xopto.materials.ri.util.fit module¶

class Fit(kind: xopto.materials.ri.util.model.base.Model, wavelengths: numpy.ndarray, n: numpy.ndarray, x0: Optional[Tuple] = None, verbose: bool = False)[source]¶

Bases: object

Fit one of the models to the measured values of the refractive index as a function of wavelength.

Parameters

kind (model.Model) –
Model of the refractive index as a function of wavelength. Use one of the predefined models:
- ”Conrady_1’: Conrady 1 model $n^{2} = A_{1} + A_{2}/\lambda + A_{3}/\lambda^{3.5}$
- ”Conrady_2’: Conrady 2 model $n^{2} = A_{1} + A_{2}/\lambda^{2} + A_{3}/\lambda^{3.5}$
- ”ConradyEx_1’: Extended Conrady 1 model $n^{2} = A_{1} + A_{2}/\lambda + A_{3}/\lambda^{A_{4}}$
- ”ConradyEx_2’: Extended Conrady 2 model $n^{2} = A_{1} + A_{2}/\lambda^{2} + A_{3}/\lambda^{A_{4}}$
- ”Exponential”: Exponential model $n = a + b*e^{-\lambda/c}$
- ”Cauchy’: Cauchy’s model $n^{2} = A_{1} + A_{2}\lambda^{2} + A_{3}/\lambda^{2} + A_{4}/\lambda^{4} + A_{5}/\lambda^{6} + A_6/\lambda^{8}$
- ”Sellmeier_1”: 1 term Sellmeier formula $n^{2} = 1 + B_{1}\lambda^{2}/(\lambda^2 - C_{1})$
- ”SellmeierEx_1”: 1 term extended Sellmeier formula $n^{2} = A + B_{1}\lambda^{2}/(\lambda^2 - C_{1})$
- ”Sellmeier_2”: 2 term Sellmeier formula $n^{2} = 1 + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2})$
- ”SellmeierEx_2”: 2 term extended Sellmeier formula $n^{2} = A + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2})$
- ”Sellmeier_3”: 3 term Sellmeier formula $n^{2} = 1 + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2}) + B_{3}\lambda^{2}/(\lambda^{2} - C_{3})$
- ”SellmeierEx_3”: 3 term extended Sellmeier formula $n^{2} = A + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2}) + B_{3}\lambda^{2}/(\lambda^{2} - C_{3})$
- ”Sellmeier_4”: 4 term Sellmeier formula $n^{2} = 1 + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2}) + B_{3}\lambda^{2}/(\lambda^{2} - C_{3} + B_{4}\lambda^{2}/(\lambda^{2} - C_{4})$
- ”SellmeierEx_4”: 4 term extended Sellmeier formula $n^{2} = A + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2}) + B_{3}\lambda^{2}/(\lambda^{2} - C_{3} + B_{4}\lambda^{2}/(\lambda^{2} - C_{4})$
- ”Sellmeier_5”: 5 term Sellmeier formula $n^{2} = 1 + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2}) + B_{3}\lambda^{2}/(\lambda^{2} - C_{3} + B_{4}\lambda^{2}/(\lambda^{2} - C_{4} + B_{5}\lambda^{2}/(\lambda^{2} - C_{5})$
- ”SellmeierEx_5”: 5 term extended Sellmeier formula $n^{2} = A + B_{1}\lambda^{2}/(\lambda^2 - C_{1}) + B_{2}\lambda^{2}/(\lambda^{2} - C_{2}) + B_{3}\lambda^{2}/(\lambda^{2} - C_{3} + B_{4}\lambda^{2}/(\lambda^{2} - C_{4} + B_{5}\lambda^{2}/(\lambda^{2} - C_{5})$
- ”Herzberger_3_2”: Herzberger 4 x 2 formula $n = A + B\lambda^{2} + C\lambda^{4} + D/(\lambda^{2} - 0.028) + E/(\lambda^{2} - 0.028)^{2}$
- ”HerzbergerEx_3_2”: Extended Herzberger 4 x 2 formula $n = A + B\lambda^{2} + C\lambda^{4} + D/(\lambda^{2} - E) + F/(\lambda^{2} - G)^{2}$
- ”Herzberger_4_2”: Herzberger 4 x 2 formula $n = A + B\lambda^{2} + C\lambda^{4} + D\lambda^{6} + E/(\lambda^{2} - 0.028) + F/(\lambda^{2} - 0.028)^{2}$
- ”HerzbergerEx_4_2”: Extended Herzberger 4 x 2 formula $n = A + B\lambda^{2} + C\lambda^{4} + D\lambda^{6} + E/(\lambda^{2} - F) + G/(\lambda^{2} - H)^{2}$
wavelengths (np.ndarray vector) – Wavelengths at which the refractive index was measured. The wavelengths must be ordered in ascending order.
n (np.ndarray vector) – The measured values of refractive index at the specified wavelengths.
x0 (list, tuple or np.ndarray vector) – Initial guess of the model parameters.
verbose (bool) – Verbose mode.

