XMF API

Summary table

xmf.fit_convex_ellipsoid_height(x2d, y2d, ...)	Fit the convex ellipsoid parameters from a measured height map.
xmf.fit_concave_ellipsoid_height(x2d, y2d, ...)	Fit the concave ellipsoid parameters from a measured height map.
xmf.fit_convex_elliptic_cylinder_height(x2d, ...)	Fit the convex elliptic cylinder parameters from a measured height map.
xmf.fit_concave_elliptic_cylinder_height(...)	Fit the concave elliptic cylinder parameters from a measured height map.
xmf.fit_convex_ellipse_height(x1d, z1d, ...)	Fit the convex ellipse parameters from a measured height profile.
xmf.fit_concave_ellipse_height(x1d, z1d, ...)	Fit the concave ellipse parameters from a measured height profile.
xmf.fit_convex_ellipse_slope(x1d, sx1d, ...)	Fit the convex ellipse parameters from a measured slope profile.
xmf.fit_concave_ellipse_slope(x1d, sx1d, ...)	Fit the concave ellipse parameters from a measured slope profile.
xmf.fit_concave_hyperboloid_height(x2d, y2d, ...)	Fit the concave hyperboloid parameters from a measured height map.
xmf.fit_convex_hyperboloid_height(x2d, y2d, ...)	Fit the convex hyperboloid parameters from a measured height map.
xmf.fit_convex_hyperbolic_cylinder_height(...)	Fit the convex hyperbolic cylinder parameters from a measured height map.
xmf.fit_concave_hyperbolic_cylinder_height(...)	Fit the concave hyperbolic cylinder parameters from a measured height map.
xmf.fit_concave_hyperbola_height(x1d, ...)	Fit the concave hyperbola parameters from a measured height profile.
xmf.fit_convex_hyperbola_height(x1d, ...)	Fit the convex hyperbola parameters from a measured height profile.
xmf.fit_convex_hyperbola_slope(x1d, sx1d, ...)	Fit the convex hyperbola parameters from a measured slope profile.
xmf.fit_concave_hyperbola_slope(x1d, sx1d, ...)	Fit the concave hyperbola parameters from a measured slope profile.

Full API Details

xmf.fig_compare_1d_height(x1d, z1d_generation, z1d_standard, str_title)

Compare two 1D height data sets and plot the difference

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_generation (numpy.ndarray) – 1D array of height data from generation

• z1d_standard (numpy.ndarray) – 1D array of height data from standard

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_compare_1d_slope(x1d, sx1d_generation, sx1d_standard, str_title)

Compare two 1D slope data sets and plot the difference

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• sx1d_generation (numpy.ndarray) – 1D array of slope data from generation

• sx1d_standard (numpy.ndarray) – 1D array of slope data from standard

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_fitting_height(x1d, z1d_measured, z1d_fit, z1d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 1D fitting plot with measured data, fitted data, residuals, input parameters, and optimized parameters

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_measured (numpy.ndarray) – 1D array of measured z-values

• z1d_fit (numpy.ndarray) – 1D array of fitted z-values

• z1d_res (numpy.ndarray) – 1D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_fitting_slope(x1d, sx1d_measured, sx1d_fit, sx1d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 1D fitting plot with measured slopes, fitted slopes, residuals, input parameters, and optimized parameters

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• sx1d_measured (numpy.ndarray) – 1D array of measured slopes

• sx1d_fit (numpy.ndarray) – 1D array of fitted slopes

• sx1d_res (numpy.ndarray) – 1D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_height(x1d, z1d_quad_sln, z1d_expression, str_title)

Show a 1D plot of height data from quadratic equation solution and expression

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_quad_sln (numpy.ndarray) – 1D array of height data from quadratic equation solution

• z1d_expression (numpy.ndarray) – 1D array of height data from expression

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_slope(x1d, sx1d, str_title)

xmf.fig_show_2d_fitting_map(x2d, y2d, z2d_measured, z2d_fit, z2d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 2D fitting map with colorbar, input parameters, fitted parameters, and residuals

