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HW08_in_class.jl
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HW08_in_class.jl
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### 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)
# 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
# ╔═╡ 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
For a slab waveguide the fundamental modes can be calculated analytically.
If the single mode is launched into the fibre, the mode should not change while propagating through the fibre.
A derivation can be found [here](https://faculty.kfupm.edu.sa/ee/ajmal/files/Ajmal_Thesis_Chap2.pdf).
Equation (2.25) of the reference tells the condition when an eigenmode exists.
This equation cannot be analytically solved.
One simple method is to calculate for different `n_eff` the values of left hand side and right hand side of the equation.
A solution is found when they are equal.
All relevant parameters are already defined below.
But, copy past them into a `plot_condition()` function where you calculate the variables for different `n_eff`. Find all values for `n_eff` which result in eigenmodes.
Show plots of two of the eigenmodes and show that they do not change while propagating through the fibre.
Also analyze the $V$ parameter of the fibre and explain it.
"""
# ╔═╡ 571aa67d-3fb5-4051-8d99-781b68b4ce34
λ = 633e-9
# ╔═╡ e1291ca1-1b2e-4d82-ba36-6ece02896488
Ly = 100e-6
# ╔═╡ 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
# ╔═╡ f660db4f-dc0b-4e00-8e5d-1028d178526f
n_cladding = 1.44
# ╔═╡ 6453caf5-6b5e-4810-93a5-7abb7c4f81ac
n_core = 1.443
# ╔═╡ 54fd2a9b-6af9-4a9b-86b6-3d5c42bb53a2
z = range(0, 1000f-6, 1_000)
# ╔═╡ 3451100c-9c50-44c1-a89c-7f823e2543c9
dz = z[2] - z[1]
# ╔═╡ c3623162-dfff-405c-99ce-803bfef73256
k = 2π / λ
# ╔═╡ 5ab43b44-d1db-46fa-9192-29bf6480f585
A = 1
# ╔═╡ 527783e0-9b7f-4a4d-8412-04fb92d3ed05
d = 5e-6
# ╔═╡ 8cdd1db5-2a6f-491f-af92-acaa6ee7a781
make_fibre(x) = x < 0 ? n_cladding : x < 2*d ? n_core : n_cladding
# ╔═╡ 154a75fa-ba28-44c7-be5e-59871a8dee10
fibre = repeat(make_fibre.(y), 1,1, length(z));
# ╔═╡ 980990b8-a7e3-4296-a1da-43af6e323937
NA = sqrt(n_core^2-n_cladding^2)
# ╔═╡ 832b1c19-9a26-498e-9068-4971f83d5a3d
V = 2π / λ * d * NA
# ╔═╡ ed1d3e2f-c365-43d2-91a2-3e255fcd5bd7
M = V^2 / 2
# ╔═╡ 27e9e20a-6cb2-453d-ba74-83465d397dd3
n_eff = 1.44208
# ╔═╡ 00c44e0c-56be-462d-a525-eba14496e03f
phase_shift = cis.(2π ./ λ .* dz .* (fibre .- n_eff));
# ╔═╡ d30440ea-0ca2-40b8-b490-5df43cd027d9
β = k * n_eff
# ╔═╡ aa50dd5b-a83c-4272-9c3d-0da821bfa830
p = sqrt(β^2-k^2 * n_cladding^2)
# ╔═╡ 9c0b9ad6-e88e-460d-a8ce-d450254ef3b4
r = p
# ╔═╡ 86a61201-a91b-40ff-8f53-f5a1375ca279
q = sqrt(k^2 * n_core^2 - β^2)
# ╔═╡ 41756530-ce4e-42a1-9e97-b688cf937c92
B = r * A / q
# ╔═╡ dc4d9a25-b70c-4b13-88bd-8c8f7c181dfd
tan(2*d*q)
# ╔═╡ 41bf8d2e-1a66-4757-ad14-a0af6b7c955b
q * (p + r) / (q^2 - p * r)
# ╔═╡ 68d9786d-c08e-4d1a-ba49-e9705829131d
Ey(x) = x < 0 ? A * exp(r * x) : x < 2*d ? A * cos(q*x) + B * sin(q*x) : (A * cos(2*d*q) + B * sin(2*d*q)) * exp(-p*(x-2*d))
# ╔═╡ 7f5d53cc-3249-4dd3-b818-69322621c9f2
function plot_condition()
