1. PHYS 408 Applied Optics (Lecture 16)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
The stability criterion for curved-mirror cavities is less stringent than the criterion for the cavity to actually support a resonant mode(s).
Either ray optics or Gaussian beam optics, both involving the ABCD matrix elements, can be used to derive the cavity stability criterion.
Simple ABCD q conversion can be used to easily show that a curved mirror with a radius of curvature matching a Gaussian mode’s curvature at some location, will reflect that Gaussian into a counter-propagating version of exactly the same Gaussian.

3. A Gaussian may or may not be “supported” or “conserved” by two mirrors of radii R1 and R2 separated by a distance d.
As drawn, for this Gaussian, is d unique? Ans: No, could put either mirror on either side of the respective z0 positions.
More generally, given two mirrors with R1 and R2, and some separation d, one may or may not be able to find a Gaussian that matches the curvatures. Review last day’s discussion, and watch for Homework assignment.

4. Review up to here:
Given R1 and R2, can solve for range of d’s where there can be a stable cavity using the stability criterion
For a given {d, R1,R2} can figure out the Gaussian(s) for which the radius of curvatures a distance d apart are equal to R1 and R2 (see homework)
one approach is to use q=(Aq+B)/Cq+D) where ABCD correspond to a “round trip” ABCD matrix for the cavity. Solve for q at some z position (say one of the two mirror positions) in terms of the round trip ABCD matrix elements
another approach is to use fact that know R(z1m)=R1 and R(z2m)=-R2, and these two values can be used to solve for q(z) (equivalently z0)
Next?

5. Additional criterion for actual modes
What is the additional criterion?
What is x(z) for our Gaussian beam?
Discuss physical interpretation: either plane wave or Gaussian, can bounce back and forth keeping Gaussian form, but phase will in general randomize at all points, so net field will be zero: equivalent to not satisfying boundary conditions.

6. Mode criterion What is q and where does it come from?
Use these to derive the condition for the existence of a true mode. (Twice Eqn must be an integer multiple of 2 pi.
What is q, and how is it arrived at? It is the integer of free spectral ranges that the mode frequency corresponds to (after accounting for the \Delta \zeta offset.
What is q and where does it come from?
Do any of these terms depend on the curvature of the mirrors?

7. Free spectral range and allowed mode frequencies
What is q for the HeNe laser experiment? ~ 2d/lambda, ~ 60 cm/0.6 microns ~ one million

8. Yet again!
Fabry-Perot (plane mirrors)
What is q for these modes? ~ 7

9. Modified Gaussian Modes (Hermite-Gaussian)

10. Hermite-Gaussian beams (con’t)
How would you solve for the functions X and Y?
Ans: substitute this function into the paraxial Helmholtz equation and get 3 separate differential equations for X, Y and Z functions. X and Y solutions independent, but Z solution depends on X and Y solutions.

11. Finally Where is the Z function in this U? Ans: (l+m)\zeta(z))
Note the DE for X and Y yield H functions, but when multiplied by the regular Gaussian amplitude factor, these convert to G functions and you lose the regular Gaussian explicitly (it is included in the Gs)

12. Intensity distributions

13. Impact on Free Spectral Range and Allowed Mode Frequencies?
DxuF/p
Recognize this if you have done the cavity lab?
DxuF/p

14. 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?
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)