EM Mediums#
Overview#
The EM medium describes the electromagnetic properties of the material. In conjunction with Maxwell’s equations and the constitutive equations, these properties determine how EM waves propagate through and interact with the medium. Tidy3D supports a diverse set of medium types to model various material characteristics.
In the Tidy3D python API, medium classes are split into six broad categories:
Non-dispersive medium: Constant relative permittivity \(\varepsilon_r\) and conductivity \(\sigma\)
Dispersive medium: Optical property varies with frequency
Anisotropic medium: Optical property varies with direction of light propagation
Spatially-varying medium: Optical property varies with spatial coordinates
Perturbation medium: Optical property is perturbed by the result of a multiphysics (heat/charge) simulation
`Metallic`_: Metals in the shallow skin depth regime
Combinations are possible, e.g. a spatially-varying, dispersive, anisotropic medium. In addition to these categories, users can define additional modifiers on some mediums using medium specifications:
Nonlinearity: Optical property depends on electric field
Spatio-temporal modulation: Optical property is modulated by a function of time and spatial coordinates
The material library also provides a growing list of commonly used material models.
Non-Dispersive#
Dispersionless medium. |
A simple, non-dispersive optical medium can be described by constant relative permittivity \(\varepsilon_r\) and conductivity \(\sigma\) (units: S/um). Alternatively, it can also be defined using the real (\(n\)) and imaginary (\(k\)) components of the refractive index, but note that this is exact only at a single specified frequency (units: Hz). Thus the latter option is only appropriate for narrow-band simulations.
# A lossless medium with constant permittivity
my_lossless_medium = Medium(permittivity=3.8)
# A lossy medium with constant permittivity and conductivity
my_lossy_medium = Medium(permittivity=4.0, conductivity=1.0)
# Alternatively, use refractive index n and k at a specified frequency
my_medium_nk = Medium.from_nk(n=1.5, k=1e-3, freq=150e12)
Note
To specify a constant \(n\) and \(k\) over a wide frequency range, it is required to use a dispersive model. Please see the section on Dispersive mediums.
Note
To specify a lossy dielectric using a constant loss tangent model, it is required to use a dispersive model. Please see the section on Dispersive mediums.
Dispersive#
A dispersive medium described by the pole-residue pair model. |
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A dispersive medium described by the Lorentz model. |
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A dispersive medium described by the Sellmeier model. |
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A dispersive medium described by the Drude model. |
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A dispersive medium described by the Debye model. |
There are many different models that can be used to describe dispersive mediums. For detailed usage instructions, please visit their respective documentation page.
Tool for fitting refractive index data to get a dispersive medium described by |
Alternatively, the FastDispersionFitter plugin can be used to generate a dispersive medium from external data. The data can be provided as a local text file or a web URL from the materials database refractiveindex.info.
# importing from a local file
my_imported_medium = FastDispersionFitter.from_file("path/to/file.csv", skiprows=1, delimiter=",")
# importing from a URL
my_imported_medium_fromurl = FastDispersionFitter.from_url(
url_file="https://refractiveindex.info/data_csv.php?datafile=database/data-nk/main/Ag/Johnson.yml",
delimiter=",",
)
To define a material with constant loss tangent, use the constant_loss_tangent_model() method.
# constant loss tangent
my_losstan_medium = FastDispersionFitter.constant_loss_tangent_model(
eps_real=2.2,
loss_tangent=5e-4,
frequency_range=(1e9, 5e9)
)
Use the convenience methods plot() and eps_model() to access the optical properties of the dispersive material model.
