0fc9707f-5384-49ce-863d-e53228ef428a 50 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

You need that one for scaling grayscale image M lo hi Scale hi lo - (M M Min - M Max M Min - 106 -^+ / (* lo + :

Fourier Quantum demo

]]>

Part 1: Mask pattern square centered aperture

t0 1 time :

A 10 : (matrix mask ≡ R 1 : (radius center ≡ d 2 : (holes side ≡ size 50 : (canvas size ≡ 41 line

Fourier Quantum

Adapted from Mathcad 11, mathengjmg [web site].

Written permission from Frank Rioux.

http://www.users.csbsju.edu/~frioux/workinprogress.html

M is the mask pattern in preparation for diffraction.

]]>

Aperture position i 0 A range : x R 2 π * i * sin * vectorize : y R 2 π * i * cos * vectorize : 31 line

Coordinates 
space wave function k 0 size range : xx A R * - 2 A * R * k * size / + vectorize : yy A R * - 2 A * R * k * size / + vectorize : 31 line

Position wave
function Δ A : N 50 : i 0 N range : px Δ - 2 Δ * i * N / + : py Δ - 2 Δ * i * N / + : 51 line

The 'mask pattern of the space wave function xx yy Ψ 1 A sqrt / 1 d / xx x m 1 + el d 2 / - (≥ (xx x m 1 + el d 2 / + ≤ (* yy y m 1 + el d 2 / - (≥ (* yy y m 1 + el d 2 / + ≤ (* 0 cases m 1 A sum 11 line * :

Display mask i 1 size range j 1 size range M i j el xx i el yy j el Ψ abs (2^eval : for for

M M 0255 Scale :

iVBORw0KGgoAAAANSUhEUgAAADIAAAAyCAIAAACRXR/mAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAA7SURBVFhH7c6xCQAxEAPB679o84mCw1+AMMykm+wAAADwvvOT0JWXJaErL0tCV16WhK68LAkAAADcZj63BVlNV8tPbAAAAABJRU5ErkJggg==

Mask pattern

1 time t0 - min 0.5

Part 2: Fourier transform integral

Fourier integral transform position wave function into momentum representation.
This system is "scalar" at the algo kernel level ... Smath does not recognize ?
... or: I don't know how to make it "scalar interpretable" ! px py Ψ 12 π * d * A sqrt * / i px x * (* - exp x x m el d 2 / - x m el d 2 / + int i py y * (* - exp y y m el d 2 / - y m el d 2 / + int * (m 1 A sum 11 line * :

510 Ψ #

x y px py d i Clear 1

Fourier symbolic integral transform i px * x * - exp x a b int i py * y * - exp y c d int * simplify maple i b * px * - exp - i a * px * - exp + (i d * py * - exp i c * py * - exp - (* px py * /

Restore system Δ A : N 51 : (odd ≡ i 0 N range : px Δ - 2 Δ * i * N / + : py Δ - 2 Δ * i * N / + : 51 line

Restore system i 0 A range : x R 2 π * i * sin * vectorize : y R 2 π * i * cos * vectorize : d 2 : 41 line

a x m el d 2 / - (≡ b x m el d 2 / + (≡ c y m el d 2 / - (≡ d y m el d 2 / + (≡ 14 mat

i Clear 1

Optimiz => SYMBOLIC ... a bit faster px py Ψ 12 π * d * A sqrt * / i x m el d 2 / + (* px * - exp i x m el d 2 / - (* px * - exp - (i y m el d 2 / + (* py * - exp - i y m el d 2 / - (* py * - exp + (* px py * 1012 -^+ / m 1 A sum 11 line * :

Sanity check 5050 Ψ abs 2^55 Ψ abs (2^11 Ψ abs 2^31 mat 7.683 1010 -^* 0.001 0.508 31 mat

px py * 1801.3072

Memo Vector Algebra:

The dot product of two vectors is the vector sum of their product.

Each NUM component is a vector, their dot product is a number

... as well as DENOM [QED].

]]>

t0 1 time :

Optimiz => Numeric P m 1 N range n 1 N range Fo m n el px m el py n el Ψ abs (2^eval : for for Fo 21 line :

P P : spot P 0255 Scale : 21 line

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 spot - 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 spot

1 time t0 - min 2.4

m 2 :

i Clear 1

BUG => does not vectorize complex argument.

Most interesting & puzzling: it does not affect the

Fourier momentum transform integral !!!

]]>

V i x m el d 2 / - (* px * - :

i x m el d 2 / - (* px * - exp vectorize #

V exp vectorize 0.839 - 0.544 i * + 0.983 - 0.182 i * + 0.978 - 0.208 i * - 0.825 - 0.566 i * - 0.546 - 0.838 i * - 51 mat

V sin vectorize 1.101 1010 -^* 11013.233 i * - 7.441 1011 -^* 7440.52 i * - 4.524 1011 -^* 5026.801 i * - 31 mat

i x m el d 2 / - (* px * - sin vectorize #