HW04.jl

### A Pluto.jl notebook ###
# v0.19.40

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

# ╔═╡ 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; 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.

.
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.

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
	# medium in m
	dz = z[2] - z[1]

	# field parameters
	Nx = size(field, 1)
	dx = Lx / Nx
	x  = range(-Lx/2, Lx/2, Nx)
	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 .* λ0^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
	else
		H = exp.(1im .* sqrt.(1 .+ 0im .- λ0^2 .* fx.^2 .- λ0^2 .* 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=0.8) : 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

# ╔═╡ 7e1ef564-80b4-4b97-908d-cece154abfd6
md"# 1. Lens as a Scaled Fourier Transform

In this task we will simulate the propagation of a grating through a lens.
We show that the propagation of the field through the lens is equivalent to a scaled Fourier transform.

For that, define an array `grating` which is a 2D array with a rectangular pattern.
You can use a `sin` function to create the grating and then apply a threshold at 0 (`.> 0`) to obtain the rectangular pattern.

The grating is along the `x` axis and the period is `Λ = 247e-6`.
The grating is illuminated with a wavelength `λ = 633e-9` and the grating is `L = 4.931e-3` long.

We will propagate the grating through a lens with a focal length `f = 150e-3`.
"

# ╔═╡ 0e63fa75-5e61-4032-9630-762e77a16f8e
λ = 633e-9

# ╔═╡ ca1ba8bf-a741-495f-a319-9a857d7667fe
L = 4.931e-3

# ╔═╡ 7235f0cf-ce39-4e19-8739-57dcdcb271c1
z1 = range(0, 150e-3, 150)

# ╔═╡ 733dca27-a8dd-45fa-9318-970b3a76516d
z2 = range(0, 200e-3, 200)

# ╔═╡ 45511244-a5c1-4905-a98a-deb501760ba5
N = 256

# ╔═╡ ff5fb4d0-c880-430a-a85b-d83214c55f6b
y = fftpos(L, N, CenterFT)

# ╔═╡ 3c7d9658-17ef-4ce9-affa-f5e0e2f72b1d
x = y'

# ╔═╡ 2431a58a-d3d9-4577-a49f-2a7eb6ab3595
Λ = TODO

# ╔═╡ 80a0876e-dbba-4d63-b4e2-ddf47bf41c9a
grating = TODO;

# ╔═╡ c6ae5bac-b1d9-469e-b21c-912699367783
heatmap(y[:], x[:], abs.(grating))

# ╔═╡ 8585dc8b-a7e7-42ed-b755-65b8e6a410b8
prop1 = TODO # propagation with z1

# ╔═╡ 5f1f2d1c-9e8c-4bf7-a388-202d0394b9ac
focal_length = 150e-3

# ╔═╡ 21f3517f-e408-4b7e-b780-2d7095af41d7
field_after_lens = prop1[:, :, end] .* exp.(-1im .* π ./ λ ./ focal_length .* (x.^2 .+ y.^2));

# ╔═╡ 5298ae72-11ca-4be7-bf49-f529d0655c68
prop2 = TODO# propagation with z2

# ╔═╡ cf0b7836-93ca-402c-a078-3b1235e536b8
total_field = abs2.(cat(prop1, prop2, dims=3));

# ╔═╡ c1e99bfd-efe5-46d0-8f88-095016984b90
heatmap(vcat(z1, z1[end] .+ z2), y, abs.(total_field[129, :, :]).^0.4, cmap=:turbo, xlabel="propagation distance in m", ylabel="size in m",)

# ╔═╡ b0e04815-2279-416c-b3da-565940e0c026
md"""## Question
Look at the plot below.
Why does the result of `total_field` and `ft` match almost perfectly? The peaks are located at the correct positions!
Didn't we tell you before that a lens is a scaled Fourier transform? Why is there no scaling along the positional axis?

## Answer
TODO
"""

# ╔═╡ 3c29f4f1-7192-4465-bb57-b7c827066f2f
begin
	plot(x[:], abs.(total_field[129, :, 300]), xlabel="distance in mm")
	plot!(x[:], abs.(ft(grating)[129, :]))
end

# ╔═╡ 1938a0dd-ffec-478e-870c-18d14f458f6e
md"# 2. Imaging with a 4f system

As seen in the image below, a 4f system consists of two lenses.
The object is placed 1f in front of the first lens and the image is located 1f behind the second lens.
In the middle, the Fourier plane is located.

Given the electrical field at the input plane, we can perform a Fourier transform to obtain the Fourier plane. At the Fourier plane we can apply the aperture function (pointwise multiplication `.*`).
A final Fourier transform provides us the electrical field at the exit.
At the image  plane we take `abs` to obtain the amplitude.



If the fast discrete Fourier transform (FFT) is applied, there happens a scaling between the fields:

$$U'(x', y') = \mathcal{F}[U(x,y)](x', y')$$

where $x' = \frac{\lambda \cdot f \cdot N}{L^2} \cdot x$.


$f$ is the focal length of the lens, $\lambda$ the wavelength, $L$ the field size and $N$ the discretization along one axis.
"

# ╔═╡ 8812ba45-091d-4b07-b941-0c51a6f8d198
urldownload("https://d3i71xaburhd42.cloudfront.net/9d8bdf497732e41e31f871b438d58263ab5a4b54/2-Figure2-1.png")

# ╔═╡ d41eca1c-76af-45d1-8e26-e4d57bc30702
md"## 2.1 Circular Aperture
In this part, create a circular aperture and multiply it in Fourier space to the spectrum.
Take a final `ft` and take `abs.` for the amplitude.

