SFGAnalysis

Setup(ω, β, n1, n2)

Create an object representing the setup.

Arguments

• ω: Tuple{Real,Real} containing $\omega_1$ and $\omega_2$ so that $\omega = \omega_1 + \omega_2$.
• β: Tuple{Real,Real} containing the incident angles corresponing to $\omega_1$ and $\omega_2$.
• n1: Tuple{Number,Number,Number} containing the refractive indices of medium 1 corresponding to $\omega$, $\omega_1$ and $\omega_2$.
• n2: Tuple{Number,Number,Number} containing the refractive indices of medium 2 corresponding to $\omega$, $\omega_1$ and $\omega_2$.

angleofrefraction(n1, n2, β)

Compute the angle of refraction.

Arguments

• n1: refractive index of the medium in which the wave is propagating towards the interface
• n2: refractive index of the other medium
• β: incident or reflected angle

effective_susceptibility(d::Distribution, s::Setup, p::Symbol; <kwargs>)

Same as above. In this case the tilt angle θ is expressed by a Distribution d.

effective_susceptibility(θ::Real, s::Setup, p::Symbol; <kwargs>)

Compute the effective susceptibility for a moiety with tilt angle θ for a Setup s in polarization p.

Arguments

• θ: tilt angle of the moiety
• s: Setup object
• p: polarization combination (:ppp or :ssp)

Keyword Arguments

• pointgroup::Symbol: point group of the moiety (possible values: :c3v)
• mode::Symbol: vibrational mode (possible values: :ss (symmetric stretch), :as (asymmetric stretch))
• R::Real: the hyperpolarizability ratio $R = \beta_{aac} / \beta_{ccc} = \beta_{bbc} / \beta_{ccc}$. Has only to be provided for point group :c3v and mode :ss.
• beta::Real: hyperpolarizability
  • $\beta_{ccc}$ for :c3v and :ss
  • $\beta_{aca} = \beta_{bcb}$ for :c3v and :as
• N::Real: number of oscillators

effective_susceptibility_ppp(n1, n2, n′, β, χ)

Compute the effective susceptibility in ppp polarization.

Arguments

• n1: refractive indices of the medium in which the waves are propagating towards the interface
• n2: refractive indices of the other medium
• n′: refractive indices of the interface
• β: incident or reflected angles
• χ: susceptibilities as an array [χ_xxz, χ_xzx, χ_zxx, χ_zzz]

The arguments n1, n2, n′ and β have to be arrays where the first element corresponds to the property for the sum frequency, the second to $\omega_1$ and the third to $\omega_2$.

https://doi.org/10.1080/01442350500225894

effective_susceptibility_ppp(s::SFGAnalysis.Setup, χ)

Arguments

• s: setup object
• χ: susceptibilities as an array [χ_xxz, χ_xzx, χ_zxx, χ_zzz]

effective_susceptibility_pss(n1, n2, n′, β, χ_yyz)

Compute the effective susceptibility in pss polarization.

Arguments

• n1: refractive indices of the medium in which the waves are propagating towards the interface
• n2: refractive indices of the other medium
• n′: refractive index of the interface for $\omega_2$
• β: incident or reflected angles
• χ_yyz: susceptibility

The arguments n1, n2, and β have to be arrays where the first element corresponds to the property for the sum frequency, the second to $\omega_1$ and the third to $\omega_2$.

https://doi.org/10.1080/01442350500225894

effective_susceptibility_pss(s::SFGAnalysis.Setup, χ_zyy)

Compute the effective susceptibility in pss polarization.

Arguments

• s: setup object
• χ_zyy: susceptibility

effective_susceptibility_sps(n1, n2, n′, β, χ_yyz)

Compute the effective susceptibility in sps polarization.

Arguments

• n1: refractive indices of the medium in which the waves are propagating towards the interface
• n2: refractive indices of the other medium
• n′: refractive index of the interface for $\omega_2$
• β: incident or reflected angles
• χ_yyz: susceptibility

The arguments n1, n2, and β have to be arrays where the first element corresponds to the property for the sum frequency, the second to $\omega_1$ and the third to $\omega_2$.

https://doi.org/10.1080/01442350500225894

effective_susceptibility_sps(s::SFGAnalysis.Setup, χ_yzy)

Compute the effective susceptibility in sps polarization.

Arguments

• s: setup object
• χ_yzy: susceptibility

effective_susceptibility_ssp(n1, n2, n′, β, χ_yyz)

Compute the effective susceptibility in ssp polarization.

Arguments

• n1: refractive indices of the medium in which the waves are propagating towards the interface
• n2: refractive indices of the other medium
• n′: refractive index of the interface for $\omega_2$
• β: incident or reflected angles
• χ_yyz: susceptibility

The arguments n1, n2, and β have to be arrays where the first element corresponds to the property for the sum frequency, the second to $\omega_1$ and the third to $\omega_2$.

https://doi.org/10.1080/01442350500225894

effective_susceptibility_ssp(s::SFGAnalysis.Setup, χ_yyz)

Compute the effective susceptibility in ssp polarization.

Arguments

• s: setup object
• χ_yyz: susceptibility

freq2wl(ω)

Compute the wavelength of a field with frequency ω in 1/s. The wavelength will be given in nm.

fresnel_x(n1, n2, β)

Compute the diagonal elements $L_{xx}$ of the fresnel factor.