Parameters:

• x2d (numpy.ndarray) – 2D array of x-coordinates

• y2d (numpy.ndarray) – 2D array of y-coordinates

• z2d_measured (numpy.ndarray) – 2D array of measured z-values

• z2d_fit (numpy.ndarray) – 2D array of fitted z-values

• z2d_res (numpy.ndarray) – 2D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_2d_map(x2d, y2d, z2d_quad_sln, z2d_expression, str_title)

Show a 2D map of height data with colorbar

Parameters:

• x2d (numpy.ndarray) – 2D array of x-coordinates

• y2d (numpy.ndarray) – 2D array of y-coordinates

• z2d_quad_sln (numpy.ndarray) – 2D array of height data from quadratic equation solution

• z2d_expression (numpy.ndarray) – 2D array of height data from expression

• str_title (str) – Title of the plot

Return type:

None

xmf.fit_concave_ellipse_height(x1d: ndarray, z1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipse parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_ellipse_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipse parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_ellipsoid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipsoid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (dict) – The input parameters dictionary containing p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_elliptic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave elliptic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperbola_height(x1d: ndarray, z1d_measured: ndarray, input_params_dict: dict, opt_1d_cylinder_height: dict)

Fit the concave hyperbola parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperbola_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperbola parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperbolic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperbolic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperboloid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperboloid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_ellipse_height(x1d: ndarray, z1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipse parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_ellipse_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipse parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_ellipsoid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipsoid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_elliptic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex elliptic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperbola_height(x1d: ndarray, z1d_measured: ndarray, input_params_dict: dict, opt_1d_cylinder_height: dict)

Fit the convex hyperbola parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperbola_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperbola parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperbolic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperbolic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperboloid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperboloid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.generate_1d_height(standard_height_function: LambdaType, x1d: array, p: float, q: float, theta: float, x_i: float, z_i: float, beta: float, z1d_measured: array = None)

Geneate 1D height map with (p, q, theta, x_i, y_i, z_i, alpha, beta)

Parameters:

• standard_height_function (function) – The standard height function

• x1d (numpy.ndarray) – The 1D x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z1d_measured (numpy.ndarray) – The measured height map

Returns:

z1d – The height map

Return type:

numpy.ndarray

xmf.generate_1d_slope(standard_slope_function: LambdaType, x1d: array, p: float, q: float, theta: float, x_i: float, beta: float)

Generate 1D slope map with (p, q, theta, x_i, y_i, z_i, alpha, beta)

Parameters:

• standard_slope_function (function) – The standard slope function

• x1d (numpy.ndarray) – The 1D x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• sx1d_measured (numpy.ndarray) – The measured slope map

Returns:

sx1d – The slope map

Return type:

numpy.ndarray

xmf.generate_2d_curved_surface_height(standard_height_function: LambdaType, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, x_i: float, y_i: float, z_i: float, alpha: float, beta: float, gamma: float, z2d_measured: ndarray = None)

Geneate 2D curved surface height map with (p, q, theta, x_i, y_i, z_i, alpha, beta, gamma)

Parameters:

• standard_height_function (function) – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• y_i (float) – y-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the y-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z2d_measured (numpy.ndarray) – The measured height map

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.generate_2d_cylinder_height(standard_height_function: LambdaType, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, x_i: float, z_i: float, alpha: float, beta: float, gamma: float, z2d_measured: ndarray = None)

Geneate 2D cylinder surface height map with (p, q, theta, x_i, y_i, z_i, alpha, beta, gamma)

Parameters:

• standard_height_function (function) – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z2d_measured (numpy.ndarray) – The measured height map

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.standard_concave_ellipsoid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D concave ellipsoid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_concave_elliptic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard convex elliptic cylinder height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_concave_elliptic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard concave elliptic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_concave_hyperbolic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard 2D concave hyperbolic height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The 2D x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_concave_hyperbolic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard concave hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_concave_hyperboloid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D concave hyperboloid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_convex_ellipsoid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D convex ellipsoid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_convex_elliptic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard convex elliptic cylinder height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z – The height

