final_exam_solved.jl

### A Pluto.jl notebook ###
# v0.19.42

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ d1980bd2-babf-11ee-1dbb-dfefbdbdb36d
using PlutoUI, Plots, ImageShow, TestImages, FFTW, NDTools, IndexFunArrays, FileIO, FourierTools, SpecialFunctions, UrlDownload, ImageMagick

# ╔═╡ 38d09abb-9830-4902-b6e1-d863ec08149e
using JLD2

# ╔═╡ 2c34cea0-d419-4155-b4c2-44c4c7237e28
using LinearAlgebra

# ╔═╡ 17adc747-9822-42be-91c1-ad8a2e1532bc
md"# 0. Load packages"

# ╔═╡ d2c6102c-7a62-4899-9c5c-b08ce9a5baa8
FFTW.set_num_threads(4)

# ╔═╡ 8e6ec43c-9135-4829-8146-a56b8624750a
const TODO = nothing

# ╔═╡ 7331d6a5-dd03-42c7-9ab3-c84a641296bc
TableOfContents()

# ╔═╡ f326cd36-9c05-4b1c-81c0-56c954cd51f3
gauss_R(z::T, z_R) where T = iszero(z) ? T(Inf) : (1 + (z_R / z)^2)

# ╔═╡ 7c6b4ddd-ca5a-4b23-96f7-e896fb1cda6d
gauss_ψ(z, z_R) = atan(z, z_R)

# ╔═╡ dd16735b-c1d6-4977-a396-a3e23823ee68
gauss_w(z, z_R, w_0) = w_0 * sqrt(1 + (z / z_R)^2)

# ╔═╡ 8e04ae26-5474-4c2c-9fda-dbb7efd77acd
"""
	gauss_beam(y, x, z, λ, w_0)


Returns the eletrical field of a Gaussian beam at position `(y, x)` at optical axis position `z` with respect to the beam waist `w_0`.
Wavelength is `λ`.
"""
function gauss_beam(y, x, z, λ, w_0)
    k = π / λ * 2
    z_R = π * w_0^2 / λ
	r² = x ^ 2 + y ^ 2
    # don't put exp(i * k * z) into the same exp, it causes some strange wraps
	return w_0 / gauss_w(z, z_R, w_0) * exp(-r² / gauss_w(z, z_R, w_0)^2) * 
            exp(1im * k * z) * 
            exp(1im * (k * r² / 2 / gauss_R(z, z_R) - gauss_ψ(z, z_R)))
end

# ╔═╡ 5f21a849-0272-4bca-9659-54cd38068e09
"""
	bpm(field, λ0, Lx, Ly, z, n; window=true, paraxial=true, amplitude_array)

Propagates the array `field` with wavelength `λ0` and the filed size in meter size
`(Lx, Ly)`. The propagation distance `z` should be a vector of distances.
`n` is the average refractive index of the propagation medium.

The returned array is a three dimensional array where `size(arr, 3) == size(z, 1)`.


If `window=true` we apply a Hann window function to dampen the boundaries.
A keyword `amplitude_array` can be provided, which multiplies with the field at each point. This allows to include obstacles or to shift the phase.

If `paraxial=true` the Fresnel approximation is applied.
"""
function bpm(field, λ0, Lx, Ly, z, n=1; window=true, amplitude_array=ones(size(field)..., length(z)), paraxial=true)
	# free space wavenumber in m-1
	k0 = 2 * π / λ0
	# medium wavenumber m-1
	k = n * k0
	λ = λ0 / n
	# medium in m
	dz = z[2] - z[1]

	# field parameters
	Nx = size(field, 2)
	dx = Lx / Nx
	x  = Nx > 1 ? range(-Lx/2, Lx/2, Nx) : zero(typeof(Lx))
	fx = reshape(fftfreq(Nx, 1 / dx), (1, Nx))

	Ny = size(field, 1)
	dy = Ly / Ny
	y  = range(-Ly/2, Ly/2, Ny)
	fy = fftfreq(Ny, 1 / dy)

	if paraxial
		# important step, this calculates the Fourier space kernel
		H = exp.(-1im .* k .* λ^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
	else
		H = exp.(1im .* sqrt.(1 .+ 0im .- λ^2 .* fx.^2 .- λ^2 .* fy.^2) .* k .* dz) .* ((λ0^2 .* fx.^2 .+ λ0^2 .* fy.^2) .< 1)
	end
	# 3d output fields we save
	# third dimensions stores the different z propagation distances
	out_field = zeros(ComplexF64, (Ny, Nx, size(z, 1)))
	# first entry corresponds to z[1] = 0
	out_field[:, :, 1] = field
	
