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:
Match radius curvature of beam with that of mirror
Make sure the beam size less than the mirror size
-Assume stable cavity
-Given radii of mirror and their separation, find Rayleigh length$\text{ }{z}_{o}$ , spot size$\text{ }{w}_{o}$ and location$\text{ }\to$$\text{ }R\left({z}_{1}\right)$ and$\text{ }R\left({z}_{2}\right)$ fit the radius of the mirrors at$\text{ }{z}_{1}$ and$\text{ }{z}_{2}$
-Find beam sizes at mirrors$\text{ }\to$ 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$\text{ }\to$ check the size of beam at the mirror and stability of the cavity using$\text{ }|\left(A+D\right)/2|<1$
-Apply self-consistent criterion to q parameter --$\text{ }{q}_{s}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{q}_{s+1}$ , s - cell index;$\text{ }{q}_{s+1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(A{q}_{s}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}B\right)/\left(C{q}_{s}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}D\right)$
• R and w inside a cavity:
From the quadratic equation in q,$\text{ }1/R\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(D-A\right)/2B$ and$\text{ }\pi {w}^{2}/\lambda \text{\hspace{0.17em}}=\text{\hspace{0.17em}}B/\sqrt{1-\left[\left(A+D\right)/2{\right]}^{2}}$
where A, B, C and D are the elements of the ray matrix for a round trip.
Note: To locate min spot size,$\text{ }R\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\infty$ , 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,$\text{ }{\phi }_{mnp}\left(z\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}_{p}z\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\left(n+m+1\right){\text{tan}}^{-1}\left(z/{z}_{o}\right)$
The first term relates to the longitudinal mode ($\text{ }{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$\text{ }TE{M}_{mn}$ will have the same R and satisfy the cavity resonant condition for phase:$\text{ }\left[{\phi }_{pmn}\left({z}_{2}\right)\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\phi }_{pmn}\left({z}_{1}\right)\right]\text{\hspace{0.17em}}=\text{\hspace{0.17em}}p\pi$ ;
longitudinal mode spacing$\text{ }\Delta {\nu }_{FSR}$ or$\text{ }{\nu }_{F}$ ; transverse mode spacing --$\text{ }\Delta {\nu }_{t}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Delta {\nu }_{FSR}\text{\hspace{0.17em}}\left[{\text{tan}}^{-1}\left({z}_{2}/{z}_{o}\right)-{\text{tan}}^{-1}\left({z}_{1}/{z}_{o}\right)\right]$$/ \pi$

 More on resonator (Ch. 10) Nov. 6, 2017

• Density of modes$\text{ }M\left(\nu \right)$ (number of modes per freq per length): For FP resonator, mode spacing is given by$\text{ }\Delta {\nu }_{FSR}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{o}/2nd.$ Number of modes$\text{ }#\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\nu /\Delta {\nu }_{FSR}$ where 2 is accounted for two polarizations for each mode frequency. Number of modes / freq$\text{ }d#/d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}2/\Delta {\nu }_{FSR}.$ Density of modes$\text{ }M\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2/\left(\Delta {\nu }_{FSR}d\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}4n/{c}_{o}.$
Similarly calculation can be performed with wavenumber$\text{ }k\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\pi \nu /c$ 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$\text{ }{w}_{o}.$
In this situation, Fresnel number$\text{ }{N}_{F}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{a}^{2}/\left(\lambda d\right)$ is used to measure Fresnel diffraction between 2 mirrors.
Notice that we cannot assume Gaussian or HG modes inside a cavity if$\text{ }{N}_{F}$ is too small.
Two dimensional resonators:
Rectangular -- assume square geometry$\text{ }\left(\pi /d{\right)}^{2}\left({q}_{y}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{q}_{z}^{2}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}^{2}$ and${\nu }_{q}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Delta {\nu }_{FSR}\sqrt{{q}_{y}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{q}_{z}^{2}}$ where$\text{ }{q}_{y}$ and$\text{ }{q}_{z}$ are non-zero integers.
Density of modes (number of modes per freq per area)$\text{ }M\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}4\pi \nu /{c}^{2}.$
Circular -- whispering gallery modes (WGM) excited and$\text{ }\Delta {\nu }_{FSR}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}c/\left(2\pi a\right).$
• Three dimensional resonators:
Similar to 2D case,$\text{ }{\pi }^{2}\left(\left({q}_{x}/{d}_{x}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left({q}_{y}/{d}_{y}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left({q}_{z}/{d}_{z}{\right)}^{2}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}^{2}$ and${\nu }_{q}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{\left(\Delta {\nu }_{FSR}{\right)}_{x}^{2}{q}_{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\Delta {\nu }_{FSR}{\right)}_{y}^{2}{q}_{y}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\Delta {\nu }_{FSR}{\right)}_{z}^{2}{q}_{z}^{2}}$ where$\text{ }{q}_{x}$ ,$\text{ }{q}_{y}$ and$\text{ }{q}_{z}$ are non-zero integers.
Number of modes$\text{ }#\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{3}\pi {\left(\frac{2d}{c}\right)}^{3}{\nu }^{3}.$ Number of modes per freq$\text{ }=\text{\hspace{0.17em}}d#/d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\pi {\left(\frac{2d}{c}\right)}^{3}{\nu }^{2}.$ Number of modes per freq per vol$\text{ }M\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}8\pi {\nu }^{2}/{c}^{3}.$
• Microresonators: As dimension d of the cavity decreases,$\text{ }\Delta {\nu }_{FSR}$ increases.
Micropillar -- small FP cavity with Bragg gratings as mirrors.
Microdisk -- WGM.
Microtoroid -- fiber ring resonator.

Microsphere -- droplets, low-loss silica spheres have very high Q$\text{ }\text{≈}\text{\hspace{0.17em}}1{0}^{10}.$
Photonic-crystal microcavities -- 2D version of micropillar. It has a periodic structure with a defect. The periodic structure form a grating that acts as a mirror, forbidding propagation of certain freq. That frequency is only allowed in the region with the defect and trapped.