Return type:

numpy.ndarray

xmf.standard_convex_elliptic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard convex elliptic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_convex_hyperbolic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard 2D convex hyperbolic height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The 2D x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_convex_hyperbolic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard convex hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_convex_hyperboloid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D convex hyperboloid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

XMF package

xmf.fig_show module

xmf.fig_show.add_colorbar(im: _ScalarMappable, title: str = None, aspect: int = 10, pad_fraction: float = 1.5, **kwargs)

Add a vertical color bar to an image plot

Parameters:

• im (matplotlib.cm.ScalarMappable) – The image to be described by the colorbar

• title (str) – The title of the colorbar

• aspect (int) – The aspect ratio between width and height

• pad_fraction (float) – The padding fraction

Returns:

cbar – The added colorbar

Return type:

matplotlib.pyplot.colorbar

xmf.fig_show.fig_compare_1d_height(x1d, z1d_generation, z1d_standard, str_title)

Compare two 1D height data sets and plot the difference

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_generation (numpy.ndarray) – 1D array of height data from generation

• z1d_standard (numpy.ndarray) – 1D array of height data from standard

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.fig_compare_1d_slope(x1d, sx1d_generation, sx1d_standard, str_title)

Compare two 1D slope data sets and plot the difference

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• sx1d_generation (numpy.ndarray) – 1D array of slope data from generation

• sx1d_standard (numpy.ndarray) – 1D array of slope data from standard

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.fig_show_1d_fitting_height(x1d, z1d_measured, z1d_fit, z1d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 1D fitting plot with measured data, fitted data, residuals, input parameters, and optimized parameters

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_measured (numpy.ndarray) – 1D array of measured z-values

• z1d_fit (numpy.ndarray) – 1D array of fitted z-values

• z1d_res (numpy.ndarray) – 1D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.fig_show_1d_fitting_slope(x1d, sx1d_measured, sx1d_fit, sx1d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 1D fitting plot with measured slopes, fitted slopes, residuals, input parameters, and optimized parameters

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• sx1d_measured (numpy.ndarray) – 1D array of measured slopes

• sx1d_fit (numpy.ndarray) – 1D array of fitted slopes

• sx1d_res (numpy.ndarray) – 1D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.fig_show_1d_height(x1d, z1d_quad_sln, z1d_expression, str_title)

Show a 1D plot of height data from quadratic equation solution and expression

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_quad_sln (numpy.ndarray) – 1D array of height data from quadratic equation solution

• z1d_expression (numpy.ndarray) – 1D array of height data from expression

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.fig_show_1d_slope(x1d, sx1d, str_title)

xmf.fig_show.fig_show_2d_fitting_map(x2d, y2d, z2d_measured, z2d_fit, z2d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 2D fitting map with colorbar, input parameters, fitted parameters, and residuals

Parameters:

• x2d (numpy.ndarray) – 2D array of x-coordinates

• y2d (numpy.ndarray) – 2D array of y-coordinates

• z2d_measured (numpy.ndarray) – 2D array of measured z-values

• z2d_fit (numpy.ndarray) – 2D array of fitted z-values

• z2d_res (numpy.ndarray) – 2D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.fig_show_2d_map(x2d, y2d, z2d_quad_sln, z2d_expression, str_title)

Show a 2D map of height data with colorbar

Parameters:

• x2d (numpy.ndarray) – 2D array of x-coordinates

• y2d (numpy.ndarray) – 2D array of y-coordinates

• z2d_quad_sln (numpy.ndarray) – 2D array of height data from quadratic equation solution

• z2d_expression (numpy.ndarray) – 2D array of height data from expression

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show.reg_exp_rep(s)

Replace scientific notation in a string with LaTeX format

Parameters:

s (str) – The string to be processed

Returns:

