Polarization and Modern Optics

Polarizing Beam-Splitter CubeHalf-wave Plate

Quarter-wave PlateElectro-Optic Modulator

Charles Bernando

Polarization

Property of waves that describes the orientation of their oscillationsExample: electromagnetic wave (light)

φ = phase difference1 0 cos( . )E E k y wt 2 0 cos( . )E E k y wt

Linear Polarization Circular Polarization

φ = 0, ±π, ±2π, .. φ = ±π/2, ±3π/2, ..

Elliptical Polarization

1 0 cos( . )E E k y wt 2 0 cos( . )E E k y wt

1 0 cos( . )E E k y wt

1 0 cos( . )E E k y wt

2 0 0sin( . ) cos .2

E E k y wt E k y wt

2 0 cos( . )E k y wt

Polarizing Beam-splitter Cube• Split randomly polarized beams into 2 orthogonal,

linearly, polarized components (s-polarized & p-polarized). S-polarized is reflected at 900 angle, while P-polarized light is transmitted

• S-wave: electric field perpendicular to plane of incidence• P-wave: electric field parallel to plane of incidence

• AR (antireflection) coating: produce max transmission efficiency

• Cement: bind 2 prisms• BS (beam-splitter) coating or dielectric coating or thin

film coating: consists of thin layers of transparent dielectric materials with different refractive indexes, which are deposited on a substrate. It will produce phase shift due to reflections and reinforce certain wavelengths. By controlling the thickness and number of the layers, the reflectivity & transmissivity can be adjusted

= vector of beam-splitter interface = vector of incident light = state of polarization of s-wave = state of polarization of p-wave

For light incident along z-axis,

J. L. Pezzaniti and R. A. Chipman. Angular dependence of polarizing beam-splitter cubes. Appl. Opt. 33(10), pp. 1916-1929. 1994.

11,0, 1

2n

0,0,1k

n ks

n k

p s k

0,1,0s

1,0,0p

n

k

sp

Half-wave Plate• Shift the phase between 2 perpendicular

components of electric field (the phase shift is π rad)• Change linear polarized wave into another linear

polarized wave• Material: quartz, calcite• Optical axis = fast axis/extraordinary axis is

the axis with the smaller index of refraction• Slow axis = ordinary axis (perpendicular to fast axis)

is the axis with the higher index of refractionThe light component along the fast axis will travel faster than the light component along the slow axis

Due to velocity difference, the light component along fast axis and slow axis will be shifted until the outcome phase shift is π rad. The phase shift depends on the thickness of a material.

φ = phase shiftne = extraordinary index

no = ordinary index

L = thickness of materialλ = wavelength

2 .( ).e on n L

Quarter-wave Plate• Shift the phase between 2 perpendicular components of

electric field (the shift phase is π/2 rad)• Change linear polarized wave into circular polarized

wave when the incident light makes 450 angle with the fast axis

Electro Optic Modulator• A device used for controlling the polarization of a laser

beam with an electrical control signal• By applying an appropriate voltage to the RF input, a

corresponding change in the extraordinary refractive index will occur

• The voltage required for inducing a phase change of π is called the half-wave voltage

Jones Matrix• Linear horizontal polarizer• Linear vertical polarizer• Half wave plate

• Quarter wave plate

Example: linearly polarized light 450 (basis ) is incident on a horizontal polarizer. The outcome is

horizontally polarized

1 0

0 0

0 0

0 1

cos 2 sin 2

sin 2 cos 2

2 2

2 2

cos sin (1 )sin cos

(1 )sin cos sin cos

i i

i i

1 0 1 1

0 0 1 0

1

1

For beam splitter, please see: A. L. Fymat. Jones's matrix representation of optical instruments. I: Beam splitters. Appl. Opt. 10(11), pp. 2499-2505. 1971.

Conclusion• Beam splitter is used to resolve a beam of light

into 2 beams by reflecting the s-wave and transmitting the p-wave

• The wave plates (half and quarter wave plate) are designed to change the state of polarization of incoming light

• The Electro Optic Modulators use a potential difference to change and adjust the extraordinary refractive index to get the expected phase shift between 2 components of light