Ray Tracing in Cavities Oct. 9, 2017

Stability of cavity (pp. 29-33, pp. 378-381): stable$\text{ }\to$ beam trapped by mirrors and the only exit is the output coupler (one of the partially reflective mirrors) with nonzero transmission.
unstable$\text{ }\to$ beam escapes when its divergences > mirror size, again escaped radiation is the output.
• Obtain the position and slope of ray at the ends of the unit cell (the location selected by your choice according to your interest) for successive iterations by solving$\text{ }{r}_{m+2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\left(A+D\right){r}_{m+1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ where m is the cell index.
• Stable cavity has ray oscillating around the cavity axis,$\text{ }{r}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{r}_{\text{max}}\text{sin}\left(m\theta \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\alpha \right)$ where$\text{ }\text{cos}\theta \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(A+D\right)/2$ and$\text{ }\alpha$ and$\text{ }{r}_{\text{max}}$ can be determined by initial$\text{ }r$ and$\text{ }r\prime$
Note:$\text{ }{r}_{\text{max}}$ may not appear in ray tracing location and the ray may not return to the initial position after one round trip. Precise number of round trip m for returning to the original position is determined by$\text{ }m\theta \text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\pi q$ where q is an integer.
• Stability criterion:$\text{ }|\left(A+D\right)/2|\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}1$ ; for cavity with two mirrors --$\text{ }0\le {g}_{1}{g}_{2}\le 1$ where$\text{ }{g}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}d/{R}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}d/\left(2{f}_{i}\right)$ and$\text{ }d$ distance between mirrors,$\text{ }{R}_{i}$ is radius for mirror i = 1 or 2.
• Unstable cavity has ray moving further away after each iteration, e.g.$\text{ }{r}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{r}_{a}{F}_{1}^{m}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}_{b}{F}_{2}^{m}$ with$\text{ }|{F}_{i}|\text{\hspace{0.17em}}>\text{\hspace{0.17em}}1$

