HW06_in_class.jl

### A Pluto.jl notebook ###
# v0.19.41

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, 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)
	@assert ndims(field) == 2 "field needs to be 2 dimensional"
	# 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, 2)
	dx = Lx / Nx
	x  = Nx > 1 ? range(-Lx/2, Lx/2, Nx) : zero(typeof(Lx))
	fx = reshape(fftfreq(Nx, 1 / dx), (1, Nx))

	# this access the size of the first dimension
	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 .* n^2 .* 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

# ╔═╡ 1d6a9432-2784-4902-ba96-f082be62fa78
"""
	bpm(field, λ0, Lx, Ly, z, n_1, n_2; 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_1` is the refractive index of region 1 and `n_2` the refractive index of region 2.
`n_array` is an array filled with either `n_1` or `n_2` and the algorithm stiches the regions together.

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_split(field, λ0, Lx, Ly, z, n1=1, n2=1; window=true, n_array=ones(size(field)..., length(z)), paraxial=true)
	# free space wavenumber in m-1
	k0 = 2 * π / λ0
	# medium wavenumber m-1
	k1 = n1 * k0
	k2 = n2 * k0
	λ1 = λ0 / n1
	λ2 = λ0 / n2
	# 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
		H1 = exp(1im * k1 * dz) .* exp.(-1im .* k1 .* λ1^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
		H2 = exp(1im * k2 * dz) .* exp.(-1im .* k2 .* λ2^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
	else
		H1 = exp.(1im .* sqrt.(1 .+ 0im .- λ1^2 .* fx.^2 .- λ1^2 .* fy.^2) .* k1 .* dz) .* ((λ1^2 .* fx.^2 .+ λ1^2 .* fy.^2) .< 1)
		H2 = exp.(1im .* sqrt.(1 .+ 0im .- λ2^2 .* fx.^2 .- λ2^2 .* fy.^2) .* k2 .* dz) .* ((λ2^2 .* fx.^2 .+ λ2^2 .* 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_1 = out_field[:, :, z_index - 1] .* window_f
		
		u1_1 = invp * ((p * u0_1) .* H1)
		u1_2 = invp * ((p * u0_1) .* H2)

		out_field[:, :, z_index] .= u1_1 .* (n_array[:, :, z_index] .≈ n1) .+ u1_2 .* (n_array[:, :, z_index] .≈ n2)
	end
	return out_field
end

# ╔═╡ 79246d3c-4cdd-4752-85c5-3526d2bb2fd9
md"""# 1. Single Mode Slab Waveguide

Single-mode fibers are optical fibers designed to transmit only a single mode of light, providing high bandwidth and low signal distortion over long distances. They consist of a core, which is the central region through which light propagates, and a cladding, which surrounds the core and helps confine the light within the core.

The core of a single-mode fiber typically has a higher refractive index than the cladding, ensuring that light remains confined within the core through total internal reflection. Total internal reflection occurs when light traveling within the core encounters the interface with the cladding at an angle greater than the critical angle, causing the light to reflect back into the core rather than refracting out into the cladding.

The refractive index of the core is higher than that of the cladding, ensuring that light remains confined within the core through total internal reflection. This refractive index contrast between the core and cladding, along with careful design of the fiber geometry, enables efficient transmission of light signals through the fiber with minimal loss. The numerical aperture (NA) of the fiber, which is a measure of the acceptance angle of light into the fiber, also plays a crucial role in determining the efficiency of light coupling into the fiber and affects the total internal reflection process.


"""

# ╔═╡ 40336b8f-8b2f-4196-8f63-7fe763802843
urldownload("https://www.fibreoptic.com.au/wp-content/uploads/2021/02/single-mode-vs-multimode-FIBERS.jpg")