n_eff = range(n_cladding + 1e-8, n_core - 1e-8, 1000)
β = k .* n_eff
p = sqrt.(β .^ 2 .- k .^ 2 .* n_cladding .^ 2)
q = @. sqrt(k^2 * n_core^2 - β^2)
r = p
c1 = @. q * (p + r) / (q^2 - p * r)
c2 = @. tan(2*d*q)
p = plot(n_eff, c1, ylim=(-40, 40), xlabel="n_eff")
plot!(n_eff, c2)
return p
end
# ╔═╡ dd2d9b09-5731-4ec6-bb24-304ebdaf3441
plot_condition()
# ╔═╡ 08b5e24f-7f0b-440f-b0d2-8765ac43b09d
eigenmode = reshape(Ey.(y), (:, 1));
# ╔═╡ 454faa14-3f08-45b5-849d-2de0573840a6
prop_mode = real.(bpm(eigenmode, λ, Lx, Ly, z, n_eff, amplitude_array=phase_shift, paraxial=true));
# ╔═╡ d07ac6d0-cd50-45ed-ae42-865d3ef2afe8
@bind iz Slider(axes(prop_mode,3 ), show_value=true, default=500)
# ╔═╡ 9b6eb2e3-261b-4479-ae58-08debc3de071
begin
plot(y, prop_mode[:, 1, iz], label="rect free space", title="Distance in $(round(z[iz]*1000, digits=2)) mm", ylabel="real value of the mode")
plot!([0, 0], [0, 5], linestyle=:dash, color=:gray, label="Fibre")
plot!([2*d, 2*d], [0, 5], linestyle=:dash, color=:gray)
plot!(y, real.(phase_shift)[:, 1,1])
end
# ╔═╡ 9574a6df-937a-48b2-9f0e-efc51c9b331e
md"## Answer
"
# ╔═╡ e7e1e2cf-917e-46c5-82b1-d6e2a54a55f5
md"# 2. Optical Interconnect
In this case the light is propagating in the bottom fibre.
We want to achieve that after the close encounter of the fibres the light only propagates in the top fibre and leaves the bottom fibre.
Try to find values for `n_core2` and `n_cladding2` which are the refractive indices of the upper fibre and the cladding region such that this happens.
Explain why such a device would be useful.
"
# ╔═╡ 8bddb717-4b72-4fdc-8086-b29243e7e39e
N2 = 512
# ╔═╡ 05b654f0-3872-4a09-8c2b-b13292963fff
sz2 = ((N2, 1, 1000))
# ╔═╡ 0ab28a39-aa1a-4d06-82da-d0d1744f16b5
L2 = 50f-6
# ╔═╡ 6888d5c2-66d9-46c3-8fbf-5e99fe425dd4
n_core2 = 1.45
# ╔═╡ cfa6773f-ac93-415d-8f0a-bfc1069c2909
n_core2_2 = 1.45
# ╔═╡ c31a9a93-a22e-49e0-a9d3-2d73c51e68b6
n_cladding2 = 1.441
# ╔═╡ f3e52175-396b-4e10-b8e6-4d77ab4c425f
n_eff2 = 1.44
# ╔═╡ b7ed6382-90fa-4d9f-8ed7-3e3eef0f9082
y2 = fftpos(L2, sz2[1], CenterFT)
# ╔═╡ 7c0b2a6a-9a9b-4670-be47-de781aeeab67
z2 = fftpos(L2 * 15, sz2[3])
# ╔═╡ 90fa8397-9d2c-4fe9-a171-719b11ee3ba8
begin
fibre2 = ((rr(sz2, offset=(-800, 1, 256)) .<= 1000) .-
(rr(sz2, offset=(-800, 1, 256)) .<= 990)) .* (n_core2_2 .- n_cladding2) .+
((rr(sz2, offset=(1200, 1, 256)) .<= 1000) .-
(rr(sz2, offset=(1200, 1, 256)) .<= 990)) .* (n_core2 .- n_cladding2) .+ n_cladding2;
beam2 = circshift(gaussian((N2, 1), sigma=8), (-96,)) .* cis.(0.02 .* xx((N2, 1)));
end;
# ╔═╡ 68e5fb09-63e1-4c51-ad52-04a36b5c0877
phase_shift2 = cis.(2π ./ λ .* dz .* (fibre2 .- n_eff2));
# ╔═╡ 40dc5147-d200-47f1-83cc-fc3ff75a1576
beam_prop2 = abs.(bpm(beam2, λ, L2 / N2, L2, z2, n_eff2, amplitude_array=phase_shift2 , paraxial=true));
# ╔═╡ 9bfb002f-c22e-42bb-9c0a-aad052e9e9db
heatmap(z2, y2, beam_prop2[:, 1, :], title="intensity of the propagated light")
# ╔═╡ c756d6f6-611c-42a9-a5e2-6cc6e9b8aa94
heatmap(z2, y2, fibre2[:, 1, :], title="refractive index of the fibres")
# ╔═╡ 9e6cd1de-3d9a-4a07-8a24-071c903ea327
md"## Answer
"
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