Ray Tracing in Cavities

Oct. 9, 2017

Stability of cavity (pp. 29-33, pp. 378-381): stable beam trapped by mirrors and the only exit is the output coupler (one of the partially reflective mirrors) with nonzero transmission.
unstable beam escapes when its divergences > mirror size, again escaped radiation is the output.
• Obtain the position and slope of ray at the ends of the unit cell (the location selected by your choice according to your interest) for successive iterations by solvingrm+2(A+D)rm+1+rm=0 where m is the cell index.
• Stable cavity has ray oscillating around the cavity axis,rm=rmaxsin(mθ+α) wherecosθ=(A+D)/2 andα andrmax can be determined by initialr andr
Note:rmax may not appear in ray tracing location and the ray may not return to the initial position after one round trip. Precise number of round trip m for returning to the original position is determined bymθ=2πq where q is an integer.
• Stability criterion:|(A+D)/2|1 ; for cavity with two mirrors --0g1g21 wheregi=1d/Ri=1d/(2fi) andd distance between mirrors,Ri is radius for mirror i = 1 or 2.
• Unstable cavity has ray moving further away after each iteration, e.g.rm=raF1m+rbF2m with|Fi|>1

Interference & Fabry-Perot resonator

Oct. 9, 2017

• Interference (pp. 58-60):E1=Eoejk1r1 andE2=Eoejk2r2
After interference,ET=E1+E2=Eoej(k1r1+k2r2)/22cos((k1r1k2r2)/2)
Interference pattern is a power distribution, i.e.|ET|2
Result of interference depends on phase differenceΔφ=k1r1k2r2
Destructive whenΔφ=pπ where p is odd integer.
Constructive whenΔφ=qπ where q is even integer.
Observation -Δφ is controlled by refractive index and/ or distance.
Applications - interferometer (filter) (e.g. Mach-Zehnder, Michelson, Sagnac), antireflective coatings.
• Fabry-Perot (FP) resonator (pp. 62-66, 371-376): Also called etalon, FP interferometer and FP cavity. It consists of two parallel reflective surfaces. A wave with appropriateλo undergoes constructive interference (resonance) after multiple reflections between the two surfaces. Transmission and reflection peak atλo
• Transfer function of FP resonator: A wave inside a gain/lossy medium with lengthd can be expressed asE=Ae(αg)d/2ejkd
Note -α andg are attenuation and gain coefficients for POWER.
At x=0,E=A and at x=d, transmitted field after 1st passE0=AT1T2e(αg)d/2ejkd where power transmittivityT1=1R1 & power reflectivityR1 are for the front surface, power transmittivityT2=1R2 & power reflectivityR2 are for the back surface. One more pass, i.e. after a round trip (traveling2d ), field becomesE1=E0hejφ whereh=R1R2e(αg)d is change in amplitude andφ=2kd is change in phase.Etot=E0+E1+E2+...AT1T2e(αg)d/2ejkd/(1hejφ)
Power transmittance=|Etot|2/A2T1T2e(αg)d/[(1h)2+4hsin2(φ/2)] orT=(1R1)(1R2)e(αg)d(1R1R2e(αg)d)2+4R1R2e(αg)dsin2(2kd/2)
• Resonant conditions: Similar to the transfer function of a comb filter or bandpass filter in DSP.
Freq condition -φ/2=qπλo=2dn/q for integer q which gives resonant freq. in terms of multiplier of freq spacing between resonances called free spectral range (FSR)ΔνFSR=νF=co/(2nd)
Amplitude condition -h=1R1R2e(αg)d=1 (gain and losses break even, !!lasing threshold!!)
We can express h in log scale in terms of gaing and effective loss (αr ) coefficients for a round trip, i.e.(e(αrg)2d)1/2=1 whereαr=α+12dln(1R1R2)
• Tunable by incident angleθg andd : Resonant freq. affected by length changeΔν/ν=Δd/d
For oblique incident angleθg inside the etalon,φ=2kodncosθg
Follow the resonant condition for freq.ν=qco/(2ndcosθg)
• Finesse: Another measurement of spectral widthδν or linewidthδλ at FWHM (i.e.0.5Tmax points withTmax=T1T2/(1h)2 ) relates to Q factor of a filter.
By definition,FΔνFSR/δν andQν/δν where the resonant freq.ν=νq=qΔνFSR
HenceQ=qF
High loss low F or Q factor. Forg=α=0 ,F=π(R1R2)1/4/(1R1R2) ,
• Photon lifetime: Another concept relates toδν is the decay rate of photons over time in a cavity known as photon lifetimeτp
Fraction of power loss per round trip time is defined as1/τp
Relate toδν=1/(2πτp) that givesτpn/(coαr)


Last Modified: Oct. 9, 2017
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