PHYS 408 Applied Optics (Lecture 21) Jan-April 2016 Edition Jeff Young AMPEL Rm 113

Quiz #11 The two dimensional Fourier amplitude of a forward-propagating harmonic electric field distribution in an x-y plane directly gives one amplitude of a propagating 3D plane wave with an in-plane wavevector corresponding to the 2D Fourier component, and a kz value given by : T/F In the paraxial limit, the z dependent phase of each 3D plane wave has a quadratic dependence on kx and ky, and this is fundamentally why a Gaussian field distribution propagates as a Gaussian field distribution in the paraxial limit: T/F Adding up all of the propagating plane waves with the appropriate amplitude weighting function effectively performs an inverse 2D Fourier transform of the original field distribution’s Fourier transform, with additional, z-dependent phase factor: T/F The polarization of an electromagnetic wave at a fixed location is aligned with the electric field vector at that location: T/F

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, each of these 3D plane waves, propagates at an angle

Today: How a thin lens renders the Fourier transform of the field at an input plane at its focal plane How diffraction limits the resolution one can achieve with light beams A half-hour crash course on polarization!

Effect of a lens

Physics approach What does a lens do with a plane wave incident at some (small) angle q? What fields are emanating from the f(x,y) plane? So within a phase factor, the field at some location in the focal plane of the lens is proportional to what? Sketch plane wave and lens on board. Use 2 rays from ray optics to sort of figure it out…what principle would you use to prove all rays end up there? So, regardless of a phase factor, what is the intensity at each point in the focal plane of the lens proportional to?

Rigorous approach Show that a lens focusses a plane wave incident at some (small) angle q, to a point at fq from the optic axis on the screen? Get them to say in words what each equation represents. Pause after 2nd equation and get them to formulate the propagation integral, then show handwritten notes (page 3 of Lens Fourier Transform II.jnt) l- l+ s Hint: propagate a phase front at the input to the lens of to the focal plane, knowing the transmission function of the thin lens is

Next, keep track of the relative phases of the plane waves that have propagated from input plane a distance d1 to the thin lens Show pages 4 and 5 of .jnt file Since we know how propagates through lens and to screen, just have to substitute so as to pick up the correct weighting for each Fourier Component.

How does diffraction influence resolution? Think back to our thought-experiment: If make ideal image out of only the propagating plane waves (excluding evanescent waves), what is the wavelength of the highest frequency spatial frequency component (wavevector)? So even if you used/collected every propagating plane wave, best possible resolution would be ~l/2.

How does diffraction influence resolution? Think back to our thought-experiment: What is the impact of using a lens of diameter D, a distance d away from the screen? The maximum value of k// is then . So then sharpest edge will be then roughly

Repeat slightly more rigorously for circular distribution of plane waves focussed from a collimated beam through lens diameter D

Polarization Should really spend 3 lectures on this topic, but prefer to use last 3 lectures to do a mini design project. Proper treatment can be developed using Stokes vectors or Jones vectors, or the Poincare sphere, or …

Basics For a plane wave propagating in the positive z direction: Jones vector

Special plane wave polarizations Linear: Ax and Ay =? Circular: Ax and Ay =? Ax Ay t Before showing 2 and 3, ask what defines a circularly polarized wave? Right or Left as show in blue?

Generalize Simple way to generalize is to use plane wave expansion, and let each individual plane wave component have some specified plane wave polarization Jones vector. Example: Gaussian (0,0) mode with each kx=0 plane wave component polarized in the x direction, and each ky=0 component polarized in the x-z plane: