The SPIDER Technique

SPIDER is a technique which characterizes the electric field of ultrashort optical pulses, based on shearing interferometry in the optical frequency domain. As square law detectors are not sensitive to the phase, the measurement of the intensity (whether it is spatial or spectral) is an easy task but the measurement of the phase needs indirect solutions.

Spatial vs. Spectral Shearing

Spatial shearing interferometry relies on the measurement of interferences between two replicas of the wavefront one wants to characterize which are displaced by a small amount X along the spatial axis. For an electric field

, which interferes with a spatially shifted replica of itself,

the interference pattern as recorded by a square-law detector is

. The measurement of the intensity at a point x is directly related to the phase difference between the phase of the initial wavefront at point x and at a point x+X,

. This phase difference can be easily and algebraically extracted from the interference pattern using standard Fourier processing techniques. The spatial phase of the field,

, can then be reconstructed by concatenation. This principle can be transposed to the characterization of the ultrashort optical pulses using a shear along the frequency axis. In this case, we measure the spectral interference between two replicas of the unknown pulse when one of the replicas has been shifted slightly in frequency (spectrally sheared). This interference pattern can be written as

where

is the electric field of the pulse in the frequency domain and is the spectral shear. The spectral phase,

, can be extracted in the same way as in the case of spatial shearing.

Figure 1 : principle of shearing interferometry. The initial field (fullline) is replicated and sheared (dotted line).

Creating the Spectral Shear

A direct way to generate a spectral shear is to use a linear temporal phase modulation. Such modulation can be obtained by driving a phase modulator with a sine wave and synchronizing the short pulse to be modulated with the zero-crossing of the modulation. This has the advantage of being a linear technique, and therefore highly sensitive. However, the availability of suitable modulators has restricted the use of this direct approach up to now. Nonlinear optics provides a general approach to globally shearing a complex spectral amplitude by a large frequency

. This can be done by sum frequency generation between a monochromatic frequency,

, and the field you want to shear,

, resulting in a sheared field,

. It is convenient to obtain the monochromatic frequency from a strongly chirped ultrashort pulse for which the instantaneous frequency, which is approximately a linear function of time, does not vary during the nonlinear interaction with the short pulse. As shown in Figure 2, the delay between the chirped pulse and the short pulse determines which monochromatic frequency of the chirped pulse the short pulse is sheared by. If the chirped pulse interacts in the nonlinear medium with two replicas of the short pulse are delayed with respect to each other, the sheared fields will have a spectral shear between them. The interference of these two fields is given by

, where

is the time delay between the pulse replica. This is just a spectral shearing interferogram with an overall frequency shift.

Figure 2 : generation of two sheared replicas of the input pulse by non-linear interaction with a chirped pulse

As mentioned earlier, the phase difference between the two spectrally sheared replicas can be extracted easily and algebraically using methods taken from Fourier Transform Spectral Interferometry (FTSI). The extraction of the phase difference is performed using a Fast Fourier Transform, a filtering of one of the interference terms, and a Fast Fourier Transform back to the initial experimental points. The phase of the initial pulse is obtained from the spectral phase difference in three steps: subtraction of the linear term , a global shift by along the frequency axis (these two actions only need calibrated quantities), and concatenation of the phase at steps of . This inversion algorithm is thus totally algebraic and does not rely on iterative procedures.

Advantages of SPIDER

From an experimental point of view, the set up is simple. SPIDER does not involve any moving parts.
The interferogram is measured with a 1-dimensional spectrometer, which brings low cost, high acquisition rates and simplicity.
The acquisition of the experimental trace is done in a single-shot.
From a theoretical point of view, the technique is clear and relies on solid concepts.
Because the processing from the experimental trace (the 1-D interferogram) to the retrieved spectral phase is algebraic (it is composed of a predefined number of algebraic operations and not on iterative techniques), the reconstruction of the pulse shape from the experimental data is quick and shows low sensitivity to the noise. This technique has currently demonstrated the highest update rates (>1 kHz) when characterizing short pulses. These rates are only limited by the acquisition of the interferogram since the reconstruction from the interferogram to the spectral phase can be done in a couple of milliseconds on a personal computer.

Experimental Implementation

Although various setups have been used for different applications, the implementation of SPIDER always relies on the elements summarized on the following figure.

Figure 3 : main elements in a SPIDER setup

Generation of a chirped pulse. This chirped pulse is usually part of the initial pulse which goes into a dispersive delay line (either a block of dispersive material, or a double-pass two-grating compressor) as in Figure 3, but can also be at a different wavelength, like in a cross-correlation setup.
Generation of two replicas of the initial pulse, separated by a delay . This can be done using a conventional Michelson interferometer, or a glass etalon. The etalon is a very simple solution, and provides a very stable delay. Experimentally, the two replicas are generated by reflection on the two faces of the etalon; the remaining transmitted energy can be sent to the dispersive delay line for the generation of the chirped pulse.
Nonlinear interaction. This is usually an upconversion in a nonlinear crystal, although downconversion has also been used for the characterization of blue pulses. A type II crystal has two advantages. First, it allows a collinear background-free measurement if the two replicas and the chirped pulse have different polarizations. Secondly, in these crystals, the phase matching function is not symmetric for different polarization; there is a very broad spectral acceptance only along one of the axes. Because the process of the nonlinear conversion is non-symmetric (each upconversion takes place between a short pulse, having a broad spectrum, and a monochromatic frequency), it is possible to exploit the large spectral acceptance by setting the axis of the crystal correctly. Therefore it is feasible to use thicker nonlinear crystals, whereas most other techniques, which rely on symmetric interaction, need a thin nonlinear crystal. Experimentally, it is convenient to use a non-collinear setup with a type II crystal.
Measurement of the interferogram. A monodimensional spectrometer is used to measure the interferogram. The resolution needed is accessible even by cheap commercial spectrometers.