"

# ╔═╡ 50358b73-9736-4195-98a0-b5f79b8cd150
begin
	img = sqrt.(Float32.(testimage("resolution_test_512")));
	img[490:512, :] .= 0.5 .* (1 .+ sin.((fftpos(512, 512)) .* π .* 2 / 8))'
end;

# ╔═╡ 64fba384-d96e-4895-9e2a-1bc03fb839ef
simshow(img)

# ╔═╡ c7bdd3a3-59f2-451f-8ccb-0fdff26abc53
L2 = 1e-3

# ╔═╡ ff42a725-4756-4d71-835c-274b8940c138
N2 = size(img, 1)

# ╔═╡ c5093acf-bb4e-402c-a1fa-430cafd9b523
λ2 = λ

# ╔═╡ 0c936df2-e9bf-48e0-bb9a-42728b14af43
f2 = 100e-3

# ╔═╡ 092b0931-8bc7-440c-8e59-105c4aedb3e1
y2 = range(-L2 / 2, L2 / 2, N2)

# ╔═╡ 036e035a-2266-4893-b1a9-ef5be72ca0ab
y2_fourier = y2 .* λ2 .* N2 .* f2 ./ L2.^2

# ╔═╡ 030c2b45-f39c-40c9-bd08-4cdac52b4473
# perform the imaging operation described in the text
img_aperture = TODO

# ╔═╡ bc1256d4-d482-483c-b755-266cb193029d
heatmap(y2, y2, img_aperture, yticks=1:20:512)

# ╔═╡ 9c898ea1-4a1e-424b-acba-e6ff73345394
md"radius aperture= $(@bind radius Slider(range(0, 10e-3, 100), show_value=true, default=0.005))"

# ╔═╡ 7ecf1776-f711-4ac1-8de2-741a9bf61686
# define a circular aperture. Use `rr` for that and compare if smaller (.<) than radius
circ_aperture = rr(size(img), scale=f2 * λ2 * N2 / L2^2 * (L2 / N2)) .<= radius;

# ╔═╡ 76312730-1911-4b80-8491-3399de005b0e
# show the aperture
heatmap(y2_fourier, y2_fourier, circ_aperture)

# ╔═╡ 05822651-21b8-4d67-8755-db6cfd7b21ae
plot(y2[1:50], (img_aperture)[5, 1:50], ylim=(0, 2.5e5))

# ╔═╡ c8b49998-0b73-4dc5-b164-e2f75d472368
md"### Question

We want to observe the sinusoidal pattern at the bottom of the image.
It's period is 8 pixels, corresponding to $(round(L2 / N2 * 8 * 1000, digits=3))mm.

What minimum physical size does the aperture need to have, to let the sinusoidal pattern still transmit through the system?
Select the correct radius at Slider and provide your calculation below.
"

# ╔═╡ 2e2dbf7f-394a-4d41-bc1c-1b84f725f1cb
md"### Answer
TODO
"

# ╔═╡ 68cd875d-d844-49f5-b914-99c3a5d7b953
md"## 2.2 Annulus Aperture
Instead of the circular aperture, we can use the annulus aperture as shown below.

Pick a suitable set of `radius_inner` and `radius_outer` and comment the obtained image.
"

# ╔═╡ 0a9670fe-9e9a-4337-a2ac-240f8cceaa56
urldownload("https://irsa.ipac.caltech.edu/data/SPITZER/docs/dataanalysistools/tools/mopex/mopexusersguide/images/MOPEX_Users_Guide090.png")

# ╔═╡ 276045a0-aa06-4b54-a0a7-b190212f06e1
@bind radius_outer Slider(1:256, show_value=true, default=150)

# ╔═╡ 4fb435eb-4283-4fe5-9349-0ec965ba824a
@bind radius_inner Slider(0:radius_outer, show_value=true, default=30)

# ╔═╡ 5eb45850-fa25-49f9-b6c6-8ad8337ee714
annulus = TODO# TODO

# ╔═╡ 64a2924f-011d-477c-8931-fa7cee6c899b
simshow(annulus)

# ╔═╡ c3893fd5-9ae3-49f5-bf2e-8ce4119703cd
img_annulus = TODO 

# ╔═╡ 48075c22-0a33-4dbd-b200-f6df9fcafc7c
simshow(img_annulus)

# ╔═╡ 2d07963b-8026-4772-a6ee-caad2427423d
md"### Task
For those parameters (`radius_outer=150` and `radius_inner=30`) the Siemens star on the left looks like a donut.
Why is there hole in the middle and why is it black on the outside?


### Answer
TODO

"

# ╔═╡ 98e84036-e7d9-49a2-b2c4-07b1ef39b31d
md"# 3. Observing Lac Leman from the Moon

Assume you would be the first astronaut from Switzerland on the Moon.
You look back to the Earth and you spot Switzerland and the Lac Leman.
Is this a true story? Or is it just a dream? Can you see the Lac Leman from the Moon?

Hint: Consider the diffraction limit of the human eye
"

# ╔═╡ 38bb85bf-dd1b-4e6d-88ae-b2dd4f8f8670
md"## Answer
TODO

"

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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"
"""

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