nonlin-opt-sigmoid.jl

### A Pluto.jl notebook ###
# v0.19.9

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

# ╔═╡ 93233800-f607-11eb-3ef6-9bc20acb1e57
begin
	using DiffEqOperators, OrdinaryDiffEq, SpecialFunctions, Interpolations, Plots, Plots.PlotMeasures, PlutoUI, NPZ, ThreadPools
	#import PyPlot
	#pyplot()
end;

# ╔═╡ faa8b77f-dfd8-41d6-884c-d4a3775fc086
md"""
# Sigmoid Activation Function for Neural Networks based on Nonlinear Optics

*Supplementary to "All-Photonic Artificial Neural Network Processor Via Non-linear Optics" by Basani, Heuck, Englund, and Krastanov, 2022*

Can be found at:

- Repository github.com/Krastanov/all-photonic-nn
- Interactive at pluto.krastanov.org/nonlin-opt-sigmoid.html
- Archived together with the publication
"""

# ╔═╡ 18d01b35-04d1-4408-bf56-087f7e17d46d
md"""

# Interaction in the nonlinear waveguide

Solving

$\frac{n}{c}\partial_tE_n + \partial_zE_n
=-\kappa E_s^2 - \alpha E_n$

$\frac{n}{c}\partial_tE_s + \partial_zE_s
=\kappa E_n E_s^* - \alpha E_s$

We will work in units of length $z_0$ and units of time $t_0=\frac{n}{c}z_0$. $E_s$ and $E_n$ will be measured in separate units $\varepsilon_s$ and $\varepsilon_n$, leading to dimensionless equations (we use hat to denote dimensionless):

$\partial_\hat{t}\hat{E}_n + \partial_\hat{z}\hat{E}_n
=-\kappa z_0 \frac{\varepsilon_s^2}{\varepsilon_n} \hat{E}_s^2 - \alpha z_0 \hat{E}_n$

$\partial_\hat{t}\hat{E}_s + \partial_\hat{z}\hat{E}_s
=\kappa z_0 \varepsilon_n \hat{E}_n \hat{E}_s^* - \alpha z_0 \hat{E}_s$

The initial conditions are $\hat{E}_n = e^{ -\frac{z^2}{4w^2} } e^{-i\varphi_0}$ and $\hat{E}_s = e^{ -\frac{z^2}{4w^2} }$.

We can simplify the problem additionally by introducing:

- new variables $\partial_\hat{\eta}=\partial_\hat{t}+\partial_\hat{z}$ and $\partial_\hat{\xi}=\partial_\hat{t}-\partial_\hat{z}$,
- i.e. $\hat{t} = \hat{\eta}+\hat{\xi}$ and $\hat{z}=\hat{\eta}-\hat{\xi}$,
- i.e. $2\hat{\eta}=\hat{t}+\hat{z}$ and $2\hat{\xi}=\hat{t}-\hat{z}$.

Boundary conditions like $f_{\hat{t}\hat{z}}(0,z)$ transform into $f_{\hat{\eta}\hat{\xi}}(\frac{z}{2},-\frac{z}{2})=f_{\hat{t}\hat{z}}(0,z)$. The equations to solve are:

$\partial_\hat{\eta}\hat{E}_n =-\kappa z_0 \frac{\varepsilon_s^2}{\varepsilon_n} \hat{E}_s^2 - \alpha z_0 \hat{E}_n$

$\partial_\hat{\eta}\hat{E}_s =\kappa z_0 \varepsilon_n \hat{E}_n \hat{E}_s^* - \alpha z_0 \hat{E}_s$

