Polarization

OBLIQUE INCIDENCE – PROPAGATING WAVES

When a plane EM wave incident at an oblique angle on a dielectric interface, there are two cases to be considered: incident electric field has polarization parallel to the plane of incidence, and incident electric field has polarization that is perpendicular to the plane of incidence.

1. Parallel polarization:

The incident, reflected, and transmitted electric field vectors lie in the plane of incidence: the x-z plane.

Note: the angles are measured with respect to normal.

Oblique incidence – propagating waves

Incident wave:

Reflected wave:

Transmitted wave:

1 sin cos0 cos sin i ijk x z

i i x i zE E u u e

1 sin cos0 cos sin r rjk x z

r r x r zE E u u e

2 sin cos0 cos sin t tjk x z

t t x t zE E u u e

1 sin cos0

1

i ijk x zi y

c

EH u e

Z

1 sin cos0

1

r rjk x zr y

c

EH u e

Z

2 sin cos0

2

t tjk x zt y

c

EH u e

Z

Oblique incidence – propagating waves

From the boundary conditions: i.e., continuity of tangential electric and magnetic fields at the interface z = 0, we derive:

1 21

1 21

sin sinsin

sin sinsin

1 1 2

cos cos cosi tr

i tr

jk x jk xjk xi r t

jk x jk xjk x

c c c

e e e

e e e

Z Z Z

To satisfy these conditions, the Snell’s laws of reflection and refraction must hold:

1 2; sin sini r i tk k

Oblique incidence – propagating waves

For parallel polarization, a special angle of incidence exists, known as the Brewster’s angle, or polarizing angle, i = B, for which the reflection coefficient is zero: = 0.

It happens when

2 1cos cosc t c iZ Z

1

2

1sin

1B

Therefore:

If the incidence angle equals to Brewster’s angle, the reflected field will be polarized perpendicularly to the plane of incidence.

Oblique incidence – propagating waves

1. Perpendicular polarization:The incident, reflected, and transmitted electric field vectors are perpendicular to the plane of incidence: the x-z plane.

Incident wave:

Reflected wave:

Transmitted wave:

1 sin cos0

i ijk x zi yE E u e

1 sin cos0

r rjk x zr yE E u e

2 sin cos0

t tjk x zt yE E u e

1 sin cos0

1

cos sin i ijk x zi i x i z

c

EH u u e

Z

1 sin cos0

1

cos sin r rjk x zr r x r z

c

EH u u e

Z

2 sin cos0

2

cos sin t tjk x zt t x t z

c

EH u u e

Z

Oblique incidence – propagating waves

From the continuity of tangential electric and magnetic fields at the interface z = 0, we again derive:

1 21

1 21

sin sinsin

sin sinsin

1 1 2

cos coscos

i tr

i tr

jk x jk xjk x

jk x jk xjk xi tr

c c c

e e e

e ee

Z Z Z

As before, the Snell’s laws of reflection and refraction must hold:

1 2; sin sini r i tk k

These simplifications lead to very similar expressions:

2 1

2 1

2

2 1

cos cos

cos cos

2 cos

cos cos

c i c t

c i c t

c i

c i c t

Z Z

Z Z

Z

Z Z