	# FFT plan for calculating FFTs
	# It's a more efficient syntax: p * x == fft(x)
	p = plan_fft(field, (1,2))
	window_f = window ? IndexFunArrays.window_hanning(size(out_field)[1:2], border_in=0.9) : 1
	# inverse FFT
	invp = inv(p)
	for z_index in 2:size(out_field, 3)
		u0 = out_field[:, :, z_index - 1] .* window_f .* amplitude_array[:, :, z_index - 1]
    	u1 = invp * ((p * u0) .* H)
		out_field[:, :, z_index] .= u1
	end
	return out_field
end

# ╔═╡ 66f6d122-3ab9-4f44-b702-4a526c0700db
"""
	free_space_propagation(field, z, λ, L)

Propagate a `field` with wavelength `λ` and field size `L` with free space bandlimited angular spectrum over a distance `z`.
"""
function free_space_propagation(field, z, λ, L)
	@assert size(field, 1) == size(field, 2) "Requires quadratic field"
	@assert ndims(field) == 2 "Requires a 2D array"
	# physical parameters
	Lp = 2 * L
	field_p = select_region(field, M=2)
	Np = size(field_p, 1)
	f_y = fftfreq(Np, inv(Lp / Np))
	f_x = f_y'
	k = 2 * π / λ

	# matsushima bandlimit
	Δu = 1 / Lp
	u_limit = 1 / (sqrt((2 * Δu * z)^2 + 1) * λ)
	W = ((abs2.(f_y) ./ u_limit .^2 .+ abs2.(f_x) * λ^2) .< 1) .*
        ((abs2.(f_x) ./ u_limit .^2 .+ abs2.(f_y) * λ^2) .< 1)

	# H is the Fourier space kernel
	H = W .* exp.(1im .* k .* z .* sqrt.(1 .+ 0im .- abs2.(f_x .* λ) .- abs2.(f_y .* λ)))

	# do wraparound free convolution
	field_propagated = select_region(fftshift(ifft(fft(ifftshift(field_p)) .* H)), M=0.5)

	return field_propagated
end

# ╔═╡ 83f49548-e22f-4bf4-9e22-235fdd8b0ad2
"""
	free_space_propagation(field, z, λ, L)

Propagate a `field` with wavelength `λ` and field size `L` with free space bandlimited angular spectrum over a distance `z`.
"""
function free_space_propagation_1D(field, z, λ, L)
	# physical parameters
	Lp = 2 * L
	field_p = select_region(field, M=2)
	Np = size(field_p, 1)
	f_y = fftfreq(Np, inv(Lp / Np))
	k = 2 * π / λ

	# matsushima bandlimit
	Δu = 1 / Lp
	u_limit = 1 / (sqrt((2 * Δu * z)^2 + 1) * λ)
	W = ((abs2.(f_y) ./ u_limit .^2 .+ abs2.(0) * λ^2) .< 1)

	# H is the Fourier space kernel
	H = W .* exp.(1im .* k .* z .* sqrt.(1 .+ 0im .- abs2.(f_y .* λ)))

	# do wraparound free convolution
	field_propagated = select_region(fftshift(ifft(fft(ifftshift(field_p)) .* H)), M=0.5)

	return field_propagated
end

# ╔═╡ a5e74e57-501d-4113-adaf-586647e500e5
md"# 1. Superresolution

The classic aldiffraction limited resolution for a 4f system is based on a sharp cut-off in the Fourier domain due to the finite aperture in the Fourier plane.
In practice, this sharp cut-off is not osberved due to the finite aperture of the input.  
In this exercises you explore this phenomenon.
The optical system is shown below.

The input is multiplied with an aperture `A` and in Fourier space with aperture `B`.

The size of aperture `B` is `40µm`.


1.  Measure the intensity which propagates through the optical system for different frequencies. For that, use the provided `frequencies`. Create a 1D plane wave and send it through the optical system. Sum (integrate) the intensity at the detector such that each plane wave input gives you a single intensity number at the end each. Finally calculate and plot the frequency response (integrated output intensity as a function of frequency) for two different sizes of `A`. For `200µm` and `500µm`. Explain the difference.