**TODO: implement the $\eta\xi$ solution as it should be much faster.**
"""

# ╔═╡ 12b6d3a2-e266-4491-8e57-d41c0bdfefe7
begin
	const T = 1. # All simulations would be evolved for one unit of time
	const Z = 5. # All simulations will span a few units of space
	# The above two can be simplified if we rewrite the equations wrt t+z
	const w = 0.1 # All pulses will have width of 0.1 units
	nknots = 800 # Make smaller for interactive use and larger for higher accuracy
	dz = Z/(nknots+1)
	const knots = range(dz, step=dz, length=nknots)
	ord_deriv = 1
	ord_approx = 2 # Make smaller for interactive use and larger for higher accuracy
	const Δ = CenteredDifference(ord_deriv, ord_approx, dz, nknots)
	const bc = Dirichlet0BC(Float64)
	
	function step!(∂ₜÊ, Ê, p, t)
		(kz₀εₛ²εₙ⁻¹, kz₀εₙ, αz₀) = p
	    ∂ₜÊₙ = @view ∂ₜÊ[:,1]
	    ∂ₜÊₛ = @view ∂ₜÊ[:,2]
	      Êₙ = @view   Ê[:,1]
	      Êₛ = @view   Ê[:,2]
	    ΔÊₙ  = Δ*bc*Êₙ
		ΔÊₛ  = Δ*bc*Êₛ
	    ∂ₜÊₙ .= .-ΔÊₙ .- kz₀εₛ²εₙ⁻¹ .* Êₛ .^ 2         .- αz₀ .* Êₙ
	    ∂ₜÊₛ .= .-ΔÊₛ .+ kz₀εₙ      .* Êₙ .* conj.(Êₛ) .- αz₀ .* Êₛ
		return ∂ₜÊ
	end
	
	Ê₀param(z,w;φ₀=0) = exp( -(z-5*w)^2 / (2*w^2) ) * exp(-im*φ₀) # 5σ should be enough to avoid clipping
end;

# ╔═╡ 25074b34-1361-4795-904f-f9624dca3e74
begin
	md"""
	`kz₀εₛ²εₙ⁻¹`: $(@bind p1_kz₀εₛ²εₙ⁻¹ Slider(0:0.1:5; default=1, show_value=true))
	
	`kz₀εₙ_____`: $(@bind p1_kz₀εₙ      Slider(0:0.1:5; default=1, show_value=true))

	`αz₀_______`: $(@bind p1_αz₀        Slider(0:0.1:5; show_value=true))
	
	`φ₀________`: $(@bind p1_φ₀         Slider(0:0.1:2pi; show_value=true))
	"""
end 

# ╔═╡ 3686615d-c3cd-45cf-897d-791852a64fe9
let
	Êₙ₀ = Ê₀param.(knots, w; φ₀=p1_φ₀)
	Êₛ₀ = Ê₀param.(knots, w; φ₀=0)
	Ê₀ = hcat(Êₙ₀, Êₛ₀)
	
	params = (p1_kz₀εₛ²εₙ⁻¹,p1_kz₀εₙ,p1_αz₀)
	εₛ²εₙ⁻² = (p1_kz₀εₛ²εₙ⁻¹/p1_kz₀εₙ)
	global p1_εₛ²εₙ⁻² = εₛ²εₙ⁻²
	prob = ODEProblem(step!, Ê₀, (0, T), params)
	alg = Tsit5()
	sol = solve(prob, alg)
	
	Ê₀ₛₒₗ = sol(0.)
	Ê₁ₛₒₗ = sol(1.)
	global p1_Ê₀ₛₒₗ = Ê₀ₛₒₗ 
	global p1_Ê₁ₛₒₗ = Ê₁ₛₒₗ 
	mask₀ = sum(abs.(Ê₀ₛₒₗ),dims=2)[:,1] .> 1e-2
	mask₁ = sum(abs.(Ê₁ₛₒₗ),dims=2)[:,1] .> 1e-2
	plot( knots[mask₀], abs.(Ê₀ₛₒₗ[mask₀,:]), c=[1 2], ls=[:solid :dash], lw=4,
		  label=["N @ t=0" "S @ t=0"])
	plot!( knots[mask₀], sqrt.(sum(abs.(Ê₀ₛₒₗ[mask₀,:]).^2 .*[1 εₛ²εₙ⁻²],dims=2)), c=:black, ls=:dot, lw=2,
		  label="Tot @ t=0")
	plot!(knots[mask₁], abs.(Ê₁ₛₒₗ[mask₁,:]), c=[1 2], ls=[:solid :dash], lw=4,
		  label=["N @ t=1" "S @ t=1"])
	plot!( knots[mask₁], sqrt.(sum(abs.(Ê₁ₛₒₗ[mask₁,:]).^2 .*[1 εₛ²εₙ⁻²],dims=2)), c=:black, ls=:dot, lw=2,
		  label="Tot @ t=1")
	plot!(xlabel="space - time", ylabel="Abs pulse value", legend=:outertopright)
end

