Ray Matrices for Paraxial Optical Elements

1. "Free space" region, index n₀, length L

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill L/n_0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]}$

2. Thin lens, focal length f (f>0 for converging lens)

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -1/f\hfill & \hfill 1\hfill \end{array}\right]}$

3. Curved mirror, radius R, normal incidence (R>0 for concave mirror)

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -2/R\hfill & \hfill 1\hfill \end{array}\right]}$

4. Curved mirror, arbitrary incidence
   R_e = R cosθ in the plane of incidence ("tangential")
   R_e = R/cosθ ⊥ to the plane of incidence ("sagittal")

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -2/R_e\hfill & \hfill 1\hfill \end{array}\right]}$

5. Curved dielectric interface, normal incidence (R>0 for concave surface)

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill (n_2-n_1)/R\hfill & \hfill 1\hfill \end{array}\right]}$

6. Curved interface, arbitrary incidence, tangential plane
   R>0 for concave surface, n₁ sinθ₁ = n₂ sinθ₂
   Δn_e = (n₂ cosθ₂ - n₁ cosθ₁) / cosθ₁cosθ₂

   ${\displaystyle \left[\begin{array}{cc}\hfill \frac{\cos {\theta }_2}{\cos {\theta }_1}\hfill & \hfill 0\hfill \\ \hfill \Delta n_e/R\hfill & \hfill \frac{\cos {\theta }_1}{\cos {\theta }_2}\hfill \end{array}\right]}$

7. Curved interface, arbitrary incidence, sagittal plane
   R>0 for concave surface, n₁ sinθ₁ = n₂ sinθ₂
   Δn_e = (n₂ cosθ₂ - n₁ cosθ₁)

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill \Delta n_e/R\hfill & \hfill 1\hfill \end{array}\right]}$

8. "Duct" (radially varying index and gain)
   n(x) = n₀ - ½ n₂ x², γ² ≡ n₂ / n₀

   ${\displaystyle \left[\begin{array}{cc}\hfill \cos \gamma z\hfill & \hfill (n_0{\gamma }^{-1})\sin \gamma z\hfill \\ \hfill -(n_0\gamma )\sin \gamma z\hfill & \hfill \cos \gamma z\hfill \end{array}\right]}$