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This article answers the frequently asked question: What is the normalization radius used by diffractive surfaces for?
Authored By Mark Nicholson
Introduction
The normalization radius (R) is generally set to be equal to the semi-diameter of a surface and is used to scale the X and Y intercept coordinates so that the polynomial terms that make up the surface sag are dimensionless. It is related directly to the phase coefficients of a diffractive surface. As long at the two are carried together, then the value for the normalization radius should not matter. That is, so long as the relationship holds, the phase profile at the diffractive surface should be the same for any value of R - assuming the rest of the system remains unchanged.
This article will use a Binary 2 surface to define the normalization radius.
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 ρ. For example, the Binary Optic 2 surface adds phase using the following equation:
$$\Phi = M\sum_{i=1}^N A_i \rho^{2i}$$
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 ρ, 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 ρ = 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 Ai coefficients. The sample lens file “Normalization Radius Example.ZMX” (shown below) is available as an attachment.
This 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.
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.
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.