15. Polarization. Linear, circular, and elliptical polarization. Mathematics of polarization. Uniaxial crystals. Birefringence.

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1 15. Polarization Linear, circular, and elliptical polarization Mathematics of polarization Uniaial crstals Birefringence Polarizers

2 Notation: polarization near an interface Parallel ("p") polarization Perpendicular ("s") polarization These are onl defined relative to an interface between two media. But even when there is no interface around, we still need to consider the polarization of light waves.

3 Polarization of a light wave We describe the polarization of a light wave (without an interface nearb) according to how the E-field vector varies in a projection onto a plane perpendicular to the propagation direction. For convenience, the propagation direction is generall assumed to be along the positive z ais. Here are two possibilities: E-field variation over time (and space) r ( E r, t ) = E ˆ ep ( ) 0 j kz t [ ω ] r ( E r, t ) = E ˆ ep ( ) 0 j kz t [ ω ] In these diagrams, the propagation direction is out of the screen at ou.

4 45 Polarization { [ ω ]} E ( z, t) = Re E0 ep j( kz t) E ( z, t) = Re E0 ep j( kz t) { [ ω ]} and the total wave is: r E( z, t) = E ˆ ˆ + E Here, the comple amplitude, E 0 is the same for each component. So the components are alwas in phase.

5 Arbitrar-Angle Linear Polarization Here, the -component and the -component have the same phase, but different magnitudes. { ( α) [ ω ]} E ( z, t) = Re E0 cos ep j( kz t) E ( z, t) = Re E0 sin ep j( kz t) { ( α) [ ω ]} and (as alwas) the total wave is: r E( z, t) = E ˆ ˆ + E E-field variation over time (and space) α

6 The Mathematics of Polarization Define the polarization state of a field as a 2D vector Jones vector containing the two comple amplitudes: For man purposes, we onl care about the relative values: E E 1 = E E E E = E Some specific eamples: 0 linear () polarization: E /E = 0 90 linear () polarization: E /E = 45 linear polarization: E /E = 1 Arbitrar linear polarization: E E sinα = = tanα cosα

7 Jones vectors - a common mistake NOTE: the Jones vector contains the comple amplitudes onl. Its components do not depend on,,z, or t. E ( ωt) j kz E e = j( kz ωt) Ee This is wrong!

8 Circular (or Helical) Polarization E ( z, t) = E cos( kz ωt) 0 E ( z, t) = E sin( kz ωt) 0 or, in comple notation: { 0 [ ω ]} [ ω ] E ( z, t) = Re E ep j( kz t) { 0 } E ( z, t) = Re je ep j( kz t) Here, the comple amplitude of the -component is -j times the comple amplitude of the - component. So the components are alwas 90 out of phase. The resulting E-field rotates clockwise around the k-vector (looking along k). This is called a right-handed rotation.

9 Right vs. Left Circular (or Helical) Polarization E ( z, t) = E cos( kz ωt) 0 E ( z, t) = E sin( kz ωt) or, more generall: 0 { 0 [ ω ]} [ ω ] E ( z, t) = Re E ep j( kz t) { 0 } E ( z, t) = Re + je ep j( kz t) Here, the comple amplitude of the -component is +j times the comple amplitude of the -component. So the components are alwas 90 out of phase, but in the other direction. E-field variation over time (and space) kz-ωt = 90 kz-ωt = 0 Note: In this drawing, the z ais is coming out of the screen at ou. So ou are looking in the opposite direction from the k-vector, which is wh it rotates clockwise according to the arrow - but we refer to this as a counter-clockwise rotation. The resulting E-field rotates counterclockwise around the k-vector (looking along k). This is a left-handed rotation.

10 Circular Polarization - the movie Question: is this cartoon showing right-handed or lefthanded circular polarization?

11 Unequal Arbitrar-Relative-Phase Components ield "Elliptical Polarization" E ( z, t) = E cos( kz ωt) { 0 [ ω ]} [ ω ] E ( z, t) = Re E ep j( kz t) 0 E ( z, t) = E cos( kz ωt θ ) where 0 E 0 0 { 0 } E ( z, t) = Re E ep j( kz t) where E0 and E0 are arbitrar comple amplitudes. E or, in comple notation: E-field variation over time (and space) The resulting E-field can rotate clockwise or counter-clockwise around the k-vector.

