PHYS 408 Applied Optics (Lecture 19) JAN-APRIL 2016 EDITION JEFF YOUNG AMPEL RM 113

Quick review of key points from last lecture The internal field intensity varies by orders of magnitude when exciting a high Finesse cavity on or far off resonance. The 2D Fourier transform of the Electric field distribution in some input plane, which is a function of in- plane spatial frequencies k x and k y, is very closely related to the spatial distribution (variables x, and y) of the light intensity pattern on a screen placed a long distance away from (and parallel to) the input plane. The connection is made by recognizing what 3D plane wave ( ) is associated with each of the in-plane Fourier components ( ) of the input E field, noting that.

Fourier Optics: empirical approach DxDx d

Analyze kxkx ? E slit x 0D x /2 -D x /2

So have: E slit x 0D x /2 -D x /2 kxkx ? DxDx d x

Describe in words what is happening here: assume infinitely thin, perfectly reflecting mask

What things still need to be explained? A) How does a single plane wave incident on a mask generate a continuum of forward propagating 3D plane waves? B) How do we rigorously explain the one-to-one relationship between the 2D Fourier Transform of the field in the slit (argument of k // ), and the intensity distribution of the light on a screen placed “a long way” from the slit (argument of x)? [To realize why B) is non-trivial, think what the distribution of light would roughly look like on a screen very close to the slit, and as you gradually move the screen away from the slit]

Zeroth order analysis of A) What function describes ?

Zeroth order analysis of A) con’t To zeroth order, what is the polarization density generated by the single incident plane wave in the mask?

What E fields are generated by a 2D sheet of spatially and temporally harmonic polarization density? Pitch  /k // k // k=  /c  fixed by frequency of driving field, and k // fixed by phase matching in Maxwell Equations. What is k z  k  

So by deduction, what must be the net effect of these polarization-driven fields? Cancel the incident field in the forward z direction (on the other side of the mask) Generate diffracted, out-going plane waves, with weights governed by the Fourier components of the mask’s susceptibility (essentially determined by its geometry) Some of these out-going plane waves will be evanescent, and so not observed in the far-field.

What would happen if you reversed the direction of the diffracted plane waves, keeping their relative phases in tact?

Now to make the rigorous mathematical connection What is k z ? At z=0 what is this E(x,y)? At some arbitrary z, what is this E(x,y)? (think what happens on a screen as you move it from the slit to far away)

Large distance limit Rearrange to make look more like a potential FT… For what values of k x and k y will you get the biggest contributions?

Finally…the connection made! What is ? to contribute significantly to the integral, so effectively sample at

So have: DxDx d E slit x 0D x /2 -D x /2 kxkx ? x