4.2. Descripción de la propuesta
4.2.3. Efecto neto de la propuesta
From time dependent perturbation theory, it is seen that the phase of the input electric fields are imparted on the electric field signal. Therefore, characterization of the phase of input electric fields is needed to isolate phase changes originating from the signal itself. In the frequency domain, the group delay is expanded in a Taylor series as
0 0 0 2 0 0 1 2 T (3.4)where , , and are the derivatives of the phase and are known as the group delay, chirp, and third-order dispersion, respectively. The group delay changes the time a pulse arrives but does not change the shape of a pulse. Chirp is a linear dependence of the group velocity with respect to frequency, and third-order dispersion possesses a quadratic dependence with frequency. Positive chirp is defined as lower frequencies travelling faster than higher ones which is the type of chirp imparted by glass. Pulse duration depends primarily on chirp and to a lesser extent on higher order phases. This is seen by calculating the effects of broadening from glass. After passing through 1 mm of BK7 glass, a pulse centered at 600 nm will accumulate 71 fs2 of group velocity dispersion and 27 fs3 of third order dispersion. At a bandwidth of 1200 cm-1 (pulse duration 12.3 fs), this corresponds to a broadening of 36 fs and 5.5 fs from chirp and third order dispersion, respectively.
Pulse compression occurs when chirp and higher order phases are minimized. A useful parameter for determining compression is the time bandwidth product which gives the theoretical limit of compression. It is computed as the width of the intensity of a pulse in time multiplied by its corresponding spectral width intensity. Assuming the spectrum of the intensity of an electric field is Gaussian and contains a flat phase, the time bandwidth product is given as 0.44. For example, a 1200 cm-1 full width at half maximum Gaussian intensity profile compresses to an electric field duration of no less than 12.3 fs by this formula. Pulses with nonzero phase will have higher values of the time bandwidth product. The time bandwidth product also implies that bandwidth and pulse duration are inversely related, so a pulse will need twice the bandwidth to compress to (ideally) twice as short.
3.3.2.Pulse Compression
Conventional methods of pulse compression include the use of prisms and gratings8. These methods are useful for their ease of use. The drawback for prisms is that their ratio of compensation for chirp and third-order dispersion is fixed so that pulses of large bandwidth cannot be compressed over their entire frequency range2. Grating compressors can adjust this ratio but they provide low throughputs. Specially engineered dielectric mirrors, deformable mirror-prism pairs, and pulse shapers provide optimal pulse compression but are complex and a prism compressor is sufficient for the systems of interest in this work9,10.
The amplified seed from the OPA is compressed using prism pairs mounted on delay stages placed approximately 100 cm apart from tip to tip. Prisms are rotated to Brewster’s angle to prevent loss from reflections. The second prism is larger than the first to accommodate the large bandwidth dispersed by the first prism. A mirror redirects the pulse through the prism pair to recombine all frequencies. The prism spacing is large enough to
impart an excessive amount of negative chirp. During compression, the prisms are inserted into the pulse using the delay stages to provide a fine adjustment of positive chirp.
3.3.3.Frequency Resolved Optical Gating
Characterization of the electric field for compression typically involves using the field to measure itself. Pulses are mixed in a fast responding (i.e., off-resonant) medium and their arrival times are scanned. The resulting signal is spectrally resolved. For a transient grating signal with an instantaneously responding medium, the signal of the electric field is approximated by Esig
t, E t E t E t1
2 3 . The frequency resolved optical gating (FROG)8 spectrogram is given by
2 2 , exp TG FROG I E t E t i t dt
(3.5)where the delay between pulses is given by and it is assumed that all pulses are the same. Two pulses, E t
2, act as a gating function for the third field. Since the spectrogram is resolved in frequency and time, the phase of the pulse can easily be estimated by observation. For instance, frequencies which are linearly delayed in time (showing “tilt”) in the spectrogram come from chirp (Figure 3.3). The value of transient grating FROG over a two pulse autocorrelation is that FROG yields the sign of the chirp and third order dispersion. Also, since the gating pulse is the square modulus of the electric field, the phase of the gating pulse does not change the shape of the investigated field in the frequency domain. This makes it possible to use a pulse other than the investigated pulse as the gate pulse. Lengthening the gate pulse will cause loss of information in the time domain that manifests as an equal broadening in time for all frequencies in the spectrogram. FROG does notmeasure the absolute phase or group delay. Interferometric measurements outlined in the next chapter provide a way to determine the absolute phase which is necessary to determine the real and imaginary components of the electric field signal.
Figure 3.3: Example of a spectrogram taken using transient grating frequency resolved optical gating. The prism pair used to compress this pulse has minimized chirp but does not compensate for third order dispersion which is seen as the quadratic dependence of peak amplitude verses frequency. Contours are spaced at 10% intervals.
3.4.HETERODYNE FOUR WAVE MIXING