Snapshot: It is now possible to measure the complete temporal intensity and phase of even the shortest of optical pulses. The authors describe the revolution in measurement techniques that have made this possible.

A stroboscope records fast events by illuminating them for an interval shorter than the timescale of their dynamics. In the middle years of this century, Harold Edgerton at MIT raised this technology to a fine art, producing some of the most dramatic technical photographs ever taken, such as the one above of a bullet slicing the Jack of Hearts. Today, physicists, chemists, and biologists use the same concept in studying the dynamics of atoms, electrons, and other bits of matter, but on a much faster time scale.

They use pulses from the current generation of ultrafast lasers, which are as short as a few femtoseconds ( 10^-15sec). These are the shortest events ever created by man. The mind-boggling brevity of these durations can be appreciated by applying the well-known formula that time is money. If 1 second corresponds to the current debt of the U.S. government (about $5 trillion), then 10 femtoseconds is the equivalent of 1 nickel (5 cents).

How do we know that these pulses are so short? This simple question turns out to be a difficult problem, one whose solution has eluded researchers for many years and has only recently been found. To appreciate how difficult this problem is, remember that, in principle, to measure an event, you need a faster event with which to strobe it. But, since ultrashort laser pulses are the fastest man-made events, you don't have a faster event. The solution has only recently been found and turns out to require the synthesis of parts of several branches of physics and even draws on ideas from music.

In this article, we describe the revolution that has occurred in ultrashort-pulse measurement and some of the new capabilities it has made available.

The goal: measurement of the intensity and phase vs. time or frequency

The goal is to measure the variations in time, not only of the pulse intensity, but also of its color. These variations are embodied in the pulse electric field, E(t), which oscillates rapidly in time. Rather than deal with the field directly, it's easier to extract out the rapidly varying part and think in terms of the relatively slowly varying envelope, E(t):

(1)

where is the carrier frequency. E(t), which is complex, can be written

(2)

where I(t) is the intensity, and the pulse's variations in color are contained in the phase, . The phase determines the variation of the pulse's frequency from . Quantitatively, the frequency, or color, of the pulse at time t is defined by:

(3)

The variation of the frequency with time is called "chirp" (named after the rising or falling frequency with time in a bird's chirp). Positive chirp is an increasing frequency with time, and negative chirp is a decreasing frequency with time. But more complex chirps are also quite common. Indeed, understanding the complicated chirps present in the shortest of ultrashort pulses plays an important role in understanding the pulse-formation physics in modelocked lasers and is the key to making even shorter pulses.

The pulse can equally well be characterized in the frequency domain by taking the Fourier transform of Eq. (2) to obtain

(4)

where is the frequency relative to the carrier. By analogy, is the spectral intensity (or just the spectrum) and the spectral phase. Likewise one may compute the derivative of the spectral phase with respect to frequency, , by analogy to Eq.(3). This quantity has the units of time and represents the delay in the arrival time of a particular slice of the spectrum at frequency (compared to that of the carrier frequency) at a particular location. It is called the group delay, . If is constant, all frequencies arrive at the same time, and the pulse is as short as possible.

The autocorrelation

So how do we measure these incredibly short events? It's not possible using electronics because these pulses are five orders of magnitude faster than oscilloscopes, three orders of magnitude faster than photodiodes, and an order of magnitude faster than the fastest streak cameras. Early on, it was realized that the only event remotely fast enough to measure an ultrashort pulse was the pulse itself. This gave birth to the now-standard method of measurement: the intensity autocorrelation (AC).[1] The AC is measured by crossing the pulse and a delayed replica in a second-harmonic-generation (SHG) crystal [or some other nonlinear medium, such as a two-photon absorber] and detecting the SHG energy as a function of delay. This is maximized when the two pulses are temporally coincident, so that the range of delays over which the signal is detected is approximately the pulse duration (see Fig. 1). Mathematically, the autocorrelation is given by:

The AC gives the approximate pulse length, but not much more. Because it uses the pulse to measure itself - and that's not quite fast enough to resolve the pulse - it smears out structure in the pulse, so a pulse with several closely spaced intensity peaks, for example, has a smooth, structureless autocorrelation. Thus the AC alone does not fully determine I(t). Moreover, it contains no information about . Measurement of the spectrum , , gives some additional information, but can only tell you only that [and ] is or is not constant. But many experiments require knowledge of the specific variations in the intensity and phase.