Arguments

• n1: refractive index of the medium in which the wave is propagating towards the interface
• n2: refractive index of the other medium
• β: incident or reflected angle

https://doi.org/10.1103/PhysRevB.59.1263

fresnel_y(n1, n2, β)

Compute the diagonal elements $L_{yy}$ of the fresnel factor.

Arguments

• n1: refractive index of the medium in which the wave is propagating towards the interface
• n2: refractive index of the other medium
• β: incident or reflected angle

https://doi.org/10.1103/PhysRevB.59.1263

fresnel_z(n1, n2, n′, β)

Compute the diagonal elements $L_{yy}$ of the fresnel factor.

Arguments

• n1: refractive index of the medium in which the wave is propagating towards the interface
• n2: refractive index of the other medium
• n′: refractive index of the interface
• β: incident or reflected angle

https://doi.org/10.1103/PhysRevB.59.1263

Documentation

ninterface(n2)

Calculate the index of refraction of the interfacial layer according to

$\left(\frac{1}{n^\prime}\right)^2 = \frac{4n_2^2 + 2}{n_2^2(n_2 ^2+5)}$.

Arguments

• n2: refractive index of the medium in which the wave is not propagating towards the interface

https://doi.org/10.1103/PhysRevB.59.1263

sfangle(ω, ω1, ω2, β1, β2)

Compute the angle of the reflected sum frequency field. In a counter-propagating geometry one of the βs gets negative.

Arguments

• ω: sum frequency
• ω1: frequency of wave 1
• ω2: frequency of wave 2
• β1: incident angle for field with frequency ω1
• β2: incident angle for field with frequency ω2

sfgintensity(ω, β, χ2_eff; <keyword arguments>)

Compute the intentensity of the sum frequency in the reflected direction according to

$I_\mathrm{SF} = \frac{8\pi^3\omega^2\sec^2\beta}{c^3 n n_1 n_2} \left| \chi_\mathrm{eff}^{(2)} \right|^2 I_1 I_2$.

Arguments

• ω: sum frequency
• β: reflection angle of the sum frequency field
• χ2_eff: effective susceptibility

Keyword Arguments

• n: refractive index of the bulk medium for ω
• n1: refractive index of the bulk medium for ω1
• n2: refractive index of the bulk medium for ω2
• I1: intensity of ω1
• I2: intensity of ω2

https://doi.org/10.1103/PhysRevB.59.12632

sfgintensity(s::Setup, χ2_eff::Number; <keyword arguments>)

Same as sfgintensity(ω, β, χ2_eff; <keyword arguments>).

Arguments

• s: Setup object
• χ2_eff: effective susceptibility

Keyword Arguments

• I1: intensity of ω1
• I2: intensity of ω2

https://doi.org/10.1103/PhysRevB.59.12632

If no phase information and non-resonant background is passed to the function, we set it to 0.

If no phase information and but the non-resonant background is passed it's assumed that the relative phase of the resonances is +/- π, which can be controlled via the sign of A.

If just one phase is passed, we assume that this belongs to the non-resonant background.

sfspectrum(x::Number, A::T, ω::T, Γ::T[, φ::AbstractArray, χnr]) where {T<:AbstractArray}

Calculate the sum frequency spectrum according to

$I = \left| \chi_{NR} + \sum_q \frac{A_q}{x - ω_q - i\Gamma_q} \right|^2$

Arguments

• x the frequency/wavenumber at which to evaluate the sfspectrum
• A oscillator strength
• ω resonances
• Γ damping coefficients

Keyword Arguments

Examples

x = range(2800, 3000, length=201)
A = [1, 1]
ω = [2880, 2930]
Γ = [8, 7]

In order to use the broadcast functionality we have to wrap the parameters in Refs:

y = sfspectrum.(x, Ref(A), Ref(ω), Ref(Γ))

Alternatively use array comprehensions:

y = [sfspectrum(_x, A, ω, Γ) for _x in x]

susceptibility(θ::Real, t::Symbol; pointgroup, mode, R, beta, N; <kwargs>)

Compute the susceptibility for angle θ and tensor element t. Tensor elements important for specific polarization combinations:

• ssp: :yyz
• sps: :yzy
• pss: :zyy
• ppp: :xxz, :xzx, :zxx and :zzz

Arguments

• θ: title angle of the moiety
• t: tensor element (see above)

Keyword Arguments

• pointgroup::Symbol: point group of the moiety (possible values: :c3v)
• mode::Symbol: vibrational mode (possible values: :ss (symmetric stretch), :as (asymmetric stretch))
• R::Real: the hyperpolarizability ratio $R = \beta_{aac} / \beta_{ccc} = \beta_{bbc} / \beta_{ccc}$. Has only to be provided for point group :c3v and mode :ss.
• beta::Real: hyperpolarizability
  • $\beta_{ccc}$ for :c3v and :ss
  • $\beta_{aca} = \beta_{bcb}$ for :c3v and :as
• N::Real: number of oscillators

Susceptibility for tensor elements xxz and yyz.

Susceptibility for tensor elements xzx, zxx, yzy and zyy.

Susceptibility for tensor element zzz

Susceptibility for tensor elements xxz and yyz.

Susceptibility for tensor elements xzx, zxx, yzy and zyy.

Susceptibility for tensor element zzz

wl2freq(λ)

Compute the frequency of a field with wavelength λ in nm.