Matrix-Optics Reference Tables ============================== This page recreates the four principal summary tables in Gerrard and Burch as Sphinx-native tables. Notation has been regularized and short verification notes have been added. The diagrams are original SVG companions rather than scans of the printed pages. Ray-transfer matrices --------------------- Use the reduced ray vector :math:`\mathbf r=(y,V)^T=(y,nv)^T`, surface power :math:`P=(n_2-n_1)/r`, and translation :math:`T(t,n)=(1,t/n;0,1)`. For a mirror the outgoing index changes sign, so :math:`P=-2n/r`. .. list-table:: Recreated Table 1 — common ray-transfer matrices :header-rows: 1 :widths: 8 35 37 20 * - No. - Situation - Matrix - Quick check * - 1 - Translation through thickness :math:`t` in index :math:`n` - :math:`\begin{bmatrix}1&t/n\\0&1\end{bmatrix}` - Heights shear; :math:`V` is unchanged. * - 2 - Refraction at one spherical surface - :math:`\begin{bmatrix}1&0\\-P&1\end{bmatrix}`, :math:`P=(n_2-n_1)/r` - Height is continuous. * - 3 - Reflection at one spherical surface - :math:`\begin{bmatrix}1&0\\2n/r&1\end{bmatrix}` - Plane mirror: :math:`r\to\infty` gives :math:`I`. * - 4 - Thin lens in air, focal length :math:`f` - :math:`\begin{bmatrix}1&0\\-1/f&1\end{bmatrix}` - Parallel ray crosses axis after :math:`f`. * - 5 - Between the principal planes of a lens system - :math:`\begin{bmatrix}1&0\\-1/f&1\end{bmatrix}` - Same reduced action as a thin lens. * - 6 - Between the two focal planes - :math:`\begin{bmatrix}0&f\\-1/f&0\end{bmatrix}` - Height and angle exchange roles. * - 7 - Imaging between conjugate planes with lateral magnification :math:`m` - :math:`\begin{bmatrix}m&0\\-1/f&1/m\end{bmatrix}` - :math:`B=0`; object height alone fixes image height. * - 8 - Afocal system with lateral magnification :math:`m` - :math:`\begin{bmatrix}m&0\\0&1/m\end{bmatrix}` - :math:`C=0`; parallel input remains parallel. Every matrix has determinant one. For a compound system, multiply in reverse order of encounter so the rightmost factor acts first. .. figure:: /_static/knowledge_base/worked_exercises/introduction_matrix_methods_optics/ray_matrix_elements.svg :alt: Eight panels illustrating translation, refraction, reflection, thin lens, principal planes, focal planes, conjugate imaging, and an afocal telescope. :align: center :width: 100% Optical meaning of the eight matrices in the recreated ray-transfer table. Resonator and Gaussian-beam relations -------------------------------------- Let :math:`M=(A,B;C,D)` be a real unimodular round-trip matrix and define :math:`s=(A+D)/2`. .. list-table:: Recreated Table 2a — eigenvalue and stability summary :header-rows: 1 :widths: 22 26 28 24 * - Regime - Criterion - Eigenvalues - Interpretation * - Positive unstable branch - :math:`s>1` - :math:`\lambda_\pm=e^{\pm\tau}`, :math:`\cosh\tau=s` - One eigenray expands while its reciprocal contracts. * - Negative unstable branch - :math:`s<-1` - :math:`\lambda_\pm=-e^{\pm\tau}`, :math:`\cosh\tau=-s` - Expansion/contraction plus parity reversal. * - Stable - :math:`|s|<1` - :math:`\lambda_\pm=e^{\pm i\theta}`, :math:`\cos\theta=s` - Bounded ray orbit and confined Gaussian eigenmode. * - Marginal - :math:`|s|=1` - Repeated :math:`+1` or :math:`-1` - Stability boundary; diffraction/apertures decide behavior. For the unstable branches, the real eigenvector curvature may be written .. math:: R=\frac{\lambda-D}{C}, \qquad \frac1R=\frac{\lambda-A}{B}. For the stable branch, choose the fixed point with the physical imaginary sign: .. math:: q=\frac{A-D}{2C}+i\frac{\sin\theta}{C}, \qquad \frac1q=\frac{D-A}{2B}+i\frac{\sin\theta}{B}. With the book's convention :math:`1/q=1/R+i\lambda/(\pi w^2)`, the associated beam data are: .. list-table:: Recreated Table 2b — Gaussian eigenbeam parameters :header-rows: 1 :widths: 34 38 28 * - Parameter - Formula at the reference plane - Interpretation * - Wavefront curvature - :math:`R=2B/(D-A)` - Infinite when :math:`A=D`. * - Spot radius - :math:`w^2=\lambda B/(\pi\sin\theta)` - Select the eigenvalue branch giving :math:`w^2>0`. * - Neck location - :math:`z=(A-D)/(2C)` - Signed distance from the reference plane. * - Neck