 Interference & Fabry-Perot resonator Oct. 9, 2017

• Interference (pp. 58-60):$\text{ }{E}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{o}{e}^{-j{k}_{1}{r}_{1}}$ and$\text{ }{E}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{o}{e}^{-j{k}_{2}{r}_{2}}$
After interference,${E}_{T}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{E}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{o}{e}^{-j\left({k}_{1}{r}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{k}_{2}{r}_{2}\right)/2}2\text{cos}\left(\left({k}_{1}{r}_{1}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{k}_{2}{r}_{2}\right)/2\right)$
Interference pattern is a power distribution, i.e.$\text{ }|{E}_{T}{|}^{2}$
Result of interference depends on phase difference$\text{ }\Delta \phi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}_{1}{r}_{1}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{k}_{2}{r}_{2}$
Destructive when$\text{ }\Delta \phi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}p\pi$ where p is odd integer.
Constructive when$\text{ }\Delta \phi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}q\pi$ where q is even integer.
Observation -$\text{ }\Delta \phi$ is controlled by refractive index and/ or distance.
Applications - interferometer (filter) (e.g. Mach-Zehnder, Michelson, Sagnac), antireflective coatings.
• Fabry-Perot (FP) resonator (pp. 62-66, 371-376): Also called etalon, FP interferometer and FP cavity. It consists of two parallel reflective surfaces. A wave with appropriate$\text{ }{\lambda }_{o}$ undergoes constructive interference (resonance) after multiple reflections between the two surfaces.$\to$ Transmission and reflection peak at$\text{ }{\lambda }_{o}$
• Transfer function of FP resonator: A wave inside a gain/lossy medium with length$\text{ }d$ can be expressed as$E\text{\hspace{0.17em}}=\text{\hspace{0.17em}}A{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d/2}{e}^{-jkd}$
Note -$\text{ }\alpha$ and$\text{ }g$ are attenuation and gain coefficients for POWER.
At x=0,$\text{ }E\text{\hspace{0.17em}}=\text{\hspace{0.17em}}A$ and at x=d, transmitted field after 1st pass$\text{ }{E}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}A\sqrt{{T}_{1}{T}_{2}}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d/2}{e}^{-jkd}$ where power transmittivity$\text{ }{T}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{R}_{1}$ & power reflectivity${R}_{1}$ are for the front surface, power transmittivity$\text{ }{T}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{R}_{2}$ & power reflectivity$\text{ }{R}_{2}$ are for the back surface. One more pass, i.e. after a round trip (traveling$\text{ }2d$ ), field becomes$\text{ }{E}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{0}\text{\hspace{0.17em}}h{e}^{j\phi }$ where$\text{ }h\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{R}_{1}{R}_{2}}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d}$ is change in amplitude and$\text{ }\phi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}2kd$ is change in phase.$\text{ }{E}_{tot}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{E}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{E}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{E}_{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}...$$\text{ }\to \text{\hspace{0.17em}}A\sqrt{{T}_{1}{T}_{2}}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d/2}{e}^{-jkd}\text{\hspace{0.17em}}/\left(1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}h{e}^{j\phi }\right)$
Power transmittance$\text{ }=\text{\hspace{0.17em}}|{E}_{tot}{|}^{2}/{A}^{2}$$\text{ }\to \text{\hspace{0.17em}}{T}_{1}{T}_{2}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d}\text{\hspace{0.17em}}/\left[\left(1-h{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4h{\text{sin}}^{2}\left(\phi /2\right)\right]$ or$\text{ }T\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\left(1-{R}_{1}\right)\left(1-{R}_{2}\right){e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d}\text{\hspace{0.17em}}}{\left(1-\sqrt{{R}_{1}{R}_{2}}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4\sqrt{{R}_{1}{R}_{2}}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d}{\text{sin}}^{2}\left(2kd/2\right)}$
• Resonant conditions: Similar to the transfer function of a comb filter or bandpass filter in DSP.
Freq condition -$\text{ }\phi /2\text{\hspace{0.17em}}=\text{\hspace{0.17em}}q\pi$$\text{ }\to \text{ }{\lambda }_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2dn/q$ for integer q which gives resonant freq. in terms of multiplier of freq spacing between resonances called free spectral range (FSR)$\text{ }\Delta {\nu }_{FSR}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\nu }_{F}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{o}/\left(2nd\right)$
Amplitude condition -$\text{ }h\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$$\to \text{ }\sqrt{{R}_{1}{R}_{2}}{e}^{-\left(\alpha \text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ (gain and losses break even, !!lasing threshold!!)
We can express h in log scale in terms of gain$\text{ }g$ and effective loss ($\text{ }{\alpha }_{r}$ ) coefficients for a round trip, i.e.$\text{ }\left({e}^{-\left({\alpha }_{r}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}g\right)2d}{\right)}^{1/2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ where$\text{ }{\alpha }_{r}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{2d}\text{ln}\left(\frac{1}{{R}_{1}{R}_{2}}\right)$
• Tunable by incident angle$\text{ }{\theta }_{g}$ and$\text{ }d$ : Resonant freq. affected by length change$\text{ }\Delta \nu /\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\Delta d/d$
For oblique incident angle$\text{ }{\theta }_{g}$ inside the etalon,$\text{ }\phi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}2{k}_{o}dn\text{cos}{\theta }_{g}$
Follow the resonant condition for freq.$\text{ }\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}q{c}_{o}/\left(2nd\text{cos}{\theta }_{g}\right)$
• Finesse: Another measurement of spectral width$\text{ }\delta \nu$ or linewidth$\text{ }\delta \lambda$ at FWHM (i.e.$\text{ }0.5{T}_{\text{max}}$ points with$\text{ }{T}_{\text{max}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{T}_{1}{T}_{2}/\left(1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}h{\right)}^{2}$ ) relates to Q factor of a filter.
By definition,$\text{ }F\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\Delta {\nu }_{FSR}/\delta \nu$ and$\text{ }Q\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\nu /\delta \nu$ where the resonant freq.$\text{ }\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\nu }_{q}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}q\Delta {\nu }_{FSR}$
Hence$\text{ }Q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}qF$
High loss$\to$ low F or Q factor. For$\text{ }g\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ ,$\text{ }F\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\pi \left({R}_{1}{R}_{2}{\right)}^{1/4}/\left(1-\sqrt{{R}_{1}{R}_{2}}\right)$ ,
• Photon lifetime: Another concept relates to$\text{ }\delta \nu$ is the decay rate of photons over time in a cavity known as photon lifetime$\text{ }{\tau }_{p}$
Fraction of power loss per round trip time is defined as$\text{ }1/{\tau }_{p}$
Relate to$\text{ }\delta \nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}1/\left(2\pi {\tau }_{p}\right)$ that gives$\text{ }{\tau }_{p}\text{\hspace{0.17em}}\text{≈}\text{\hspace{0.17em}}n/\left({c}_{o}{\alpha }_{r}\right)$