# ╔═╡ ecb69589-5021-4742-ac72-d27bf7977617
begin
	en0 =              sum(abs.(p1_Ê₀ₛₒₗ[:,1]).^2)*dz
	es0 = p1_εₛ²εₙ⁻² * sum(abs.(p1_Ê₀ₛₒₗ[:,2]).^2)*dz
	en1 =              sum(abs.(p1_Ê₁ₛₒₗ[:,1]).^2)*dz
	es1 = p1_εₛ²εₙ⁻² * sum(abs.(p1_Ê₁ₛₒₗ[:,2]).^2)*dz
	
	
	
	md"""
	The (εₛ/εₙ)² ratio is $(p1_εₛ²εₙ⁻²)
	
	|time |energy in Êₙ  | energy in Êₛ| total energy |
	| --- | --- | --- | --- |
	|0|$(round(en0,digits=3))|$(round(es0,digits=3))|$(round(en0+es0,digits=3))|
	|1|$(round(en1,digits=3))|$(round(es1,digits=3))|$(round(en1+es1,digits=3))|
	
	"""
end

# ╔═╡ 76902710-23dc-45fe-95dc-5fcc779bfcb6
md"""

# Capture

Solving

$\frac{{\rm d}A}{{\rm d}t}=-\frac{\gamma+\gamma_h}{2}A + \sqrt \gamma S_i$

$S_o=S_i - \sqrt \gamma A$

where we impose $S_o = 0$ leading to $\sqrt\gamma = \frac{S}{A}$ and

$\frac{{\rm d}A}{{\rm d}t}=-\frac{\gamma_h}{2}A + \frac{S_i^2}{2A}$

and for an Gaussian input $S_i = S e^{-\frac{(t-t_0)^2}{2w^2}}$.

It seems this is a special case of [Bernoulli's equation](https://mathworld.wolfram.com/BernoulliDifferentialEquation.html).

The solution ([from Wolfram Alpha](https://www.wolframalpha.com/input/?i=solve+f%27%28t%29+%3D+-kappa%2F2*f%28t%29+%2B+S%5E2%2F2*exp%28-%28t-t0%29%5E2%2Fw%5E2%29%2Ff%28t%29)): 

$A(t) = \sqrt{C_1 e^{-\gamma_h t} - \frac{1}{2}\sqrt\pi S^2 w e^{\frac{\gamma_h}{2} (\frac{\gamma_h w^2}{2} + 2 t_0 - 2 t)} {\rm erf}\left(\frac{\frac{\gamma_h w^2}{2}  + t_0 - t}{w}\right)}$

Finding $C_1$ to fullfil $A(0)=0$ leads to:

$A(t) = S\sqrt{\sqrt\pi w e^{\frac{\gamma_h}{2} (\frac{\gamma_h w^2}{2} + 2 t_0 - 2 t)}\frac{\left( 1 + {\rm erf}\left(\frac{\frac{-\gamma_h w^2}{2} - t_0 + t}{w}\right)\right)}{2}}$

"""

# ╔═╡ 1462c607-a4c0-4b65-90b0-9b728711276f
begin
	Sᵢ(w,t₀,S,t) = S * exp(-0.5*(t-t₀)^2/w^2)
	A(γₕ,w,t₀,S,t) = S * sqrt(√π * w
		* exp(γₕ/2*(γₕ/2*w^2 + 2*t₀ - 2*t))
	    * 0.5 * (1-erf((γₕ/2*w^2 + t₀ - t)/w))
	)
	function γ(γₕ,w,t₀,t)
		if abs(t₀-t)/w > 6
			return 0.0
		else
			return (Sᵢ(w,t₀,1.0,t)/A(γₕ,w,t₀,1.0,t))^2
		end
	end
end;

