PHYS 408 Applied Optics (Lecture 11) JAN-APRIL 2016 EDITION JEFF YOUNG AMPEL RM 113

Quiz #5 1)You can use M and S matricies either from input to output, or from output to input T/F 2)The input to output direction allows one to evaluate the internal field distribution in a fairly straight forward fashion T/F 3)When two high reflectivity dielectric stacks face each other, light between them is trapped forever at any wavelength T/F 4)There is a midterm exam in this course this coming Friday T/F

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.

Cavities and Gaussian Beams In this module, we are going to consider non-planar waves and how they interact with optical elements. An important class of non-planar waves that find many applications are Gaussian Beams.

Recall n1n1 d1d1 n1n1 d1d1 n2n2 d2d2 n1n1 d1d1 n2n2 d2d2 n3n3 d3d3 n1n1 d1d1 n2n2 d2d2 … … nlayers-1

Cavity Modes n 1 =2; n 2 =sqrt(12); n 3 =4 d 1 =300 nm; d 2 =173 nm; d 3 =10*d 2 10 periods SYMMETERIZED

Cavity Modes n 1 =2; n 2 =sqrt(12); n 3 =4 d 1 =300 nm; d 2 =173 nm; d 3 =20*d 2 10 periods SYMMETERIZED

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

Cavity defined by two perfectly reflecting planar mirrors Boundary conditions? Form of allowed fields inside? Allowed solutions? z 0 d vacuum

Planar cavity modes: standing waves Equations to satisfy boundary conditions?

Observations? Amplitude unrestricted…make sense? Consistent with results shown earlier?

Generalize/application 3 dimensional cavities

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.

Does this remind you of anything we did earlier in the term? 

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

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

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.

… 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 di ff raction-type e ff ects can be mimicked by an appropriate ray ensemble. … …

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