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Homework 4

EEC 243

Winter 2023

Due 11:59pm, Saturday, February 11 in Canvasc

This  homework  requires  Matlab  programming.  Please  submit  your  matlab  codes  and  other associated files together with the homework.  Please provide enough comments in your matlab files and name your files in the following format:

HW4_ YourName.m

1. Computer-generated hologram and Gerchberg-Saxton algorithm [50 pt]

We want to generate a hologram through Gerchberg-Saxton algorithm. Specifically, we have a uniform coherent light source illuminating a 2f system. We want to design a phase plate so that we can project a desired intensity pattern in the projection plane. The phase plate does not modulate the light amplitude, but only its spatial phase. The phase profile on this phase plate is called hologram.

Please set the grid at the phase plate and projection plane to be both 128x128, and choose any desired/target intensity pattern at the projection plane (please take a photo using your cell phone, convert it to gray scale and 128x128 size) to calculate the hologram in the phase plate.

In your solution, please show:

(1) The desired/target intensity pattern.

(2) Calculated hologram (i.e. the phase profile on the phase plate).

(3) Derived intensity pattern using the calculated hologram.

(4) Mean square error between the target intensity pattern and derived intensity pattern versus iteration number. Do you see the result converges with large enough iteration?

Note 1: You may want to first convert the desired intensity pattern into a desired amplitude pattern, and use that in the Gerchberg-Saxton algorithm.

Note 2: You may refer to the following Wikipedia page.

https://en.wikipedia.org/wiki/Gerchberg%E2%80%93Saxton_algorithm

Note 3: Please submit your target intensity pattern together with the homework and matlab code.

2. 3D Optical transfer function [50pt]

For a circular aperture, the 3D optical transfer function is expressed as

0TF() = √1 − ( + )2

Note that the expression is only valid for transverse spatial frequencies bounded by x⊥ ≤ Δx⊥ and for axial spatial frequencies bounded by |κz | ≤ (Δx⊥  − x⊥ ). Outside this boundary, OTF is 0.

Also note that

Δκ⊥ = 2K = 2入(N)0(A)0

where NA0 is the NA defined at the object side, and 入0 is the vaccum wavelength.

On a 2D contour plot, please plot 0TF(K⊥ , Kz). Please use color to indicate the value of the OTF. Plot this for NA0=0.25 and NA0=0.5, with 入0 = 500 nm, and refractive index n=1. What do you

see the difference, and what does it imply on the xyz extend of the corresponding 3D PSF.

Based on this plot, explain why we could not resolve the depth of a uniform self-illuminated plane.