The processed string with LaTeX format

Return type:

str

xmf.layer_01_standard module

xmf.layer_01_standard.standard_concave_ellipsoid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D concave ellipsoid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_concave_elliptic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard convex elliptic cylinder height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_concave_elliptic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard concave elliptic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_concave_hyperbolic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard 2D concave hyperbolic height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The 2D x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_concave_hyperbolic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard concave hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_concave_hyperboloid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D concave hyperboloid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_convex_ellipsoid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D convex ellipsoid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_convex_elliptic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard convex elliptic cylinder height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z – The height

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_convex_elliptic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard convex elliptic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_convex_hyperbolic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard 2D convex hyperbolic height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The 2D x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_convex_hyperbolic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard convex hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_convex_hyperboloid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D convex hyperboloid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_quadric_cylinder_height(x: ndarray, p: float, q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard quadric cylinder height with (p, q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z – The height

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_quadric_cylinder_xslope(x: ndarray, p: float, q: float, theta: float)

The standard quadric cylinder slope with (p, q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.layer_01_standard.standard_quadrics_height(x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard quadrics height with (p, q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d_quad_sln – The 2D height map

Return type:

numpy.ndarray

xmf.layer_02_generation module

xmf.layer_02_generation.compose_transformation_matrix(alpha: float, beta: float, gamma: float, x_i: float, y_i: float, z_i: float)

The standard concave hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• y_i (float) – y-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the y-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

Returns:

T – The x-slope”

Return type:

numpy.ndarray

xmf.layer_02_generation.generate_1d_height(standard_height_function: LambdaType, x1d: array, p: float, q: float, theta: float, x_i: float, z_i: float, beta: float, z1d_measured: array = None)

Geneate 1D height map with (p, q, theta, x_i, y_i, z_i, alpha, beta)

Parameters:

• standard_height_function (function) – The standard height function

• x1d (numpy.ndarray) – The 1D x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z1d_measured (numpy.ndarray) – The measured height map

Returns:

z1d – The height map

Return type:

numpy.ndarray

xmf.layer_02_generation.generate_1d_slope(standard_slope_function: LambdaType, x1d: array, p: float, q: float, theta: float, x_i: float, beta: float)

Generate 1D slope map with (p, q, theta, x_i, y_i, z_i, alpha, beta)

Parameters:

• standard_slope_function (function) – The standard slope function

• x1d (numpy.ndarray) – The 1D x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• sx1d_measured (numpy.ndarray) – The measured slope map

Returns:

sx1d – The slope map

Return type:

numpy.ndarray

xmf.layer_02_generation.generate_2d_curved_surface_height(standard_height_function: LambdaType, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, x_i: float, y_i: float, z_i: float, alpha: float, beta: float, gamma: float, z2d_measured: ndarray = None)

Geneate 2D curved surface height map with (p, q, theta, x_i, y_i, z_i, alpha, beta, gamma)

Parameters:

• standard_height_function (function) – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• y_i (float) – y-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the y-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z2d_measured (numpy.ndarray) – The measured height map

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.layer_02_generation.generate_2d_cylinder_height(standard_height_function: LambdaType, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, x_i: float, z_i: float, alpha: float, beta: float, gamma: float, z2d_measured: ndarray = None)

Geneate 2D cylinder surface height map with (p, q, theta, x_i, y_i, z_i, alpha, beta, gamma)

Parameters:

• standard_height_function (function) – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z2d_measured (numpy.ndarray) – The measured height map

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.layer_02_generation.iter_generate_height(standard_height_function, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, tf: ndarray, z2d_measured: ndarray = None, thr_rms_dxy: float = 1e-09)

The height generation with iterations(p, q, theta, thr_rms_dxy)

Parameters:

• standard_height_function – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• tf (numpy.ndarray) – The transformation matrix

• z2d_measured (numpy.ndarray) – The measured height map

• thr_rms_dxy (float) – The threshold of RMS

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.layer_03_optimization module

xmf.layer_03_optimization.check_input_params(input_params_dict: dict, x: ndarray, y: ndarray, v: ndarray)

Function to check input parameters.

