Ray Tracing in Cavities |
Oct. 9, 2017 |
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• (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
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.
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}}`
`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 `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}`.