Jean-Claude Diels introduced a partial remedy to this problem by placing the SHG crystal at the output of a Michelson interferometer. In this case interference fringes appear in the autocorrelation signal, leading to the name interferometric autocorrelation (IAC). This contains some information about but again does not fully determine it [or I(t)]. For example, it cannot distinguish positive from negative chirp (see Fig. 1).

Figure 1. The intensity vs. time, the frequency vs. time, the intensity autocorrelation vs. delay, the interferometric autocorrelation vs. delay, and spectrograms (or sonograms) of negatively chirped, unchirped and positively chirped Gaussian-intensity pulses. In the spectrograms, the vertical axis is frequency and the intensity is color-coded. Note that the autocorrelation and interferometric autocorrelation cannot distinguish positive from negative chirp, while the spectrogram and sonogram can.

The time-frequency domain

The current revolution actually started 25 years ago with an idea by Brian Treacy. [2] He introduced the notion of measuring the intensity vs. time for different spectral slices of an ultrashort laser pulse, as was often done for acoustic waves. His measurements thus provided simultaneously some time and some frequency information and had to be thought of in a hybrid "time-frequency" domain. Although this concept seems at first counter-intuitive, it is not: a well-known example is the musical score, which shows the frequencies present in an acoustic waveform during a given time interval, and is thus a plot of the waveform's frequency vs. time. Additional marks at the top - pianissimo or fortissimo - indicate the intensity vs. time. A mathematically rigorous form of the musical score is the "spectrogram"
[3]:

(6)

where is a variable-delay time-gate function. The spectrogram, like the musical score, is the set of the spectra of all temporal slices of the field. Two examples of musical scores and their corresponding spectrograms are shown in Figure 2.

Figure 2. Spectrograms (or sonograms), frequencies vs. time, and equivalent musical scores for two different pulses (one better described in the time domain and the other better described in the frequency domain). Note that the spectrogram and sonogram graphically follow the pulse frequency vs. time (left) or the group delay vs. frequency (right).

An analogous quantity - the "sonogram" - can be defined, which is the intensity vs. time for all frequency slices of the pulse (See Fig. 3). It is mathematically equivalent to the spectrogram, but it involves gating in frequency, rather than in
time:

(7)

and is the quantity that Treacy measured. Although several others - notably Yuzo Ishida, Eric Ippen, Andrew Wiener and J. P. Likforman - made measurements of various time-frequency domain quantities in the 1980's, Treacy's method did not find wide application until Juan Chilla and Oscar Martinez [4] showed in 1991 that it could be used reconstruct the full intensity and phase of the pulse. They realized that it was possible under certain circumstances to measure the approximate group delay as a function of frequency from the sonogram, and thus to obtain by integration. Since is readily available, this solved the pulse measurement problem for many pulses. This technique was labelled by them frequency domain phase measurement, or FDPM, and by Treacy the dynamic spectrogram. Experimentally it is simple to measure: a portion of the pulse spectrum is selected by a spectrometer and the cross-correlation of the selected slice with the input pulse is taken by crossing the two in an SHG crystal.

The difficulty with the Chilla-Martinez recipe is that it requires the frequency gate to be extremely narrowband, making the filtering step quite inefficient. Only when the pulse spectral phase is well behaved can the gate be made wide enough to allow through enough energy to detect and simultaneously narrow enough that the inversion recipe works. In addition, it is sometimes the case that, for a given frequency, the sonogram has two or more peaks in time, so that it is not possible to define uniquely a group delay. In this case the pulse cannot be reconstructed using this simple algorithm. Even for relatively well-behaved pulses, the intensity vs. time at a given frequency need not be symmetric in time, and this again raises the question as to how one defines the group delay for this frequency.