12 The Mathematics of Circular and Elliptical Polarization Circular polarization has an imaginar Jones vector -component: Right circular polarization: E 1 E= = ± E j E / E = j A clockwise rotation, when looking along the propagation direction. Left circular polarization: E / E =+ j counterclockwise rotation. For elliptical polarization, the two components have different amplitudes, and ma even be comple: E / E = a+ jb We can calculate the eccentricit and tilt of the ellipse if we feel like it.

13 = 1+ j An eample 1 This Jones vector is equivalent to: j 4 ( 1+ j) = 2e π E What is the polarization of this wave? r E( z, t) = E ˆ 0 + ( 1+ j) ˆ e Using we find, at z = 0: E E ωt = 0 E0 E0 ωt = π/4 E E = E cos t 0 0 ( ωt) j kz ( ω ) ( ω π ) E = 2E cos t+ 4 ωt = π/2 0 E0 ωt = 3π/4 E E0 ωt = π E0 E0 left-handed elliptical polarization

14 A polarizer is a device which filters out one polarization component The light can ecite electrons to move along the wires. These moving charges then emit light that cancels the input light. This cannot happen if the E-field is perpendicular to the wires, since the current can onl flow along the wires. Such polarizers are commonl used for long-wave infrared radiation, because the wire spacing has to be much smaller than the wavelength (and so it is easier to manufacture if the wavelength is longer).

The light can ecite electrons to move along the wires, just as in the case

15 Polmer-based polarizers A polmer is a long chain molecule. Some polmers can conduct electricit (i.e., the can respond to electric fields similar to the wa a wire does). The light can ecite electrons to move along the wires, just as in the case of the polmer chains. This is how polarized sunglasses work.

16 Crossed polarizers block light Blocking both and polarizations means that ou have blocked everthing. Inserting a third polarizer between the two crossed ones can allow some light to leak through.

17 Wh sunglasses are polarized no sunglasses sunglasses Brewster s angle revisited

$The - and -polarizations can see different refractive inde curves.$

18 Birefringence The molecular "spring constant" can be different for different directions. The - and -polarizations can see different refractive inde curves. Hence, the refractive inde of a material can depend on the orientation of the material relative to the polarization ais!

$Uniaial crstals have an optic ais Uniaial crstals have one refractive inde for light polarized along the optic ais (n e ) and another for light polarized in either of the two directions perpendicular$

19 Uniaial crstals have an optic ais Uniaial crstals have one refractive inde for light polarized along the optic ais (n e ) and another for light polarized in either of the two directions perpendicular to it (n o ). Light polarized along the optic ais is called the etraordinar ra, and light polarized perpendicular to it is called the ordinar ra. These polarization directions are the crstal principal aes. Light with an other polarization must be broken down into its ordinar and etraordinar components, considered individuall, and recombined afterward.

$of different polarizations will refract b different amounts at an interface. o-ra n o n e e-ra$

20 Birefringence can separate the two polarizations into separate beams Due to Snell's Law, light of different polarizations will refract b different amounts at an interface. o-ra n o n e e-ra

21 Birefringent Materials Calcite, CaCO 3 Calcite is one of the most birefringent materials known.

22 Some polarizers use birefringence. optic ais optic ais E-ra O-ra For eample, a Wollaston prism: Combine two calcite prisms, rotated so that the ordinar polarization in the first prism is etraordinar in the second (and vice versa). The ordinar ra in the first prism becomes the etraordinar ra in the second one. Since n e < n o, the E-ra is refracted awa from the normal to the interface. The opposite happens for the O-ra.

Waves, Polarization, and Coherence

05-0-4 Waves, Polarization, and Coherence Lecture 6 Biophotonics Jae Gwan Kim jaekim@gist.ac.kr, X 0 School of nformation and Communication ngineering Gwangju nstitute of Sciences and Technolog Outline