Transformation of s-Reflection Coefficient between Normal and Oblique Incidence

|z|

arg(z)

r = 0.90 δ = 60.00°

If

w

denotes an interface Fresnel reflection or transmission coefficient for

s

- or

p

-polarized light at an oblique angle of incidence

ϕ

, and

z

denotes the same coefficient at normal incidence, then it can be shown that w is an analytic function of

z

that depends parametrically on the angle of incidence

ϕ

. The mapping

between the complex

z

and

w

planes is illustrated here by one of the Fresnel coefficients (for s reflection) at normal incidence and one oblique angle of incidence (45°). Here

z=|r|exp(jδ)

, where

|r|

and

δ

are the normal-incidence amplitude reflectance and phase shift.

This figure shows that the orthogonal (polar) set of straight lines and circles through and around the origin in the

z

plane is mapped onto orthogonal sets of curves in the

w

plane.