Page 36 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 17

and the x axes, respectively, we can now get an easy expression for
the projection along an axis that is tilted through an angle .

2
|F (x)| = W F (x, u) du

= W f (x cos − u sin ,x sin + u cos ) du

= W f ( ,u) ( cos + u sin − x) d du (1.35)

1
We thus conclude that not only are the marginals for = 0 and =
2
correct, but in fact any marginal for an arbitrary angle is correct. We
observe a strong connection between the Wigner distribution W f (x, u)
2
and the intensity |F (x)| of the signal’s fractional Fourier transform.
Note also the relation to the Radon transform.
Since the ambiguity function is the two-dimensional Fourier trans-
2
form of the Wigner distribution, we could also represent |F (x)| in
the form 54–56

2
|F (x)| = A F ( sin , − cos ) exp(−i2 x ) d (1.36)

and we conclude that the values of the ambiguity function along the
line defined by the angle and the projections of the Wigner distri-
bution for the same angle are related to each other by a Fourier
transformation. Note that the ambiguity function in Eq. (1.36) is rep-
resented in a quasi-polar coordinate system ( , ).
We recall that the signal f (x) =| f (x)| exp[i2 (x)] can be recon-
structed by using the intensity profiles of the fractional Fourier trans-
form F (x) for two close values of the fractional angle . 56 The recon-
struction procedure is based on the property 54–56

2
∂|F (x)| d 2 d (x)
=− | f (x)| (1.37)
∂ dx dx
=0
which can be proved by first differentiating Eq. (1.35) with respect to
and using the identity

∂ ( cos + u sin − x)

∂
=0

= (− sin + u cos ) ( cos + u sin − x) = u ( − x)
=0