# ╔═╡ 49f3d21a-a8fd-4ed2-ae6b-409a34c57d67
md"And example plot with unitless time."

# ╔═╡ 0e0041b3-8641-481b-b5bf-821f54dcb604
let
	ts = 0:0.1:20
	w = 1.
	t₀ = 9.
	S = 1.
	γₕ = 0.5
	mask = ts.>(t₀-2w)
	plot(ts, A.(0,w,t₀,S,ts), label="A at no loss", c=1)
	plot!(ts, A.(γₕ,w,t₀,S,ts), label="A at γₕ = $(γₕ)", c=2)
	plot!(ts[mask], γ.(0,w,t₀,ts)[mask]*(√π*w), label="γ at no loss", c=1, ls=:dot, lw=3)
	plot!(ts[mask], γ.(γₕ,w,t₀,ts)[mask]*(√π*w), label="γ at γₕ = $(γₕ)", c=2, ls=:dot, lw=3)
    plot!(ts, Sᵢ.(w,t₀,S,ts)*√(√π*w), label="Sᵢₙ (w = $(w))", color=:black, linestyle=:dash)
	plot!(legend=:bottomright, yticks=nothing,xlabel="time")
end

# ╔═╡ 1dccf41e-fe5d-487e-9679-58aa08f552ed
md"A quick sanity check comparing to numerical solutions:"

# ╔═╡ 025a3d16-3548-4ba3-9468-e0f67c1c55fb
begin
	function dA(A, p, t)
		(γfunc, Sfunc, γₕ) = p
		γ = γfunc(t)
		S = Sfunc(t)
		dA = -(γₕ+γ)*A/2 + √γ * S
        return dA
	end;
	function dAelimγ(A, p, t)
		(Sfunc, γₕ) = p
		S = Sfunc(t)
		dA = -γₕ*A/2 + S^2/A/2
		return dA 
	end;
end

# ╔═╡ 783f9b75-de7d-45cb-8ee4-a424184d438a
let
	ts = 0:0.1:40
	w = 2.
	t₀ = 9.
	S = 1.0 + 1.0im
	γₕ = 0.2
	A0 = 1e-6 + 0.0im
	
	plot(xlabel="time",title="Numerical vs Analytical solution")
	plot!(ts, A.(γₕ,w,t₀,S,ts), label="A(t) analytical", lw=8)

	γfunc(t) = γ(γₕ,w,t₀,t)
	Sfunc(t) = Sᵢ(w,t₀,S,t)
	param_test_capture = (γfunc, Sfunc, γₕ)
	prob = ODEProblem(dA, A0, (0, ts[end]), param_test_capture)
	alg = KenCarp4(autodiff=false)
	sol = solve(prob, alg)
	plot!(sol, label="A(t) semi-numerical", lw=6)

	param_test_capture_elimγ = (Sfunc, γₕ)
	prob_elimγ = ODEProblem(dAelimγ, A0, (0, ts[end]), param_test_capture_elimγ)
	alg = KenCarp4(autodiff=false)
	sol_elimγ = solve(prob_elimγ, alg)
	plot!(sol_elimγ, label="A(t) numerical", lw=3)
	plot!(ylabel="Im", zlabel="Re")
end

# ╔═╡ c23699e9-0a2e-49e8-8f32-3968fa7893f7
md"# Full simulation of the nonlinear activation function"

# ╔═╡ c14acc74-f848-4b00-a3d1-a0ff1c152b96
begin
	const offset = 5 # 7σ should be enough to avoid clipping
	function nonlinsim(;w=0.1,
			            kz₀εₛ²εₙ⁻¹ = 0.0,
						kz₀εₙ = 1.0,
		                αz₀=0,
						γₕ = 0.1,
			            φ₀ = 0.0,
						use_sub = false
		)

