HW01.jl

### A Pluto.jl notebook ###
# v0.19.37

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

# ╔═╡ 7d8a9001-31cd-4e57-ae24-b76c0f423193
using PlutoTest

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

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

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

# ╔═╡ 67d711ec-a6ab-4b8a-b5aa-7342531fd6bb
md"""
# 1.Beam propagation method - Notebook

This first notebook presents the general concepts and the numerical implementation of beam propagation method (BPM) in free space. 
The propagation of a monochromatic wave in an homogeneous medium is described by the scalar Helmoltz equation (wave equation):

$$\frac{\partial^2E}{\partial x^2} + \frac{\partial^2E}{\partial y^2} + \frac{\partial^2E}{\partial z^2} + k^2E=0$$


where $E$ is the scalar field and $k=\frac{2\pi \cdot n}{\lambda}$ is the wavenumber in the propagation medium with refractive index $n$.

In the BPM we assume that the wave is propagating along a specific axis, in the following $z$, and we factorize the field as a product of an enveloppe $A(x,y,z)$ and a fast oscillating term along the propagation direction $\exp(-jkz)$. By plugging $E=A\exp(-jkz)$ in the wave equation we can write in the wave equation we get:

$$
2jk\frac{\partial A}{\partial z} = \frac{\partial^2A}{\partial x^2} + \frac{\partial^2A}{\partial y^2} + \frac{\partial^2A}{\partial z^2}
$$

If we now consider that the dependence of $A$ on $z$ is slow because the fast variation along this axis is taken into account by the fast oscillating term $\exp(-jkz)$ we can neglect the second order derivative along $z$, better knwon as slowly varying envelope (SVE) approximation. We finally get to the BPM equation in free space

$$\frac{\partial A}{\partial z} = \hat{A}A$$

where we labelled with $\hat{A}$ the diffraction operator:

$$\hat{A} = \frac{j}{2k}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)$$

In practice it is more convenient to solve this first order partial differential equation in the spatial frequency domain. Indeed if we take the Fourier transform of $A$ with respect to $x$ and $y$, labelled $\tilde{A}$ hereafter, we get:

$$\frac{\partial \tilde{A}}{\partial z}=-\frac{j}{2k} \left(K_x^2+K_y^2\right)\tilde{A} $$

This is a differential equation in $z$ with general solution:

$$\tilde{A}(x,y,z+\Delta z) = \tilde{A}(x,y,z)\exp(D\Delta z)$$

with 

$$D = -\frac{i}{2k} \left(K_x^2+K_y^2\right) $$

being the diffraction operator in the frequency domain. 
In summary, the SVE propagation describing diffraction can be solved in the inverse space and then transformed back in the direct space:

$$A(x,y,z+\Delta z) = \text{IFT}\left\{\text{FT}[A(x,y,z)]\exp(D\Delta z)\right\}$$

where $\text{FT}$ and $\text{IFT}$ denote Fourier transform and its inverse operation, respectively.

Note that this is an approximation, and a rigorous solution is the angular spectrum method of plane waves.
"""

# ╔═╡ 43ae81bb-8a6b-47be-8e64-129c19f045be
md"# 2. Load Initial Field"

# ╔═╡ fb52f064-67d6-4771-8d81-5e7226c32910
testimage("mandril_gray")

# ╔═╡ ed2480ca-0c0b-490d-b470-97760f976d71
field = 0im .+ select_region(Float64.(testimage("mandril_gray")[begin:4:end, begin:4:end]), new_size=(300, 300));

# ╔═╡ d9f1e0c0-a815-419f-a3d3-4611a28a4e38
md"#
This is a complex image. The phase is encoded as color (here constant) and the amplitude as brightness of the image
"

