Classical cavity

I first want to consider a classical description of an optical cavity. For simplicity, I’ll start by mod- eling the cavity as a pair of flat mirrors that are parallel to one another and are separated by a distanceL, then later I’ll consider the effects of mirror curvature.

Suppose a plane wave with wavelength λ is incident on one of the mirrors, and that the coeffi- cients of transmission and reflection for both mirrors aretandr. We want to solve for the electric field Ec(z) inside the cavity at a distancez from the input mirror. Since the light that enters the cavity bounces back and forth between the two mirrors, we can expressEc(z) as sum of the fields for each bounce:

Ec(z) = tEieikz+trEieik(2L−z)+tr2Eieik(2L+z)+· · · = tEi(eikz+r e−ikz)(1 +r2e2ikL+r4e4ikL+· · ·) = tEi(eikz+r e−ikz)(1−r2e2ikL)−1

where k = 2π/λ. If we assume that the mirrors are highly reflective (r ≃ −1), then we can approximate this as

Ec(z) = 2itEi(1−r2e2ikL)−1 sinkz

I will define R =r2 _{and T = t}2_{. Assuming the mirrors do not absorb any of the light, R and T} satisfy the conservation equationR+T = 1. The intensity inside the cavity is then

Ic(z) = |Ec(z)/Ei|2Ii

= 4T Ii(1 +R2−2Rcos 2kL)−1sin2kz

= 4T Ii(1 +R2_{−2R+ 4Rsin}2_kL)−1_sin2_kz = 4T Ii(T2+ 4Rsin2kL)−1sin2kz

SinceR≃1, we may approximate this as

Ic(z) = (4/T)Ii(1 + (4/T2_{) sin}2_kL)−1_sin2_kz

Because the mirrors are highly reflective, the 4/T2_{factor in the denominator is very large, so light} is only coupled into the cavity if its wavelength is tuned such thatkL=nπfor some integern. We can understand this condition as follows. If the mirrors were perfectly reflective, then the cavity would have normal modes at integer multiples of the free spectral rangeνF SR = 1/2L. If we now allow the mirrors to be slightly transmissive, we expect light to be coupled into the cavity when it is tuned into resonance with one of these modes. The width of the resonance can be determined as follows. Assume the light is nearly resonant with mode n, so the detuning of the light from the mode is small compared to the free spectral range. The detuning is given by

In terms of the detuning,

kL= 2πL/λ= ∆L+nπ

and

Ic(z) = (4/T)Ii(1 + (4/T2_{) sin}2_∆L)−1_sin2_kz If we expand around ∆ = 0, we find

Ic(z) = (4/T)Ii(1 + (2∆/κ)2)−1sin2kz

where κ=T /L is the full width at half maximum of the resonance. According to this definition,

κgives the energy decay rate for the entire cavity. Note that some authors define κdifferently, so that it represents an amplitude decay rate, or so that it gives the decay rate for an individual mirror.

I now want to solve for the transmitted and reflected fields. The electric field at the face of the output mirror is

Et = t2EieikL+t2r2Eie3ikL+· · · = t2Ei(1−r2e2ikL)−1eikL

Thus, making the same approximations as before, we find that the transmitted intensity is

It=Ii(1 + (4/T2) sin2kL)−1

The reflected field can be obtained from this by using the conservation equationIr+It=Ii.

The results obtained thus far were derived by treating the mirrors as flat planes. If we now in- clude the curvature of the mirrors, the results are modified in several ways. Because of the mirror curvature, light is confined in the transverse direction, resulting in a set of transverse modes. If we drive only the lowest order transverse mode (the gaussian mode) then the intra-cavity intensity is

Ic(~r) = (4/T)(1 + (2∆/κ)2₎−1_|ψ(~r)|2_I i

where

ψ(~r) = sinkz e−(x2+y2)/w20

describes the mode shape. The mode radiusw0 is related to the mirror radiusR by

w20=

λ

2π(L(2R−L))

ForR≫L, this may be approximated as

w20=

λ

2π(2RL)

1/2

For our cavity,R= 20 cm andL= 45µm, sow0= 25µm at the FORT wavelength ofλ= 895 nm. We can define an effective mode volume by

V = Z |ψ(~r)|2d3r=λ 8(2RL) 1/2_L=1 2AL

whereAis an effective area for the mode, obtained by integrating over the transverse mode profile:

A= Z Z e−2(x2+y2)/w02dx dy= π 2w 2 0

To couple light into the cavity, the input light must be spatially mode matched to a cavity mode. The input intensityIi is related to the input powerPi byIi =Pi/A, whereAis the effective area. Usually Pi is less than the total power in the input beam, since not all the input power is mode matched into the cavity. We can measure Pi in the lab by tuning the input beam to resonance and measuring the output power, and then including a correction factor to account for light that is absorbed in the mirrors. Note that we can express the intensity inside the cavity in terms ofPi:

Ic(~r) = (2/κV)(1 + (2∆/κ)2)−1|ψ(~r)|2Pi

The total energy inside the cavity is therefore

E=

Z

Ic(~r)d3r= (2/κ)(1 + (2∆/κ)2)−1Pi