PHYS 408 Applied Optics (Lecture 8) Jan-April 2016 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture One can fairly easily keep track of all multiply reflected wavefronts in a thin-film transmission problem involving incident plane wave, including both their amplitude and phase. Summing the infinite series of interfering waves yields an interesting, non-trivial wavelength dependence to the transmission and reflection properties of the film. The most natural quantities to describe the behaviour include the Fresnel reflection and transmission coefficients associated with both interfaces, and the thickness of the film.

A reminder 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.

The solution Show pdf of handwritten algebra

The solution Condition (equation) for zero reflection? 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 Condition (equation) for zero reflection?

Vary e for fixed d=600 nm d=600 nm e=13 d=600 nm e=4 d=600 nm e=2.4

Moving forward Cast our result in the form of a matrix equation that yields the reflected and transmitted wave amplitudes (the out-going wave amplitudes) when the incident wave amplitude from the left hand medium is specified.

Moving forward Cast our result in the form of a matrix equation that yields the reflected and transmitted wave amplitudes (the out-going wave amplitudes) when the incident wave amplitude from the left hand medium is specified.

Moving forward If you had an incident wave from the right hand side, how might you express the corresponding matrix representation of the out-going field amplitudes?

Moving forward Can you combine these to generate a matrix equation that yields the outgoing waves for the general case when you have two in-coming waves, one from each side?

The S and M matricies Would this S matrix help you easily solve for the overall reflected and transmitted fields if you had multiple dielectric layers up against each other? What would the ideal “transfer matrix” be that would allow you to get the overall transmission simply by multiplying matrices for each boundary?

The S and M matricies Can you obtain this desired matrix, call it the M matrix, from the S matrix?

The S and M matricies What is then the algorithm for obtaining the overall transmission and reflection amplitude for plane waves incident on an arbitrary number of dielectric films arranged in a planar stack?

Let’s play! … … n1 d1 n2 d2 n1 d1 n2 d2 n1 d1 n3 d3 n1 d1 n2 d2 n1 d1 nlayers-1 nlayers-1