1. PHYS 408 Applied Optics (Lecture 11)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
M matricies can be used to propagate either left to right or right to left (input to output or output to input).
The output to input formulation is both a bit more numerically friendly, and much more useful if you want to explore the field distribution inside the dielectric stack.
High reflectivity mirrors surrounding a “defect” region (uniform region of arbitrary thickness) can “trap” light of certain wavelengths, which corresponds to light bouncing back and forth many times between the mirrors before “leaking out”.
This is perhaps the simplest form of an optical cavity.

3. Cavities and Gaussian Beams
In this module, we are going to consider non-planar waves and how they interact with optical elements.
Will start with standing waves made up of plane waves, then quickly transition to an important class of non-planar waves that find many applications: Gaussian Beams.

4. Recall … … n1 d1 n2 d2 n1 d1 n3 d3 n1 d1 n2 d2 n1 d1 n2 d2 nlayers-1

5. Cavity Modes n1=2; n2=sqrt(12); n3=4 d1=300 nm; d2=173 nm; d3=10*d2
10 periods SYMMETERIZED

6. Cavity Modes n1=2; n2=sqrt(12); n3=4 d1=300 nm; d2=173 nm; d3=20*d2
10 periods SYMMETERIZED

7. What is a “cavity mode”? First ask what is a “photonic mode”?
- Stationary solution of Maxwell Equations in a lossless medium for non-zero w
What is a “stationary solution”?
Yes, plane waves in uniform media
- One that is harmonic in time…i.e. proportional to exp-iwt, where w is real
Have we already encountered examples?

8. In a region where the dielectric landscape is non-trivial, how does this dielectric landscape influence the “photonic modes”?
Surrounding Regions perfect conductors:
Boundary conditions?
Form of allowed fields inside?
Allowed solutions?
vacuum z d

9. Planar cavity modes: standing waves
Equations to satisfy boundary conditions?
Why not m=0?
Only exist for discrete values of k!!

10. Observations? Consistent with results shown earlier?
n1=2; n2=sqrt(12); n3=4
d1=300 nm; d2=173 nm; d3=10*d2
10 periods SYMMETERIZED
Consistent with results shown earlier?
Amplitude unrestricted…make sense?
Why aren’t m=1, 2 etc. showing up? (2 ways to check, number of nodes, and integer number of 1/nd
Because mirror R not large everywhere.
What m’s doe these 3 modes correspond to?
Subtle difference in mode spacing has to do with the mirror having a finite thickness, and well-defined texture.
Discrete frequencies means? Either very quiet, or very loud inside cavity, depending on the frequency you are sensitive to. Essentially sweeping up many vacuum modes, and concentrating them all into disrete frequency “cavity modes”.
Intuitive way to understand why fields can’t “exist” away from resonant contition: if imagine launching a wave in that region, each round trip would have phase slightly off, so with r=1, would always average to zero except right at resonance. If r<1, then can avoid going exactly to zero near resonance since only add up a finite number before leaking out. Note this round trip constructive interference condition exactly the same as from the perfect BCs.
Plane wave amplitude unrestricted….in general, but what about…
Quantum optics/photons??
What does it mean to have a single photon in a mode? Impact on E_{max}^{single photon}?

11. Generalize/application
3 dimensional cavities
Sketch gain profile of laser medium and discuss “pure” single mode operation

12. Stability/sensitivity issues
If you can’t ensure both mirrors are perfectly parallel, modes mess up very quickly (eg. HeNe laser lab)
Lateral size of the mode not so easy to control using macroscopic mirrors/gain media.
Any “stationary solutions” in planar case?

13. Does this remind you of anything we did earlier in the term?q

14. Gaussian Beams
Can you guess why these Gaussian beams may be relevant to the modes that can be supported by two curved mirrors?

15. Next lectures: Geometric ABCD matrix and Gaussian Beam propagation
There is something quite profound/pathological about Gaussian beams, so they deserve some serious consideration.

16. Magic!
It turns out that the propagation properties of Gaussian beams that satisfy the “paraxial approximation” (essentially the slowly varying envelope approximation), through collections of optical elements (thin films, prisms, lenses, mirrors etc.), can be derived using simple equations that describe classical ray optics (which totally ignore the wavelike properties of the light). The diffraction properties are “magically” taken account of.

17. … from Steck
It seems amazing that a solution to the wave equation can be propagated using classical rules. But really
what this is saying is that, in some sense, wave phenomena are absent in the paraxial approximation.
As we will see, this transformation rule applies in more general situations as long as the paraxial
approximation holds. But let痴 focus on the Gaussian beam for concreteness. The Gaussian beam stays
Gaussian through any optical system as long as the paraxial approximation is valid. As soon as the paraxial
approximation breaks down (i.e., nonlinear terms become important), the beam will become non-Gaussian. …
beam optics is really just ray optics; Gaussian beams do not show any manifestly wave-like behavior, at least
within the paraxial approximation. All the diffraction-type effects can be mimicked by an appropriate ray
ensemble. …

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