# ╔═╡ 7d659569-4bc3-44cb-8366-7c0a737200d8
md"We can also just plot it as a heatmap. The y axis is swapped in this case"

# ╔═╡ 664fabff-7205-47e0-b076-6f26824d0cad
heatmap(abs.(field))

# ╔═╡ cf3f3ec0-4c9c-4089-ba0a-bbca193ad3f7
heatmap(angle.(field))

# ╔═╡ a015a2af-557e-442f-b92a-37455b93e393
md"# 3. Define Helper Variables"

# ╔═╡ 237ce060-85ba-4bde-b08c-b29425ae95ac
# points along the optical axis
Nz = 100 

# ╔═╡ 9d02e310-287d-48a1-b172-1de96b9a9107
Nx = size(field, 2)

# ╔═╡ 6e88f054-7ce5-454b-97ee-b3841636a4dc
Ny = size(field, 1)

# ╔═╡ c89a2539-58d1-4639-88f1-5166a8ef6283
# width of field
Lx = 200e-6

# ╔═╡ 5b4648b6-4d41-4581-91fe-de0e10fe8565
Ly = Lx

# ╔═╡ 77c5991f-5bd5-410b-ac29-055717af6280
# propagation distances
Lz = 1000e-6

# ╔═╡ 7fcb26f9-1fea-4ec0-ba63-e2f0b70ea7c8
y  = range(-Ly /2, Ly / 2, Ny)

# ╔═╡ 13c6547f-42cc-49b6-a748-e65422180eae
x  = range(-Lx /2, Lx / 2, Nx)

# ╔═╡ a0303640-8adf-468f-85f2-a6a12ec85107
# background refractive index
n0 = 1

# ╔═╡ 90f4e2b0-8d71-4a0d-9be4-2f2b713fea17
# free space wavelength
λ = 532e-9

# ╔═╡ 116286e2-6787-4ce9-b2cd-40b1a9317894
z = range(0, Lz, Nz)

# ╔═╡ 0cebf599-48cd-4d86-a3e6-0bf9eaae4bdb
md"# 4. BPM Function"

# ╔═╡ 5f21a849-0272-4bca-9659-54cd38068e09
"""
	bpm(field, λ0, Lx, Ly, z)

Propagates the array `field` with wavelength `λ0` and in meter size
`(Lx, Ly)`. The propagation distance `z` is a vector of distances.
.
The returned array is a three dimensional array where `size(arr, 3) == size(z, 1)`.
"""
function bpm(field, λ0, Lx, Ly, z, n=1)
	# 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)
	kx = 2π .* reshape(fftfreq(Nx, 1 / dx), (1, Nx))

	Ny = size(field, 1)
	dy = Ly / Ny
	y  = range(-Ly/2, Ly/2, Ny)
	ky = 2π .* fftfreq(Ny, 1 / dy)

	# important step, this calculates the Fourier space kernel
	H = exp.(-1im .* (kx.^2 .+ ky.^2) ./ (2*k) * dz)

	# 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 sytanx: p * x == fft(x)
	p = plan_fft(field, (1,2))
	
	# inverse FFT
	invp = inv(p)
	for z_index in 2:size(out_field, 3)
		u0 = out_field[:, :, z_index - 1]
    	u1 = invp * ((p * u0) .* H)
		out_field[:, :, z_index] .= u1
	end
	return out_field
end

# ╔═╡ 5d095823-5b49-4f0e-994e-70b4600f5db5
md"# 5. Experiments"

# ╔═╡ 8a1ac052-66ed-4388-8769-034f15f3d950
out_field = bpm(field, λ, Lx, Ly, z);

# ╔═╡ d5c7d05c-0080-4ee9-84ab-c4738a641df5
@bind iz PlutoUI.Slider(eachindex(z))

# ╔═╡ f2427112-dc6b-4345-a3ec-ec6dbb297428
heatmap(x[:] .* 1000, y[:] .* 1000, abs2.(out_field[:, :, iz]), xlabel="x in mm", ylabel="y in mm", title="intensity of the propagated field by distance $(round(z[iz] * 1000, digits=3))mm")

# ╔═╡ 60de432a-dded-4f6c-b5c2-6316d0a35028
heatmap(z .* 1000, x[:] .* 1000, abs2.(out_field[:, 150, :]), xlabel="z in mm", ylabel="y in mm")

# ╔═╡ 273d6dae-ffc8-4d35-afd6-6307ef7228d3
md"# Homework

Please solve homework 2 and 3. 
If you see a `nothing` you have to replace it with the required expression
to solve the homework
"

# ╔═╡ f08de158-f6e6-4766-b9af-6af7c400e2a2
md"## 1.1. Talbot effect
The [Talbot effect](https://en.wikipedia.org/wiki/Talbot_effect) is a optical phenomenon that a repetitive sample self images itself without the use of a lens.

As an example, play with the sliders and find the distance at which the mandrils reoccur again.
"

