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Optical Resonators
The microwave resonators are metal boxes or pipes (1) to build up large field intensity with moderate input power, (2) to act as a special and frequency filter selectively to fields, (3) to be used in spectral analyses.
The number of modes with a frequency between ν and ν +dν per unit volume in a three dimensional resonator is
ν
πν d c
n N (^) 3
For a box of 1cm^3 , a microwave resonator contains at most one mode at a frequency of 10^9 Hz. However, at an optical frequency of 10^14 Hz , the number of modes per unit volume can be as high as 10^9 for a bandwidth of 10^10 Hz, unless the dimension of the resonator is reduced to the size of a wavelength. This problem can be overcome by using open resonators with a pair of mirrors with small areas. Only the mode propagating normal to the mirrors have high Q for meaningful resonance.
In a resonator of distance l between mirrors, the beam from one of the mirrors of size a 1 must be smaller than a 2 to avoid losses.
2 1 2 1 1
l
aa l a or a λ
λ
Fabry-Perot resonator with plane parallel mirrors)
It can be proved, by considering multiple reflections between the mirrors, that, for an incoming electromagnetic radiation of wavelength k 0 , the transmission intensify is given by
( 1 ) 4 sin ( )
0
2 2 1 2
2 2 1 2
4 1 2 0 0
0
RR e RR e n k l
R R e
T
r
nkl nkl
nkl
i i
i
where the R’s are the reflectance of the mirrors and the index of refraction is a complex number.
For ni =0,
( 1 ) 4 sin ( )
2 1 2
2 1 2
1 2 RR RR kl
R R
T
Case I : R 1 =R 2 =R
) sin ( )
sin ( ) ( 1 )
( 1 ) 4 sin ( )
2 2 2 2
1 2 2
2 1
F
F
kl R
R R kl R
R
T
ν
πν π
Where the Finesse is defined by
( 1 R )
R
F
π
The linewidth is given by
F
δν =^ ν^ F
Meaning?
Case II : R 1 =R 2 and ni ≠0. Case III: R 1 exp(2ni k 0 l) ≈ 1
Finesse and Q-value
Factors affecting linewidth:
- Reflectivity of mirrors
- Length of resonators
- Parallelism of mirrors
- Diffraction losses
- Imperfections in optical materials
- Coherence length
Resonator modes:
Plane-parallel mirrors of finite sizes cannot confine electromagnetic radiation. Stable confinement always involves spherical mirrors.
Hermite Gaussian mode:
From (33) of Lecture 1, the solution for propagating beam in a monogenous medium of index n are:
Resonance frequency: p 335
Unfolding a resonator
Consider a resonator formed by two mirrors of radii of curvature R1 and R2 separated by a distance R This system can be unfolded into periodic lens sequence
d
R (^1) R 2
Using the ABCD matrix to represent the position and slop after m round trips
^ =
0
0
y C D
y A B m
m
m (^) (10)
Consider the relation between consecutive round trips
m m m
m m m Cy D
y Ay B θ θ
θ = +
1
1 (11)
By eliminating the angles using the relation,
B
y (^) m Aym m
θ = +^1
B
y (^) m Aym m 1 1 1
We have
y (^) m bym F y m 2
Where
F AD BC M
A D
b
det
The beam position parameter after m round trips can be related to the initial by
y m = y max Fm sin( m ϕ + ϕ 0 ) (14)
where the parameters ymax , and ϕ 0 are to be determined by the initial condition, and
F
1^ b ϕ = cos− (15)
For the round trip effect is to reproduce the original position, then F=1 and
b ≤ 1 or
1 2
R
d R
d (16)
It is customary to express the condition for stability as 0 ≤ g 1 g 2 ≤ 1
where 1
R
d g = +
2
R
d g = +
Concave mirrors: R<0.
Discussion of resonator stability diagram.
g 1
g 2
symmetrical
confocal
1
1 Planar planar
concentric Symmetric stable resonator