It's also possible to measure the spectrogram of an ultrashort pulse. This is the basis of frequency-resolved optical gating, or FROG, developed by Daniel Kane, Ken DeLong, and Rick Trebino. [5] In this case, gating must occur in time, rather than frequency, followed by measurement of the spectrum of each time slice. Of course, no fast time gate is available, so one must gate the pulse with itself. For example, using SHG to gate, the measured FROG spectrogram is

Experimentally, FROG is also quite simple: the FROG spectrogram is just the spectrum of the autocorrelation. Since experimenters already routinely measure pulse autocorrelations and spectra, they need only move their spectrometer to the output of their autocorrelator in order to make a FROG trace, as shown in Figure 3. In fact, single-shot operation of FROG is straightforward. In addition, almost any nonlinear optical process can be used to generate a self-time gate, and a number have been demonstrated.

Measuring a spectrogram doesn't avoid the problems of inversion described above, however. If anything, they are worse: because the gate is the pulse itself, it can never be narrow enough to accurately obtain the frequency vs. time directly from the trace.

Figure 3. Apparatuses for the measurement of (a) the spectrogram and (b) the sonogram, as a function a delay t and frequency . Key: L = lens, M = mirror, BS = beamsplitter, G = diffraction grating, F = filter. The symmetry between the experimental implementations is indicated by the block diagrams, in which the order of the spectrometer and SHG time-gate are reversed in (b) from that in (a).

Phase retrieval

The next development occurred when Rick Trebino and Daniel Kane realized that the problem of determining the pulse intensity and phase from a spectrogram was essentially equivalent to the two-dimensional "phase retrieval" problem in image science and astronomy. Phase retrieval is the problem of finding a function knowing only the magnitude (but not the phase) of its Fourier transform. Phase retrieval for a function of one variable is impossible. For example, knowledge of a pulse spectrum does not fully determine the pulse-many different pulses have the same spectrum. But, a decade ago, image scentists found that phase retrieval for a function of two variables is possible. Knowledge of only the magnitude of a two-dimensional Fourier transform of a function of two variables essentially uniquely determines the function (provided that the function is of finite extent). Interestingly, these results follow directly from the existence of the Fundamental Theorem of Algebra for polynomials of one variable and its nonexistence for polynomials of two variables. [6]

Measurement of a spectrogram (or sonogram), that is, the Fourier transform of a function of two variables, thus frames the ultrashort-pulse measurement problem in a form that allows a rigorous and general solution. This realization lead to the introduction of iterative inversion algorithms. [6,7] The general prescription is that one seeks to find a test field that minimizes the difference between the measured spectrogram and the test spectrogram [obtained from Eq. 3 by replacing E(t) with ]. An initial guess is refined through iteration by continually comparing the test and measured spectrograms and then using the difference between them to determine how to alter the test field. The important point is that any algorithm that takes into account all the NxN data points of the spectrogram, rather than N data points in the time domain and N data points in the frequency domain, produces a better estimate of the pulse, since it has much more material with which to work. Moreover, if the algorithm does not converge, there is some systematic error in the measurement. For example, convergence may not occur if different parts of the beam have different pulse shapes.
 