Parameters:

• input_params_dict (dict) – The p, q, theta, x_i (optional), y_i (optional), z_i (optional), alpha (optional), beta (optional) and gamma (optional) target parameters, suggested in unit of [m] [m] [rad] [m] [m] [m] [rad] [rad] [rad]

• x (numpy.ndarray) – The measured x-coordinate in in unit of [m] as a suggestion

• y (numpy.ndarray) – The measured y-coordinate in in unit of [m] as a suggestion

• v (numpy.ndarray) – The measured slope or height in [rad] or [m] as a suggestion

Returns:

init_params – The used initial parameters.

Return type:

numpy.ndarray

xmf.layer_03_optimization.optimize_parameters(surface_generation_function: LambdaType, standard_surface_shape_function: LambdaType, x: ndarray, y: ndarray, v: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Basic function to provide a convenient way to optimize the surface parameters.

Parameters:

• surface_generation_function (function) – The function to generate surface (1D or 2D, slope or height)

• standard_surface_shape_function (function) – The function handle for a standard surface shape

• x (numpy.ndarray) – The measured x-coordinate in in unit of [m] as a suggestion

• y (numpy.ndarray) – The measured y-coordinate in in unit of [m] as a suggestion

• v (numpy.ndarray) – The measured slope or height in [rad] or [m] as a suggestion

• input_params_dict (dict) – The p, q, theta, x_i (optional), y_i (optional), z_i (optional), alpha (optional), beta (optional) and gamma (optional) target parameters, suggested in unit of [m] [m] [rad] [m] [m] [m] [rad] [rad] [rad]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

Returns:

• v_res (numpy.ndarray) – The residual (1D or 2D)

• v_fit (numpy.ndarray) – The fitting result (1D or 2D)

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

• init_params (numpy.ndarray) – The used initial parameters.

xmf.layer_03_optimization.optimize_parameters_with_opt(surface_generation_function: LambdaType, standard_surface_shape_function: LambdaType, x: ndarray, y: ndarray, v: ndarray, input_params_dict: dict, opt_dict: dict)

Basic function to provide a convenient way to optimize the surface parameters with optimization flag.

Parameters:

• surface_generation_function (function) – The function to generate surface (1D or 2D, slope or height)

• standard_surface_shape_function (function) – The function handle for a standard surface shape

• x (numpy.ndarray) – The measured x-coordinate in in unit of [m] as a suggestion

• y (numpy.ndarray) – The measured y-coordinate in in unit of [m] as a suggestion

• v (numpy.ndarray) – The measured slope or height in [rad] or [m] as a suggestion

• input_params_dict (dict) – The p, q, theta, x_i (optional), y_i (optional), z_i (optional), alpha (optional), beta (optional) and gamma (optional) target parameters, suggested in unit of [m] [m] [rad] [m] [m] [m] [rad] [rad] [rad]

• opt_dict (dict) – The structure to set whether optimization is needed for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

Returns:

• v_res (numpy.ndarray) – The residual (1D or 2D)

• v_fit (numpy.ndarray) – The fitting result (1D or 2D)

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

• init_params (numpy.ndarray) – The used initial parameters.

xmf.layer_03_optimization.optimize_parameters_with_tol(surface_generation_function: LambdaType, standard_surface_shape_function: LambdaType, x: ndarray, y: ndarray, v: ndarray, input_params_dict: dict, tol_dict: dict)

Basic function to provide a convenient way to optimize the surface parameters with tolerances.

