment. The pulse is once again split into two replicas, where one is used as a strong pump, depleting the carriers in a semiconductor optical amplifier, therefore temporally modulating the gain and phase of the SOA. The second replica, with a lower power, is used as a probe, and one spectrally resolves this replica after the SOA. This approach is extremely sensitive, as it usually suffices to have peak powers of the order of $1\mathrm{mW}$ to induce significant changes in the SOA transmission [132]. Spectrograms measured by using a SOA depleted by a strong pump pulse as the gate acting on a probe pulse from a different source have also been measured, leading to the characterization of dynamical processes in SOAs [133].

Two-photon absorption frequency-resolved optical gating. The high sensitivity of the two-photon absorption in an indium phosphide crystal has been used to characterize pulses in the telecommunication environment [134, 135]. This is a two-pulse mixing, but it is not background free, since the pump induces absorption on the probe. A background-free trace can be obtained by proper modulation of the interacting beams.

Cross-correlation frequency-resolved optical gating. Nonlinear mixing of the pulse under test with another optical pulse has also been used. In this cross-correlation FROG (X-FROG), the pulse and the gate do not have an explicit mathematical link, and the X-FROG trace can be written as function of the field of the pulse $E\left(t\right)$ and the field of the gate $E^{\prime }\left(t\right)$ as

Single-shot frequency-resolved optical gating. The single-shot acquisition of a FROG trace relies on encoding the time and frequency variables into the two transverse spatial coordinates, making use of a two-dimensional detector to record the spectrogram [142]. Mapping the optical frequency to one spatial coordinate is performed with a conventional grating-based spectrometer, which can record the optical spectrum of a pulse in a single shot by using a CCD array. A relative delay between two pulses can be mapped into the other spatial coordinate by using a geometry similar to the single-shot autocorrelator [143]. In this configuration, the convergence of two unfocused beams means that the delay is a linear function of the distance across the interaction plane of the beams. The field at spatial position x after nonlinear interaction is $E\left(t\right)E(t-\alpha x)$. This plane can be reimaged onto the input slit of an imaging spectrometer, which then allows the measurement of the optical spectrum as a function of the optical frequency ω and spatial position x. The measured intensity is therefore $S(\tau ,\omega )$, where the relation between x and τ has been calibrated. Since delay is now coded in a transverse spatial coordinate, it is necessary to avoid spatial distortions of the beam, as they can appear as temporal distortions on the FROG trace. Some corrections of the FROG trace taking into account the beam spatial profile can in general be performed [144]. Single-shot operation of FROG has been used to characterize the output of large laser systems based on chirped-pulse amplification [10].

3.5c. Poor Man’s Frequency-Resolved Optical Gating and GRENOUILLE

In FROG, the stationary filter allowing the optical spectrum measurement after a nonlinear interaction is usually a standard optical spectrum analyzer. It is, however, known that nonlinear interactions in a thick nonlinear crystal having limited spectral acceptance lead to a coupling between the optical frequency of the converted field and its wave vector. That is, the upconversion of a particular frequency occurs only in a particular direction. It was pointed out that such coupling could be used to provide angular dispersion of the upconverted spectrum, allowing, by means of suitable imaging optics and a detector, implementation of the frequency-resolving element of a FROG diagnostic [103, 104]. This property can be used in concert with the time-to-space mapping configuration of single-shot autocorrelators and FROG to provide complete mapping of the two transverse spatial coordinates into the time and frequency coordinates of the FROG trace (Fig. 22). Such an arrangement, nicknamed grating eliminated no-nonsense observation of ultrafast incident laser light electric fields (i.e., GRENOUILLE, the French word for “frog”), is composed of a cylindrical lens, a Fresnel biprism, a thick nonlinear crystal, and a combination of two cylindrical lenses, so that a spatially extended input pulse is focused in the vertical direction in the crystal, where the spatially dependent phase matching leads to angular dispersion in the upconverted beam [104]. To characterize shorter pulses some of the transmissive optics may be replaced by reflective optics [145]. The biprism generates two beams that cross one another from a single input beam, so that the relative time between the two waves is mapped into the horizontal spatial coordinate. The two cylindrical lenses after the crystal are used to map the horizontal position and the vertical dispersion into the horizontal and vertical coordinates in a plane where a two-dimensional detector can be set. The GRENOUILLE trace is fundamentally a SHG-FROG trace; it therefore suffers from a direction-of-time ambiguity, and the trace does not encode chirp information in a direct visual manner. Nonetheless, it has been shown that some types of space–time coupling can be evaluated by using some prior assumptions [146].

