tidy3d.PoleResidue#

class PoleResidue[source]#

Bases: DispersiveMedium

A dispersive medium described by the pole-residue pair model.

Parameters:

attrs (dict = {}) – Dictionary storing arbitrary metadata for a Tidy3D object. This dictionary can be freely used by the user for storing data without affecting the operation of Tidy3D as it is not used internally. Note that, unlike regular Tidy3D fields, attrs are mutable. For example, the following is allowed for setting an attr obj.attrs['foo'] = bar. Also note that Tidy3D` will raise a TypeError if attrs contain objects that can not be serialized. One can check if attrs are serializable by calling obj.json().
name (Optional[str] = None) – Optional unique name for medium.
frequency_range (Optional[Tuple[float, float]] = None) – [units = (Hz, Hz)]. Optional range of validity for the medium.
allow_gain (bool = False) – Allow the medium to be active. Caution: simulations with a gain medium are unstable, and are likely to diverge.Simulations where ‘allow_gain’ is set to ‘True’ will still be charged even if diverged. Monitor data up to the divergence point will still be returned and can be useful in some cases.
nonlinear_spec (Union[NonlinearSpec, NonlinearSusceptibility] = None) – Nonlinear spec applied on top of the base medium properties.
modulation_spec (Optional[ModulationSpec] = None) – Modulation spec applied on top of the base medium properties.
viz_spec (Optional[VisualizationSpec] = None) – Plotting specification for visualizing medium.
heat_spec (Union[FluidSpec, SolidSpec, SolidMedium, FluidMedium, NoneType] = None) – DEPRECATED: Use td.MultiPhysicsMedium. Specification of the medium heat properties. They are used for solving the heat equation via the HeatSimulation interface. Such simulations can beused for investigating the influence of heat propagation on the properties of optical systems. Once the temperature distribution in the system is found using HeatSimulation object, Simulation.perturbed_mediums_copy() can be used to convert mediums with perturbation models defined into spatially dependent custom mediums. Otherwise, the heat_spec does not directly affect the running of an optical Simulation.
eps_inf (Union[PositiveFloat, Box] = 1.0) – [units = None (relative permittivity)]. Relative permittivity at infinite frequency (\(\epsilon_\infty\)).
poles (Tuple[Tuple[Union[tidy3d.components.types.tidycomplex, tidy3d.components.types.ComplexNumber, autograd.tracer.Box], Union[tidy3d.components.types.tidycomplex, tidy3d.components.types.ComplexNumber, autograd.tracer.Box]], ...] = ()) – [units = (rad/sec, rad/sec)]. Tuple of complex-valued (\(a_i, c_i\)) poles for the model.

Notes

The frequency-dependence of the complex-valued permittivity is described by:

\[\epsilon(\omega) = \epsilon_\infty - \sum_i \left[\frac{c_i}{j \omega + a_i} + \frac{c_i^*}{j \omega + a_i^*}\right]\]

Example

>>> pole_res = PoleResidue(eps_inf=2.0, poles=[((-1+2j), (3+4j)), ((-5+6j), (7+8j))])
>>> eps = pole_res.eps_model(200e12)

See also

CustomPoleResidue:

A spatially varying dispersive medium described by the pole-residue pair model.

Notebooks

Fitting dispersive material models

Lectures

Modeling dispersive material in FDTD

Attributes

`loss_upper_bound`	Upper bound of Im[eps] in frequency_range
`attrs`

Methods

`compute_derivatives`(derivative_info)	Compute adjoint derivatives for each of the `fields` given the multiplied E and D.
`eps_model`(frequency)	Complex-valued permittivity as a function of frequency.
`from_admittance_coeffs`(a, b[, eps_inf, pole_tol])	Construct a `PoleResidue` model from an admittance function defining the relationship between the electric field and the polarization current density in the Laplace domain.
`from_lo_to`(poles[, eps_inf])	Construct a pole residue model from the LO-TO form (longitudinal and transverse optical modes).
`from_medium`(medium)	Convert a `Medium` to a pole residue model.
`imag_ep_extrema`(poles)	Extrema of Im[eps] in the same unit as poles.
`lo_to_eps_model`(poles, eps_inf, frequency)	Complex permittivity as a function of frequency for a given set of LO-TO coefficients.
`to_medium`()	Convert to a `Medium`.