Spectrographic and sonographic methods

Note that nonlinear optics plays an important role in all methods for characterizing ultrashort pulses. The reason is simply that it is the only way to make a time gate with sufficiently fast response. Note also that both spectrograms and sonograms contain a frequency gate, in the form of a spectrometer, in addition to a time gate, except in the reverse order. This leads to differences in the experimental implementation of each method, and thus in the sorts of experiments in which each is useful, but the two methods are, in principle, quite similar. In fact the pleasing symmetry between the two is suggestive of some deeper lying truth: simultaneous time and frequency information of one form or another is always needed to obtain full information about the field. [8]

Measuring ultrashort, ultraweak pulses

Because a nonlinear-optical process is required in these techniques, their applicability is limited to pJ pulses or greater in multishot measurements and pJ pulses or greater in single-shot measurements. But, if a fully characterized reference pulse is available, there are a number of options for characterizing other, much weaker pulses. A particularly simple and sensitive technique is spectral interferometery (SI), first developed by Froehly and coworkers, [9] in which the reference and an unknown pulse are combined at a beamsplitter, and the spectrum of the combined pulse pair is measured. From this measurement the difference in the spectral phases of the two pulses can be found. Since the reference-pulse spectral phase is known, that of the unknown pulse is easily found. An important advantage of this method is that the unknown pulse can be almost arbitrarily weak. In a recent demonstration by David Fittinghoff and colleagues, the unknown pulses each contained about 40 zeptoJoules, or 0.2 photons, on average, although the signal was obtained by integrating over some pulses. They used FROG to measure the reference pulse and refer to the combination of FROG and SI as temporal analysis by dispersing a pair of light E-fields, or TADPOLE.

If the test pulse contains wavelengths not present in the reference pulse, then spectral interferometry doesn't work. However, the test pulse can be characterized using upconversion as shown by Likformann and Manuel Joffre.

Applications

With this new-found capability, a number of otherwise impossible experiments are now possible. It has been known for may years that a knowledge of the electric field of the pulses at the output of a modelocked laser could provide significant information about the physical mechanisms responsible for generating short pulses. Work along these lines by Eric Ippen, Jean-Claude Diels, Philippe Fauchet and their colleagues using the IAC, and Mark Beck and Ian Walmsley using FDPM, provided some significant insights into the role of different pulse shaping mechanisms in the colliding-pulse-modelocked dye laser. More recently, Margaret Murnane, Henry Kapteyn, Greg Taft and Andy Rundquist used FROG to measure the sub-ten-femtosecond pulses emitted by a self-modelocked Ti:Sapphire laser. [10] Two different theories existed for the precise shape of these pulses, both predicting identical spectra and ACs, and both agreeing with experimental measurements, so it was not possible to decide which was correct. The FROG measurement resolved the question decisively (see Fig. 4), demonstrating that the main limitation to making shorter pulses is group-delay dispersion (the tendency of different colors to experience different delays in passing through optical elements), and not the finite bandwidth of the gain medium.

These techniques have also had impact in materials characterization. For instance, Antoinette Taylor and Traci Sharp-Clement of Los Alamos National Labs are using FROG to measure the nonlinear refractive index of glasses. The key here is that a pulse propagating through a medium with a nonlinear refractive index has its temporal phase altered in a manner that depends on the intensity of the pulse, among other things. For a given pulse spectrum there are an infinitely large number of possible pulse shapes, and thus one cannot infer uniquely the refractive index of the sample from a measurement of the change in the pulse spectrum on transmission through the sample - the complete pulse shape is
needed.

Another obvious arena in which the knowledge of an exciting electric field plays an important role is that of quantum control. Central to this field is the idea that the way in which the electrons and nuclei in atoms and molecules are driven strongly affects their future behavior. This means that the electric field that drives the atomic or molecular dipole must be precisely controlled using shaped ultrashort pulses. Of course, it is also essential to measure these pulses. Bern Kohler at Ohio State and Kent Wilson of the University of California San Diego have used FROG to characterize the complex pulses they are using in molecular coherent control experiments. [11]

Figure 4. FROG traces of 10 fs pulses. Left: the theoretical FROG trace of a pulse with distortions introduced by coherent ringing in the laser gain medium (calculated by John Harvey and coworkers). Center: the theoretical FROG trace of a pulse with distortions introduced by higher-order group-velocity dispersion (calculated by Henry Kapteyn and coworkers). Right: experimental trace of a ~ten-fs pulse, measured by the group of Margaret Murnane and Henry Kapteyn, showing that the pulse distortions are most likely due to higher-order group-velocity dispersion.

