PHYS 408 Applied Optics (Lecture 19) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture The 2D Fourier transform of the Electric field distribution in some input plane, which is a function of in- plane spatial frequencies kx and ky, 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 Dx d Propagation through a slit aperture, as per last day. Clearly a relationship between the Fourier transform of the aperture function and the intensity distribution in the far field, near the optical axis.

Analyze kx Eslit x Dx/2 -Dx/2 ? Review from last day

So have: Dx d x Eslit x Dx/2 -Dx/2 kx ? Review from last day

Describe in words what is happening here: assume infinitely thin, perfectly reflecting mask Start with “A plane wave is incident on a perfectly conducting, infinitely think mask with a slit in it….” …end up with: At exit surface of the mask, z=0, the electric field is harmonically oscillating at frequency omega with an approximately constant phase front. At z=0 you can then expand, using 2D Fourier transforms, the constant, but localized field, as a well defined superposition of 2D plane waves with each k// having a very particular amplitude. Knowing that the 3D field can be expanded as a summation of 3D plane waves, then each of the 2D Fourier components must in fact be “attached to”, or be “part of” a 3D plane wave with wavevector kx x +ky y + kz z, where kz=+/-sqrt(omega^2/c^2-kx^2-ky^2). The diagram shows the forward propagating set of these 3D plane waves travelling towards the screen. X

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) X y What function describes ? Sketch inverse tophat function

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? The incident plane wave induces, to zeroth order, a 2D sheet of oscillating (at frequency omega) polarization density that radiates. We can “easily” understand the radiation produced by this polarization sheet by Fourier transforming it and understanding how each harmonic sheet of polarization radiates plane waves at a particular angle. (see next slide)

What E fields are generated by a 2D sheet of spatially and temporally harmonic polarization density? Pitch L=2p/k// w fixed by frequency of driving field, and k// fixed by phase matching in Maxwell Equations. k=w/c k=w/c k// k// What is kz(k//,w)? For each k//, knowing omega, and hence k, can figure out the angle of the plane wave that the particular sheet of polarization radiates. The amplitude of that radiated plane wave is proportional to the Fourier amplitude of P(k//). Note that there will be non-zero P(k//) amplitudes for k// values larger than omega/c, in which case kz is imaginary, so the plane wave does not propagate away from the plane, it “evanescently” decays away from it. These evanescent fields are very real, they just can’t bee detected further than a wavelength or so from the sheet.

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? Something missing compared to generating a slit pattern of light…what? Can’t reproduce the evanescent components of the field from far away, so this reversed propgating plane wave would only generate a partial inverse Fourier transform, missing high k// components, so the sharp edges of the slit would not be reproduced. The reversed propagating plane waves would make an image in the plane of the slit that is like the slit, but with rounded, or smoothed edges. The smoothing range is at best on the order of a wavelength.

Now to make the rigorous mathematical connection What is kz? Get them to see that at some large z value, somehow it must reduce again to a FT…what approximation might we make in that case? 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)

Limit of a large distance between aperture plane and screen, near optical axis on screen Rearrange to make look more like a potential FT… Get them to think about it, suggesting reverse engineering, knowing the answer. For what values of kx and ky will you get the biggest contributions?

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

Integrand and its average Essentially a FT of unity (or a pure phase function), i.e. a delta function kx

So have: Dx d Eslit x Dx/2 -Dx/2 x kx ?