Note

For better convergence of the fit process use an appropriate preprocessor with the supplied refractive index model. A preprocessor that scales the wavelengths by the inverse of the first (shortest) wavelength usually produces good results, e.g. Sellmeier_1(pp=Scale(1.0/wavelengths[0]))

error() → Tuple[numpy.ndarray, numpy.ndarray][source]¶

Fit error at the wavelengths passed to the constructor.

Returns

err (np.ndarray of length wavelengths.size) – Fit errors defined as $(n_{fit} - n_{measured})$ .
rel_err (np.ndarray of length wavelengths.size) – Relative fit error defined as $(n_{fit} - n_{measured})/n_{measured}$ .

property model: xopto.materials.ri.util.model.base.Model¶: Model instance obtained by the fit to the measured data.

property n: numpy.ndarray¶: The measured values of refractive index at the specified wavelengths. The values are returned as passed to the constructor.

visualize(wavelengths: Optional[numpy.ndarray] = None, show=False)[source]¶

Plots the quality of fit.

Parameters

wavelengths (np.ndarray vector) – Wavelengths (m) at which to visualize the fit. If None, the wavelengths passed to the constructor are used.
show (bool) – If nonzero, calls pp.show().

property wavelengths: numpy.ndarray¶: Wavelengths at which the refractive index was measured Returns values as passed to the constructor.

class ThermalFit(kind: xopto.materials.ri.util.model.base.Model, wavelengths: numpy.ndarray, n: numpy.ndarray, temperatures: numpy.ndarray, pk_order: int = 2, x0: Optional[Tuple] = None)[source]¶

Bases: object

Fit a temperature dependent refractive index model. Each parameter of the selected refractive index model is modelled as a polynomial function of the temperature.

Parameters

kind (model.Model) – Refractive index model instance.
wavelengths (np.ndarray vector) – Wavelengths of light (m). The first wavelength in the vector is used by the model preprocessor to normalize/divide all the wavelengths.
n (numpy_ndarray of shape (wavelengths.size, temperatures.size)) – The measured refractive indices.
temperatures (np.ndarray vector) – Temperatures (should be in K, but accepts any units).
pk_order (int) – Order of the polynomial that models the temperature dependence of the refractive index model parameters.
x0 (list, tuple ndarray vector) – Initial guess of the refractive index model parameters.

error() → Tuple[numpy.ndarray, numpy.ndarray][source]¶

Fit error at the wavelengths and temperatures passed to the constructor.

Returns

err (np.ndarray of shape (wavelengths.size, temperatures.size)) – Fit errors defined as $(n_{fit} - n_{measured})$ .
rel_err (np.ndarray of shape (wavelengths.size, temperatures.size)) – Relative fit error defined as $(n_{fit} - n_{measured})/n_{measured}$ .

property n: numpy.ndarray¶: The measured values of refractive index as passed to the constructor.

polynomial_coefficients() → numpy.ndarray[source]¶

Returns the polynomial coefficients that model the temperature dependence of the refractive index model parameters. Use np.polyval to determine the values of polynomial coefficients at a given temperature (use the same temperture units as for the temperatures that were passed to the constructor).

Returns: pk – The polynomial coefficients of each model parameter are stored in columns.
Return type: np.ndarray of shape (pk_order + 1, num_terms*2)

property temperatures: numpy.ndarray¶: The temperatures at which the refractive index was measured. Returns the values as passed to the constructor.

visualize(wavelengths: Optional[numpy.ndarray] = None, show: bool = False)[source]¶

Plots the quality of fit.

Parameters

wavelengths (np.ndarray vector) – Wavelengths (m) at which to visualize the fit. If None, the wavelengths passed to the constructor are used.
show (bool) – If nonzero, calls pp.show().