PHYS 408 Applied Optics (Lecture 20) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture The key to diffraction is realizing that if you do a 2D Fourier Transform of the field distribution in some x-y plane, and assume that you can relate that to a forward, z-propagating overall field, then each of the 2D Fourier components, maps uniquely onto a 3D planewave propagating normal to the plane. The wavevector in the propagation direction of each of these 3D planewaves is given by , and the relative amplitudes of each of these 3D planewaves is given directly by the 2D Fourier amplitude of the corresponding in-plane component of the field in the aperture. In the paraxial limit (meaning that the sin or tan of the angle on some screen far from the aperture where the intensity of the diffracted field is significant, can be approximated as just the angle), each of these 3D plane waves, propagating at an angle Empasized that these are the essential principles of diffraction, and related equations won’t be provided on formula sheets. This fundamental concept must be thoroughly understood.

Today: Explaining the “magic” behind Gaussian beams Contrast with slit diffraction: “Shape” of pattern on screen changes dramatically from just behind screen to the far field for the slit, but for the Gaussian, it remains exactly the same, just scales: why? A hint must come from the far field limit: in the slit case, and in general, the far field pattern is related to the FT of the field in the aperture. For a Gaussian, the FT of a Gaussian is a Gaussian….this explains ~ 80% of the “magic”…but what about in between (the Fresnel zone?)…let’s see.

Gaussian beams and Fourier Optics What is the 2D Fourier Transform of our (0,0) Gaussian beam at the beam waist? Write the expression for the diffracted field for z>0 based on a 3D planewave decomposition at the beam waist. See accompanying handwritten notes. Bottom line, is that the far field diffraction pattern results in an analytic expression for our Gaussian (in the “q parameterization” forumulation), that we know applies for all z, so long as the Gaussian satisfies the paraxial approximation. So why does this far field analysis work even for small z? Answer: the Gaussian mode was derived from the paraxial Helmholtz Eqn, which requires that all plane waves that make up the solution satisfy the paraxial approximation, everywhere. So the paraxial approximation we make in the far field analysis in the attached notes, is actually valid, for Gaussian beams that satisfy the paraxial approximation, at all z. Getting back to the comparison with the slit case, here the reason why the Gaussian is so “simple”, and satisifies the parxial approximation everywhere, is because it is everywhere, even at the beam waist, “smoothly varying”, which limits the number of 2D Fourier components that need to be used to describe it. Using the paraxial approximation show that this diffracted field is exactly our generalized Gaussian beam!

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