Parameters:

• surface_generation_function (function) – The function to generate surface (1D or 2D, slope or height)

• standard_surface_shape_function (function) – The function handle for a standard surface shape

• x (numpy.ndarray) – The measured x-coordinate in in unit of [m] as a suggestion

• y (numpy.ndarray) – The measured y-coordinate in in unit of [m] as a suggestion

• v (numpy.ndarray) – The measured slope or height in [rad] or [m] as a suggestion

• input_params_dict (dict) – The p, q, theta, x_i (optional), y_i (optional), z_i (optional), alpha (optional), beta (optional) and gamma (optional) target parameters, suggested in unit of [m] [m] [rad] [m] [m] [m] [rad] [rad] [rad]

• tol_dict (dict) – The structure to set the tolerances for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

Returns:

• v_res (numpy.ndarray) – The residual (1D or 2D)

• v_fit (numpy.ndarray) – The fitting result (1D or 2D)

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

• init_params (numpy.ndarray) – The used initial parameters.

xmf.layer_04_fit module

xmf.layer_04_fit.fit_concave_ellipse_height(x1d: ndarray, z1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipse parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_ellipse_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipse parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_ellipsoid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipsoid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (dict) – The input parameters dictionary containing p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_elliptic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave elliptic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_hyperbola_height(x1d: ndarray, z1d_measured: ndarray, input_params_dict: dict, opt_1d_cylinder_height: dict)

Fit the concave hyperbola parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_hyperbola_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperbola parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_hyperbolic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperbolic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_concave_hyperboloid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperboloid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_ellipse_height(x1d: ndarray, z1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipse parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_ellipse_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipse parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_ellipsoid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipsoid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_elliptic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex elliptic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_hyperbola_height(x1d: ndarray, z1d_measured: ndarray, input_params_dict: dict, opt_1d_cylinder_height: dict)

Fit the convex hyperbola parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_hyperbola_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperbola parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_hyperbolic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperbolic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.layer_04_fit.fit_convex_hyperboloid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperboloid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

Module contents

xmf.fig_compare_1d_height(x1d, z1d_generation, z1d_standard, str_title)

Compare two 1D height data sets and plot the difference

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_generation (numpy.ndarray) – 1D array of height data from generation

• z1d_standard (numpy.ndarray) – 1D array of height data from standard

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_compare_1d_slope(x1d, sx1d_generation, sx1d_standard, str_title)

Compare two 1D slope data sets and plot the difference

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• sx1d_generation (numpy.ndarray) – 1D array of slope data from generation

• sx1d_standard (numpy.ndarray) – 1D array of slope data from standard

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_fitting_height(x1d, z1d_measured, z1d_fit, z1d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 1D fitting plot with measured data, fitted data, residuals, input parameters, and optimized parameters

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_measured (numpy.ndarray) – 1D array of measured z-values

• z1d_fit (numpy.ndarray) – 1D array of fitted z-values

• z1d_res (numpy.ndarray) – 1D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_fitting_slope(x1d, sx1d_measured, sx1d_fit, sx1d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 1D fitting plot with measured slopes, fitted slopes, residuals, input parameters, and optimized parameters

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• sx1d_measured (numpy.ndarray) – 1D array of measured slopes

• sx1d_fit (numpy.ndarray) – 1D array of fitted slopes

• sx1d_res (numpy.ndarray) – 1D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_height(x1d, z1d_quad_sln, z1d_expression, str_title)

Show a 1D plot of height data from quadratic equation solution and expression

Parameters:

• x1d (numpy.ndarray) – 1D array of x-coordinates

• z1d_quad_sln (numpy.ndarray) – 1D array of height data from quadratic equation solution

• z1d_expression (numpy.ndarray) – 1D array of height data from expression

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_1d_slope(x1d, sx1d, str_title)

xmf.fig_show_2d_fitting_map(x2d, y2d, z2d_measured, z2d_fit, z2d_res, input_params_dict, opt_params_dict, opt_params_ci_dict, str_title)

Show a 2D fitting map with colorbar, input parameters, fitted parameters, and residuals

Parameters:

