17. Jones Matrices & Mueller Matrices

Transcription

1 7. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates - the rotation matrix Stokes Parameters and unpolarized light Mueller Matrices R. Clark Jones (96-24) Sir George G. Stokes (89-93) Hans Mueller (9-965)

2 Jones vectors describe the polarization state of a wave Define the polarization state of a field as a 2D vector Jones vector containing the two complex amplitudes: = = x x x 2 2 y y x + y x y (normalized to length of unity) A few examples: linear (x) polarization: y / x = linear (arbitrary angle) polarization: y / x = tan α right or left circular polarization: y / x = ±j tanα ± j

3 To model the effect of a medium on light's polarization state, we use Jones matrices. Since we can write a polarization state as a (Jones) vector, we use matrices, A, to transform them from the input polarization,, to the output polarization,. This yields: =A = a + a x x 2 y = a + a y 2 x 22 y A a 2 = a 2 a22 a This should be thought of as a transfer function. For example, an x-polarizer can be written: A x = So: x x = A = = y x

4 Other Jones matrices A y-polarizer: A y = A half-wave plate: A HWP = A half-wave plate rotates 45-degreepolarization to -45-degree, and vice versa. R. Clark Jones (96-24) = = A quarter-wave plate: A QWP = ± j = ± ± j j

5 The orientation of a wave plate matters. Remember that a quarter-wave plate only converts linear to circular if the input polarization is ±45. If it sees, say, x polarization, the input is unchanged. or 9 Polarizer Wave plate w/ axes at or 9 Jones matrices are an extremely useful way to keep track of all this. = j A QWP Note: this little cube is a cartoon representation of a polarizer. Cube polarizers are commonly used in optics.

6 A wave plate example What does a quarter-wave plate do if the input polarization is linear but at an arbitrary angle? tan = ( ) tan j α j ( α ) A QWP in out For arbitrary α, this is an elliptical polarization. α = 3 α = 45 α = 6

7 Jones Matrices for standard components Horizontal (x) linear polarizer: Quarter-wave plate, fast axis vertical: e jπ 4 j Vertical (y) linear polarizer: Quarter-wave plate, fast axis horizontal: e jπ 4 j Linear polarizer at 45 degrees: 2 Right circular polarizer: 2 j j Linear polarizer at 45 degrees: 2 Left circular polarizer: 2 j j

8 Rotated Jones matrices What about when the polarizer or wave plate responsible for the transfer function A is rotated by some angle, θ? Rotation of a vector by an angle θ means multiply by the rotation matrix: rotated Jones vector of the input where: ( θ ) ( θ ) ( θ ) ' = R and ' = R R cos( θ ) sin( θ ) = sin( θ ) cos( θ ) rotated Jones vector of the output Rotating by θ and inserting the identity matrix R(θ) - R(θ), we have: ( θ ) ( θ ) A ( θ ) A ( θ ) ( θ ) ' = R = R = R R R ( θ ) ( θ ) ( θ ) ( θ ) ( θ ) R A R R R A R A = = ' = ' ' Thus: ( θ ) A ( θ ) A ' = R R

9 Rotated Jones matrix for a polarizer xample: apply this to an x polarizer. A ' ( θ ) A ( θ ) = R R A x ( θ ) cos( θ ) sin( θ ) cos( θ ) sin( θ ) = sin( θ ) cos( θ ) sin( θ ) cos( θ ) cos( θ ) sin( θ ) cos( θ ) sin( θ ) = sin( θ ) cos( θ ) So, for example: A x ( o 45 ) 2 cos ( θ ) cos( θ )sin( θ ) = 2 cos( θ )sin( θ ) sin ( θ ) / 2 / 2 = / 2 / 2 A x ( ψ ) ψ ψ for a small angle ψ

10 To model the effect of many media on light's polarization state, we use many Jones matrices. The aggregate effect of multiple components or objects can be described by the product of the Jones matrix for each one. input transfer function output } } A A 2 A 3 =AAA 3 2 The order may look counter-intuitive, but order matters!

