1. PHYS 408 Applied Optics (Lecture 15)
Jan-April 2017 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, with a diameter just large enough to capture ~ 99% of the input power, 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 various fields)?
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.
In this example that we have already looked at, stability was not addressed (implicitly it is stable as only considered normal incidence and infinite lateral dimensions).

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
Upper left quadrant should be shaded; in d) R1=-d.
Note various particularly symmetric cavity types labelled.
What does this equation and analysis really translate to?
Ans: for a given R1 and R2, there may or may not be a continuous range or discrete ranges of d’s for which you can use them to set up a stable cavity.
Ask: does the fact that the signs of g1 and g2 need be the same mean that the signs of R1 and R2 need be the same?
Note, needing the signs of g1 and g2 to be the same doesn’t mean that the signs of R1 and R2 need be the same!

8. Can do same with Gaussian beam
First prove that the following geometry is self-consistent.
Ask them to figure out an algorithm at least. What happens if send a general Gaussian rightward, onto the right hand mirror?
Ans: it will reflect as some “other shaped” Gaussian. So question really is, what condition must be met to have it reflect as the same shaped Gaussian, just going in the backward direction?
See hand written notes…keep track of direction of z1 and z2!
How?

9. What about stability though?
Show this stability criterion using Gaussian beams.
How could you approach this problem?
Or another way of looking at it is, for a given R1 and R2, can you always find a physical Gaussian to match those radii for an arbitrary d?
First do qualitative analysis:
Go back to look at previous slide and consider what happens if move the mirrors further apart:
Ans: have to decrease radius of curvature at larger z, so looking at R(z) equation, can do so to some extent by reducing z_0, but limited (since proportional to (1+(z_0/z)^2), so eventually won’t be able to reduce the curvature enough.
Mathematical analysis/answer on following page, based on using ABCD matrix elements to evaluate z0 as a function of d, R1 and R2. z0 has to be real (since if imaginary, w(z) becomes imaginary at large enough z), and something suspicious happens when z_0 approaches zero (w_0 imaginary for z_0<0).

10. Be systematic!
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

11. Stable? Have we completely determined the Gaussian parameters?
Need to know lambda to know everything about the Gaussian, though this doesn’t impact the stability criterion.
Stability criterion?

12. Stability?
Planar cavity analogy: Just because you know the field can be expanded as plane waves doesn’t mean they satisfy the boundary conditions, and hence satisfy the mode conditions (which, recall, is equivalent to satisfying the round-trip constructive interference condition).
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?