Figure P.6.29 shows two identical concave spherical mirrors...

Figure P.6.29 shows two identical concave spherical mirrors forming a so-called confocal cavity. Show, without first specifying the value of $d$, that after traversing the cavity two times the system matrix is $$ \left[\begin{array}{cc} \left(\frac{2 d}{r}-1\right)^{2}-\frac{2 d}{r} & \frac{4}{r}\left(\frac{d}{r}-1\right) \\ 2 d\left(1-\frac{d}{r}\right) & 1-2 \frac{d}{r} \end{array}\right] $$ Then for the specific case of $d=r$ show that after four reflections the system is back where it started and the light will retrace its original path.

Figure P.6.29 shows two identical concave spherical mirrors forming a so-called confocal cavity. Show, without first specifying the value of $d$, that after traversing the cavity two times the system matrix is
$$
\left[\begin{array}{cc}
\left(\frac{2 d}{r}-1\right)^{2}-\frac{2 d}{r} & \frac{4}{r}\left(\frac{d}{r}-1\right) \\
2 d\left(1-\frac{d}{r}\right) & 1-2 \frac{d}{r}
\end{array}\right]
$$
Then for the specific case of $d=r$ show that after four reflections the system is back where it started and the light will retrace its original path.

Key Concepts

Ray Transfer Matrix Method

This is an analytical tool in geometrical optics in which optical systems are represented by 2×2 matrices (often called ABCD matrices) that describe the linear relationship between the height and angle of a light ray at the input and output of an optical system. It simplifies the analysis of complex optical setups by reducing them to matrix multiplication, particularly useful when dealing with cascaded optical elements like lenses, free-space propagation, and mirrors.

Optical System Matrix Composition

The system matrix is obtained through the multiplication of individual matrices representing each optical component (such as propagation through free space and reflections from mirrors). This composite matrix provides a complete description of how a light ray is transformed as it passes through the entire system. The property that the overall matrix can be raised to powers or composed in sequences is essential in analyzing periodic or resonant systems, like optical cavities.

Concave Spherical Mirrors

Concave spherical mirrors are curved reflective surfaces characterized by a constant radius of curvature. Their geometry determines the focusing properties and the transformation of rays, which in the paraxial approximation is described by specific reflection matrices. They are key components in many optical systems, including optical resonators, where their reflective properties lead to the convergence or divergence of light rays.

Confocal Optical Cavity

A confocal optical cavity is a resonator setup where two identical spherical mirrors are arranged so that each mirror's center of curvature coincides with the other mirror's focal point. This configuration has special imaging properties and stability characteristics, allowing light to retrace its path after a specific number of reflections. It is widely used in laser cavities and optical resonator designs due to its unique resonant behavior.

Periodic Ray Paths and Reversibility

This concept refers to the condition under which, after a specific number of traversals through an optical system, a ray returns to its initial position and direction. In the context of a confocal cavity, the analysis of the overall system matrix can show that after a finite number of reflections (or matrix multiplications), the system behaves as an identity transformation, leading the ray to retrace its path. This phenomenon is critical for understanding mode stability and resonance in optical cavities.

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