Ray Tracing in Cavities Oct. 9, 2017



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.



• 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.



• 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.



• 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.



• 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.



 Interference & Fabry-Perot resonator Oct. 9, 2017



• 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.



• 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.



• 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)



• 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}).



• 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}.



• 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}}.



• 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}.