# ╔═╡ 571aa67d-3fb5-4051-8d99-781b68b4ce34
λ = 633e-9

# ╔═╡ e1291ca1-1b2e-4d82-ba36-6ece02896488
Ly = 100e-6 * 2

# ╔═╡ a018317d-cd5c-4d1d-9d57-34d939b3b61b
N = 512

# ╔═╡ 5d05491c-5563-4be5-90e2-d5c3eedc9b65
y = fftpos(Ly, N, CenterFT)

# ╔═╡ d125073b-4d5e-4a8a-aecf-d9bae776f952
Lx = Ly / N

# ╔═╡ 411e58f5-c412-4bfe-82c5-a5cb1fb625ec
width = 10f-6

# ╔═╡ f660db4f-dc0b-4e00-8e5d-1028d178526f
n_cladding = 1.45

# ╔═╡ 6453caf5-6b5e-4810-93a5-7abb7c4f81ac
n_core = 1.48

# ╔═╡ 89cb950b-21b4-457c-b594-13792e1b7cd3
md"""## 1. Top hat Beam
Propagate a top heat beam with the size of the fibre through the fibre.
Use the `bpm` and the `box` function.

Plot the top hat beam in free space and the top hat beam in the fibre.
"""

# ╔═╡ 54fd2a9b-6af9-4a9b-86b6-3d5c42bb53a2
z = range(0, 10_000f-6, 10_000)

# ╔═╡ e3f254f7-8892-4a90-8a3e-c6d2d07f1905
fibre = repeat(box((N, ), (width / Ly * N, )), 1, 1, length(z)) .* (n_core - n_cladding) .+ n_cladding;

# ╔═╡ 23f31642-986e-43d2-b6f8-18f703b60d1f
plot(y, fibre[:, 1, 1], ylabel="refractive index", xlabel="position in m")

# ╔═╡ 3451100c-9c50-44c1-a89c-7f823e2543c9
dz = z[2] - z[1]

# ╔═╡ 820b00bf-2856-4dda-9464-1c1699f9ed5f
width / Ly * N

# ╔═╡ ec7fefe8-b818-4205-940e-88b99bffe728
beam_rect = box((N, 1), (26, 1))

# ╔═╡ 34b64a24-ffcf-4467-8688-4667f9b3812e
size(zeros((10,)))

# ╔═╡ 361e4d1f-4263-41f1-a090-d6d8daa1860d
plot(y, beam_rect)

# ╔═╡ 00c44e0c-56be-462d-a525-eba14496e03f
phase_shift = cis.(2π ./ λ .* dz .* (fibre .- n_cladding));

# ╔═╡ 9fa39708-c158-4e58-816a-9036ad99bfd5
prop_rect = abs2.(bpm(beam_rect, λ, Lx, Ly, z));

# ╔═╡ b8c2b94c-1407-439b-9512-ae94162a5288
prop_rect_fibre = abs2.(bpm(beam_rect, λ, Lx, Ly, z, n_cladding, amplitude_array=phase_shift));

# ╔═╡ d07ac6d0-cd50-45ed-ae42-865d3ef2afe8
@bind iz Slider(axes(prop_rect,3 ), show_value=true, default=500)

# ╔═╡ 9b6eb2e3-261b-4479-ae58-08debc3de071
begin
	plot(y, prop_rect[:, 1, iz], label="rect free space", title="Distance in $(round(z[iz]*1000, digits=2)) mm")
	plot!(y, prop_rect_fibre[:, 1, iz], label="rect fibre")
	plot!([- width / 2, -width/2], [0, 1.8], linestyle=:dash, color=:gray, label="Fibre")
	plot!([width / 2, width/2], [0, 1.8], linestyle=:dash, color=:gray, label=nothing)
end

# ╔═╡ c68beb3e-8c35-4b5a-b4c1-dc57ac3d98e4
md""" ## 1.2 Gaussian beam in Fibre
Now propagate a Gaussian beam with a σ half the size of the `width` of the fibre.
Use `gaussian` for it.

