PHYS 408 Applied Optics (Lecture 6) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture Point-like dipoles have a very characteristic radiation pattern that can be derived using the vector potential A(r). The radiation is fundamentally a spherical wave, with angular modulation of the E and H vectors’ amplitudes. There is no radiation along the axis of the dipole, and most of the power is radiated with a polarizaiton aligned with the dipole, in the equatorial plane.

New Module: Light encounters with Optical Elements What happens when one of our plane waves strikes a flat interface between two media with different dielectric constants? What is the boundary condition for field components parallel to surface? qin qr n=n1 kin kr kt z x Write down the relevant equations, starting with the general expressions for the 3 waves What happens if no incident light? Go through what happens when there is incident light (drives polarization in proportion to field, polarization generates fields, total field is the sum, so self-consistent problem best suited for an integral formulation of ME). In practice, use symmetry and boundary conditions to simplify solutions in special cases. In this case, if E in y direction (s or TE polarization), from symmetry (discuss) total field must everywhere be along y, and the z component of the wavevector must be shared by all. See from phase of polarization, also from having to match BCs at all z points, also refer to group theory and translational invariance/momentum conservation. n=n2 qt

Set up the problem Why, given Einc, can we assume this form? How do we use the boundary conditions? Only in y direction since all induced polarization in y, and from symmetry and our knowledge of the fields radiated by dipoles, all E fields generated by this P must be along y. Not obvious necessary, what the various angles of reflection and transmission are…

Use continuity of E field across boundary Specify the k’s… Get them to get the k’s

Conclusion “Phase matching” the 3 wavefronts. Key concept, can’t have any z dependence on right hand side. So have “derived” the Snell’s law of reflection and refraction “Phase matching” the 3 wavefronts.

New Module: Light encounters with Optical Elements n=n1 z x n=n2

Fresnel equations (you will derive rs) Where k// is the conserved in-plane wavevector This formulation is more general; can always convert to cos and sec if you want, but in general these are complex in nano-optical/near field situations What does this mean for Show that n1rs2 cos(q1) + n2 ts2 cos(q2)=1

Thin film optics z x What do you notice about the transmitted wave? What do you think the amplitude of the transmitted wave would be if the incident wave had amplitude 1? They will hopefully suggest t12 x t21 , work them through n1=1, n2=n, then n1=n, n2=1 Then get them to include the phase factor from slide Start by asking them what the general form for the field inside the film is 1 2 d

What about power conservation now? Reflected wave? If power was conserved at the single interface, it wouldn’t be here. Wherein lies the problem?

Multiple reflections In reality, there are an infinite number of reflections and transmissions of a harmonic beam that encounters a series of parallel dielectric discontinuities Sketch the reflected fields, with addition of multiple phase-shifted cosines and note what the phase shift is, and that it is the same for each successive reflection May look daunting, but in your assignment, will see it is actually fairly easy to keep track of all the terms and do the sum analytically

Two quite different approaches to solving problem Can evaluate the series directly In class we will solve it in a different way Discuss the general approach here (expansion of field in homogeneous solutions in each space) Set up the approach, but likely won’t solve in this lecture. Get them to think though how many unknowns Where do they get the 4 required equations? Combination of them deriving/solving them for normal incidence, and showing the answer