Ray Tracing in Cavities

Oct. 9, 2017

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Stability of cavity (pp. 29-33, pp. 378-381): stable `->` beam trapped by mirrors and the only exit is the output coupler (one of the partially reflective mirrors) with nonzero transmission.
unstable `->` beam escapes when its divergences > mirror size, again escaped radiation is the output.

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• 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 `r_{m+2}-(A+D) r_{m+1} + r_m =0` where `m` is the cell index.

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• Stable cavity has ray oscillating around the cavity axis, `r_m = r_{max} sin ( m theta + alpha )` where `cos theta = {A+D}/2`, `alpha` and `r_{max}` can be determined by initial `r` and `r'`.
Note:`r_{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 `m theta = 2 pi q` where `q` is an integer.

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• Stability criterion: `{|A+D|}/2 <= 1`; for cavity with two mirrors -- `0 <= g_1 g_2 <= 1` where `g_i=1 - d/{R_i} = 1 - d /{2 f_i}` and `d` is distance between mirrors, `R_i` is radius for mirror `i` = 1 or 2.

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• Unstable cavity has ray moving further away after each iteration, e.g. `r_m = r_a F_1^m + r_b F_2^m` with `|F_i| >1`.

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Interference & Fabry-Perot resonator

Oct. 9, 2017

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• Interference (pp. 58-60): `E_1 = E_o e^{-jk_1 r_1}` and `E_2 = E_o e^{-j k_2 r_2}`.
After interference, `E_T=E_1 + E_2=E_o e^{-j{k_1 r_1+k_2 r_2 }/2}2cos ({k_1 r_1-k_2 r_2 }/2)`.
Interference pattern is a power distribution, i.e. `|E_T|^2`.
Result of interference depends on phase difference `Delta phi = k_1 r_1-k_2 r_2`.
Destructive when `Delta phi = p pi` where `p` is an odd integer.
Constructive when `Delta phi = q pi` where `q` is an even integer.
Observation `- Delta phi` is controlled by refractive index and/ or distance.
Applications `-` interferometer (filter) (e.g. Mach-Zehnder, Michelson, Sagnac), antireflective coatings.

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• 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 `lambda_o` undergoes constructive interference (resonance) after multiple reflections between the two surfaces `->` transmission and reflection peak at `lambda_o`.

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• Transfer function of FP resonator: A wave inside a gain/lossy medium with length `d` can be expressed as `E= A e^{-{alpha - g}/2 d}e^{jkd}`.
Note `- alpha` and `g` are attenuation and gain coefficients for POWER.
At x=0, `E = A` and at `x=d`, transmitted field after 1st pass `E_0 = A sqrt{T_1 T_2} e^{-{alpha - g}/2 d} e^{-jkd}` where power transmittivity `T_1 = 1 - R_1` & power reflectivity `R_1` is for the front surface, power transmittivity `T_2 = 1 - R_2` & power reflectivity `R_2` is for the back surface. One more pass, i.e. after a round trip (traveling `2d`), field becomes `E_1=E_0 h e^{j phi}` where `h = sqrt{R_1 R_2} e^{-(alpha-g)d}` is change in amplitude and `phi = 2 kd` is change in phase. `E_{\t\ot}=E_0+E_1+E_2+ ... -> {A sqrt{T_1 T_2} e^{-{alpha - g}/2 d} e^{-jkd}} /{1-h e^{j phi}}`
Power transmittance `T = {|E_{\t\ot}|^2} /A^2 -> {T_1 T_2 e^{-(alpha - g)d}}/ {(1-h)^2 + 4 h sin^2 ( phi / 2)}` or `T = {(1-R_1)(1-R_2)e^{-(alpha - g)d}}/ {(1-sqrt{R_1R_2}e^{-(alpha - g)d})^2+4sqrt{R_1R_2} e^{-(alpha - g)d} sin^2({2kd}/2)`

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• Resonant conditions: Similar to the transfer function of a comb filter or bandpass filter in DSP.
Freq condition ` - phi /2 = q pi -> lambda_o = {2dn}/q` for integer `q` which gives resonant freq. in terms of multiplier of freq spacing between resonances called free spectral range (FSR) `Delta nu_{FSR} = nu_F = {c_o}/{2nd}`.
Amplitude condition ` - h = 1 -> sqrt{R_1 R_2} e^{-(alpha - g)d} =1` (gain and losses break even, !!lasing threshold!!).
We can express `h` in log scale in terms of gain `g` and effective loss (`alpha_r`) coefficients for a round trip, i.e. `(e^{-(alpha_r - g)d})^{1/2} =1` where `alpha_r = alpha + 1/{2d} ln (1/{R_1 R_2})`.

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• Tunable by incident angle `theta_g` and `d`: Resonant freq. affected by length change `{Delta nu} / nu = - {Delta d} /d`.
For oblique incident angle `theta_g` inside the etalon, `phi = 2k_o d n cos theta_g`.
Follow the resonant condition for freq. `nu = {q c_o} / {2nd cos theta_g}`.

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• Finesse: Another measurement of spectral width `delta nu` or linewidth `delta lambda` at FWHM (i.e. `0.5 T_{max}` points with `T_{max}= {T_1 T_2 } /{(1-h)^2}`) relates to `Q` factor of a filter.
By definition, `ccF -= {Delta nu_{FSR}}/{delta nu}` and `Q -= nu /{delta nu}` where the resonant freq. `nu = nu_q = q Delta nu_{FSR}`.
Hence, `Q = q ccF`.
High loss `->` low `ccF` or `Q` factor. For `g = alpha = 0`, `ccF = {pi (R_1 R_2)^{1/4}}/ {1 - sqrt{R_1 R_2}}`.

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• Photon lifetime: Another concept relates to `delta nu` is the decay rate of photons over time in a cavity known as photon lifetime `tau_p`.
Fraction of power loss per round trip time is defined as `1 / {tau_p}`.
Relate to `delta nu =1 / {2 pi tau_p}` that `tau_p = n / {c_o alpha_r}`.


Last Modified: Oct. 9, 2017
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