		# Waveguide propagation parameters
		# kz₀εₛ²εₙ⁻¹
		# kz₀εₙ
		# αz₀
		# Capture parameters
		# γₕ = 0.1

		# Some implementation details constants
		t₀ = 0.0
		A0 = 1e-6+0.0im
		Tcapt = (-offset*w, offset*w)

		# Prepare wave before interaction with the subharmonic
		Ê₀param(z,φ₀) = exp(-im*φ₀) * exp( -(z-offset*w)^2 / (2*w^2) )
		Êₙ₀ = Ê₀param.(knots,φ₀)
		Êₛ₀ = Ê₀param.(knots,0.0)
		Ê₀ = hcat(Êₙ₀, Êₛ₀)

	    # Solve the interaction with the subharmonic
		params_nonlin = (kz₀εₛ²εₙ⁻¹,kz₀εₙ,αz₀)
		prob = ODEProblem(step!, Ê₀, (0, T), params_nonlin)
		alg = Tsit5()
		sol = solve(prob, alg)

		# Time-offset the wave before capture to make the equation simpler
		# Remember that time and space have a different direction on the plots
		Ê₁ₛₒₗ = sol(T)
		neuronpulse = use_sub ? Ê₁ₛₒₗ[:,2] : Ê₁ₛₒₗ[:,1]
		Sᵢₙ(t) = LinearInterpolation(knots,neuronpulse)(-t+offset*w+T)

		# Solve the capture
		γfunc(t) = γ(γₕ,w,t₀,t)
		param_test_capture = (γfunc, Sᵢₙ, γₕ)
		prob = ODEProblem(dA, A0, Tcapt, param_test_capture)
		alg = Tsit5()
		capturesol = solve(prob, alg)
		
		(; Ê₀, Sᵢₙ, γfunc, capturesol)
	end
	
	normsim = nonlinsim(w=w,kz₀εₛ²εₙ⁻¹=0.0,kz₀εₙ=1.0,αz₀=0.0,γₕ=0.0,φ₀=0.0)
	Anorm = normsim.capturesol[end]
end;

# ╔═╡ 9d773718-ff18-404c-8213-41da8062294f
begin
	md"""
	`kz₀εₛ_`: $(@bind p2_kz₀εₛ      Slider(0:0.1:5; default=0.2, show_value=true))
	
	`εₙεₛ⁻¹`: $(@bind p2_εₙεₛ⁻¹     Slider(0:0.1:5; default=1, show_value=true))

	`αz₀___`: $(@bind p2_αz₀        Slider(0:0.1:5; show_value=true))

	`γₕ____`: $(@bind p2_γₕ         Slider(0:0.1:5; show_value=true))
	
	`φ₀____`: $(@bind p2_φ₀         Slider(0:0.1:2pi, show_value=true))
	
	angle of view: $(@bind p2_cam   Slider(0:1:90, default=20, show_value=false))
	"""
end 

# ╔═╡ be709699-4e82-4e1f-b3b0-f40a9f2cab2d
let
	kz₀εₛ = p2_kz₀εₛ
	εₙεₛ⁻¹ = p2_εₙεₛ⁻¹
	kz₀εₛ²εₙ⁻¹ = kz₀εₛ/εₙεₛ⁻¹
	kz₀εₙ = kz₀εₛ*εₙεₛ⁻¹
	#kz₀εₛ²εₙ⁻¹ = 0.0
	#kz₀εₙ = 1.0
	αz₀ = p2_αz₀
	γₕ = p2_γₕ
	φ₀ = p2_φ₀

	global p2_sim = nonlinsim(w=w,kz₀εₛ²εₙ⁻¹=kz₀εₛ²εₙ⁻¹,kz₀εₙ=kz₀εₙ,αz₀=αz₀,γₕ=γₕ,φ₀=φ₀)
end;