Inherited Common Usage

eps_inf#

poles#

eps_model(frequency)[source]#: Complex-valued permittivity as a function of frequency.

__str__()[source]#: string representation

classmethod from_medium(medium)[source]#

Convert a Medium to a pole residue model.

Parameters:: medium (Medium) – The medium with permittivity and conductivity to convert.
Returns:: The pole residue equivalent.
Return type:: PoleResidue

to_medium()[source]#

Convert to a Medium. Requires the pole residue model to only have a pole at 0 frequency, corresponding to a constant conductivity term.

Returns:: The non-dispersive equivalent with constant permittivity and conductivity.
Return type:: Medium

static lo_to_eps_model(poles, eps_inf, frequency)[source]#

Complex permittivity as a function of frequency for a given set of LO-TO coefficients. See from_lo_to in PoleResidue for the detailed form of the model and a reference paper.

Parameters:

poles (Tuple[Tuple[float, float, float, float], ...]) – The LO-TO poles, given as list of tuples of the form (omega_LO, gamma_LO, omega_TO, gamma_TO).
eps_inf (pd.PositiveFloat) – The relative permittivity at infinite frequency.
frequency (float) – Frequency at which to evaluate the permittivity.

Returns:

The complex permittivity of the given LO-TO model at the given frequency.

Return type:

complex

classmethod from_lo_to(poles, eps_inf=1)[source]#

Construct a pole residue model from the LO-TO form (longitudinal and transverse optical modes). The LO-TO form is \(\epsilon_\infty \prod_{i=1}^l \frac{\omega_{LO, i}^2 - \omega^2 - i \omega \gamma_{LO, i}}{\omega_{TO, i}^2 - \omega^2 - i \omega \gamma_{TO, i}}\) as given in the paper:

M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61, 8187 (2000).

Parameters:

poles (Tuple[Tuple[float, float, float, float], ...]) – The LO-TO poles, given as list of tuples of the form (omega_LO, gamma_LO, omega_TO, gamma_TO).
eps_inf (pd.PositiveFloat) – The relative permittivity at infinite frequency.

Returns:

The pole residue equivalent of the LO-TO form provided.

Return type:

PoleResidue

static imag_ep_extrema(poles)[source]#

Extrema of Im[eps] in the same unit as poles.

Parameters:: poles (Tuple[PoleAndResidue, ...]) – Tuple of complex-valued (a_i, c_i) poles for the model.

property loss_upper_bound#: Upper bound of Im[eps] in frequency_range

compute_derivatives(derivative_info)[source]#: Compute adjoint derivatives for each of the fields given the multiplied E and D.

classmethod from_admittance_coeffs(a, b, eps_inf=1, pole_tol=0.01)[source]#

Construct a PoleResidue model from an admittance function defining the relationship between the electric field and the polarization current density in the Laplace domain.

Parameters:

a (np.ndarray) – Coefficients of the numerator polynomial in increasing monomial order.
b (np.ndarray) – Coefficients of the denominator polynomial in increasing monomial order.
eps_inf (pd.PositiveFloat) – The relative permittivity at infinite frequency.
pole_tol (pd.PositiveFloat) – Tolerance for the pole finding algorithm in Hertz. Two poles are considered equal, if their spacing is closer than ``pole_tol`.

Returns:

The pole residue equivalent.

Return type:

PoleResidue

Notes

The supplied admittance function relates the electric field to the polarization current density in the Laplace domain and is equivalent to a frequency-dependent complex conductivity \(\sigma(\omega)\).

\[J_p(s) = Y(s)E(s)\]

\[Y(s) = \frac{a_0 + a_1 s + \dots + a_M s^M}{b_0 + b_1 s + \dots + b_N s^N}\]

An equivalent PoleResidue medium is constructed using an equivalent frequency-dependent complex permittivity defined as

\[\epsilon(s) = \epsilon_\infty - \frac{1}{\epsilon_0 s} \frac{a_0 + a_1 s + \dots + a_M s^M}{b_0 + b_1 s + \dots + b_N s^N}.\]

__hash__()#: Hash method.

tidy3d.PoleResidue

Contents

tidy3d.PoleResidue#