1. PHYS 408 Applied Optics (Lecture 15)
Jan-April 2016 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
Using ABCD matrix elements to propagate Gaussian beams through optical systems is formally straight forward, but you have to be very aware of coordinate systems and conventions to get the right results.
One can transform either q(z) or 1/q(z) using the ABCD matrix elements in slightly different ways. Both can provide useful insight and/or efficient ways of understanding the effect of the optical element on a Gaussian beam.
A double-convex thin lens at the focus of one Gaussian input beam generates a second output Gaussian beam with a beam waist located not quite at the focal length away from the lens, and the beam waist is approximately equal to l f#, where the f# is the focal length f divided by the diameter of the lens.

3. Resonators/Cavities
What is a resonator? Examples? From earlier in the course?
Fabry-Perot (plane mirrors)
Preface this change of topics with the fact we now have a bunch of tools that can be used to analyze resonators/cavities in more generality.

4. Generalize General considerations? Stability?
Since paraxial, lateral size of mirrors limited, so for plane mirrors, any angular mismatch would be fatal, for example
General considerations?
Stability?
Stability necessary, but is it sufficient? What else?
Resonance
Wavefunction/Eigenmode shape?

5. Stability Criterion
How can we formulate a stability analysis using tools we have acquired?
Like a differential equation, but called a difference equation.

6. Stability con’t

7. Stability con’t

8. Can do same with Gaussian beam
First prove that the following geometry is self-consistent.
See hand written notes…keep track of direction of z1 and z2!
How?

9. Actually stable?
R(z1) is the Gaussian convention and negative since assume Gaussian positive z direction is to the right. R1 uses ray optics convention and is negative also for a concave mirror, so they are both negative.
R(z2) is positive, but R2 is still negative, hence the negative sign in 7.64

10. Stable? Have we completely determined the Gaussian parameters?
Stability criterion?

11. Stability?
Just because you know the field can be expanded as plane waves doesn’t mean they satisfy the boundary conditions.
Does satisfying this stability criterion mean that any Gaussian with the “right parameters” would actually be a solution of the Maxwell Equations, for any l?
Think of limit of plane mirrors?