# ╔═╡ eda4d017-d65f-4c60-a45d-6bbb37a28ff1
let
	span = range(-offset*w,offset*w,length=100)
	plot()
	plot!(-(knots.-offset*w), p2_sim.Ê₀, c=[1 2], ls=[:solid :dash], lw=4,
		  label=["N@t=0" "S@t=0"])
	plot!(span, p2_sim.Sᵢₙ.(span), c=3, ls=:solid, lw=4,
		  label="N@t=1")
	plot!(span, p2_sim.capturesol.(span)/Anorm, label="capture", lw=2)
	plot!(xlabel="time", xlim=(span[begin],span[end]), legend=:topleft)
	plot!(ylabel="Re", zlabel="Im", camera=(p2_cam,40))
end

# ╔═╡ f803aeec-2ce2-45ca-899f-fe6c31eb3bda
md"""## At φ₀=0"""

# ╔═╡ b40d4754-85c1-4edd-9793-926cc1a912e3
let
	αz₀ = 0.0
	γₕ = 0.0
	
	
	global kz₀εₛs = [0, 0.2, 0.3]
	global εₙεₛ⁻¹_scale = 20.0
	global εₙεₛ⁻¹s = ((-1:0.2:1) .+ 1e-2)*εₙεₛ⁻¹_scale

	if isfile("nonlins_phi=0.npz")
		f = npzread("nonlins_phi=0.npz")
		global Ass_cat = f["allA"]./εₙεₛ⁻¹s
	else
		Ass = qmap(kz₀εₛs) do kz₀εₛ
			As = qmap(εₙεₛ⁻¹s) do εₙεₛ⁻¹
				kz₀εₛ²εₙ⁻¹ = kz₀εₛ/εₙεₛ⁻¹
				kz₀εₙ = kz₀εₛ*εₙεₛ⁻¹
				sim = nonlinsim(w=w,kz₀εₛ²εₙ⁻¹=kz₀εₛ²εₙ⁻¹,kz₀εₙ=kz₀εₙ,
								αz₀=αz₀,γₕ=γₕ,φ₀=0.0)
				A = sim.capturesol[end]
				return A/Anorm
			end
			return As
		end
		global Ass_cat = hcat(Ass...)
		npzwrite("nonlins_phi=0.npz",
			kze=kz₀εₛs,en_over_es=collect(εₙεₛ⁻¹s),
			allA=Ass_cat.*εₙεₛ⁻¹s)
	end
end;

# ╔═╡ d8ad2e68-886e-4c4a-abc8-073a97d45c2e
begin
	plot()
	for (kz₀εₛ,As) in zip(kz₀εₛs,eachcol(Ass_cat))
	    plot!(εₙεₛ⁻¹s,real.(As.*εₙεₛ⁻¹s),marker=true,label=kz₀εₛ)
	end
	plot!(εₙεₛ⁻¹s,εₙεₛ⁻¹s,color=:gray,alpha=0.5,label=false)
	plot!(εₙεₛ⁻¹s,0*εₙεₛ⁻¹s,color=:gray,alpha=0.5,label=false)
	plot!(aspect_ratio=:equal,
		xlim=(εₙεₛ⁻¹s[begin],εₙεₛ⁻¹s[end]),
		ylim=(εₙεₛ⁻¹s[begin],εₙεₛ⁻¹s[end]),
	    legend_title="kz₀εₛ",legend=:outertopright,
		xlabel="input equivalent εₙ in units of (εₛ)",
		ylabel="real output equivalent εₙ in units of (εₛ)",
	)
end

# ╔═╡ ef69b43b-313d-4770-8654-0cd480470c2b
md"## At arbitrary φ₀"

# ╔═╡ 691659c2-90fd-4077-9014-b60640f2843c
let
	αz₀ = 0.0
	γₕ = 0.0
	
	global φs = range(0,2pi,length=36)
	global εₙεₛ⁻¹s_pos = ((0:0.05:1) .+ 1e-2)*εₙεₛ⁻¹_scale