# ╔═╡ 37f4c7da-1b3b-4ddc-a5e1-b03539892769
mandril = abs.(resample(Float32.(testimage("mandril_gray")), (64, 64)));

# ╔═╡ 940600d3-5632-4aa2-b1d2-96842020d305
mandril_large = select_region(repeat(mandril, 4,4), new_size=(256,256));

# ╔═╡ d3a1dfc1-f480-4b43-ae88-10104ceaa185
simshow(mandril_large)

# ╔═╡ 7c4e702a-6292-47f6-b475-9d2d3aafe649
z2 = range(0, 10f-3, 300)

# ╔═╡ df5870c1-78a4-4f04-9fec-40bbc311a8e0
mandril_prop = bpm(mandril_large, λ, Lx, Ly, z2);

# ╔═╡ 4d7453b1-66d9-4b58-a361-dcacc52610e9
x2 = range(-Lx/2, Lx/2, size(mandril_large, 1))

# ╔═╡ bc28ee52-22fa-4dc0-a0fd-b58b196c346d
y2 = range(-Ly/2, Ly/2, size(mandril_large, 1))

# ╔═╡ 1c1d36b2-1d2d-4b81-af44-34951569628c
@bind iz2 PlutoUI.Slider(eachindex(z2))

# ╔═╡ 19822b65-a4c8-4e11-8607-1009fb180f4c
md"
propagation distance z = $(round(z2[iz2] * 1000, digits=3))mm
"

# ╔═╡ 45a4261b-f8b0-4c1b-b65d-4f9e0e62c91e
heatmap(y2, x2, abs.(mandril_prop[:, :, iz2]), xlabel="in mm", ylabel="in mm")

# ╔═╡ 829baabc-1462-480f-b879-b0ca453d0e9f
md"## 1.2 Talbot length
Can you derive/find the correct Talbot length for the given situation?

"

# ╔═╡ 8ba00f7c-13a1-49ac-8a91-6cf21bceb787
talbot_length = nothing

# ╔═╡ 05be96ef-6e66-4972-98b4-8241d6eec4c1
md"""## 2. Propagation through aperture

Use BPM to propagate a plane wave that goes through the input apertures described below 

- A rectangular aperture of $20\mathrm{\mu m} \times 200\mathrm{\mu m}$

Note: For the BPM code use a window of $2\mathrm{mm} \times 2\mathrm{mm}$, propagation distance 40 mm and λ=532nm

Provide the amplitude x-z plot and comment on your results.

Hint: use the `box` function
"""

# ╔═╡ f48628e5-1825-486b-9ced-03fede0da7a9
# access the help with typing ?box

# ╔═╡ 0b4d696e-8218-413f-be22-eab08829efbf
Lrec = nothing

# ╔═╡ 4c7096ad-8b6d-4808-938c-9d5cc5b272cb
Nrec = nothing

# ╔═╡ a49957f5-b120-4e7f-a6fb-c562e0aebd80
aperture_size = nothing

# ╔═╡ fef13208-7aad-4e9f-9140-ccc59e91d9ae
aperture = nothing

# ╔═╡ 6095b2ae-b516-47ba-8301-159749f659d9
simshow(aperture)

# ╔═╡ 8ce4e662-1143-4fd8-98b8-ba0df79335ae
z_aperture = nothing

# ╔═╡ c3b363ee-5489-4e86-8849-14fbcac80a19
aperture_p = nothing

# ╔═╡ 4b703f6f-7bfd-4c07-8011-696b4a0f5731
@bind iz3 PlutoUI.Slider(eachindex(z))

# ╔═╡ 7b478e05-c748-48bf-9037-d499474968e9
md"
propagation distance z = $(round(z_aperture[iz3] * 1000, digits=3))mm
"

# ╔═╡ 27d270b9-5f3a-48eb-b5c8-79602bcbf25b
heatmap(abs.(aperture_p[:, :, iz3]), xlabel="in mm", ylabel="in mm")

# ╔═╡ 48073d7a-2257-4134-aead-aa9e5238f60e
md"""## 3. Grating

Repeat the above question with 5 rectangular apertures (20 μm by 200 μm) with a spacing of 50 μm each in x-axis. What do you observe?
Then, repeat this experiment for 200-μm separation as well. Comment on your results.


Hint: use again the `box` function. And then use `circshift`.
"""

# ╔═╡ dec9d570-8759-4455-bec6-52660cac36da
spacing_N = nothing

# ╔═╡ 8eacfde4-935f-457b-92fe-5cd84edc3857
grating = nothing

# ╔═╡ d18e9f18-d523-401f-8cca-fe9251656e09
grating_p = bpm(grating, λ, Lrec, Lrec, z_aperture);

# ╔═╡ e87820dd-b62b-4e08-81b6-abb97540e07d
@bind iz4 PlutoUI.Slider(eachindex(z))

# ╔═╡ a8588063-bcfb-44d3-8617-ad7bf79bef37
md"
propagation distance z = $(round(z_aperture[iz4] * 1000, digits=3))mm
"

# ╔═╡ 7496534a-54b7-4a8a-b09d-72eb5efa7835
heatmap(abs.(grating_p[:, :, iz4]), xlabel="in mm", ylabel="in mm")

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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 = "93284c28274d9e75218a416c65ec49d0e0fcdf3d"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.40+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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