Page 33 - Phase Space Optics Fundamentals and Applications

14 Chapter One

t t
we can construct the three-dimensional vector [j ,j z ] , which is known
r

as the geometrical vector flux. 47 The total radiant flux 42 j z (r) dr fol-
lows from integrating the radiant emittance over the space variable r.
More on radiometry can be found in Chap. 7 by Arvind Marathay.
1.4.4 Instantaneous Frequency
The Wigner distribution W f (r, q) satisfies the nice property that for
a coherent signal f (r) =| f (r)| exp[i2 (r)], the instantaneous fre-
quency d /dr =∇ (r) follows from W f (r, q) through

q W f (r, q) dq
d
= (1.29)
dr
W f (r, q) dq

To prove this property, we proceed as follows. From f (r) =
| f (r)| exp[i2 (r)], we get ln f (r) = ln | f (r)|+ i2 (r), hence
Im{ln f (r)}= 2 (r), which then leads to the identity

d (r) d ln f (r) ∇ f (r)
2 = Im = Im
dr dr f (r)

∗
1 ∇ f (r) ∇ f (r)
= −
2i f (r) f (r)
∗
1 [∇ f (r)] f (r) − f (r)[∇ f (r)] ∗
=
2i f (r) f (r)
∗

1 ∂
=−i f r + r f ∗ r − r
1
1
2
| f (r)| ∂r 2 2
r =0

On the other hand, we have the identity

2 q W f (r, q) dq

t
= 2 f r + r f ∗ r − r exp(−i2 q r ) dr q dq
1
1

2 2

t
1
1
= f r + r f ∗ r − r 2 q exp(−i2 q r ) dq dr
2 2

= i f r + r f ∗ r − r [∇ (r )] dr
1
1
2 2

∂
1
=−i f r + r f ∗ r − r
1
2
2
∂r
r =0