2. Measure the system matrix $M$. For that, propagate different plane waves (as basis functions) through the optical system, use `frequencies` again. Obtain the system matrix by calculating $Y \cdot \text{pinv}(X)$ where $Y$ is the complex measurement matrix and $X$ are the complex plane waves as input. 

3. Try to invert  `object_blurry = optical_system_1(object1, A, B)` by multiplying the pseudoinverse `pinv(M)`. Are you able to recover the original object in the case of $A_L=200\mu m$? And how about in the case $A_L=500\mu m$? Explain the differences and why there is a difference. Use the frequencies response for that. Plot your obtained reconstructions. Also plot `object_blurry`. Also use the frequency spectrum of `object1` to argue.
"

# ╔═╡ c24ec9ec-03e6-491b-8b15-ae619d922eec
urldownload("https://felix.sumpi.org/task3.png")

# ╔═╡ d2e57f72-4ce0-4e89-bbec-f98cb3c7a604
# total field size in image space
L1 = 500e-6

# ╔═╡ aabf5012-5714-4cf8-887f-7e35bdc99bd5
# size of aperture A
A_L1 = 500e-6

# ╔═╡ 9b95e9e0-e56c-4655-8535-a66937c35db9
A_L2 = 200e-6

# ╔═╡ 347cac27-8b58-4949-8d7b-daf46fad0f7d
# size of the aperture B
B_L = 40e-6

# ╔═╡ 3f389836-1bcd-4256-889f-f873b12a7992
λ1 = 633e-9

# ╔═╡ 7bc47b96-1002-4bca-8888-28a32d2a2fec
N1 = 200

# ╔═╡ 7ad54d1b-effa-4348-8a4e-a36f62c7219c
dx1 = L1 / N1

# ╔═╡ 3e8118c9-852a-45e6-b5e3-c9968e5dbc7a
xs1 = fftpos(L1, N1, CenterFT)

# ╔═╡ 5075554c-7bd5-4852-8569-222ac55b6eea
# focal length
f1 = 1e-3

# ╔═╡ 0d784e71-60d9-4c0b-a6c2-d84f23aa22af
# the total field size in Fourier space
L_ft = λ1 * N1 * f1 / L1

# ╔═╡ 43f4933c-7285-4537-b342-ed506a347b58
x_ft1 = fftpos(L_ft, N1, CenterFT)

# ╔═╡ 546dc74a-77ac-4a1d-a8fa-6ca098162d6d
A1 = abs.(xs1) .< (A_L1 / 2)

# ╔═╡ 89406e0f-dc15-4130-95e5-6e07c45c9665
dx = L1 / size(A1, 1)

# ╔═╡ 539f7f4f-8d46-456b-b303-3c279efb56df
frequencies = fftshift(fftfreq(200, inv(dx)))

# ╔═╡ 8d175756-adb3-4c45-a173-af180f20043f
A2 = abs.(xs1) .< (A_L2 / 2)

# ╔═╡ 8488d10f-0cc1-44f1-90c6-75565defe274
B = abs.(x_ft1) .< (40e-6 / 2)

# ╔═╡ 25b3ec9a-cca9-44fe-89fc-945f81cd0220
function optical_system_1(object, A, B)
	x_4 = ift(ft(object .* A) .* B)
	return x_4
end

# ╔═╡ 7daec515-b788-44a1-bbb7-7c69f0e22a0d
function measure_frequency_response(frequencies, A)
	return [sum(abs.(optical_system_1(exp.(1im .* xs1 .* f .* 2π), A, B))) for f in frequencies]
end