3.5d. Linear Spectrograms

A spectrogram can be measured by gating the test pulse with a fast temporal modulator that acts as the nonstationary filter [23, 147]. There are several advantages to this approach, which is particularly well suited to optical telecommunication pulse durations and wavelengths [148]. The technique is highly sensitive, since no nonlinear interaction is involved, and pulse trains with as low as $100\mathrm{nW}$ of average power have been characterized. The choice of appropriate electroabsorption modulator renders the method fairly insensitive to both the polarization and the wavelength. Another technical advantage is that both the electric field of the pulse and the transfer function of the modulator can be retrieved by using the principal component generalized projection algorithm . Therefore, such an experiment can also be used to characterize the amplitude and phase transfer function of an unknown modulator. Since the gate is independent of the pulse, deconvolution of the spectrogram can be performed with a known or approximate transfer function for the gate. While the gate must have a bandwidth similar to the bandwidth of the pulse under test, this condition is rather loose, and subpicosecond pulses have been characterized by using a $30\mathrm{ps}$ gate. The action of the modulator must be synchronized to the pulse under test. In the telecommunication environment, a clock synchronized to the source of pulses is usually available, and can therefore be used to drive the modulator. This approach has been implemented, for example, with an electroabsorption modulator or an electro-optic modulator [149, 150] driven by a RF sine wave, in which the relative delay between the pulse under test and the gate was controlled by using a RF phase shifter, so that no free-space optical delay line was needed [Fig. 23(a)]. Such an arrangement was also used to characterize the output of various pulse carvers [78, 151, 152], to characterize various pulse shapers [153, 154], and to characterize subpicosecond pulses [147, 155]. Figure 23(b) presents the spectral representation of the $2.4\mathrm{ps}$ pulse from a mode-locked laser diode. This pulse was sent in a nonlinear fiber, where self-phase modulation was used to broaden its spectrum, followed by a dispersive fiber to compensate for the nonlinear chirp. The output pulse [represented in Fig. 23(c)] is recompressed to $900\mathrm{fs}$. The characterization of the temporal behavior of a gain-depleted SOA was also performed by gating a probe pulse with the SOA under test [133]. It is possible to operate a linear spectrogram system close to video-rate retrieval, effectively in real time, by using a Fabry–Perot etalon to measure the optical spectra after modulation has demonstrated update rates of the order of $10\mathrm{Hz}$ [24].

3.6. Conclusions

Spectrographic techniques have been used for a large variety of pulses in very diverse configurations. Nonlinear optics plays a central role in most implementations to provide a gating process for subpicosecond pulses, although linear implementations have also been used to characterize longer pulses, e.g., in optical telecommunications. Spectrographic techniques lead in some cases to intuitive experimental traces capable of displaying the time-to-frequency correlation in a chirped pulse. In most cases, the electric field reconstruction must be iterative.

4. Tomography and Imaging Concepts

4.1. Introduction

Means for measuring the phase of the spatial electric field describing a quasi-monochromatic beam of light have been known for a long time. The concepts developed for imaging and wavefront sensing have inspired approaches to characterizing the temporal field variation of short optical pulses. For example, imaging of a pulse using temporal magnification is analogous to imaging of a spatially localized, pointlike object with spatial magnification by means of a conventional imaging system. Similarly, approaches to the measurement of spatially extended wave field by means of tomography have parallels in temporal pulse characterization. In this section the analogy between the spatial and the temporal domains in wave propagation is outlined. We then focus our attention on specific aspects of imaging and wave-field tomography as applied to the characterization of short optical pulses. Finally, we describe in detail various experimental implementations of these concepts.

