OpticalFibers.jl

Material(B1::Real, B2::Real, B3::Real, C1::Real, C2::Real, C3::Real)

Create a Material with Sellmeier coefficients B₁, B₂, B₃, C₁, C₂, and C₃, where C₁, C₂, and C₃ must be given in square micrometers.

sellmeier_equation(material::Material, wavelength_μm::Real)

Compute the refractive index, n, of the material for light with a free space wavelength of wavelength_μm, which must be given in micrometers.

The Sellmeier equation reads

\[n^{2}(λ) = 1 + \sum_{i} \frac{B_{i} λ^{2}}{λ^{2} - C_{i}},\]

where $B_{i}$ and $C_{i}$ are constants that depend on the material, and $λ$ is the wavelength of light in vacuum.

fiber_equation(u, parameters)

Compute the value of the characteristic fiber equation with all terms moved to the same side of the equal sign, where u is the propagation constant, and parameters contains the fiber radius, refraction index, and frequency in said order.

Used as an input to the non-linear solver to find the propagation constant.

The characteristic fiber equation for single mode cylindrical fibers reads Fam Le Kien and A. Rauschenbeutel, Phys. Rev. A 95, 023838 (2017)

\[\frac{J_{0}(p a)}{p a J_{1}(pa)} + \frac{n^{2} + 1}{2 n^{2}} \frac{K_{1}'(q a)}{q a K_{1}(q a)} - \frac{1}{p^2 a^2} + \Biggl[\biggl( \frac{n^2 - 1}{2 n^2} \frac{K_{1}'(q a)}{q a K_{1}(q a)} \biggr)^2 + \frac{\beta^2}{n^2 k^2} \biggl( \frac{1}{p^2 a^2} + \frac{1}{q^2 a^2} \biggr)^2 \Biggr]^{1 / 2} = 0,\]

where $a$ is the fiber radius, $k$ is the free space wave number of the light, $n$ is the refractive index of the fiber, $p = \sqrt{n^2 k^2 - \beta^2}$, and $q = \sqrt{\beta^2 - k^2}$. Futhermore, $J_n$ and $K_n$ are Bessel functions of the first kind, and modified Bessel functions of the second kind, respectively, and the prime denotes the derivative.

_propagation_constant(a::Real, n::Real, ω::Real)

Compute the propagation constant of a fiber with radius a, refactive index n and frequency ω by solving the characteristic equation of the fiber as written as eq. (1A) in [PRA 95, 023838].

_propagation_constant_derivative(a::Real, n::Real, ω::Real; dω = 1e-9)

Compute the derivative of the propagation constant with respect to frequency evaluated at ω of a fiber with radius a, and refactive index n.

radius(fiber::Fiber)

Return the radius of the fiber in micrometers.

Examples

julia> fiber = Fiber(0.1, 0.4, Material(0.6961663, 0.4079426, 0.8974794, 0.0684043^2, 0.1162414^2, 9.896161^2));

julia> radius(fiber)
0.1

wavelength(fiber::Fiber)

Return the wavelength of the fiber mode in micrometers.

Examples

julia> fiber = Fiber(0.1, 0.4, Material(0.6961663, 0.4079426, 0.8974794, 0.0684043^2, 0.1162414^2, 9.896161^2));

julia> wavelength(fiber)
0.4

frequency(fiber::Fiber)

Return the frequency of the fiber mode.

material(fiber::Fiber)

Return the material of the fiber.

refractive_index(fiber::Fiber)

Return the refractive index of the fiber for light with the same wavelength as the fiber mode.

propagation_constant(fiber::Fiber)

Return the propagation constant of the fiber for light with the same wavelength as the fiber mode.

propagation_constant_derivative(fiber::Fiber)

Return the derivative of the propagation constant of the fiber evaluated at the the wavelength of the fiber mode.

normalized_frequency(fiber::Fiber)

Return the normalized frequency of the fiber mode.

effective_refractive_index(fiber::Fiber)

Return the effective refractive index of the fiber.

guided_mode_normalization_constant(a::Real, n::Real, β::Real, h::Real, q::Real, K1J1::Real, s::Real)

Compute the normalization constant of an electric guided fiber mode.