# ╔═╡ 91476a41-c2ea-4f05-8dd0-6be4f7b99a49
object1 = load(download("felix.sumpi.org/object1.jld2"), "object1");

# ╔═╡ eb32c36d-d11d-44cc-8f8e-83573838d410
plot(object1)

# ╔═╡ 693e2cbf-f638-498c-aeaa-1bd495733b9a


# ╔═╡ fa59e914-f444-4218-8627-94b324c17c5b
mfr1 = measure_frequency_response(frequencies, A1)

# ╔═╡ ac5e9c05-8aa2-41e0-9569-b91c36557a34
mfr2 = measure_frequency_response(frequencies, A2)

# ╔═╡ c3ae80a7-8221-4d61-8771-f523b82c6ba4
begin
	plot(frequencies, mfr1)
	plot!(frequencies, mfr2)
end

# ╔═╡ 62113fc2-4d5c-4391-be17-05fcf2a98d5e


# ╔═╡ f06b70b1-147b-4914-95a4-7d078b6de131
function build_matrix(frequencies, A)
	N = length(A)
	K = length(frequencies)
	Y = zeros(ComplexF64, (N, K))
	X = zeros(ComplexF64, (N, K))

	for k in 1:K
		x = exp.(1im .* xs1 .* frequencies[k] .* 2π)
		X[:, k] .= x
		Y[:, k] = optical_system_1(x, A, B)
	end

	return Y * pinv(X)
end

# ╔═╡ f0afb312-211f-4879-9cda-ffe4f1b1c659
M1 = build_matrix(frequencies, A1);

# ╔═╡ 41e2ef2f-5087-4db9-a9cf-002988648dcd
M2 = build_matrix(frequencies, A2);

# ╔═╡ 6cb21efd-cb8b-4b31-8221-813ee84b093e


# ╔═╡ 5dbd0e3e-a8bd-42cb-b8b4-df938c220c77
object_blurry1 = real.(optical_system_1(object1, A1, B))

# ╔═╡ 26098235-1b90-46fc-9b60-784f6cb0312c
object_blurry2 = real.(optical_system_1(object1, A2, B))

# ╔═╡ f0cd96d7-6c3e-4403-916d-d4b5ee64a637
begin
	plot(xs1, object_blurry1)
	plot!(xs1, object_blurry2)
end

# ╔═╡ 7415c2b7-294a-4540-b607-9bc1d4afb30f
begin
	plot(xs1, real.(pinv(M1) * object_blurry1))
	plot!(xs1, real.(pinv(M2) * object_blurry2))
	plot!(xs1, object1)
end

# ╔═╡ 124ba386-473a-461a-bf4b-0110df4883e4
begin
	# TODO plot
end

# ╔═╡ 970a2236-947b-4911-a56d-aad0f4104a88
md"## Answers

The finite aperture of A allows that some frequencies pass through the optical system even though there is a hard cut off in the Fourier plane.
This can be seen from the frequency response curve.


The matrix is capturing this information and allows to revert it.
"

# ╔═╡ 32b1e651-5abe-4e79-8482-84eb38a2e9ba


# ╔═╡ ee96bf68-bdfa-45e1-9300-53b300ed6dfe


# ╔═╡ 5bdc18e1-be5b-4f4e-a15b-cd9b637adaeb
md"# 2. Ptychography

Ptychography is a computational method of microscopic imaging closely related to the Gerchberg Saxton algorithm for extracting phase of intensity measurements.
It generates images by processing the intensities of multiple coherent diffraction patterns that have been scattered from an object of interest. 
Its defining characteristic is translational invariance, which means that the diffraction patterns are generated by one constant function (e.g. a field of illumination) moving laterally by a known amount with respect to another constant function (the specimen itself). 
At each step, only a small part of the sample is illuminated (the so called *tile*).
The diffraction patterns occur some distance away from the sample, so that the scattered waves spread out and fold into one another as shown in the figure.
Assume that in this case the Fraunhofer approximation is valid.


To recover the specimen we can apply a modified form of the Gerchberg Saxton algorithm.
Instead of looping between the input and the detector planes, we have an additional _for loop_ over the different tiles (the illuminated patch of the sample).
The tile positions are given in `positions` where the entries indicate the center of each tile. 
The corresponding diffraction pattern intensities are given in `measurements`. 
The third dimension of `measurements` has 81 entries because we have 81 different scan positions.
Each tile has a size of `(80, 80)` pixels.

Use the following sketch for the algorithm:
Two nested for loops, the outer one over the iterations, the inner one over the tiles (this part is already provided).
The inner *for loop* over the tiles is like a Gerchberg Saxton for a single tile.
The outer loop takes advantage of the fact that the far field patterns overlap for different tiles and this helps obtaining a better reconstruction.
Since the tiles are overlapping, as you go over the tiles the diffraction of the latest tile overwrites the overlapping areas.