• x2d (numpy.ndarray) – 2D array of x-coordinates

• y2d (numpy.ndarray) – 2D array of y-coordinates

• z2d_measured (numpy.ndarray) – 2D array of measured z-values

• z2d_fit (numpy.ndarray) – 2D array of fitted z-values

• z2d_res (numpy.ndarray) – 2D array of residuals

• input_params_dict (dict) – Dictionary of input parameters

• opt_params_dict (dict) – Dictionary of optimized parameters

• opt_params_ci_dict (dict) – Dictionary of optimized parameters confidence intervals

• str_title (str) – Title of the plot

Return type:

None

xmf.fig_show_2d_map(x2d, y2d, z2d_quad_sln, z2d_expression, str_title)

Show a 2D map of height data with colorbar

Parameters:

• x2d (numpy.ndarray) – 2D array of x-coordinates

• y2d (numpy.ndarray) – 2D array of y-coordinates

• z2d_quad_sln (numpy.ndarray) – 2D array of height data from quadratic equation solution

• z2d_expression (numpy.ndarray) – 2D array of height data from expression

• str_title (str) – Title of the plot

Return type:

None

xmf.fit_concave_ellipse_height(x1d: ndarray, z1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipse parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_ellipse_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipse parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_ellipsoid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave ellipsoid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (dict) – The input parameters dictionary containing p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_elliptic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave elliptic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperbola_height(x1d: ndarray, z1d_measured: ndarray, input_params_dict: dict, opt_1d_cylinder_height: dict)

Fit the concave hyperbola parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperbola_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperbola parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperbolic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperbolic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_concave_hyperboloid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the concave hyperboloid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_ellipse_height(x1d: ndarray, z1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipse parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_ellipse_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipse parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_ellipsoid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex ellipsoid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_elliptic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex elliptic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperbola_height(x1d: ndarray, z1d_measured: ndarray, input_params_dict: dict, opt_1d_cylinder_height: dict)

Fit the convex hyperbola parameters from a measured height profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• z1d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• z1d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperbola_slope(x1d: ndarray, sx1d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperbola parameters from a measured slope profile.

Parameters:

• x1d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• sx1d (numpy.ndarray) – The measured slope in the suggested unit of [m/m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z1d_residual (numpy.ndarray) – The height residual after the best fit.

• sx1d_fit (numpy.ndarray) – The fitted slope

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperbolic_cylinder_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperbolic cylinder parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.fit_convex_hyperboloid_height(x2d: ndarray, y2d: ndarray, z2d: ndarray, input_params_dict: dict, opt_or_tol_dict: dict)

Fit the convex hyperboloid parameters from a measured height map.

Parameters:

• x2d (numpy.ndarray) – The x coordinate in the suggested unit of [m]

• y2d (numpy.ndarray) – The y coordinate in the suggested unit of [m]

• z2d (numpy.ndarray) – The z coordinate in the suggested unit of [m]

• input_params_dict (numpy.ndarray) – The the p, q, theta, x_i (optional), and y_i (optional) in the suggested unit of [m] [m] [rad] [m]

• opt_or_tol_dict (dict) – The structure to set whether optimization flag or tolerance for p, q, theta, x_i, y_i, z_i, alpha, beta, gamma.

• init_params (numpy.ndarray) – The used initial parameters.

Returns:

• z2d_residual (numpy.ndarray) – The height residual after the best fit.

• z2d_fit (numpy.ndarray) – The fitted height

• opt_params_dict (dict) – The optimized parameters in dictionary

• opt_params_ci_dict (dict) – The confidence intervals of the optimized parameters in dictionary

xmf.generate_1d_height(standard_height_function: LambdaType, x1d: array, p: float, q: float, theta: float, x_i: float, z_i: float, beta: float, z1d_measured: array = None)

Geneate 1D height map with (p, q, theta, x_i, y_i, z_i, alpha, beta)

Parameters:

• standard_height_function (function) – The standard height function

• x1d (numpy.ndarray) – The 1D x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z1d_measured (numpy.ndarray) – The measured height map

Returns:

z1d – The height map

Return type:

numpy.ndarray

xmf.generate_1d_slope(standard_slope_function: LambdaType, x1d: array, p: float, q: float, theta: float, x_i: float, beta: float)