# plotting within applicable frequency range
my_freqs = np.linspace(3e14, 5e14, 201)
_ = my_dispersive_medium.plot(freqs=my_freqs)
# evaluate complex-valued permittivity at a frequency point
eps = my_dispersive_medium.eps_model(3e14)
See also
For more in-depth discussion and examples, please see the following learning center articles:
For a background introduction to modeling dispersive materials in FDTD, please see the following FDTD101 resource:
Anisotropic#
Diagonally anisotropic medium. |
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Fully anisotropic medium including all 9 components of the permittivity and conductivity tensors. |
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2D diagonally anisotropic medium. |
An anisotropic medium has different optical properties depending on the direction of light propagation. Its relative permittivity is thus specified in the form of a 3x3 tensor. For diagonally anisotropic mediums, use AnisotropicMedium:
# specify mediums for the xx, yy, zz diagonal components
medium_xx = Medium(permittivity=4.0)
medium_yy = Medium(permittivity=4.1)
medium_zz = Medium(permittivity=3.9)
my_anisotropic_medium = AnisotropicMedium(xx=medium_xx, yy=medium_yy, zz=medium_zz)
Note that xx, yy, and zz can support dispersive mediums as well. To specify all 9 components of the tensor, use FullyAnisotropicMedium:
# specify the full 3x3 tensor
perm = [[2, 0, 0], [0, 1, 0], [0, 0, 3]]
cond = [[0.1, 0, 0], [0, 0, 0], [0, 0, 0]]
my_anisotropic_medium = FullyAnisotropicMedium(permittivity=perm, conductivity=cond)
Currently, FullyAnisotropicMedium only supports non-dispersive mediums.
The Medium2D class is used to simulate 2D materials without an out-of-plane response.
See also
For more in-depth discussion with examples, please see the following learning center articles:
Example applications:
Metallic/PEC/PMC#
Perfect electrical conductor class. |
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Perfect magnetic conductor class. |
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Lossy metal that can be modeled with a surface impedance boundary condition (SIBC). |
At lower frequencies, the EM field typically does not penetrate very far into the metallic medium. In this regime, metallic structures are commonly modeled as boundary conditions. In Tidy3D, a metallic medium is assigned to a structure and the corresponding boundary conditions are automatically applied to its geometric boundaries.
# lossless metal
my_pec = PECMedium()
# lossy metal
my_lossy_metal = LossyMetalMedium(conductivity=58, freq_range=(1e9, 10e9))
The LossyMetalMedium class implements the surface impedance boundary condition (SIBC). It can also accept surface roughness specifications using the Hammerstad or Huray models. Please refer to its documentation page for details.
Note
For lossy metallic mediums, always be sure to check the skin depth — if the skin depth is not negligible compared to the structure size, then LossyMetalMedium may be not accurate. In that case, use a regular dispersive medium instead.
Spatial Variation#
Spatial distribution. |
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The CustomMedium class is used to define a spatially-varying non-dispersive medium. The spatial distribution of the optical property is defined using the SpatialDataArray class. Below, we define an example spherical profile \(n(r) = \sqrt(2-(r/R)^2)\) as in a Luneburg lens:
# spatial grid
R = 20
Nx, Ny, Nz = 100, 100, 100
X = np.linspace(-R, R, Nx)
Y = np.linspace(-R, R, Ny)
Z = np.linspace(-R, R, Nz)
Xg, Yg, Zg = np.meshgrid(X, Y, Z, indexing='ij')
rg = np.sqrt(Xg**2 + Yg**2 + Zg**2)
# n profile
n_profile = np.ones((Nx, Ny, Nz))
n_profile[rg <= R] = np.sqrt(2-(rg[rg<=R]/R)**2)
# define SpatialDataArray
n_data = SpatialDataArray(n_profile, coords=dict(x=X, y=Y, z=Z))
# define CustomMedium
my_custom_medium = CustomMedium.from_nk(n_data)
Analogous classes also exist for spatially-varying dispersive and anisotropic mediums.
