FP Transmission of a Fabry--Perot interferometer
by Chuck DiMarzio Northeastern University May 2009
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Contents
Equation
Parameters
Fresnel reflection coefficients
lambda=(600:0.1:650)*1e-9; % wavelength in meters c=299792458; % m/s f=c./lambda; k=2*pi./lambda; %
Transmissions of two etalons with 10--micrometer spacing
Delta=10e-6; % Spacing of etalon reflectors R1=0.9;R2=0.9; % Power reflectivities rho1=sqrt(R1);rho2=sqrt(R2); % Assume field coefficients are real T1=1-R1;T2=1-R2; T90=T1*T2./(abs(1-exp(1i*k*2*Delta)*rho1*rho2)).^2; % % Lower reflecivity % R1=0.5;R2=0.5; % Power reflectivities rho1=sqrt(R1);rho2=sqrt(R2); % Assume field coefficients are real T1=1-R1;T2=1-R2; T50=T1*T2./(abs(1-exp(1i*k*2*Delta)*rho1*rho2)).^2; % % fig1=figure;plot(f/1e12,T90,'-',f/1e12,T50,'--');grid on; hold on; for lambdaplot=(610:20:650); plot(c/(lambdaplot*1e-9)/1e12,1.1,'k^'); text(c/(lambdaplot*1e-9)/1e12,1.1,[sprintf('%5.0f',lambdaplot),' nm']); end; hold off; xlabel('f, Freq., THz.'); ylabel('T, Trannsmission');

Laser cavity
Gain line assumed lorentzian
fcenter=c/514.5e-9; % Argon ion laser green line hwhm=5e10/2; % Gain linewidh % % New frequency axis f1=fcenter+(-4:0.001:4)*1e10; lambda=c./f; k=2*pi./lambda; % g=1./(1+((f1-fcenter)/hwhm).^2); % Delta=30e-2; % Spacing of etalon reflectors % % The Fabry--Perot lines are too narrow to plot, so we will just plot % their locations. % The fundamental frequency is f0=c/(2*Delta); Ncenter=round(fcenter/f0); % Integer mode number nearest center Nlow=ceil(f1(1)/f0); % Low and highest integer mode numbers Nhigh=floor(f1(end)/f0); N=(Nlow:Nhigh); fN=N*f0; % % Note below we only plot every fifth cavity mode because they are too dense fig2=figure;plot((fN(5:5:end)-fcenter)/1e9,0.5*ones(size(N(5:5:end))),'.',... (f0*Ncenter-fcenter)/1e9,0.6,'*',... (f1-fcenter)/1e9,g,'--');grid on; xlabel('\delta f, Freq., GHz.'); ylabel('G/G_{max}');

Etalon inside a laser cavity
f2=fcenter+(-40:0.0001:40)*1e9; % Narrow the frequency plot lambda2=c./f2; k2=2*pi./lambda2; Ncenter=round(fcenter/f0); % Integer mode number nearest center Nlow=ceil(f2(1)/f0); % Low and highest integer mode numbers Nhigh=floor(f2(end)/f0); N=(Nlow:Nhigh); fN=N*f0; % recompute the gain for new frequency axis g2=1./(1+((f2-fcenter)/hwhm).^2); % Transmissions of etalon with 5--micrometer spacing % Delta=5.003e-3; % Spacing of etalon reflectors R1=0.95;R2=0.95; % Power reflectivities rho1=sqrt(R1);rho2=sqrt(R2); % Assume field coefficients are real T1=1-R1;T2=1-R2; Tetalon=T1*T2./(abs(1-exp(1i*k2*2*Delta)*rho1*rho2)).^2; % Note below we only plot every fifth cavity mode because they are too dense fig3=figure;plot((fN(5:5:end)-fcenter)/1e9,0.5*ones(size(N(5:5:end))),'.',... (f0*Ncenter-fcenter)/1e9,0.5,'*',... (f2-fcenter)/1e9,g2,'--',... (f2-fcenter)/1e9,Tetalon,'-.');grid on; xlabel('\delta f, Freq., GHz.'); ylabel('G/G_{max}, T');