4.1a. Analogy between Space and Time

The analogy between wave propagation and spatial focusing, on the one hand, and dispersion and temporal focusing, on the other, has proved a fruitful tool for understanding and development of concepts for temporal pulse-field characterization. In conventional optical imaging, a magnified image of an object is obtained by propagating the field emanating from, or scattered by, the object, through a sequence of optical elements that modify the field in such a way that a scaled version of the field at the object is formed at an image plane, where it can be recorded. In this way, a small object can be viewed by using a detector of limited spatial resolution. The process of temporal imaging follows a similar sequence, allowing very brief pulses to be viewed by using a slow detector. Temporal imaging of electrical waveforms was first studied by Caputi [156, 157, 158] and Tournois et al. [159], who experimentally demonstrated a temporal magnification equal to $-2$ [i.e., the input and output intensities of the waveform before and after the time lens imaging system were related by $I_{\text{OUTPUT}}\left(t\right)=I_{\text{INPUT}}(-t∕2)$]. The analogy between Fresnel diffraction in space and dispersive propagation in frequency for optical waves was also mentioned independently [160], and the equivalence between spatial and temporal domains has been studied extensively [161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171]. Experimental realizations with short optical pulses include the time-to-frequency converter [172, 173, 174, 175, 176], leading to the magnification of various waveforms [177, 178, 179] and various applications in optical telecommunications [180, 181, 182]. Other related works include the pinhole time camera [183], the generalization of various imaging concepts to the time domain [184], the conversion of a waveform from time to space [185, 186, 187, 188, 189], the temporal Talbot effect [190, 191, 192], and the temporal van Cittert–Zernike theorem [193]. All of this work is based on the correspondence between the pair of transverse space and transverse wave vector coordinates $(x,k_x)$ and the time–frequency coordinates $(t,\omega )$, the equivalence between spatial diffraction and chromatic dispersion, and the possibility of mimicking the action of a lens in the time domain by using a quadratic temporal phase modulation. Such equivalence is depicted in Fig. 24 in the context of an imaging system. The physics underlying this can be seen by the following argument. Fresnel diffraction over a distance L is expressed in the transverse wave-vector space as a quadratic phase factor [194],

4.1b. Wigner Formalism

Temporal imaging can be understood easily by using the Wigner formulation. The Wigner functions of a pulse before and after quadratic spectral phase modulation $\varphi {\omega }^2∕2$ are related by

The temporal and spectral intensities of the pulse are the time and frequency marginals of the Wigner function, i.e., the projections of that function onto the time and frequency axes:

4.1c. Tomography

Tomography broadly relates to the reconstruction of an object in N dimensions from a set of projections onto lower-dimensional data sets [195]. This concept has been used in medical imaging for a long time [196] and has also been applied to quantum state reconstruction [197], where a high-dimensional entity is estimated from a set of probability distributions. In imaging applications, the three-dimensional reconstruction of an object is obtained from a set of two-dimensional measurements of the attenuation of a probe beam through the object taken along different directions through the object. Restricting our attention to a two-dimensional object with attenuation specified by the function $a(x,y)$, we can define $P_{\theta }\left(u\right)$ as the projection of a nondiffracting source of uniform spatial intensity orthogonal to an axis making an angle θ with the y axis (Fig. 25). Noting that the line ${\Delta }_{\theta }\left(u\right)$ satisfies $y=x\tan \left(\theta \right)+u∕\cos \left(\theta \right)$, one has

4.2. Chronocyclic Tomography

4.2a. Principle

Implementing chronocyclic tomography for reconstructing the electric field of a short pulse requires an apparatus that can project the Wigner function of the pulse onto sets of axes defined in the chronocyclic phase space spanned by $(t,\omega )$ [198]. The approach is equivalent to rotating the Wigner function and measuring its projection on a fixed axis, usually the frequency axis. Rotation of the Wigner function by an angle θ corresponds to an axis rotation by the angle $-\theta$. Experimentally, one can easily measure the frequency marginal of the rotated Wigner function by means of a spectrometer with sufficient resolution. An arbitrary rotation in the chronocyclic phase space can be implemented by means of quadratic spectral phase modulation in series with quadratic temporal phase modulation. The Wigner function $W_{\varphi ,\psi }(t,\omega )$ of a pulse after a quadratic spectral phase modulation $\exp (i\varphi {\omega }^2∕2)$ followed by a quadratic temporal phase modulation $\exp (i\psi t^2∕2)$ is related to the input test pulse Wigner function $W(t,\omega )$ by

4.2b. Inversion for Chronocyclic Tomography

Tomographic data inversion has been studied extensively, both theoretically and experimentally. We deal here with inversion methods for a common configuration: parallel projections measured with a nondiffracting source. For data sets taken in this configuration, inversion can be performed by using the filtered backprojection algorithm. We briefly discuss this algorithm in order to illustrate its principle and to emphasize that such inversion is algebraic and noniterative. More complete treatments of tomographic imaging can be found in the literature [195]. The key to the inversion of tomographic data is the Fourier slice theorem. Using the notation of Subsection 4.1c, we can define the Fourier transform of the attenuation function $a(x,y)$ by