The fiber modes are normalized according to the condition

\[\int_{0}^{\infty} \! \mathrm{d} \rho \int_{0}^{2 \pi} \! \mathrm{d} \phi \, n^{2}(\rho) \lvert \mathrm{\mathbf{e}}(\rho, \phi) \rvert^{2} = 1,\]

where $n(\rho)$ is the step index refractive index given as

\[n(\rho) = \begin{cases} n, & \rho < a, \\ 1 & \rho > a, \end{cases}\]

and $\mathrm{\mathbf{e}}(\rho, \phi) = e_{\rho} \hat{\mathrm{\mathbf{\rho}}} + e_{\phi} \hat{\mathrm{\mathbf{\phi}}} + e_{z} \hat{\mathrm{\mathbf{z}}}$, where the components are given by electric_guided_mode_cylindrical_base_components.

electric_guided_mode_cylindrical_base_components(ρ::Real, a::Real, β::Real, p::Real, q::Real, K1J1::Real, s::Real)

Compute the underlying cylindrical components of the guided mode electric field used in the expressions for both the quasilinear and quasicircular guided modes.

These components for $\rho < a$ are given by

\[\begin{aligned} e_{\rho} &= A \mathrm{i} \frac{q}{p} \frac{K_{1}(q a)}{J_{1}(p a)} [(1 - s) J_{0}(p \rho) - (1 + s) J_{2}(p \rho)] \\ e_{\phi} &= -A \frac{q}{p} \frac{K_{1}(q a)}{J_{1}(p a)} [(1 - s) J_{0}(p \rho) + (1 + s) J_{2}(p \rho)] \\ e_{z} &= A \frac{2 q}{\beta} \frac{K_{1}(q a)}{J_{1}(p a)} J_{1}(p \rho), \end{aligned}\]

and the components for $\rho > a$ are given by

\[\begin{aligned} e_{\rho} &= A \mathrm{i} [(1 - s) K_{0}(q \rho) + (1 + s) K_{2}(q \rho)] \\ e_{\phi} &= -A [(1 - s) K_{0}(q \rho) - (1 + s) K_{2}(q \rho)] \\ e_{z} &= A \frac{2 q}{\beta} K_{1}(q \rho), \end{aligned}\]

where $A$ is the normalization constant, $a$ is the fiber radius, $\beta$ is the propagation constant, $p = \sqrt{n^2 k^2 - \beta^2}$, and $q = \sqrt{\beta^2 - k^2}$, with $k$ being the free space wavenumber of the light. Futhermore, $J_n$ and $K_n$ are Bessel functions of the first kind, and modified Bessel functions of the second kind, respectively, and the prime denotes the derivative. Lastly, $s$ is defined as

\[s = \frac{\frac{1}{p^2 a^2} + \frac{1}{q^2 a^2}}{\frac{J_{1}'(p a)}{p a J_{1}(p a)} + \frac{K_{1}'(q a)}{q a K_{1}(q a)}}.\]

electric_guided_field_cartesian_components(ρ::Real, ϕ::Real, z::Real, t::Real, f::Integer, fiber::Fiber, polarization::Polarization, power::Real)

Compute the cartesian components of a guided electric field at position $(ρ, ϕ, z)$ and time $t$ with the given $power$.

electric_guided_field_cartesian_vector(ρ::Real, ϕ::Real, l::Integer, f::Integer, fiber::Fiber, polarization_basis::CircularPolarization, power::Real)

Compute the guided electric field vector at position $(ρ, ϕ, z)$ and time $t$ with the given $power$.

vacuum_coefficients(r, d, ω₀)

Compute the dipole-dipole and decay coefficients for the master equation describing a cloud of atoms with positions given by the columns in r (in cartesian coordinates), dipole moment d, and transition frequency ω₀ coupled to the vacuum field.

guided_mode_coefficients(r, d, fiber)

Compute the guided dipole-dipole and decay coefficients for the master equation describing a cloud of atoms with positions given by the columns in r (in cartesian coordinates), and dipole moment d coupled to an optical fiber.

guided_mode_directional_coefficients(r, d, fiber)

Compute the guided dipole-dipole and decay coefficients due to the modes with direction f for the master equation describing a cloud of atoms with positions given by the columns in r (in cartesian coordinates), and dipole moment d coupled to an optical fiber.

radiation_mode_decay_coefficients(r, d, fiber; abstol = 1e-3)

