Jones calculus and Mueller matrices are powerful tools for understanding polarized light. They help us describe how light's polarization changes as it passes through different optical elements.
These mathematical methods allow us to predict and analyze complex optical systems. By using Jones vectors and matrices or Stokes vectors and Mueller matrices, we can calculate how light behaves in various setups.
Jones Calculus and Mueller Matrices
Jones calculus for polarization states
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12.8 Polarization – Douglas College Physics 1207 View original
Represents the polarization state of light using Jones vectors, which are 2x1 complex vectors
Horizontal linear polarization: [10]
Vertical linear polarization: [01]
+45° linear polarization: 21[11]
Right circular polarization: 21[1−i]
Enables calculation of the output polarization state when light passes through polarization devices represented by Jones matrices
Suitable for completely polarized light and non-depolarizing devices
Jones and Mueller matrices for polarization
Jones matrices represent the effect of polarization devices on the electric field of light
2x2 complex matrices that operate on Jones vectors
Cannot account for depolarization or partially polarized light
Examples: linear polarizers ([1000] for horizontal, [0001] for vertical), wave plates (21[1−i−i1] for quarter-wave plate with fast axis at 45°, [cos2θsin2θsin2θ−cos2θ] for half-wave plate with fast axis at angle θ)
Mueller matrices describe the effect of polarization devices on the Stokes parameters of light
4x4 real matrices that operate on Stokes vectors
Can handle depolarization and partially polarized light
Examples: linear polarizer (211cos2θsin2θ0cos2θcos22θsin2θcos2θ0sin2θsin2θcos2θsin22θ00000 with transmission axis at angle θ), quarter-wave plate (10000100000100−10 with fast axis at 0° or 90°), half-wave plate (10000cos4θsin4θ00sin4θ−cos4θ0000−1 with fast axis at angle θ)
Mueller matrices of optical elements
Represent the polarization properties of optical elements using Stokes parameters
Stokes vector: S=IQUV, where I is total intensity, Q is the difference between horizontal and vertical polarization intensities, U is the difference between +45° and -45° polarization intensities, and V is the difference between right and left circular polarization intensities
Mueller matrices can describe depolarization, unlike Jones matrices
The first row and column of a Mueller matrix relate to the total intensity, while the remaining 3x3 submatrix describes the polarization properties of the element
Polarized light in complex systems
Calculate the output polarization state and intensity of light passing through a series of polarization devices using Jones calculus
Multiply the Jones matrices of the devices in reverse order to obtain the total Jones matrix: Jtotal=Jn⋅Jn−1⋅...⋅J2⋅J1
Multiply the input Jones vector by the total Jones matrix to get the output Jones vector: Eout=Jtotal⋅Ein
Calculate the output intensity, which is proportional to the square of the absolute value of the output electric field: Iout∝∣Eout∣2=∣Jtotal⋅Ein∣2
Solve problems involving the propagation of polarized light through complex optical systems using Mueller calculus
Multiply the Mueller matrices of the optical elements in reverse order to obtain the total Mueller matrix: Mtotal=Mn⋅Mn−1⋅...⋅M2⋅M1
Multiply the input Stokes vector by the total Mueller matrix to get the output Stokes vector: Sout=Mtotal⋅Sin
Analyze the polarization properties using the output Stokes vector, such as calculating the degree of polarization (DOP): DOP=IQ2+U2+V2
Combine Jones and Mueller calculus when necessary by converting between Jones and Mueller matrices or vectors
Jones to Mueller matrix conversion: M=A(J⊗J∗)A−1, where A=1100001i001−i1−100 and ⊗ denotes the Kronecker product
Stokes to Jones vector conversion: E=[I+QI−Qeiarctan(V/U)]