Plot the propagation of the Gaussian beam in free space vs fibre.
"""

# ╔═╡ 2ab47f4a-cbf1-42ac-aab6-d8be45a58b65
beam_gauss = gaussian((N, 1), sigma=10)

# ╔═╡ 7cfe9115-3e5e-4d87-ab5e-058eff481441
prop_gauss = abs2.(bpm(beam_gauss, λ, Lx, Ly, z));

# ╔═╡ 454faa14-3f08-45b5-849d-2de0573840a6
prop_gauss_fibre = abs2.(bpm(beam_gauss, λ, Lx, Ly, z, n_cladding, amplitude_array=phase_shift));

# ╔═╡ 1eb2456a-b224-4d3f-8b2c-44b396d75deb
@bind iz2 Slider(axes(prop_gauss,3 ), show_value=true, default=500)

# ╔═╡ 14348281-a8b9-4582-b6b9-09c4dd0cefd0
begin
	plot(y, prop_gauss[:, 1, iz2], label="Gauss Free Space", title="Distance in $(round(z[iz]*1000, digits=2)) mm")
	plot!(y, prop_gauss_fibre[:, 1, iz2], label="Gauss fibre")
	plot!([- width / 2, -width/2], [0, 1.3], linestyle=:dash, color=:gray, label="fibre")
	plot!([width / 2, width/2], [0, 1.3], linestyle=:dash, color=:gray, label=nothing)
end

# ╔═╡ c263561e-bda6-4ea6-a960-301065a6dad6
md"## 1.3 Energy loss
Finally plot the total intensity of both beams over the propagation distance.
How do you calculate the energy? See `sum` for that.

Why do we observe an energy loss?
"

# ╔═╡ 3260a50f-b0d3-4998-b754-bf49d0b7e614
md"""### 1.3 Answer
todo
"""

# ╔═╡ 7fd5a666-46d2-4064-b097-bab6ee4dfa8c
size(prop_rect_fibre)

# ╔═╡ 3f7de961-6cd0-4dae-b0a5-0aa8e084a6fe
sum(prop_rect_fibre, dims=1)[1,1, :]

# ╔═╡ 4ca7ef14-f351-4663-819f-2c2d31fdbedb
begin
	plot(z, sum(prop_gauss_fibre, dims=(1,2))[:], xlabel="distance in m", ylabel="normalized intensity")
	plot!(z, sum(prop_gauss, dims=(1,2))[:])
end

# ╔═╡ 89ce5336-2c1a-4f0d-ac56-11b945b5060c
md"# 2. Multi Mode Slab Waveguide
Instead of propagating a single mode, we want to observe a higher order mode propagating in the fibre.

What initial beam can you choose to seed a higher order mode?
Choose a suitable initial beam.
"

# ╔═╡ 9e5af4ed-fa2a-4934-87c9-07e7e29c17d9
width_multi = 50f-6

# ╔═╡ 9fee0be1-fa05-4f05-bfc3-fe6ff9b9624b
fibre_multi = repeat(box((N, ), (width_multi / Ly * N, )), 1, 1, length(z)) .* (n_core - n_cladding) .+ n_cladding;

# ╔═╡ 91af8dde-55b8-4b0f-8012-52984ab8af30
#beam_multi_seed = gaussian((N,1), sigma=30) .* (1 .- box((N,1), (20,)))

# ╔═╡ 2f340646-9d8a-4b0c-b88f-61a76c48128d
beam_multi_seed = reshape((-width_multi / 2 .≤ y .≤ width_multi/2) .* sin.(y / width_multi * 2π), (N, 1));

# ╔═╡ 10207699-534c-4ef3-b832-1a45e70f153a
prop_multi_fibre = abs2.(bpm(beam_multi_seed, λ, Lx, Ly, z, n_core, amplitude_array=cis.(2π ./ λ .* dz .* (fibre_multi .- n_core)), paraxial=true));

# ╔═╡ 7a0837ad-4e2c-4583-bfe2-5235334298cc
@bind iz3 Slider(axes(prop_gauss,3 ), show_value=true)

# ╔═╡ 49d1f998-0046-493b-bf6a-5e7baf242c82
begin
	plot(y, prop_multi_fibre[:, 1, iz3], label="Multi Free Space", title="Distance in $(round(z[iz3]*1000, digits=2)) mm")
	plot!(y, prop_multi_fibre[:, 1, iz3], label="Gauss fibre")
	plot!([- width_multi / 2, -width_multi/2], [0, 1.8], linestyle=:dash, color=:gray, label="fibre")
	plot!([width_multi / 2, width_multi/2], [0, 1.8], linestyle=:dash, color=:gray)
end

# ╔═╡ d93b62b7-1d52-4042-8c31-ff264754d5fc
md"# 3. Backpropagating a Signal
In this part we consider the following situation:

We propagate a signal through a fibre and measure the complex signal (experimentally we could with holography for example).

