Linear, circular, and elliptical polarization Mathematics ...

1 15. PolarizationLinear, circular, and elliptical polarizationMathematics of polarizationPolarizersBirefringenceNotat ion: polarization near an interfaceParallel ("p") polarization Perpendicular ("s") polarizationBut even when there is no interface around, we still need to consider the polarization of light cannot define these unless the propagating wave is near an interface. polarization of a light waveWe describe the polarization of a light wave (without any interface nearby) according to how the E-field vector varies in a projection onto a plane perpendicular to the propagation convenience, the propagation direction is generally assumed to be along the positive z variation over time (and space)xyIn these diagrams, the propagation direction is out of the screen at are two possibilities: 0 ( , ) exp ( ) rr%E r t E x j kz t 0 ( , ) exp ( ) rr%E r t E y j kz t Reminder: the complex amplitude is a vector!

2 45 PolarizationHere, the complex amplitudes of the xcomponent and the ycomponent are the components are always in phase. 00( , ) Re exp ( )( , ) Re exp ( ) %%xyE z t E j kz tE z t E j kz t ( , )xyE z t E x E y rand the total wave is:Arbitrary-Angle Linear PolarizationHere, they-component and the x-component have the same phase, but different variation over time (and space)a 00( , ) Re cos exp ( )( , ) Re sin exp ( ) %%xyE z t Ej kz tE z t Ej kz ta a ( , )xyE z t E x E y rand (as always) the total wave is:where ais a (or helical ) PolarizationHere, the complex amplitude of the y-component is -jtimes the complex amplitude of the ( , ) cos( )( , ) sin( ) xyE z t E kz tE z t E kz t or, in complex notation:So the components are always 90 out of resulting E-field rotates clockwisearound the k-vector (looking along k).

3 This is called a right-handedrotation. 00( , ) Re exp ( )( , ) Re exp ( ) %%xyE z t E j kz tE z tjE j kz t The x and y components of the E-field need not be in example:Right vs. Left Circular (or helical ) PolarizationHere, the complex amplitudeof the y-component is +jtimes the complex amplitude of the variation over time (and space)kz- t = 0 kz- t = 90 00( , ) cos( )( , ) sin( ) xyE z t E kz tE z t E kz t or, more generally:So the components are always 90 out of phase, but in the other direction. 00( , ) Re exp ( )( , ) Re exp ( ) %%xyE z t E j kz tE z tjE j kz t The resulting E-field rotates counterclockwisearound the k-vector (looking along k). This is a :In this drawing, the z axis is coming out of the screen at you. So you are looking in the oppositedirection from the k-vector, which is why it rotates clockwise according to the arrow - but we refer to this as a polarization - the movieQuestion: is this cartoon showing right-handed or left-handed circular polarization ?

4 Unequal Arbitrary-Relative-Phase Components yield " elliptical polarization "The resulting E-field can rotate clockwise or counter-clockwise around the variation over time (and space) 00( , ) Re exp ( )( , ) Re exp ( )xxyyE z t E j kz tE z t E j kz t %%or, in complex notation:0xE%0yE%where and are arbitrary complex ( , ) cos( )( , ) cos()xxyyE z t E kz tE z t E kz t 00xyE E whereThe Mathematics of PolarizationFor many purposes, we only care about the relativevalues: xyEEE1 yxxEEEED efine the polarization state of a field as a 2D vector Jones vector containing the two complex amplitudes:sintancos yxEEaaaSome specific examples: 0 linear (x) polarization : Ey /Ex = 090 linear (y) polarization : Ey /Ex= 45 linear polarization : Ey /Ex= 1 Arbitrary linear polarization : 10 01 Circular polarization has an imaginary Jones vector y-component:Right circular polarization :Left circular polarization :The Mathematics of Circular andElliptical Polarization1 xyEEEj/ y xE E j/ y xE E j/ y xE E a jbFor elliptical polarization , the two components have different phases and maybe also different amplitudes, so their ratio is complex:We can calculate the eccentricity and tilt of the ellipse if we feel like clockwise rotation, when looking along the propagation vectors - a common mistakeNOTE: the Jones vector contains the complex amplitudes only.

5 Its components do notinclude the rapidly varying dependence on position and time. j kz txj kz tyE eEE e This is wrong!An example11 EjWhat is the polarization of this wave? 0 ,1j kz tE z t E x j y e r This Jones vector is equivalent to: 412 jj e Usingwe find, at z = 0: 00cos2 cos4 xyE E tE E t t = 000 EEExEy t = /402 2 0 E t = /200 E t = 3 /4002 2 2 EE t = 00 EExyleft-handed elliptical polarizationA polarizer is a device which filters out one polarization componentSuch polarizers are commonly 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 excite electrons to move along the wires. These moving charges then emit light that cancels the input light.

6 This cannot happen if the E-field is perpendicular to the wires, since the current can only flow along the polarizersA polymer is a long chain molecule. Some polymers can conduct electricity ( , they can respond to electric fields similar to the way a wire does).This is how polarized sunglasses light can excite electrons to move along the wires, just as in the case of the polymer polarizers block lightBlocking both x and y polarizations means that you have blocked a third polarizer between the two crossed ones can allow some light to leak sunglassessunglassesWhy sunglasses are polarizedBrewster s angle revisitedBirefringenceThe x- and y-polarizations can see different refractive index , the refractive index of a material can depend on the orientation of the material relative to the polarization axis!The molecular "spring constant" can be different for different crystals have an optic axisLight with any other polarization must be broken down into its ordinary and extraordinary components, considered individually, and added back together crystals have one refractive index for light polarized along the optic axis (ne) and another for light polarized in either of the two directions perpendicular to it (no).

7 Light polarized along the optic axis is called the extraordinaryray, and light polarized perpendicular to it is called the ordinaryray. These polarization directions are the crystal principal axes. Birefringence can separate the twopolarizations into separate beamsDue to Snell's Law, light of different polarizations will refract by different amounts at an is one of the most birefringent materials known. It is particularly useful because it s also transparent over the entire visible spectrum and even into the UV (~300nm).Birefringent MaterialsCalcite, CaCO3 Refractive Indices of Uniaxial Crystals(20 C; l= )Material nitrate polarizers use effectively splits the two orthogonal polarization components into two separate (diverging) output Wollaston polarizing beam splitter uses refraction in two birefringent prisms, the second rotated by 90 with respect to the x- polarization goes from e (low-n) to o (high-n) and so bends toward the normal (upward).

8 The y- polarization goes from o (high-n) to e (low-n) and so bends away from the normal (downward). Optic axisOptic axisThe x- polarization is the o-ray in the > neNormal to the interfaceCalciteThe y- polarization is the e-ray in the Polarizer0 Polarizer0 Polarizer90 PolarizerThe ideal (perfect) polarizer passes 100% of the desired polarization and 0% of the undesired polarization . Real polarizers aren t of polarizer Ext. Ratio TransmissionCostCalcite: 106> 95% $1000 - $2000 Dielectric/wire grid: 103> 99% $100 - $200 Polaroid sheet: 103~ 50%$1 - $2 Unpolarized light inputThe ratio of the transmitted irradiance through polarizers oriented parallel and then crossed is the Extinction ratioor Extinction coefficient. We d like it to be .The second polarizer is often called an analyzer. The measure of a polarizer