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PHYS 408 Applied Optics (Lecture 11) JA N - A PRIL 2 0 1 7 E D I T IO N JE F F YOUN G AMPEL RM 113 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. 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. Recall n1 n2 d1 d2 … nlayers-1 n1 n3 n1 n2 n1 n2 d1 d3 d1 d2 d1 d2 … nlayers-1 n1=2; n2=sqrt(12); n3=4 d1=300 nm; d2=173 nm; d3=10*d2 10 periods SYMMETERIZED Cavity Modes 1 0.9 0.8 Transmission 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3500 100 200 90 180 80 160 70 140 60 120 4000 4500 Wavenumber 1/ 5000 35 20 2 50 25 |E| 2 |E| |E| 2 30 100 40 80 30 60 20 40 10 20 15 10 5 0 -7 -6 -5 -4 -3 Z (cm) -2 -1 0 1 x 10 -4 0 -7 -6 -5 -4 -3 Z (cm) -2 -1 0 1 x 10 -4 0 -7 -6 -5 -4 -3 Z (cm) -2 -1 0 1 x 10 -4 n1=2; n2=sqrt(12); n3=4 d1=300 nm; d2=173 nm; d3=20*d2 10 periods SYMMETERIZED Cavity Modes 1 0.9 0.8 Transmission 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3400 3600 3800 4000 4200 4400 Wavenumber 1/ 4600 4800 5000 What is a “cavity mode”? First ask what is a “photonic mode”? - Stationary solution of Maxwell Equations in a lossless medium for nonzero w What is a “stationary solution”? - One that is harmonic in time…i.e. proportional to exp-iwt, where w is real Have we already encountered examples? In a region where the dielectric landscape is non-trivial, how does this dielectric landscape influence the “photonic modes”? vacuum Surrounding Regions perfect conductors: Boundary conditions? Form of allowed fields inside? Allowed solutions? 0 z d Planar cavity modes: standing waves ikz ikz E ( z ) ( E exp E exp ) y Equations to satisfy boundary conditions? E ( z 0) E ( z d ) 0 E ( z 0) 0 E E 0 E ( z d ) 0 (expikd exp ikd ) 0 kd m k m / d with m 1,2,.... Observations? Consistent with results shown earlier? Amplitude unrestricted…make sense? n1=2; n2=sqrt(12); n3=4 d1=300 nm; d2=173 nm; d3=10*d2 10 periods SYMMETERIZED Generalize/application 3 dimensional cavities 1 0.9 0.8 Transmission 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3400 3600 3800 4400 4200 4000 Wavenumber 1/ 4600 4800 5000 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? q 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. … … just ray optics; Gaussian beams do not show any manifestly wave-like behavior, at least beam optics is really within the paraxial approximation. All the diffraction-type effects can be mimicked by an appropriate ray ensemble. Please take 5 minutes to fill out the mid term survey to help improve the course for you https://www.surveymonkey.com/r/YKQNM6C