radius - :math:`w_0^2=-\lambda\sin\theta/(\pi C)` - Physical branch again requires positivity. * - Confocal parameter - :math:`z_0=-\sin\theta/C=\pi w_0^2/\lambda` - Half the usual confocal length under this notation. * - Mode discrimination warning - Geometrical stability alone is insufficient. - Aperture loss and gain profile select transverse modes. .. figure:: /_static/knowledge_base/worked_exercises/introduction_matrix_methods_optics/resonator_stability.svg :alt: Trace axis with stable region between minus one and one and Gaussian mode between two curved mirrors. :align: center :width: 100% The half-trace classifies the matrix; the physical fixed point supplies the Gaussian eigenmode. Mueller matrices ---------------- Use Stokes order :math:`(I,Q,U,V)^T`, :math:`C_2=\cos2\theta`, :math:`S_2=\sin2\theta`, :math:`\beta=\cos\delta`, and :math:`\mu=\sin\delta`. .. list-table:: Recreated Table 3 — ideal Mueller elements :header-rows: 1 :widths: 24 56 20 * - Device - Mueller matrix - Special cases * - Linear polarizer at :math:`\theta` - :math:`\dfrac12\begin{bmatrix} 1&C_2&S_2&0\\C_2&C_2^2&C_2S_2&0\\ S_2&C_2S_2&S_2^2&0\\0&0&0&0 \end{bmatrix}` - :math:`\theta=0` passes :math:`+Q`; :math:`\theta=\pi/2` passes :math:`-Q`. * - Linear retarder, retardance :math:`\delta`, fast axis :math:`\theta` - :math:`\begin{bmatrix} 1&0&0&0\\ 0&C_2^2+S_2^2\beta&C_2S_2(1-\beta)&-S_2\mu\\ 0&C_2S_2(1-\beta)&S_2^2+C_2^2\beta&C_2\mu\\ 0&S_2\mu&-C_2\mu&\beta \end{bmatrix}` - Quarter wave: :math:`\delta=\pi/2`; half wave: :math:`\delta=\pi`. * - Quarter-wave retarder, fast axis horizontal - :math:`\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}` - Interchanges :math:`U` and :math:`V` with the convention's signs. * - Half-wave retarder, fast axis horizontal - :math:`\operatorname{diag}(1,1,-1,-1)` - Reverses the :math:`U,V` components. * - Rotation of axes through :math:`\theta` - :math:`R_M(\theta)=\begin{bmatrix} 1&0&0&0\\0&C_2&S_2&0\\0&-S_2&C_2&0\\0&0&0&1 \end{bmatrix}` - Rotates the linear Stokes pair by :math:`2\theta`. A device rotated from its tabulated zero-angle form transforms as .. math:: \boxed{M(\theta+\phi)=R_M(-\theta)M(\phi)R_M(\theta)}. Jones matrices -------------- Use :math:`c=\cos\theta`, :math:`s=\sin\theta`. Overall nonzero complex scalars are physically irrelevant unless absolute transmission or phase is being compared. .. list-table:: Recreated Table 4 — ideal Jones elements :header-rows: 1 :widths: 25 55 20 * - Device - Jones matrix - Special cases * - Linear polarizer at :math:`\theta` - :math:`\begin{bmatrix}c^2&cs\\cs&s^2\end{bmatrix}` - :math:`\theta=0`: :math:`\operatorname{diag}(1,0)`; :math:`\theta=\pi/2`: :math:`\operatorname{diag}(0,1)`. * - Linear retarder, retardance :math:`\delta`, fast axis :math:`\theta` - :math:`\begin{bmatrix} c^2+s^2e^{-i\delta}&cs(1-e^{-i\delta})\\ cs(1-e^{-i\delta})&s^2+c^2e^{-i\delta} \end{bmatrix}` - At :math:`\theta=0`: :math:`\operatorname{diag}(1,e^{-i\delta})`. * - Quarter-wave retarder - Set :math:`\delta=\pi/2` in the general retarder. - Converts suitable linear states to circular states and conversely. * - Half-wave retarder - :math:`\begin{bmatrix}\cos2\theta&\sin2\theta\\ \sin2\theta&-\cos2\theta\end{bmatrix}` up to common phase - Rotates a linear polarization direction through twice the plate angle. * - Rotation of axes through :math:`\theta` - :math:`R_J(\theta)=\begin{bmatrix}c&s\\-s&c\end{bmatrix}` - Also represents an ideal circular retarder with the associated angle. The rotation rule is .. math:: \boxed{J(\theta+\phi)=R_J(-\theta)J(\phi)R_J(\theta)}. Jones calculus retains complex field phase but applies only to fully polarized light. Mueller calculus propagates measurable Stokes data and also handles partial or unpolarized states. .. figure:: /_static/knowledge_base/worked_exercises/introduction_matrix_methods_optics/polarization_matrix_pipeline.svg :alt: Parallel Jones and Mueller calculation pipelines through a rotated retarder and polarizer, ending in field or Stokes output. :align: center :width: 100% Jones and Mueller products use the same rightmost-first composition rule, but operate on different state spaces.