* Extract current guess of the specimen at the tile position (essentially Gerchberg Saxton on a single tile)
* Multiply the beam with the current guess of the specimen
* Propagate with a Fourier transform `ft` to get to the far field.
* Replace the obtained amplitude with the measurement.
* Backpropagate
* Divide by the beam
* Write the result to the current guess at the tile position


Tasks:

1. Reconstruct the measured image by implementing the sketched algorithm.
2. The wavelength is $\lambda=405\mathrm{nm}$. The distance between sample and detector is $100\mathrm{mm}$. The pixel size of the detector is $10\mathrm{\micro m}$. What is the average distance between the scan positions?
3. Bonus: Analyze the reconstruction with 5 iterations of the outer loop. Why are the boundaries of the image more noisy than others?

"

# ╔═╡ 8af36c7e-69c5-4ba1-80f4-6727ffc58f29
urldownload("https://upload.wikimedia.org/wikipedia/commons/6/63/Ptychography_imaging_data_collection_single_aperture.png")

# ╔═╡ 4ccc02c2-9c2a-49d3-a9d4-c13d764f362a
measurements = load(download("felix.sumpi.org/ptychography.jld2"), "measurements");

# ╔═╡ f4ef4f25-6dc1-44e4-b85d-178c48267793
size(measurements)

# ╔═╡ 3f23d0fa-abce-4aa5-9ce9-686cd164d11b
beam = load(download("felix.sumpi.org/ptychography_probe.jld2"), "probe");

# ╔═╡ 2f7d18fd-a6d0-4169-9f8c-2e13210114f1
simshow(beam)

# ╔═╡ bbf84ef7-4f01-4f3a-85d7-10b71591c024
positions = [(51, 55) (51, 84) (57, 119) (64, 148) (64, 174) (53, 205) (60, 245) (56, 270) (51, 302); (93, 63) (90, 90) (83, 122) (91, 144) (83, 173) (94, 210) (83, 241) (94, 265) (87, 304); (120, 56) (121, 86) (119, 114) (111, 155) (112, 172) (111, 202) (125, 245) (119, 274) (116, 294); (145, 58) (151, 92) (153, 113) (152, 151) (150, 185) (141, 206) (152, 235) (147, 269) (153, 294); (185, 62) (184, 93) (178, 118) (171, 145) (171, 185) (171, 209) (179, 231) (183, 272) (179, 294); (208, 59) (203, 83) (208, 117) (211, 147) (206, 179) (206, 210) (201, 245) (207, 266) (205, 297); (242, 57) (241, 93) (236, 111) (242, 142) (237, 180) (231, 212) (243, 242) (235, 273) (238, 302); (274, 55) (272, 85) (262, 112) (261, 145) (267, 185) (265, 201) (275, 236) (261, 272) (270, 298); (295, 63) (305, 83) (301, 121) (302, 143) (301, 184) (294, 205) (298, 233) (303, 270) (302, 298)][:];

# ╔═╡ 34ca6dba-0059-483d-a53f-0dd2ca202803
# access the first index of the first tile, y coordinate
positions[1][1]

# ╔═╡ 7617aa1f-efd5-45a9-94dc-cf02af9f3047
# access the second index of the first tile, x coordinate
positions[1][2]

# ╔═╡ cf0bed1c-4859-4cc2-a522-38ef1eecbc95
@bind imeas Slider(axes(measurements, 3))

# ╔═╡ 874d3d95-0767-4650-8efb-e3a725cb07c7
simshow(measurements[:, :, imeas], γ=1, cmap=:turbo)

# ╔═╡ b3f9a643-81d1-4c06-b723-325836da4aed
function reconstruct_ptychography(measurements, beam, positions, N;
								  N_iterations=30)
	reconstruction = zeros(Float64, N, N)

	for Ni in 1:N_iterations
		for (i, p) in enumerate(positions)
			img_cropped = reconstruction[p[1]-40:p[1]+39, p[2]-40:p[2]+39]

			f = ft(img_cropped .* beam)
			f = cis.(angle.(f)) .* sqrt.(measurements[:, :, i])			
			
			img_space = ift(f) ./ beam 
			
			update = abs.(img_space)
			reconstruction[p[1]-40:p[1]+39, p[2]-40:p[2]+39] .= update
		end
	end

	return reconstruction

end

# ╔═╡ 127e7753-d0d1-4944-99a7-704003239f22
reconstruction = reconstruct_ptychography(measurements, beam, positions, 400, N_iterations=5);

