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gaussian_optics_TP.jl
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### A Pluto.jl notebook ###
# v0.19.35
using Markdown
using InteractiveUtils
# ╔═╡ a87ae66e-95b1-11ee-29a5-d99cfec531f9
using ABCDMatrixOptics, Plots, PlutoUI
# ╔═╡ d9bd754b-327b-4b2b-8715-85a271fda3a9
using LsqFit
# ╔═╡ 77ecd2e5-3b58-4a17-94c2-1230d0f5e20b
TableOfContents()
# ╔═╡ c29fe8c6-1c2b-4d82-8eef-dfb7cff920da
md"""# Basics of Gaussian Beam
See also [Wikipedia](https://en.wikipedia.org/wiki/Gaussian_beam).
${E(r,z)} = E_0 \, \frac{w_0}{w(z)}\exp \left( \frac{-r^2}{w(z)^2}\right) \exp \left( -i \left(kz +k \frac{r^2}{2R(z)} - \psi(z) \right) \right)$
where the beam waist is given by
$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 }$
and the Rayleigh length
$z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}$
and the radius
$R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right]$
and the Gouy phase
$\psi(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right)$
The intensity is then given by
$I(r,z) \sim I_0 \exp \left( \frac{-2 \cdot r^2}{w(z)^2}\right) \exp \left( -2 \cdot i \left(kz +k \frac{r^2}{2R(z)}\right)\right)$
A Gaussian function is
$f(x) = \frac{1}{\sigma \sqrt 2} \exp\left(- \frac{x^2}{2 \sigma^2} \right)$
"""
# ╔═╡ e5193200-9c24-4327-83f3-4094b1686767
md"
For $w=w_0$ the intensity of the Gaussian beams falls off to $1/\textrm{e}^2$.
"
# ╔═╡ 8790ee5f-5579-444f-8508-070ada312a83
md""" # Numerical Experiment
"""
# ╔═╡ 0bb0fb09-f4bd-4ef3-ad23-58f5777f6dfd
λ = 633e-9
# ╔═╡ e881e496-2561-417c-b4a4-c600e020bdf6
w₀ = 0.4e-3
# ╔═╡ f536f41d-0dda-4d4c-8410-a4d1eee96633
begin
r = range(-3e-3, 3e-3, 500)
f_g(r) = exp.(- 2 .* r.^2 ./ w₀^2)
plot(r, f_g.(r), label="Gaussian Beam", xlabel="Position in m", ylabel="normalized intensity")
plot!([-w₀, -w₀], [0, 1], linecolor=:red, label="1/e^2 intensity")
plot!([w₀, w₀], [0, 1], linecolor=:red, label=nothing)
end
# ╔═╡ 487cca13-1710-4dc0-8823-8695137460ee
z_R = π * w₀^2 / λ
# ╔═╡ e43e3ee4-d69b-4f99-9ebd-90c674d8dd2a
w(z) = w₀ * √(1 + z^2 / z_R^2)
# ╔═╡ f43ed8ac-c4a9-40b5-9839-283fdef305e7
w(55e-3)
# ╔═╡ 381831af-a25d-47cc-8fcb-4b7f292e5e38
gb = GaussianBeam(;λ, w0=w₀)
# ╔═╡ 497182b7-6b49-4ed8-ad46-dc8776acefb5
plot([FreeSpace(10)], gb, ylabel="Spatial profile in m", title="Gaussian Beam propagated 1m")
# ╔═╡ f95e6581-0876-4154-8a74-38f2309c61c0
plot([ThinLens(4.5e-3), FreeSpace(4.50e-3)], gb, ylabel="Spatial profile in m", title="Gaussian Beam Focussed by a 4.5mm lens")
# ╔═╡ 1fc9e504-db8d-4efd-808d-0c407493b7fc
w_after_lens = ABCDMatrixOptics.w([ThinLens(4.5e-3), FreeSpace(4.5e-3)] * gb)
# ╔═╡ e69e2710-be1b-4821-8f19-a0d74e28d085
md"""
The waist given after focussing is $(round(w_after_lens, digits=7))m
"""
# ╔═╡ 3278db7c-abb7-4afb-a328-830eb81f9439
# ╔═╡ c92c2071-5569-489a-a4a4-cad4d31fec2f
md"""
# Measuring of Beam Divergence
To measure the beam divergence, you need to be far away such that the approximation
$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2}\approx w_0\frac{z}{z_\mathrm{R}}$
is valid.
In experiments, depending on $w_0$, this can be hard to achieve.
"""
# ╔═╡ 1bfae4ab-1963-4e88-a004-b2451a2618c7
zs = [10, 60, 110, 140, 180, 210, 240, 270, 290] .* 10^(-3)
# ╔═╡ 76346a70-692f-4ba4-bfe8-3609f3a4e6f3
w_z = [1.53, 1.56, 1.59, 1.61, 1.62, 1.65, 1.67, 1.68, 1.69] .* 10^(-3) ./ 2
# ╔═╡ 2d5d7a50-d29e-4807-90c2-01ad8f5b804d
md"The divergence and waist are connected with
$w(z) = \theta \cdot z + \textrm{const}$.
So if we fit the experimental measurements, the slope will be the divergence.
"
# ╔═╡ 0c3c78d4-124c-4962-a312-b59e5d738f34
model(z, p) = z .* p[1] .+ p[2]
# ╔═╡ a1d9b6fc-616a-4375-b0d1-5d36ee2eee01
fit_bounds = curve_fit(model, zs, w_z, [1.0, 1.0])
# ╔═╡ 36291ef8-b47e-48c4-b64b-25c91b688cef
ps = fit_bounds.param
# ╔═╡ ed587b8e-7f20-4740-9f08-f6bff77493d2
θ = fit_bounds.param[1]
# ╔═╡ 0ffe9473-8711-4184-be86-d35b87076a77
md"
The fitted $\theta$ = $(round(ps[1], digits=5))
"
# ╔═╡ 55a68ed4-52fb-4f54-9f97-f99732da7eac
begin
plot(zs, w_z, label="Measurements", xlabel="z in meter", ylabel="\$w(z)\$ in meter")
plot!(zs, model(zs, ps), label="Linear fit")
end
# ╔═╡ 64117772-d239-4f14-93b3-c9fc812f4c31
md"The initial beam diameter is given by"
# ╔═╡ 0a6379fc-ce9f-4cc7-beae-414c106a665d
w₀_measured = λ / π / θ
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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