A well functioning [optical cavity](Optical%20cavity.md) is able to confine light within its boundaries over many round-trips between its mirrors. _Cavity stability_ refers to a cavity's effectiveness in achieving this purpose. # Modeling cavity stability with ray optics [Optical cavities](Optical%20cavity.md) may be modeled as an abstract sequence of of [lenses](Modeling%20mirrors%20as%20lenses.md) since we may [model a mirror as a lens](Modeling%20mirrors%20as%20lenses.md). Arbitrarily large sequences of $n$ optical elements may be classified as either _stable_ or _unstable_ based on the way in which a beam moving through the sequence of lenses focuses and defocuses. If the tendency is for the light to on average de-focus across the sequence this means that the overall optical system, and thus the cavity it models is unstable. ## [Paraxial](Paraxial%20approximation.md) analysis ### ABCD matrix Sequences of [n optical elements](Ray%20transfer%20matrix.md#Sylvester's%20matrix%20theorem) (corresponding to n interactions with a cavity wall) are stable when the [trace](Trace.md) obeys the inequality $-1<\frac{1}{2}(A+D)<1.$ where ray transfer matrix for a sequence of $n$ optical elements is again given here as  In the _unstable_ case the matrix elements in transform into [hyperbolic functions](Analysis%20(index).md#Hyperbolic%20functions), modeling how the beam in the system becomes more dispersed. ### Stability diagram The diagram below, which is adapted from _Laser Beams and Resonators_ by H. Kogelnik and T. Li., is a stability diagram which plots which [cavity](Optical%20cavity.md) geometries provide a stable cavity. The horizontal and vertical axes are $g_1$ and $g_2$ respectively, which are the so called [stability parameters](Stability%20parameters%20of%20optical%20cavities.md)  Each point on this diagram corresponds to a possible cavity configuration - where the three parameters associated with each cavity are $d$, the distance between the mirrors, $R_1$, the radius of curvature of one mirror, and $R_2$, the radius of curvature of the second mirror. These parameters are combined to generate the [stability parameters](Stability%20parameters%20of%20optical%20cavities.md). The regions below the curves correspond to stable configurations while the regions outside of the curves correspond to unstable configurations. Observe the diagram to notice that the diagonal axis refers to the $R_2=R_1$ case. # Modeling cavity stability with wave-optics #Electromagnetism/Optics #Electromagnetism/Optics/waveOptics