# ╔═╡ 152287b3-5843-44c3-86f4-0285bb61e0c6
simshow(reconstruction, γ=1)

# ╔═╡ bb6a106a-8356-4d3f-ac13-2e56bc9d8f7d


# ╔═╡ 2695d736-71db-4e76-a8e2-4c42d02da622


# ╔═╡ adf2f842-26e7-420a-9e59-1fe8aafc6e75
object_space_pixel_size = 405e-9 * 100e-3 / (80 * 10e-6) / 80

# ╔═╡ b240c885-09d1-4587-b589-8416a03e7e02
average_distance = [sqrt((positions[i][2] - positions[i-1][2])^2 +(positions[i][1] - positions[i-1][1])^2) for i in 2:length(positions)]

# ╔═╡ 374baf86-1356-4c83-b24a-68ffe52bedcc
30 * object_space_pixel_size

# ╔═╡ b685eea0-b4cf-4231-8042-f05294a3547b
md"## Answers
The average distance between the tiles is roughly 19µm.


On the corners of the image are less tiles and less cross talk, hence the reconstruction quality is worse.
"


# ╔═╡ 121be5eb-4eb0-4a60-89d2-8104ed806994


# ╔═╡ 1824cd6b-8ba0-4e54-94ee-9972c758c85d


# ╔═╡ ce6e853b-e8de-4c14-9900-63236eaffce3


# ╔═╡ cc2acb93-dd22-4ff1-a085-d5db3a8d8867


# ╔═╡ e939a228-954c-43bd-a5e6-a4a9edfbac1d


# ╔═╡ 394a2921-11ca-40b1-b57f-31aec08f601e


# ╔═╡ 200141af-5d24-43c9-8167-980f8ff89a69
md"# 3. Find the transparency

Consider the optical systems shown in Figure 3. A unknown mystery transparency is placed at the input plane $P_1$.
A second known phase pattern (called `diffuser3`) transparency is placed at plane $P_2$, a given distance $z_{31}$ away. The complex field of the light transmitted through the second transparency is detected on a **finite** aperture camera at distance $z_{32}$ after the second transparency. This final field is given and called `field3`.

1. Use phase conjugation and reverse propagation to retrieve the phase of the input transparency from the measured field given the phase of the transparency in the middle. 

2. The reconstruction looks not perfect. Why do you think that is the case? What makes this optical system not perfectly reversible.

3. Bonus: Assume that you do not know the phase of the transparency in the middle but you know instead a set of measurement of complex fields with different illumination angles. How would you reconstruct the input phase in this case?

"

# ╔═╡ ca833fc1-248f-49c4-a6ea-20ee52062aa9
urldownload("https://felix.sumpi.org/task3.png")

# ╔═╡ b5b25dd3-0adb-44e6-9c17-13ee912313cd
field3 = load(download("felix.sumpi.org/phase_exam.jld2"), "phase_exam");

# ╔═╡ bb83a3a0-f9f1-4042-8290-dbfc878acf07
diffuser3 = load(download("felix.sumpi.org/diffuser_exam.jld2"), "diffuser_exam");

# ╔═╡ 868f67b2-4834-4803-9c77-c1def9ceb433
simshow(field3, γ=1)

# ╔═╡ 75267942-8e91-4c8b-9d2e-4dec9e5b9318
simshow(diffuser3)

# ╔═╡ 4f4aa449-f774-4d19-af09-b79fcc351450
z31 = 10e-3

# ╔═╡ 6b870f0d-4b69-4c42-bccd-d4ff72612000
z32 = 20e-3

# ╔═╡ 61f151c5-63bb-4448-aa51-cce91efeb088
L3 = 1e-3

# ╔═╡ a07bb98a-9437-4f72-80aa-e6aacbcf5be7
λ3 = 405e-9

# ╔═╡ 198ed99a-c0fd-49df-adbc-b8b0253a6971
TODO

# ╔═╡ a770bf76-fd85-4414-a064-902aeb0e8957


# ╔═╡ 144a7b46-3d9b-4b91-9777-9a8c292131fd


# ╔═╡ 8ba2fe0a-3c82-4846-98a9-7afb3e7cf1b1
mystery_transparency = conj.(free_space_propagation(free_space_propagation(field3, -z32, λ3, L3) .* conj.(diffuser3), -z31, λ3, L3));

# ╔═╡ 55147568-a2e1-4e1d-8cda-0572c9ed1233
simshow(mystery_transparency)

# ╔═╡ d7751773-e975-41f2-8e8b-a56f9ef8c366


# ╔═╡ b803f945-772b-4a43-8d90-39027e3f7d88
md"## Answers
## 2 
Diffraction spreads out in space. If you have a finite detector, some intensity is lost and therefore backpropagation is missing some information.


