Ultrashort Laser Pulse Phenomena, 2nd Edition

468 Diagnostic Techniques

thickness, Rayleigh range ρ

0

of the focused fundamental beams, or the overlap-

ping region of the fundamental pulses in a noncollinear geometry. The shorter

the crystal, the broader the bandwidth over which phase matched harmonic

conversion is obtained, but the lower the conversion efficiency. There is clearly a

compromise to be reached between bandwidth and sensitivity. In Eq. (9.21), the

bandwidth efficiency factor η() includes only the frequency dependence of the

phase matching condition, and not a finite response time for the harmonic gener-

ation process. It is assumed here that the response time of the second harmonic

process is much shorter than the pulses to be measured, which is a reasonable

assumption since the second-order nonlinearity of wide bandgap crystals is a

nonresonant electronic process.

Provided the pulses can be approximated by a Gaussian, a simple test can

be performed to determine whether the proper focusing and crystal thickness

has been chosen. In the case of the autocorrelation [Fig. 9.4(b)], a standard

spectrometer (a 25-cm spectrometer is generally sufficient) is used to record the

spectral intensities of the fundamental and second harmonic [2]. In the case

of perfect phase matching and zero dispersion, and for a conversion efficiency

independent of frequency, the ratio of second harmonic to the square of the

fundamental spectral intensity will be a constant in the case of Gaussian pulses,

according to Eq. (3.106). The spectrum of the second harmonic will be narrower

than the squared fundamental spectrum if the effective crystal length is too long.

As a consequence of the SH conversion efficiency being frequency dependent,

the measured correlation width will be longer than the exact correlation length.

The background-free autocorrelation function A

c

[Eq. (9.1)] for a fluctuat-

ing cw signal consists of a symmetric “bump” riding on an infinite background

(Figure 9.5). The width of the bump is a measure of the temporal width of the fluc-

tuations, and the contrast ratio (peak-to-background ratio of A

c

) is a measure of

the modulation depth. A 100% modulation depth results in a peak-to-background

ratio of 2 to 1 [19]. Any background-free signal of finite duration results in a

function A

c

of finite width [Fig. 9.5(c)]. If that signal has some fine structure

(amplitude modulation), a narrow spike will appear in the middle of the cor-

relation function [Fig. 9.5(d)]. This is the coherence spike, typical of a signal

consisting of a burst of amplitude noise [20]. These considerations do not apply

to the phase content or phase coherence of the pulse: The intensity correlations

are the same whether the pulse is at a fixed carrier frequency or has a random or

deterministic frequency modulation.

9.3.3. Single Shot Measurements

Not all lasers provide a train of identical pulses and/or work at high repetition

rate. Pulse to pulse fluctuation can be particularly severe in oscillator ampli-

fier systems. Single shot autocorrelators are therefore highly desirable. In this

Measurement Techniques 469

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Intensity

1

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246642

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Autocorrelation

2

1

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2466420

2

1

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2466420

2

1

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2466420246642

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246642

(a) (b) (c) (d)

Figure 9.5 Some typical waveforms (intensity versus time) (top) and corresponding intensity auto-

correlation A

c

(τ) (bottom). From left to right: (a) continuous signal with 100% amplitude modulation;

(b) noisy cw signal; (c) pulse; and (d) noisy pulse.

section we will discuss the simplest single shot autocorrelators. More sophisti-

cated instruments for single shot amplitude and phase retrieval will be described

in Section 9.4.

9.3.3.1. Intensity Autocorrelators

One of the first intensity autocorrelators for mode-locked lasers was a single-

shot instrument [21]. The beam to be measured is split in two beams, which

are thereafter sent with opposite propagation vector into a nonlinear medium.

The first autocorrelator was based on two photon excitation rather than SHG:

The medium (for instance a dye solution) was selected for its large two photon

absorption and subsequent fluorescence. Because of the larger optical field in

the region where the two counter-propagating pulses collide, the observed pat-

tern of two-photon fluorescence essentially displays the intensity autocorrelation

(with background). Because of the higher conversion efficiency of SHG, two-

photon fluorescence is not widely used in the fs time scale, except in the UV,

where no transparent nonlinear crystals can be found. To circumvent the diffi-

culty of spatially resolving the µm size of the two-photon fluorescence trace, the

beams are made to intersect at a small angle, thereby magnifying the fluorescence

trace [11,22].

Single-shot autocorrelators using SHG have also been designed. In an arrange-

ment developed for ps pulses by Jansky et al. [23] and Gyuzalian et al. [25] the

autocorrelation in time is transformed into a spatial intensity distribution. This

method has been applied by numerous investigators to the fs scale [24, 26, 27].

The instrument is a typical noncolinear SH autocorrelator. The nonlinear crys-

tal is oriented for phase matched type I SHG for two beams intersecting at an