PHYS 408 Applied Optics (Lecture 7) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture When plane waves interact with a planar interface, the in-plane component of the k vector (momentum) is preserved since there are no terms in the equation of motion that depend on the in-plane coordinates. This requirement for all waves in the problem (incident, reflected, and transmitted) to share the same in-plane wavevector can be thought of as “phase matching”. It can also be understood if you think of the polarization density excited by the incident wave, and how that has to act as the source for both the reflected and transmitted waves. Transmitted light is refracted (bent) due to the difference in wave speed, and hence wavelength in the material for a given frequency field. Multiple reflections are crucial to include when solving for the full reflected and transmitted field amplitudes, and the phase of the field at each pass through the film is crucial to keep track of.

Multiple reflections: superposition of many waves with well-defined relative phase and amplitudes z x Spent time on this, get them to try to roughly get the amplitudes right for a specific r and t, similar to matlab sims Ask groups to sketch results on board

First steps… Adding various harmonic waves with the same frequency but varying phases always results in another harmonic wave at the same frequency. Notice what happens when you add separate harmonic waves of the same frequency, but different relative phases… Then, step through the multiple reflections step by step, starting with the incident wave. 1st iteration, n=2.5 Vacuum

Simulation Sketch the transmitted and reflected waves at the first interface 1st iteration, n=2.5 Vacuum Qualitatively the transmitted wave will be t1-2 at the interface, then carrying on the same phase evolution (but with shorter wavelength) as incident. Reflected wave is negative r1-2 at the interface, and then also continuing on with same phase as incident, as though it kept going in the forward direction. Note r negative and t positive and <1 (2/1+n) and 1-n/1+n

Is there something wrong at the right? How can it increase?

What is special about this case? Zero-reflection! (When might that be a useful thing to have?) What is special about this situation shown? (Answer: returning waves at the first interface always transmit in phase, positive with respect to the negative reflected wave from the incident beam)

How to solve exactly? Infinite series (homework, very intuitive) More powerful approach Remind them of what they did last week at this stage for the single interface problem Write down expressions for the total E field in each region.

Equations Wait for them to get this Four unknowns: what are the four equations you need to solve for the unknowns?

The solution (see Lecture 7.pdf) Show pdf of handwritten algebra

The solution Condition (equation) for zero reflection? cm-1 cm-1 d=600 nm d=600 nm r and t from notes? No, |r|^2 and |t|^2What do you note about these results? 2 pi/lambdo_0 x n x 2d = m (integer) 2 pi, so 1/lambda_0=m/(2 n d), m= 1, 2 Because first reflection has a negative sign, and all remaining ones have positive phase at interfaces cm-1 cm-1 Condition (equation) for zero reflection?

Vary e for fixed d=600 nm cm-1 cm-1 cm-1 d=600 nm e=13 d=600 nm e=4 Describe/interpret results: Series of wavelenths where r=0 (make sense based on our previous explanation?) Always goes to zero regardless of epsilon Max reflectivity decreases as epsilon decreases cm-1 cm-1 cm-1