	if isfile("nonlins.npz")
		f = npzread("nonlins.npz")
		global Asss_withφ_cat = f["allA"]./εₙεₛ⁻¹s_pos'
	else
		Asss_withφ = qmap(kz₀εₛs) do kz₀εₛ
			Ass = qmap(εₙεₛ⁻¹s_pos) do εₙεₛ⁻¹
				As = qmap(φs) do φ₀
					kz₀εₛ²εₙ⁻¹ = kz₀εₛ/εₙεₛ⁻¹
					kz₀εₙ = kz₀εₛ*εₙεₛ⁻¹
					sim = nonlinsim(w=w,kz₀εₛ²εₙ⁻¹=kz₀εₛ²εₙ⁻¹,kz₀εₙ=kz₀εₙ,
									αz₀=αz₀,γₕ=γₕ,φ₀=φ₀)
					A = sim.capturesol[end]
					@info "φ₀=$(φ₀), εₙεₛ⁻¹=$(εₙεₛ⁻¹), kz₀εₛ=$(kz₀εₛ)"
					return A/Anorm
				end
				return As
			end
			return Ass
		end
		global Asss_withφ_cat = cat([hcat(a...) for a in Asss_withφ]...,dims=(3,))
		npzwrite("nonlins.npz",
			kze=kz₀εₛs,en_over_es=collect(εₙεₛ⁻¹s_pos),
			allA=Asss_withφ_cat.*εₙεₛ⁻¹s_pos',phis=φs);
	end
end;

# ╔═╡ c1865065-38a6-48fb-8b3c-c6c40d36b47a
let
	i = 2
	p1 = heatmap(φs,εₙεₛ⁻¹s_pos,abs.(Asss_withφ_cat[:,:,i])',
    		aspect_ratio=:equal, proj=:polar, legend=false, colorbar=true,
	        c=:broc,
	        clim=(0.,2.), colorbar_title=" \nOutput/Input Ratio", right_margin=3mm,
			yticks=[])
	p2 = heatmap(φs,εₙεₛ⁻¹s_pos,angle.(
			Asss_withφ_cat[:,:,i])',
		    aspect_ratio=:equal, proj=:polar, legend=false, colorbar=true,
            c=:romaO,
	        clim=(-pi,pi), colorbar_title="Output Phase (radians)",
		    colorbar_ticks=[-pi,0,pi],# only in pyplot
			yticks=[])
	plot(p1,p2,layout=(2,1),size=(400,600))
end

# ╔═╡ 41d7a341-95c2-4b0e-b7a5-2c8e8b4b889f
let
	data = Asss_withφ_cat[:,:,2]
	plot(φs, angle.(data), linez=εₙεₛ⁻¹s_pos',
		legend=false, colorbar=true, c=:batlow, colorbar_title="εₙεₛ⁻¹",
	#    xlim=(0,0.97*pi), ylim=(-pi,0)
	)
end

# ╔═╡ 62c3b21b-37f9-47ab-b449-928cce82f97b
let
	data = Asss_withφ_cat[1:end÷2,:,2]
	plot(εₙεₛ⁻¹s_pos, abs.(data)' .* εₙεₛ⁻¹s_pos,
		linez=φs[1:end÷2]',
		legend=false, colorbar=true, colorbar_title="input φ",
		lw=2, c=:romaO, clim=(-pi,pi),
	)
end

# ╔═╡ 648a06a2-6ce6-4cae-ba03-fc937acdf5d4
md"# Instead of the pump being a subharmonic, what if the pump is the second harmonic"

# ╔═╡ 7537e114-6e85-43e4-a526-d2ab194c5bd4
let
	αz₀ = 0.0
	γₕ = 0.0

	global φs_ = range(0,2pi,length=36)
	global kz₀εₙs = [1.0,2.0,3.0]
	εₛεₙ⁻¹_scale = 0.5
	global εₛεₙ⁻¹s = ((0:0.05:1) .+ 1e-2)*εₛεₙ⁻¹_scale

