Page 358 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 339

The analytic signal is the single-sided inverse Fourier transform of
the Fourier transform of the field
∞

1
E(t) = √ d ˜ ε( ) exp(−i t) (11.1)
2
0
where
∞

1
˜ ε( ) = √ dt ε(t) exp(i t) (11.2)
2
−∞
The electric field is considered to have compact support in the time
domain and is further assumed to have no spectral component at
= 0, so ˜ ε(0) = 0 (the electric field of a pulse propagating in a
charge-free region of space must have zero area). The analytic signal
is complex and therefore can be expressed uniquely in terms of an
amplitude and phase

E(t) =|E(t)| exp[i t (t)] exp(i 0 ) exp(−i 0 t) (11.3)
where |E(t)| is the time-dependent envelope, 0 is the carrier fre-
quency (usually chosen near the center of the pulse spectrum), t (t)
is the time dependent phase, and 0 is a constant. The square of the
2
envelope I (t) =|E(t)| is the time-dependent instantaneous power of
the pulse, which can be measured if a square-law photodetector of suf-
ficient bandwidth is available. The derivative of the time-dependent
phase accounts for the occurrence of different frequencies at different
times; that is, (t) =−∂ t /∂t is the instantaneous frequency of the
pulse that describes the oscillations of the electric field around that
time, although such interpretation can be difficult. 7,8
The frequency representation of the analytic signal is the Fourier
transform of E(t)
∞

1
˜ E( ) =| ˜ E( )| exp [i ( )] = √ dt E(t) exp(i t)
2
−∞

˜ ε( ) > 0
= (11.4)
0 ≤ 0
Here | ˜ E( )| is the spectral amplitude and ( ) is the spectral
2
phase. The square of the spectral amplitude ˜ I( ) =| ˜ E( )| is the spec-
tral intensity (strictly speaking, this quantity is the spectral density—
the quantity measured in the familiar way by means of a spectrom-
eter followed by a photodetector). The spectral phase describes the
relative phases of the optical frequencies composing the pulse, and