Now, we use this result and propagate it in the reverse direction through the fibre.
Such that the result correct is, we have to conjugate (`conj.`) the output result.
After propagating reversely through the fibre, we should obtain the same signal.
This method is called [optical phase conjugation](https://en.wikipedia.org/wiki/Phase_conjugation).


## Derivation
Mathematically, propagation through a fibre can be described by a linear operation expressed with the matrix `M`.

So what we obtain after the fibre is 

$$y = M x$$

The matrix M is actually a unitary matrix, which means $M^{-1} = M^\dagger$. $^\dagger$ is the conjugate transpose of the matrix.

So we can obtain:

$$M^\dagger y = M^\dagger M x= M^{-1} M  x= x$$
$$(y^\dagger M)^\dagger = x$$
$$(M' y^*)^* = x$$

where $^*$ indicates the conjugate operation  and $'$ the transpose.
Finally, we know $M$ looks like

$$M = D_K \cdots \mathcal{P} \cdot D_1 \cdot \mathcal{P}$$

which means that we multiply by a diagonal matrix $D_i$ to apply the slice wise phase shift. $\mathcal{P}$ is the propagation operation (free space propagation) between the slices.
Now 

$$M' = (D_K \cdots \mathcal{P} \cdot D_1 \cdot\mathcal{P})' =  \mathcal{P}' \cdot D_1 ' \cdot \mathcal{P}' \cdots D_K'$$

Since $D$ is diagonal and the free space propagation matrix $\mathcal{P}$ is symmetric, we remove the '

$$M' = \mathcal{P} \cdot D_1 \cdot \mathcal{P} \cdots D_K$$


From this derivation, it is clear that we need to conjugate the output of the fibre $y$. Then we propagate with the same beam propagation method, but we have to reverse the fibre and then finally we conjugate the result again. Then we obtain the initial $x$.

Physically, that means we propagate the signal forward through the fibre. Then we conjugate the phases and propagate in reverse direction through the fibre.

## Question

* Fill in the missing gaps and compare the final phase conjugated result with the original sample.
* Why is the final result only approximately the same as the input? Include in your reasoning the numerical aperture of the fibre.
* What is the hidden assumption in the matrix derivation above?

"

# ╔═╡ 4e404b5c-aa44-4d5b-a19b-d1930838565f
fibre_3 = repeat(box((N, ), (width_multi / Ly * N, )), 1, 1, 1000) .* (n_core - n_cladding) .+ n_cladding;

# ╔═╡ 23a3015c-15b3-42cc-826b-9aec986b2b06


# ╔═╡ a3e9d872-b806-4278-bac5-fcc2e7144ba8
z2 = range(0, 1000f-6, 1000)

# ╔═╡ 735eca86-e698-41f0-842f-9c638c45a211
begin
	beam_3 = zeros((N, 1));
	beam_3[200:220, 1] .= 1
	beam_3[250:280, 1] .= 2
	beam_3[290:310, 1] .= 3
end;

# ╔═╡ b30ae536-213a-4668-a88a-503ed08307af
prop_beam_3 = bpm(beam_3, λ, Lx, Ly, z2, n_core, amplitude_array=cis.(2π ./ λ .* dz .* (fibre_multi .- n_core)), paraxial=true);

# ╔═╡ dec1c5d6-de08-4601-965e-ccda9fb2ad5d
size(prop_beam_3)

# ╔═╡ bd8aec79-a435-4267-aa19-b79edce0c23a
prop_beam_3_back = conj.(bpm(conj.(prop_beam_3[:, :, end]), λ, Lx, Ly, z2, n_core, amplitude_array=reverse(cis.(2π ./ λ .* dz .* (fibre_multi .- n_core))), paraxial=true));