A spatially varying dispersive medium described by the pole-residue pair model. |
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A spatially varying dispersive medium described by the Lorentz model. |
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A spatially varying dispersive medium described by the Sellmeier model. |
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A spatially varying dispersive medium described by the Drude model. |
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A spatially varying dispersive medium described by the Debye model. |
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Diagonally anisotropic medium with spatially varying permittivity in each component. |
See also
For in-depth discussion with examples, see the following learning center article:
Example applications:
Perturbation (Multiphysics)#
Dispersionless medium with perturbations. |
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A dispersive medium described by the pole-residue pair model with perturbations. |
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Nedeljkovic-Soref-Mashanovich model for the perturbation of the refractive index and |
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Stores information about parameter perturbations due to different physical effect. |
When performing a multiphysics simulation, the temperature or carrier density field will result in a perturbation to the optical medium.
# define a linear heat perturbation
my_linear_heat_perturbation = LinearHeatPerturbation(coeff=1e-5, temperature_ref = 300)
# define perturbation medium
my_perturbed_medium = PerturbationMedium(
permittivity = 2.0,
permittivity_perturbation = ParameterPerturbation(heat=my_linear_heat_perturbation)
)
Multiphysics simulations require additional set up in addition to defining a perturbation medium. To get started, we recommend that you check out the example models listed below.
See also
Example applications (thermal):
Example applications (charge):
Nonlinearity#
Abstract specification for adding nonlinearities to a medium. |
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Model for an instantaneous nonlinear chi3 susceptibility. |
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Model for Kerr nonlinearity which gives an intensity-dependent refractive index of the form \(n = n_0 + n_2 I\). |
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Model for two-photon absorption (TPA) nonlinearity which gives an intensity-dependent absorption of the form \(\alpha = \alpha_0 + \beta I\). |
Nonlinear mediums are modeled as regular mediums with an added NonlinearSpec specification.
# firstly, define a nonlinear model of interest
my_nonlinear_model = NonlinearSusceptibility(chi3=1.0)
# then, define the nonlinear spec
my_nonlinear_spec = NonlinearSpec(models=[my_nonlinear_model], num_iters=5)
# finally, define the nonlinear medium
my_nonlinear_medium = Medium(permittivity=2.0, nonlinear_spec=my_nonlinear_spec)
Nonlinear problems are solved iteratively; use the num_iters parameter in NonlinearSpec to control the number of iterations. Currently, three nonlinear models are supported, NonlinearSusceptibility, KerrNonlinearity, and TwoPhotonAbsorption. For details, please refer to the corresponding reference article.
Note
Nonlinear simulations are generally quite complex and the proper approach can vary greatly on a case-by-case basis. Other than defining a nonlinear medium, additional simulation subtleties are likely present when compared to their linear counterparts.
See also
Example applications:
Spatio-temporal Modulation#
Specification adding space-time modulation to the non-dispersive part of medium including relative permittivity at infinite frequency and electric conductivity. |
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Space-time modulation applied to a medium, adding on top of the time-independent part. |
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Class describing modulation with a harmonic time dependence. |
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The modulation profile with a user-supplied spatial distribution of amplitude and phase. |
In active nanophotonic devices, the medium can be modulated both spatially and temporally. The ModulationSpec class allows the user to specify such modulations. This modulation is then passed as a parameter into one of the standard medium classes.
# define time modulation
my_time_modulation = ContinuousWaveTimeModulation(freq0=1e12, amplitude=0.1, phase=0)
# define spatial modulation
my_spatial_modulation = SpaceModulation(amplitude=1.0, phase=0)
# define space-time modulation
my_spacetime_modulation = SpaceTimeModulation(time_modulation=my_time_modulation, space_modulation=my_spatial_modulation)
# define modulation spec
my_modulation_spec = ModulationSpec(permittivity=my_spacetime_modulation, conductivity=None)
# define modulated material
my_modulated_medium = Medium(permittivity=2.0, modulation_spec=my_modulation_spec)
In the example above, the ContinuousWaveTimeModulation specifies a time-harmonic modulation with frequency, amplitude and phase given by the respective parameters. The SpaceModulation with amplitude=1.0 results in no spatial modulation. The two modulation factors are multiplied together, and the result is added to the unmodulated permittivity value. In the modulated medium, the permittivity now varies between 1.9 and 2.1.
See also
For in-depth discussion with examples, see the following learning center article: