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What is the Normalization Radius?

Question: What is the normalization radius used by diffractive surfaces for?
Mark Nicholson
Frequently Asked Questions

The Normalization Radius

Most diffractive surfaces, like the Zernike surfaces or Binary Optic surface, express the phase added by the diffractive using some equation that uses a normalized distance parameter r. For example, the Binary Optic 2 surface adds phase using the following equation:
a typical diffractive equation written in terms of the normalized surface coordinate p
where N is the number of polynomial coefficients in the series, M is the diffraction order, and Ai is the coefficient on the 2ith power of r , which is the normalized radial aperture coordinate.

The normalized radial aperture coordinate of a ray that lands at some radial height r on a surface is given by r = r/R, where R is the 'normalization radius'. We use normalized coordinates, rather than coordinates in lens units, because it makes thermal analysis, and optimization to athermalize a diffractive optic, considerably easier. For thermal analyses and optimizations of diffractive surfaces, generally only the normalization radius requires a thermal pickup solve, because thermal expansion can be considered a scaling of length. Many aspheric surfaces also use normalization radius for this same reason.

Usually R is set to be equal to the semi-diameter of the surface, or slightly larger than the working aperture of the surface, but its exact value is not important as long as it is carried along with the Aicoefficients. You can view the example here:

The example is a typical 'achromatic singlet', in which the chromatic aberration of a singlet lens is balanced by some diffractive power. Two configurations are used, with normalization radii of 30 and 50 mm respectively. 

The Extra data editor, showing the normalization radius and first three polynomial coefficients

We then optimize for best RMS spot radius in each configuration. Although the coefficients computed by the optimizer have different values between the configurations, when scaled by the normalization radius the exact same phase profile is produced in both configurations. 

Analysis...Surface...Surface Phase


In conclusion, the exact value of the normalization radius R used does not matter, as long as the normalization radius and phase coefficients are kept together.