PR Cavity with ITM lenses

These are the matrices for reflection from the ITMs, where (Subscript[D, {x,y}]) is the dimensionless lens which couples the 00 mode to the bull’s eye mode.

(Rx=rx*{{1-Dx^2/2,-I*Dx},{-I*Dx,1-Dx^2/2}})//MatrixForm

(Ry=ry*{{1-Dy^2/2,-I*Dy},{-I*Dy,1-Dy^2/2}})//MatrixForm

Here is the round trip matrix in the PRC

(Mrt=rp/2*{{1,0},{0,E^(I*2*η)}}.(E^(I*2*k*ly)*(1-ρ)*Ry+E^(I*2*k*lx)*(1-ρ)*Rx))//MatrixForm

This is the amplitude recycling factor for the PRC with only a differential ITM lens. The differential lens D=(((kw^2)ΔR)/(2(R^2))) for k as wavenumber, w as beam width radius, R as ITM radius, and ΔR as (Subscript[ITM, {x,y}]) radius difference.

(((gprc=tp*Inverse[IdentityMatrix[2]-Mrt]/.{lx->l,ly->l,rx->ri,ry->ri}/.l->π/k)/.{Dx->Dc+Dd/2,Dy->Dc-Dd/2})/.{ρ->0,Dc->0,Dd->k*w^2*ΔR/R^2/2})[[1,1]]

Plug in numbers to the full expression, and get the power recycling gain

Abs[Eigenvalues[gprc]]^2/.{ρ->250*^-6,tp->Sqrt[.029],ri->Sqrt[1-.014],rp->Sqrt[1-.029],Dx->k*w^2*dx/2,Dy->k*w^2*dy/2,η->50/180*π}/.{k->(2*π)/1064*^-9,w->53*^-3,dx->(10.92)*2*1.457/1929.3^2,dy->(-1.56)*2*1.457/1929.3^2}

That’s the power gain for the {bull’s eye mode, 00 mode}