1. PHYS 408 Applied Optics (Lecture 17)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
There are a continuum of Gaussians with different l and w0 values that can be supported by a given set of R1, R2, and d that satisfy the cavity stability criterion: i.e. that propagate self-consistently with the same q value from round trip to round trip.
For an arbitrary l, the circulating Gaussian would have a different phase every round trip at any given location in the cavity, so the field strength would be quite low due to destructive interference. Only Gaussians with wavelengths such that the round trip phase is an integer multiple of 2p, generate significant fields due to repeated constructive interference.
The resulting mode criterion is equivalent to that which one could alternately derive using boundary conditions at perfectly reflecting mirrors.
There are a host of modified Hermite-Gaussian beams that have q values that transform the same was as the simple Gaussian, but their transverse intensity patterns are very different, and characterized by the number of nodes in the x and y directions. Their z-direction phase is also different, but only slightly.

3. Impact of longitudinal phase factor for Hermite Gaussian modes on Free Spectral Range and Allowed Mode Frequencies?
DxuF/p
The transverse intensity modulation is accompanied by an axial, or longitudinal phase factor that modifies the shift of longitudinal frequencies from the an integer multiple of the FSR by an integer multiple of (the typically much smaller)
\delta \eta \nu_{FSR} \over \pi

4. Real cavities
What is the lifetime of all these modes we have solved for so far?
Answer: Infinite…as per the definition of a mode (recall?)
What has been left out of all in all of this generic mode stability and phase criteria so far, when comparing to real physical systems?
Guide them through deriving photon lifetime, FWHM, Imax/Imin, Imax/Iinc as per notes
Answer:
reflectivity of at least one of the mirrors must be <1 to allow one to use the modes (intrinsic, unitary)
Material imperfections (scattering losses, absorption etc.) (extrinsic, non-unitary)

5. Intrinsic lifetimes
What system have we already studied that must reveal the impact of non-unity reflectivity on cavity mode behaviour?
Want S matrix for entire system a b c d 1 2 3
Recall algorithm
Get them to derive t13 in terms of r
Sab=Sba=Scd=Sdc assuming the mirror is symmetric

6. Symmetric, lossless system
Only one r and one t, |r|2+|t|2=1, t/r=-(t/r)*, arg{t}-arg{r}=+/-p/2
More generally
Hint for exam…you should fully understand/be able to derive these!!
What is |t_{net}|^2?

7. Fabry-Perot Cavity Equations
How many transverse modes represented? Just one.
What is T_max for the symmetric case? Unity.
Typo is in Fig caption….not omega, rather \nu

8. Physical interpretation(s) of Finesse
What are the two most intuitive/important physical interpretations of the Finesse of a quasi-mode?
Cavity lifetime
Internal intensity enhancement
Make this pre-reading assignment for Friday
Calculate the cavity lifetime
Calculate the internal field enhancement factor

9. Extrinsic sources of loss
Suggestions?
Finite mirror size
Shape of reflected beam? Gaussian?