11 Multiplying Jones Matrices Crossed polarizers: =A A y x AyAx = = Uncrossed polarizers (by a slight angle ψ): x-pol y-pol y so no light leaks through. rotated x-pol x z AyAx A A y x ( ψ ) ( ψ ) ψ = = ψ ψ x x = y ψ = y ψ x y-pol So I out ψ 2 I in,x

12 Multiplying Jones Matrices Now, it is easy to compute how inserting a third polarizer between two crossed polarizers leads to larger transmission. x-pol 45º-pol y y-pol x z =A A A y 45 x A A A y 45 x 2 2 = = Thus: x, in = = y, in x, in 2 2 The third polarizer, between the other two, makes the transmitted wave non-zero.

13 Natural light (e.g., sunlight, light bulbs, etc.) is unpolarized The direction of the vector is randomly changing. But, it is always perpendicular to the propagation direction. polarized light natural light

14 Light with very complex polarization vs. position is "unpolarized." Light that has scattered multiple times, or that has scattered randomly, often becomes unpolarized as a result. Here, light from the blue sky is polarized, so when viewed through a polarizer it looks much darker. Light from clouds is unpolarized, so its intensity is reduced by only 5%. If the polarization vs. position is unresolvable, we call this unpolarized. Otherwise, we refer to this light as locally polarized or partially polarized.

15 When the phases of the x- and y-polarizations fluctuate, we say the light is "unpolarized." ( ω θ ( )) { } x ( z, t) = Re x exp j kz t x t { ( ( )) } ω θ y ( z, t) = Re y exp j kz t y t where θ x (t) and θ y (t) are functions that vary on a time scale slower than the period of the wave, but faster than you can measure. The polarization state (Jones vector) is: y x exp y ( ) θ ( ) jθ t j t x In practice, the amplitudes are also functions of time! As long as the time-varying relative phase, θ x (t) θ y (t), fluctuates, the light will not remain in a single polarization state and hence is unpolarized.

16 Stokes Parameters We cannot use Jones vectors to describe something that is rapidly fluctuating like this. So, to treat fully, partially, or unpolarized light, we use a different scheme. We define "Stokes parameters." Suppose we have four detectors, three with polarizers in front of them: # detects total irradiance...i # detects horizontally polarized irradiance......i #2 detects +45 polarized irradiance...i 2 #3 detects right circularly polarized irradiance....i 3 Note that these quantities are timeaveraged, so even randomly polarized light will give a welldefined answer. The Stokes parameters: S I S 2I I S 2 2I 2 I S 3 2I 3 I

17 Interpretation of the Stokes Parameters The Stokes parameters: S I S 2I I S 2 2I 2 I S 3 2I 3 I S = the total irradiance S = the excess in intensity of light transmitted by a horizontal polarizer over light transmitted by a vertical polarizer S 2 = the excess in intensity of light transmitted by a 45 polarizer over light transmitted by a 35 polarizer S 3 = the excess in intensity of light transmitted by a RCP filter over light transmitted by a LCP filter What do we mean when we say unpolarized light? All three of these excess quantities are zero

18 Degree of polarization If any of the excess quantities (S, S 2, or S 3 ) are non-zero, then the wave has some degree of polarization. We can quantify this by defining the degree of polarization : ( ) /2 Degree of polarization = S + S + S / S 2 3 = for polarized light = for unpolarized light Note that this quantity can never be greater than unity, since S is the total intensity.

19 The Stokes vector We can write the four Stokes parameters in vector form: S S S S S 2 3 The Stokes vector S contain information about both the polarized part and the unpolarized part of the wave. S = S () + S (2) unpolarized part: S ( ) S S S S polarized part: S ( ) S S S S S S 2 3

20 Stokes vectors (and Jones vectors for comparison) Sir George G. Stokes (89-93)

21 Mueller Matrices multiply Stokes vectors We can define matrices that multiply Stokes vectors, just as Jones matrices multiply Jones vectors. These are called Mueller matrices. S in M M 2 M 3 S out To model the effects of more than one medium on the polarization state, just multiply the input polarization Stokes vector by all of the Mueller matrices: S out = M 3 M 2 M S in (just like Jones matrices multiplying Jones vectors, except that the vectors have four elements instead of two)

22 Mueller Matrices (and Jones Matrices for comparison) With Stokes vectors and Mueller matrices, we can describe light with arbitrarily complicated combination of polarized and unpolarized light. Hans Mueller (9-965)