# 2.1.6: Characterization of Ultrashort Pulses

Until now, we were concerned about the ways of producing ultrashort light pulses required for time-resolved spectroscopy. In this section, we will discuss how to determine the most important parameters of such pulses. Two most important parameters of the pulse that are of interest to the experimentalist are its frequency spectrum and duration. Measuring frequency spectrum of an ultrashort light pulse is no different from measuring the spectrum of continuous source, such as a lamp: one only needs to direct the light into a spectral instrument with dispersive element (prism or diffraction grating) and record the spectrum using a photodetector. Current spectrometers usually employ CCD arrays, similar to the sensors of digital cameras. The scheme of such measurement is depicted in fig. 11A).

The duration of the pulse is trickier: it is in the range of tens of femtoseconds. The only available tools to measure such fast events are the ultrafast pulses themselves; therefore they must become their own measurement instruments. The simplest method of determining the duration of a femtosecond pulse is depicted in fig. 11B). The technique is called *autocorrelation*, whereas the device, the optical layout of which is shown in fig. 11B) is termed *autocorrelator.* The principle of operation involves a semitransparent mirror that splits the pulse into two replicas, one of which can be delayed with respect to the other. The delay is accomplished by spatially moving two mirrors reflecting the beam (these mirrors are mounted on a precise translation stage). The moving mirrors together with the translation stage are termed an *optical delay line*. After traveling their respective distances, both pulses are intersected in a nonlinear crystal, where non-collinear second harmonic generation takes place, similarly to the one shown in fig. 8B. To measure the *autocorrelation function*, one needs to record the dependence of the generated second harmonic intensity on the position of the delay line. Non-collinear second harmonic generation is only possible when both pulses overlap both in time (arrive to the crystal at the same instance) and in space (spot on the crystal). If the pulses hit the same spot, by moving the delay line we can measure the second harmonic intensity proportional to the temporal overlap integral of the two pulses. To find the second harmonic intensity as a function of time *t*, we take multiply the intensities of both moving and stationary pulses and integrate over the entire time range:

Both pulses are identical – they are replicas of the initial pulse produced by the beamsplitter. With that in mind, the intensity of the second harmonic signal is

Such integral in statistics is called a temporal autocorrelation function of a time-dependent quantity. Hence the terms autocorrelation and autocorrelator. Plugging the envelope of a Gaussian pulse into eq. and performing the integration reveals that an autocorrelation of a Gaussian pulse is also a Gaussian function, and its width is times the width of the pulse:

.

Note that if we cross-correlate two different pulses, one of which is many times shorter than the other, eq. will yield

In other words, the generated second harmonic pulse is proportional to the intensity of the slowly varying signal at the time instance, when the short pulse arrives. To put it simply, an extremely short pulse can be used to ‘time-slice’ the long pulse thereby directly establishing its time course. This insight is the basis of time-resolved spectroscopic technique called fluorescence upconversion, discussed in the further sections.

With both spectral width and time duration of the pulse known, we can compute their time-bandwidth product and see how far our pulse is from the perfect Gaussian. In practice, the product is usually higher than 0.441, i.e. the pulses are not exactly Gaussian-shaped. Nevertheless, the product is often cited as one of the pulse quality parameters.

Let us note that neither autocorrelation, nor spectrum, nor both of them provide the full information about the pulse. The full information is complex function describing electric field in time (in addition to that, there is also the distribution across two transverse spatial coordinates). We have simply assumed Gaussian time shape and spectrum and discussed how to measure them both. In many cases, this information is sufficient. When full description of the pulse field is necessary, other, more complex pulse characterization methods are employed, including FROG, SPIDER, GRENOUILLE etc. [6].