# ╔═╡ 05cdc062-7b04-4936-a9cb-2f7685a5e353
@bind iz4 Slider(axes(prop_beam_3,3 ), show_value=true)

# ╔═╡ 6ad8f2b4-6fc1-499c-883d-837f903d3313
begin
	plot(y, abs2.(prop_beam_3[:, 1, iz4]), label="Signal through fibre", title="Distance in $(round(z2[iz4]*1000, digits=2)) mm")
	plot!([- width_multi / 2, -width_multi/2], [0, 10], linestyle=:dash, color=:gray, label="fibre")
	plot!([width_multi / 2, width_multi/2], [0, 10], linestyle=:dash, color=:gray)
end

# ╔═╡ f0c08700-ed9a-473b-980e-a6b7e130794b
@bind iz5 Slider(axes(prop_beam_3,3 ), show_value=true, default=size(prop_beam_3, 3))

# ╔═╡ 39f89d46-8656-47d5-8132-71c575df0ab4
begin
	plot(y, abs2.(prop_beam_3_back[:, 1, iz5]), label="Backpropagated", title="Distance in $(round(reverse(z2)[iz5]*1000, digits=2)) mm")
	plot!(y, abs2.(prop_beam_3[:, :,1]), label="initial beam")
	plot!([- width_multi / 2, -width_multi/2], [0, 1.8], linestyle=:dash, color=:gray, label="fibre")
	plot!([width_multi / 2, width_multi/2], [0, 1.8], linestyle=:dash, color=:gray)
end

# ╔═╡ 4c950d3c-6cf9-4bfd-9efe-5c5753655625
md"## Answer
TODO
"

# ╔═╡ a6e9d81b-81bb-47eb-9aad-0c9c64cba492
md"## Answer
The NA of the fibre is calculated with

$$\text{NA} = \sqrt{n_\text{core}^2 - n_\text{cladding}^2}$$


From that we can calculate the maximum frequency the fibre transmit.
We can calculate a bandpass filter and convolved with it.
Then we see that the fibre acts as a lowpass filter.

"

# ╔═╡ 5421b519-6e2c-4e0b-982f-e578c1e08350
NA = sqrt(n_core^2 - n_cladding^2)

# ╔═╡ 51696c9f-e096-41bf-8609-f1e055a600c4
freqs = fftfreq(N, inv(Ly / N))

# ╔═╡ 82ddbdd3-e7d0-468e-871c-9fb8d88af4c7
freq_cut_off = abs.(freqs) .< NA / λ

# ╔═╡ d521af6c-11e0-4c3b-8df9-8b8770abae59
plot(fftshift(freqs), fftshift(freq_cut_off))

# ╔═╡ 7647e94e-e999-4248-9afe-8e056d90d52a
beam_3_filtered = ifft(fft(beam_3) .* freq_cut_off);

# ╔═╡ f9ac9bbe-5ce1-456f-a4b4-b3315fe79a16
@bind iz6 Slider(axes(prop_beam_3,3 ), show_value=true, default=size(prop_beam_3, 3))

# ╔═╡ aa651deb-11ff-4376-ab88-2929ede61761
begin
	plot(y, abs2.(prop_beam_3_back[:, 1, iz6]), label="Backpropagated", title="Distance in $(round(reverse(z2)[iz6]*1000, digits=2)) mm")
	plot!(y, abs2.(prop_beam_3[:, :,1]), label="initial beam")
	plot!(y, abs2.(beam_3_filtered[:,1]), label="filtered beam")

	plot!([- width_multi / 2, -width_multi/2], [0, 10], linestyle=:dash, color=:gray, label="fibre")
	plot!([width_multi / 2, width_multi/2], [0, 10], linestyle=:dash, color=:gray)
end

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[[deps.Gettext_jll]]
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[[deps.Glib_jll]]
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[[deps.Graphite2_jll]]
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    IntervalSetsRecipesBaseExt = "RecipesBase"
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[[deps.IrrationalConstants]]
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[[deps.LAME_jll]]
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[[deps.LZO_jll]]
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[[deps.LibGit2]]
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[[deps.Libgcrypt_jll]]
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[[deps.Libglvnd_jll]]
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[[deps.Libtiff_jll]]
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[[deps.Libuuid_jll]]
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[[deps.LittleCMS_jll]]
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    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
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[[deps.MbedTLS]]
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[[deps.MbedTLS_jll]]
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[[deps.MicroCollections]]
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[[deps.Missings]]
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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"
"""

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