ENE 451Fundamental of Optical Engineering

Lecture 8

A linearly polarized plane wave with Ē vector described by

is incident on an optical element under test. Describe the state of polarization of the output wave (linear, elliptrical, or circular) if the optical element is:

Example of polarizations

ˆ ˆ( )inc x yE e e

(a) A linear polarizer oriented to transmit light polarized in the ex direction.

Example of polarizations

(b) A half-wave plate with birefringence axes oriented to coincide with ex and ey.

Example of polarizations

(c) A half-wave plate with birefringence axes oriented at 45º relative to ex and ey .

Example of polarizations

(d) A quarter-wave plate with birefringence axes oriented to coincide with ex and ey .

Example of polarizations

(e) A quarter-wave plate with birefringence axes oriented at 45º relative to ex and ey .

Example of polarizations

(f) A half-wave plate with birefringence axes oriented at 25º relative to ex and ey .

Example of polarizations

(g) A quarter-wave plate with birefringence axes oriented at 25º relative to ex and ey .

Example of polarizations

A linearly polarized light propagating in the z-direction with polarization vector in the x-direction is incident on a birefringent crystal. What is the state of polarization of the light after passing through the crystal if:(a) the crystal is a quarter-wave plate with optic

axis in the xy plane oriented at 30º relative to the y-axis?

Example of polarizations

◦ (b) the crystal is a half-wave plate with optic axis in the y-direction?

Example of polarizations

◦ (c) the crystal is a half-wave plate with optic axis in the xy plane oriented at 11º relative to the x-axis?

Example of polarizations

◦ (d) the crystal is a quarter-wave plate with optic axis in the xy plane oriented at 45º relative to the y-axis?

Example of polarizations

◦ (e) the crystal is a quarter-wave plate with optic axis in the z-direction?

Example of polarizations

For a birefringent median with n0 = 1.654 and nE = 1.485 as shown in the figure. Find the length L that makes it be (a) a full wave plate (b) a half wave plate (c) a ¼ -wave plate if the wavelength is 656 nm.

Example of birefringence

Example of birefringence

Electrooptic modulator This makes use of electrooptic effect

(applied electric fields used to change the optical properties).

There are 2 kinds of electrooptic effect: linear and quadratic.

The linear electrooptic effect is called “Pockels effect”.

This refers to the change in the indices of the ordinary and extranordinary rays proportional to applied electric field.

This effect exists only in crystals without an inversion symmetry such as LiNbO3.

Pockels effect

For a crystal with an inversion symmetry, the linear electrooptic effect can not exist, while the quadratic electrooptic effect known as “Kerr effect” is observed.

This is where the induced index change is proportional to the square of applied electric field.

Kerr effect

Optical modulator

V (pi-voltage) or half-wave voltage is the applied voltage that makes the relative phase shift be in a cube of material.

In general,

= refractive index changes produced by applied voltage.

Optical modulator

2x yn n L

and x yn n

It is preferable to design L >> h to have a low applied voltage V.

After applying a voltage, indices are changed as

Optical modulatorV L

V h

( )

( )E E E

O O O

n n n V

n n n V

V for the material is 2,700 V, L = 2 cm, and h = 0.5 mm. Find applied voltage V to have Δ = (complete extinction).

Example

An electrooptic crystal has dimensions of 2x2x3 along the x,y, and z axes with nE = 1.487 and nO = 1.536. An input wave propagating in the z-direction at λ = 0.63 μm is linearly polarized at a 45º angle relative to the x- and y- axes. A voltage V applied across the crystal in the x-direction. The voltage is increased from V = 0 until, when V = 245 V, the output polarization from the crystal is the same as that observed for V = 0. Assume that optic axis is along the y-axis.◦ (a) What is the total phase retardation, in rad, for V = 0?◦ (b) What is pi-voltage for the material?◦ (c) What is the refractive index change Δnx produced by

the applied voltage of 245 V, assuming Δny =0?

Example

Recall: interference eq.

Interference in thin films

1 2

1 2

2 21 2 1 2 1 22 cos

i i

A

Ae A e

P A A A A

Assume that

Interference in thin films

2 31 2

1 2 2 3

2 2

12 23

4 round trip phase shift

amplitude of reflected wave

amplitude of incident wave

ir i i

ir i

i jij

i j

n nn nE E E e

n n n n

n t

E E A A e

n nA

n n

2 31 2

1 2 2 3

1 and 1n nn n

n n n n

Reflectance

We can consider R into 3 cases:◦ n1 = n3.

◦ n1 < n2, n2>n3 , n1 ≠ n3.

◦ n1 < n2 < n3.

Interference in thin films2

2

12 23

2 212 23 12 232 cos

ir

i

ER A A e

E

R A A A A

Case 1: n1 = n3

From a definition: Aij = Aji

Interference in thin films

2 212 21 12 21

212

2 cos

2 1 cos

R A A A A

R A

Max in R for

Min in R for

Interference in thin films

2max 12

2 1 ; 0,1,2,...

4

N N

R A

min

2 ; 0,1,2,...

0

N N

R

n1 = 1.5, n2 = 1.6, and λ = 0.63 μm. Find t2 for Rmax and Rmin.

Example

Case 2: n1 < n2, n2>n3 , n1 ≠ n3.

Interference in thin films

12 23 0A A

Max in R for

Min in R for

Interference in thin films

2

max 12 23

2 1 ; 0,1,2,...N N

R A A

2

min 12 23

2 ; 0,1,2,...N N

R A A

n1 = 1.5, n2 = 1.6, n3 = 1.4, λ = 0.63 μm. Find t2 for Rmax and Rmin.

Example

Case 3: n1 < n2 < n3

Interference in thin films

12

23

0

0

A

A

Max in R for

Min in R for

Interference in thin films

2

max 12 23

2 ; 0,1,2,...N N

R A A

2

min 12 23

2 1 ; 0,1,2,...N N

R A A

n1 = 1.5, n2 = 1.6, n3 = 1.7, λ = 0.63 μm. Find t2 for Rmax and Rmin.

Example