Compute the decay coefficients for the master equation describing a cloud of atoms with positions given by the columns in r (in cartesian coordinates), and dipole moment d coupled to the radiation modes from an optical fiber.

radiation_mode_coefficients(r, d, fiber; abstol = 1e-6)

Compute the dipole-dipole and decay coefficients for the master equation describing a cloud of atoms with positions given by the columns in r (in cartesian coordinates), and dipole moment d coupled to the radiation modes from an optical fiber.

radiation_mode_directional_coefficients(r, d, fiber, f; abstol = 1e-6)

Compute the dipole-dipole and decay coefficients for the master equation describing a cloud of atoms with positions given by the columns in r (in cartesian coordinates), and dipole moment d coupled to the radiation modes with directions f from an optical fiber.

radiative_coupling_strength(ρ, ϕ, z, d, l, f, fiber)

Compute the coupling strength between an atom and a radiation fiber mode.

Implementation of Eq. (7), bottom equation from Fam Le Kien and A. Rauschenbeutel. "Nanofiber-mediated chiral radiative coupling between two atoms". Phys. Rev. A 95, 023838 (2017).

transmission_two_level(Δes, fiber, Δr, Ωs::AbstractArray, gs, J, Γ, γ)

Compute the transmission of a cloud of two level atoms surrounding an optical fiber for each value of the detuning given by Δes.

The parameters of the fiber are given by fiber, while the atoms have light-matter coupling constants gs, dipole-dipole interaction matrix J, and cross decay rate matrix Γ.

transmission_three_level(Δes, fiber, Δr, Ω::Number, gs, J, Γ, γ)

Compute the transmission of a cloud of three level atoms surrounding an optical fiber for each value of the lower transition detuning given by Δes, where each atom experience the same Rabi frequency.

The parameters of the fiber are given by fiber, while the atoms have upper transition detuning Δr, control Rabi frequenciy Ω, pump coupling constants gs, dipole-dipole interaction matrix J, cross decay rate matrix Γ, and Rydberg to intermediate state decay rate γ.

transmission_three_level(Δes, fiber, Δr, Ωs::AbstractArray, gs, J, Γ, γ)

Compute the transmission of a cloud of three level atoms surrounding an optical fiber for each value of the lower transition detuning given by Δes, where the Rabi frequency can be different from atom to atom.

The parameters of the fiber are given by fiber, while the atoms have upper transition detuning Δr, control Rabi frequencies Ωs, pump coupling constants gs, dipole-dipole interaction matrix J, cross decay rate matrix Γ, and Rydberg to intermediate state decay rate γ.

gaussian_beam_intensity(x::Real, y::Real, z::Real, waist::Real, power::Real, wavelength::Real)

Compute the intensity at position (x, y, z) of a gaussian beam parallel to the z-axis with parameters waist, power, and wavelength.

tweezer_trap_intensity(x::Real, y::Real, z::Real, waist::Real, power::Real, wavelength::Real)

Compute the intensity at position (x, y, z) of a dipole tweezer trap consisting of two beam parallel to the gaussian beams, one along the x-axis and one along the y-axis, where both beams have parameters waist, power, and wavelength.

tweezer_trap_potential(x::Real, y::Real, z::Real, waist::Real, power::Real, wavelength::Real, stark_shift::Real)

Compute the potential at position (x, y, z) of a dipole tweezer trap consisting of two beam parallel to the gaussian beams, one along the x-axis and one along the y-axis, where both beams have parameters waist, power, and wavelength, and the starkshift per unit intensity is `starkshift`.

fiber_potential(x::Real, y::Real, z::Real, l::Integer, f::Integer, fiber::Fiber, polarization_basis::CircularPolarization, power::Real, stark_shift::Real)

Compute the potential at position (x, y, z) of a circularly polarized fiber mode along the z-axis with polarization index l, direction of propagation index f, power, and the starkshift per unit intensity is `starkshift`.

full_potential(x::Real, y::Real, z::Real, waist::Real, power_trap::Real, wavelength::Real, stark_shift_trap::Real, l::Integer, f::Integer, fiber::Fiber, polarization_basis::CircularPolarization, power_fiber::Real, stark_shift_fiber::Real)

Compute the full potential at position (x, y, z) of a circularly polarized fiber mode along the z-axis and a dipole tweezer trap consisting of two beam parallel to the gaussian beams, one along the x-axis and one along the y-axis.