	if isfile("nonlins_inverted.npz")
		f = npzread("nonlins_inverted.npz")
		global Asss_inverted_withφ_cat = f["allA"]./εₛεₙ⁻¹s'
	else
		global Asss_inverted_withφ = qmap(kz₀εₙs) do kz₀εₙ
			Ass = qmap(εₛεₙ⁻¹s) do εₛεₙ⁻¹
				As = qmap(φs_) do φ₀
					kz₀εₛ²εₙ⁻¹ = kz₀εₙ*εₛεₙ⁻¹^2
					kz₀εₙ_ = kz₀εₙ
					sim = nonlinsim(w=w,
						kz₀εₛ²εₙ⁻¹=kz₀εₛ²εₙ⁻¹,kz₀εₙ=kz₀εₙ_,
						αz₀=αz₀,γₕ=γₕ,
						φ₀=φ₀,
						use_sub=true)
					A = sim.capturesol[end]
					@info "φ₀=$(φ₀), εₛεₙ⁻¹=$(εₛεₙ⁻¹), kz₀εₛ=$(kz₀εₙ)"
					return A/Anorm
				end
				return As
			end
			return Ass
		end
		global Asss_inverted_withφ_cat = cat([hcat(a...)
											  for a in Asss_inverted_withφ]...,
											 dims=(3,))
		npzwrite("nonlins_inverted.npz",
			kze=kz₀εₙs,es_over_en=collect(εₛεₙ⁻¹s),
			allA=Asss_inverted_withφ_cat.*εₛεₙ⁻¹s',phis=φs_);
	end
end;

# ╔═╡ 594db84f-1d44-4dbb-8f40-3bf743303f0a
let
	data = Asss_inverted_withφ_cat[:,:,end]
	plot(φs_, angle.(data), linez=εₛεₙ⁻¹s',
		legend=false, colorbar=true, c=:batlow, colorbar_title="εₙεₛ⁻¹",
	    #xlim=(0,0.97*pi), ylim=(-pi,0)
	)
end

# ╔═╡ 4acbc103-7e7e-40be-b0f4-4e905c59cbe0
let
	data = Asss_inverted_withφ_cat[:,:,2]
	plot(εₛεₙ⁻¹s, abs.(data)' .* εₛεₙ⁻¹s,
		#linez=φs_[1:end÷2]',
		linez=φs_',
		legend=false, colorbar=true, colorbar_title="input φ",
		lw=2, c=:romaO, clim=(-pi,pi),
	)
end

# ╔═╡ e8f9590a-047b-4b65-a2b9-c23016e870f9
let
	i = 2 # there is a bug with the r-limits, so we manually rescale
	p1 = heatmap(φs_,εₛεₙ⁻¹s*10,abs.(Asss_inverted_withφ_cat[:,:,i])',
    		aspect_ratio=:equal, proj=:polar, legend=false, colorbar=true,
	        c=:broc,
	        clim=(0.,2.), colorbar_title=" \nOutput/Input Ratio", right_margin=3mm,
			yticks=[])
	p2 = heatmap(φs_,εₛεₙ⁻¹s*10,angle.(
			Asss_inverted_withφ_cat[:,:,i])',
		    aspect_ratio=:equal, proj=:polar, legend=false, colorbar=true,
            c=:romaO,
	        clim=(-pi,pi), colorbar_title="Output Phase (radians)",
		    colorbar_ticks=[-pi,0,pi],# only in pyplot
			yticks=[])
	plot(p1,p2,layout=(2,1),size=(400,600))
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
DiffEqOperators = "9fdde737-9c7f-55bf-ade8-46b3f136cc48"
Interpolations = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
NPZ = "15e1cf62-19b3-5cfa-8e77-841668bca605"
OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
ThreadPools = "b189fb0b-2eb5-4ed4-bc0c-d34c51242431"

[compat]
DiffEqOperators = "~4.32.0"
Interpolations = "~0.13.4"
NPZ = "~0.4.1"
OrdinaryDiffEq = "~5.64.0"
Plots = "~1.22.3"
PlutoUI = "~0.7.14"
SpecialFunctions = "~1.7.0"
ThreadPools = "~2.1.0"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

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[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.ZipFile]]
deps = ["Libdl", "Printf", "Zlib_jll"]
git-tree-sha1 = "3593e69e469d2111389a9bd06bac1f3d730ac6de"
uuid = "a5390f91-8eb1-5f08-bee0-b1d1ffed6cea"
version = "0.9.4"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[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", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[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", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+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"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[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 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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