Scikit-Qfit¶
A gradient-orthogonal Q-polynomial representation of axially symmetric optical surfaces has been used for several years by designers. It improves upon the standard polynomial form by being simpler to interpret, using fewer terms to adequately define a surface, and sometimes even offering quicker convergence in design optimization. Q-polynomials were first introduced with the publication of:
- G W Forbes, Shape specification for axially symmetric optical surfaces, Opt. Express 15, 5218-5226 (2007)
The use of Q-polynomials was extended, by the original author, to address freeform shapes through the following articles:
- Fitting freeform shapes with orthogonal bases, Opt. Express 21, 19061-19081 (2013)
- Characterizing the shape of freeform optics, Opt. Express 20(3), 2483-2499 (2012)
- Robust, efficient computational methods for axially symmetric optical aspheres, Opt. Express 18(19), 19700-19712 (2010)
The implementation of this package follows the description in “Fitting freeform shapes with orthogonal bases”.
Q-spectrum examples¶
The Q-spectrum offers a natural way to quantify and investigate surface structure on parts with circular apertures. For example, spoke structure on the surface appears in specific columns of the spectrum whereas raster patterns appear as diagonal bands with a slope of one half. Rotationally symmetric structure, such a rings etc., are all contained in the m=0 column. The spectrum is computed with an FFT-like step that suffers similarly from aliasing unless a sufficiently large range of frequencies is computed.
Generating maps¶
Interesting features in the spectrum can be extracted to create topography layers that quantify and highlight flaws in the surface. The same process can be used to monitor residuals in the data after subtracting fitted polynomial terms. In the example below the radial-azimuthal spectrum shows quite distinct columns at particular azimuthal orders that are even more evident by summing the squares of the coefficients to produce a spectral line plot. The last image shows the contribution of just a sample of the spectral lines in the original data map. The article “Fitting freeform shapes with orthogonal bases” provides more details on the types of spectral analysis that can be performed and the characteristic flaws associated with polishing techniques that employ either raster or rotational modes.
The above images were generated from data supplied by Mahr GmbH.
Usage¶
To generate a Q-freeform spectrum from a data map, pass the coordinate axis x and y and 2-D array of data with shape (x.size,y.size) as arguments to the method qspec(). The azimuthal and radial spectrum limits are set by m_max and n_max respectively.
>>> import skqfit.qspectre as qf
>>> ...
>>> qspec = qf.qspec(x, y, zmap, m_max=500, n_max=500)
To observe the contribution of spectrum components it is necessary to use the sine and cosine elements in the forward processing step. After modifying the spectrum, an inverse process creates a data map along with optional x and y derivatives.
>>> qs = qf.QSpectrum()
>>> qs.data_map(x, y, zmap)
>>> a_nm, b_nm = qs.q_fit(mmax, nmax)
>>> ... # modify spectrum
>>> nmap, dfdx, dfdy = qs.build_map(x, y, a_nm, b_nm, inc_deriv=True)
A 1D trace across the fitted data can also be generated by passing a set of xv, yv arrays that represent the (x, y) coordinates.
>>> zv, dfdx, dfdy = qs.build_profile(xv, yv, a_nm, b_nm, inc_deriv=True)
Code structure¶
The package only contains two modules. The Jacobi module provides support for normalised Jacobi polynomials that extend the range of usable parameters before an overflow condition is encountered. Even though the Q-fitting algorithm is procedural the implementation is via a Q-spectrum class as it allowed the spectrum to be iterated with different parameters without having to reload the data. The algorithm implementation refers to the relevant section of the reference documents.