Non-spectrographic measurements

It is not absolutely necessary to measure time and frequency simultaneously to characterize a pulse, and alternative techniques are now being developed. One of these is a novel method for "temporal imaging" of the pulse that was independently proposed by Pierre Tournois, Sergei Akhmanov, Athanasios Papoulis, William J. Caputi, and Brian Kolner. Relying on the symmetry between time and space in Maxwell's equations, they have shown that it is possible to stretch or compress a pulse without changing its shape using a phase modulator. As a result, this method might be used to stretch a pulse to a duration where its intensity vs. time could be measured using a photodetector. Bernard Prade, and Andre Mysyrowicz have also developed a technique that relies on phase modulation. And Mark Beck and coworkers have extended the idea to the recovery of the full intensity and phase using computer assisted tomography. More importantly it may be possible to characterize trains of non-identical pulses using this method. In this case the field of a single pulse is not a useful quantity and the statistical properties of the field variations, such as the two-time correlation function , are more important.

Another potential class of methods uses time-domain interferometry. The earliest experiments along these lines were performed in the 1980's by Joshua Rothenberg and Dan Grischkowsky, but a more modern version for the femtosecond domain has been developed by John Heritage and coworkers.

The future

What of the future? One of the big unsolved problems is the measurement of pulses that have significant temporal and spatial structure, for example, a pulse whose energy spectrum varies from point to point in the beam. Such "spatial chirp" is characteristic of beams from pulse stretchers and compressors (even those that are only slightly misaligned!) used in high-power chirped-pulse amplifiers. These distortions may be useful in certain applications, but first it will be necessary to characterize them. Another need is for pulse measurement techniques in the ultraviolet and mid-infrared, where the problem is a lack of suitable materials.

It is clear, however, that the revolution that has taken place only recently in ultrashort-pulse measurement has not only yielded powerful new laser diagnostics, but also has opened up tremendous new possibilities for ultrafast science and technology.

References

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2.E. B. Treacy, "Measurement and Interpretation of Dynamic Spectrograms of Picosecond Light Pulses," J. Appl. Phys.42, 3848-3858 (1971).

3.L. Cohen, "Time-Frequency Distributions-A Review," Proc. IEEE 77, 941-981 (1989); Time-Frequency Analysis (Prentice Hall, Englewood Cliffs, NJ, 1995).

4.J. L. A. Chilla and O. E. Martinez, "Direct Determination of the Amplitude and Phase of Femtosecond Light Pulses," Opt. Lett., 16, 39-41 (1991).

5.R. Trebino and D. J. Kane, "Using Phase Retrieval to Measure the Intensity and Phase of Ultrashort Pulses: Frequency-Resolved Optical Gating," J. Opt. Soc. Am. A 11, 2429-2437 (1993).

6.H. Stark, ed. Image recovery: Theory and Application, (Academic, Orlando, FL, 1987).

7.K. W. DeLong et al., "Phase Retrieval in Frequency-Resolved Optical Gating Based on the Method of Generalized Projections," Opt. Lett. 19, 2152-2154 (1994).

8.V. Wong and I. A. Walmsley, "Linear Filter Analysis of Methods for Ultrashort-Pulse-Shape Measurement", J.Opt. Soc. Am. B 12, 1491-1499 (1995).

9.C. Froehly et al., Nouv. Rev. Optique 4, 183 (1973).

10.G. Taft et al., "Ultrashort Optical Waveform Measurements Using Frequency-Resolved Optical Gating," Opt. Lett. 7, 743-745 (1995).

11.B. Kohler et al., "Phase and Intensity Characterization of femtosecond Pulses from a Chirped-Pulse Amplifier by Frequency-Resolved Optical Gating," Opt. Lett. 5, 483-485 (1995).