Gaussian beam in cavity (Ch. 10) |
Nov. 6, 2017 |
• Interpretation of stability: Re-cap the 3
regimes for stability --
Stable means permanently trapped beam.
Unstable means beam escaping after 1 or multiple passes
Marginally stable is the critical case where beam can escape
very easily if there is a slight misalignment.
Consequences --
Stable cavity has low loss and hence requires low gain from
lasing media.
Unstable cavity has high loss and hence requires high
optical gain.
Notices that a resonator in standing wave configuration has
2 spherical mirrors. One of them is partially reflective at
the laser wavelength (output coupler). Therefore, the
gain cannot be too small for high power laser.
In this summary, we discuss stable spherical
resonator in standing wave configuration.
• Design of stable cavity for Gaussian beam:
-Assume stable cavity
-Given radii of mirror and their separation, find Rayleigh
length `z_o`, spot size `w_o` and
location `-> R(z_1 )` and `R(z_2 )`
fit the radius of the mirrors
at `z_1` and `z_2`
-Find beam sizes at
mirrors `->` beam size < mirror size
-Check the stability of the ABCD matrix
• Self consistent criterion: a beam retraces itself in
shape and phase after a round trip
Self-consistent solution -- start with a guess; after a few
iterations with the cavity the field reaches a steady state
and stable modes will be formed.
Procedure for determine steady state field
distributions --
-Assume Hermite-Gaussian are the modes of a cavity, i.e. we
can use the ABCD law for Gaussian beam
-Pick a unit cell for the round trip of a cavity
-Usually at a
mirror `->` check the size of beam at the mirror and stability of the
cavity using `|{A+D}/2| <1`
-Apply self-consistent criterion to q parameter
-- `q_s = q_{s+1}`,
`s` - cell index; `q_{s+1} = {A q_s + B} / {C q_s + D}`
• `R` and `w` inside a cavity:
From the quadratic equation in
`q`, ` 1/R = {D-A}/{2B}`
and `{pi w^2}/ lambda = B/{sqrt{1-[{A+D}/2]^2}}`
where `A`, `B`, `C` and `D` are the elements of the ray matrix for a
round trip.
Note: To locate min spot
size, `R -> oo`, i.e. `D=A` or the min spot size can be obtained by
minimizing B.
• Modes in resonant cavity and their spacing:
Recall that the phase of HG mode solutions has a radius of
curvature term and the following phase
terms, `phi_{mnp} (z) = k_p z - (m+n+1) tan^{-1}(z/{z_o})`
The first term relates to the longitudinal mode
(`k_p` in `z` direction) while the second term relates to the
transverse mode (the spatial extent in `x` (notice `m`) and `y`
(notice `n`) ).
In other words, many modes such
as `TEM_{mn}` will have the same R and satisfy the cavity resonant
condition for
phase: `[ phi_{m n p}(z_2) - phi_{m n p}(z_1)] = p pi`;
longitudinal mode
spacing `Delta nu_{FSR}` or `nu_F`; transverse mode spacing
-- `Delta nu_t = {Delta nu_{FSR}}/ pi [ tan^{-1} ({z_2}/{z_o}) - tan^{-1} ({z_1}/{z_o}) ] `
` `
More on resonator (Ch. 10) |
Nov. 6, 2017 |
• Density of
modes `M(nu)` (number of modes per freq per length): For FP resonator,
mode spacing is given
by `Delta nu_{FSR} = c_o / {2nd}`.
Number of
modes `# = {2 nu} / {Delta nu_{FSR}}`
where 2 is accounted for two polarizations for each mode
frequency.
Number of modes / freq ` {d#}/{d nu}=2 / {Delta nu_{FSR}}`.
Density of
modes `M(nu)=2 / {Delta nu_{FSR} d} = {4 n}/{c_o}`.
Similarly calculation can be performed with
wavenumber `k = {2 pi nu} / c` and `Delta k_{FSR} = 2 pi {Delta nu_{FSR}}/c`.
• Effect of finite mirror size: If mirror radius (`a`) is
less than twice of beam radius (`2w`), modes (particularly
high order ones) will be cut off and incur high loss. This
loss is associated with diffraction since `w` at the mirror is
larger than `w_o`.
In this situation,
Notice that we cannot assume Gaussian or HG modes inside a
cavity
if `N_F` is too small.
•
Rectangular -- assume square
geometry `(pi / d)^2 (q_y^2+q_z^2) = k^2`
and `nu_q = Delta nu_{FSR} sqrt{q_y^2+q_z^2}`
where `q_y` and `q_z` are non-zero integers.
Circular -- whispering gallery modes (WGM) excited
and `Delta nu_{FSR} = c / {2 pi a}`.
• Three dimensional resonators:
Similar to 2D
case, `pi^2 ((q_x/d_x)^2+(q_y/d_y)^2 +(q_z/d_z)^2)=k^2`
and `nu_q=sqrt{(Delta nu_{FSR_x})^2 q_x^2+(Delta nu_{FSR_y})^2 q_y^2+(Delta nu_{FSR_z})^2 q_z^2}`
where `q_x`, `q_y` and `q_z` are non-zero integers.
Number of modes per
freq ` {d#}/{d nu} = pi ({2d}/c)^3 nu^2`.
Number of modes per freq per
vol `M(nu) = {8 pi nu^2}/c^3`.
• Microresonators:
Microdisk -- WGM.
Microtoroid -- fiber ring resonator.