Page 379 - Phase Space Optics Fundamentals and Applications

360 Chapter Eleven

Spectral filter:
2
−( − C )
A
˜ S ( ; C ) = exp (11.62)
2 2
Quadratic spectral phase modulator or dispersive delay line:

2
P

˜ S ( ; ) = exp i (11.63)
Q
2
Linear spectral phase modulator or delay line:
P

˜ S ( ; ) = exp(i ) (11.64)
L
The filter of Eq. (11.59) is a time gate, or shutter, where is the
time of maximum transmission and −1 is the duration for which it is
open.BoththequadraticandlineartemporalphasemodulatorsofEqs.
(11.60) and (11.61), denoted by the subscripts Q and L, respectively,
impose new time-dependent phase on the input field. Temporal phase
is imparted on the input pulses, e.g., by passing the pulses through an
electrooptic phase modulator. Quadratic phase is imposed if the input
pulses are passed through the phase modulator near one of the max-
ima of a sinusoidal driving signal, whereas linear phase results if the
input pulses arrive near a zero crossing. Therefore is proportional

t
to the modulator’s modulation depth and the square of the modu-
lator’s driving frequency while is proportional to both the modu-

t
lation depth and driving frequency. The spectral filter of Eq. (11.62)
is simply represented by an idealized spectrometer transfer function,
where C is the center frequency and is the frequency passband.
The quadratic and linear spectral phase modulators [Eqs. (11.63) and
(11.64)] impose new frequency-dependent phase on the input field.
Spectral phase is easily imparted by a delay line. Quadratic spectral
phase is imposed if the delay line is dispersive with group-delay dis-
persion of . Linear spectral phase is the result of a nondispersive

delay line of temporal delay .

Using this formulation, the output of such a general measurement
apparatus is given by

D({p k }) = d dt W(t, )W M (t, ; {p k }) (11.65)
where the set {p k } specifies the filter parameters. Therefore any ap-
paratus measures a smoothed out, positive definite version of the
Wigner function—the pulse Wigner function W(t, ) integrated with
an apparatus Wigner function W M (t, ; {p k }).