Generate 1D slope map with (p, q, theta, x_i, y_i, z_i, alpha, beta)

Parameters:

• standard_slope_function (function) – The standard slope function

• x1d (numpy.ndarray) – The 1D x coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• sx1d_measured (numpy.ndarray) – The measured slope map

Returns:

sx1d – The slope map

Return type:

numpy.ndarray

xmf.generate_2d_curved_surface_height(standard_height_function: LambdaType, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, x_i: float, y_i: float, z_i: float, alpha: float, beta: float, gamma: float, z2d_measured: ndarray = None)

Geneate 2D curved surface height map with (p, q, theta, x_i, y_i, z_i, alpha, beta, gamma)

Parameters:

• standard_height_function (function) – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• y_i (float) – y-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the y-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z2d_measured (numpy.ndarray) – The measured height map

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.generate_2d_cylinder_height(standard_height_function: LambdaType, x2d: ndarray, y2d: ndarray, p: float, q: float, theta: float, x_i: float, z_i: float, alpha: float, beta: float, gamma: float, z2d_measured: ndarray = None)

Geneate 2D cylinder surface height map with (p, q, theta, x_i, y_i, z_i, alpha, beta, gamma)

Parameters:

• standard_height_function (function) – The standard height function

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• p (float) – The p value: the distance from the source to the chief ray intersection

• q (float) – The q value: the distance from the chief ray intersection to the focus

• theta (float) – The grazing angle

• alpha (float) – The angle around x-axis

• beta (float) – The angle around y-axis

• gamma (float) – The angle around z-axis

• x_i (float) – x-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the x-position of chief ray intersection in metrology coordinates

• z_i (float) – z-translation in the conversion from standard mirror coordinates to metrology coordinates, also revealing the z-position of chief ray intersection in metrology coordinates

• z2d_measured (numpy.ndarray) – The measured height map

Returns:

z2d – The height map

Return type:

numpy.ndarray

xmf.standard_concave_ellipsoid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D concave ellipsoid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_concave_elliptic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard convex elliptic cylinder height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_concave_elliptic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard concave elliptic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_concave_hyperbolic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard 2D concave hyperbolic height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The 2D x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_concave_hyperbolic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard concave hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_concave_hyperboloid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D concave hyperboloid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_convex_ellipsoid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D convex ellipsoid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_convex_elliptic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard convex elliptic cylinder height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z – The height

Return type:

numpy.ndarray

xmf.standard_convex_elliptic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard convex elliptic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_convex_hyperbolic_cylinder_height(x: ndarray, abs_p: float, abs_q: float, theta: float, return_z_expression_as_extra: bool = False)

The standard 2D convex hyperbolic height with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The 2D x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z_expression_as_extra (bool) – If True, return the z_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray

xmf.standard_convex_hyperbolic_cylinder_xslope(x: ndarray, abs_p: float, abs_q: float, theta: float)

The standard convex hyperbolic cylinder x-slope with (abs_p, abs_q, theta)

Parameters:

• x (numpy.ndarray) – The x coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

Returns:

sx – The x-slope

Return type:

numpy.ndarray

xmf.standard_convex_hyperboloid_height(x2d: ndarray, y2d: ndarray, abs_p: float, abs_q: float, theta: float, return_z2d_expression_as_extra: bool = False)

The standard 2D convex hyperboloid height with (abs_p, abs_q, theta)

Parameters:

• x2d (numpy.ndarray) – The 2D x coordinates

• y2d (numpy.ndarray) – The 2D y coordinates

• abs_p (float) – The abs_p value: the absolute value of the distance between the source and the chief ray intersection

• abs_q (float) – The abs_q value: the absolute value of the distance between the chief ray intersection and the focus

• theta (float) – The grazing angle

• return_z2d_expression_as_extra (bool) – If True, return the z2d_expression as well

Returns:

z2d – The 2D height map

Return type:

numpy.ndarray