Page 22 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 3

1.2.1 Impulse Response and Coherent
Point-Spread Function
˜
The input-output relationship of a general linear system f i (r,t) →
˜
f o (r,t) reads

˜ ˜ ˜
f o (r o ,t o ) = h(r o , r i ,t o ,t i ) f i (r i ,t i ) dr i dt i (1.1)
˜
where h(r o , r i ,t o ,t i ) is the impulse response, i.e., the system’s response
to a Dirac function:
˜
(r − r i ) (t − t i ) → h(r, r i ,t,t i )
˜
We restrict ourselves to a time-invariant system h(r o , r i ,t o ,t i ) =:
˜ h(r o , r i ,t o − t i ), in which case the input-output relationship takes the
form of a convolution (as far as the time variable is concerned):

˜ ˜ ˜
f o (r o ,t o ) = h(r o , r i ,t o − t i ) f i (r i ,t i ) dr i dt i (1.2)
˜
The temporal Fourier transform of the impulse response h(r o , r i , )

˜
h(r o , r i , ) = h(r o , r i , ) exp(i2 ) d =: h(r o , r i ) (1.3)
is known as the coherent point-spread function; note that throughout
we omit the explicit expression of the temporal frequency .Ifthe
temporal Fourier transform of the signal exists

˜
f (r, ) = f (r,t) exp(i2 t) dt =: f (r) (1.4)
we can formulate the input-output relationship in the temporal-
frequency domain as 12

f o (r o ) = h(r o , r i ) f i (r i ) dr i (1.5)

1.2.2 Mutual Coherence Function and
Cross-Spectral Density
How shall we proceed if the temporal Fourier transform of the signal
does not exist? This happens in the general case of partially coherent
˜
light, where the signal f (r,t) should be considered as a stochastic
process. We then start with the mutual coherence function 13–16
˜
˜ ∗
˜ (r 1 , r 2 ,t 1 ,t 2 ) = E{ f (r 1 ,t 1 ) f (r 2 ,t 2 )}=: ˜ (r 1 , r 2 ,t 1 − t 2 ) (1.6)