Expanded scale
kk1=2*pi/(c/(f0*Ncenter));% Find Tetalon at the two cavity modes nearest % the peak kk2=2*pi/(c/(f0*(Ncenter-1))); tetalon1=T1*T2./(abs(1-exp(1i*kk2*2*Delta)*rho1*rho2)).^2; tetalon2=T1*T2./(abs(1-exp(1i*kk1*2*Delta)*rho1*rho2)).^2; fig4=figure;plot((fN-fcenter)/1e9,0.5*ones(size(N)),'+',... (f0*Ncenter-fcenter)/1e9,0.5,'*',... (f0*Ncenter-fcenter)/1e9,tetalon2,'o',... (f0*(Ncenter-1)-fcenter)/1e9,tetalon1,'o',... (f0*Ncenter-fcenter)/1e9*[1,1],[0,1.1],'k-',... (f0*(Ncenter-1)-fcenter)/1e9*[1,1],[0,1.1],'k-',... (f2-fcenter)/1e9,g2,'--',... (f2-fcenter)/1e9,Tetalon,'-.');grid on; axis([-3,3,0,1.1]); xlabel('\delta f, Freq., GHz.'); ylabel('G/G_{max}, T');

Graphical representations
Show how the denominator of the Fabry Perot transmission equation varies with the optical path.
unitcircle=exp(1i*(0:360)*pi/180); semicircle=exp(1i*(-90:90)*pi/180); r1=0.99;r2=0.9;r3=0.8; lox=-0.01;hix=0.2;loy=-0.2;hiy=0.2; % Bounds of expanded region fig5=figure;plot(real(1+r1*unitcircle),imag(1+r1*unitcircle),'-',... real(1+r2*unitcircle),imag(1+r2*unitcircle),'-',... real(1+r3*unitcircle),imag(1+r3*unitcircle),'-',... real((1-r1)*sqrt(2)*semicircle),... imag((1-r1)*sqrt(2)*semicircle),'-',... real((1-r2)*sqrt(2)*semicircle),... imag((1-r2)*sqrt(2)*semicircle),'-',... real((1-r3)*sqrt(2)*semicircle),... imag((1-r3)*sqrt(2)*semicircle),'-',... [lox,hix,hix,lox,lox],[loy,loy,hiy,hiy,loy],'--',... [-0.2,2.2],[0,0],'k-',[0,0],[-1.2,1.2],'k-'); axis([-0.2,2.2,-1.2,1.2]);axis image; xlabel('Re');ylabel('Im'); fig6=figure;plot(real(1+r1*unitcircle),imag(1+r1*unitcircle),'-',... real(1+r2*unitcircle),imag(1+r2*unitcircle),'-',... real(1+r3*unitcircle),imag(1+r3*unitcircle),'-',... real((1-r1)*sqrt(2)*semicircle),... imag((1-r1)*sqrt(2)*semicircle),'-',... real((1-r2)*sqrt(2)*semicircle),... imag((1-r2)*sqrt(2)*semicircle),'-',... real((1-r3)*sqrt(2)*semicircle),... imag((1-r3)*sqrt(2)*semicircle),'-',... [lox,hix],[0,0],'k-',[0,0],[loy,hiy],'k-'); axis image; axis([lox,hix,loy,hiy]); xlabel('Re');ylabel('Im');


Demonstration of adding terms in the infinite series
phid=(0:720); n=(0:300); [ph,nn]=meshgrid(phid,n); term=0.01*0.98.^(nn/2).*cosd(nn.*ph); output=(cumsum(term,1)); figure;plot(phid,term([1,2,3,10],:)); xlabel('\phi, Phase, Deg');ylabel('Term'); figure;plot(phid,output([1,2,3,10,300],:)); xlabel('\phi, Phase, Deg');ylabel('Sum');