### 3
One way would be to set up a Gerchberg Saxton algorithm which tries to reconstruct the mystery based on the obtained measurements.
Also one could construct a system matrix and then revert it based on this.
"

# ╔═╡ d156c2c1-8c1a-4a3d-8401-3fe8e177499e


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[[deps.GLFW_jll]]
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[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits"]
git-tree-sha1 = "cb76cf677714c095e535e3501ac7954732aeea2d"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.11.1"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TestImages]]
deps = ["AxisArrays", "ColorTypes", "FileIO", "ImageIO", "ImageMagick", "OffsetArrays", "Pkg", "StringDistances"]
git-tree-sha1 = "0567860ec35a94c087bd98f35de1dddf482d7c67"
uuid = "5e47fb64-e119-507b-a336-dd2b206d9990"
version = "1.8.0"

[[deps.TiffImages]]
deps = ["ColorTypes", "DataStructures", "DocStringExtensions", "FileIO", "FixedPointNumbers", "IndirectArrays", "Inflate", "Mmap", "OffsetArrays", "PkgVersion", "ProgressMeter", "UUIDs"]
git-tree-sha1 = "34cc045dd0aaa59b8bbe86c644679bc57f1d5bd0"
uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69"
version = "0.6.8"

[[deps.TranscodingStreams]]
git-tree-sha1 = "54194d92959d8ebaa8e26227dbe3cdefcdcd594f"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.10.3"
weakdeps = ["Random", "Test"]

    [deps.TranscodingStreams.extensions]
    TestExt = ["Test", "Random"]

[[deps.Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "ConstructionBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "3064e780dbb8a9296ebb3af8f440f787bb5332af"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
version = "0.4.80"

    [deps.Transducers.extensions]
    TransducersBlockArraysExt = "BlockArrays"
    TransducersDataFramesExt = "DataFrames"
    TransducersLazyArraysExt = "LazyArrays"
    TransducersOnlineStatsBaseExt = "OnlineStatsBase"
    TransducersReferenceablesExt = "Referenceables"

    [deps.Transducers.weakdeps]
    BlockArrays = "8e7c35d0-a365-5155-bbbb-fb81a777f24e"
    DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
    LazyArrays = "5078a376-72f3-5289-bfd5-ec5146d43c02"
    OnlineStatsBase = "925886fa-5bf2-5e8e-b522-a9147a512338"
    Referenceables = "42d2dcc6-99eb-4e98-b66c-637b7d73030e"

[[deps.Tricks]]
git-tree-sha1 = "eae1bb484cd63b36999ee58be2de6c178105112f"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.8"

[[deps.URIs]]
git-tree-sha1 = "67db6cc7b3821e19ebe75791a9dd19c9b1188f2b"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "3c793be6df9dd77a0cf49d80984ef9ff996948fa"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.19.0"

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    InverseFunctionsUnitfulExt = "InverseFunctions"

    [deps.Unitful.weakdeps]
    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.3"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.UrlDownload]]
deps = ["HTTP", "ProgressMeter"]
git-tree-sha1 = "758aeefcf0bfdd04cd88e6e3bc9feb9e8c7d6d70"
uuid = "856ac37a-3032-4c1c-9122-f86d88358c8b"
version = "1.0.1"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "7558e29847e99bc3f04d6569e82d0f5c54460703"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+1"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "93f43ab61b16ddfb2fd3bb13b3ce241cafb0e6c9"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.31.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "07e470dabc5a6a4254ffebc29a1b3fc01464e105"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.12.5+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "1ea2ebe8ffa31f9c324e8c1d6e86b4165b84a024"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
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# ╠═25b3ec9a-cca9-44fe-89fc-945f81cd0220
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