Reduction of Aberrations

Algebraic techniques can find sets of design parameter values that minimize image size at one or two wavelengths, but to optimize the imaging of an entire spectral range is usually so complicated that computer implementation of a design procedure is essential. MKS has developed a set of proprietary computer programs to design and analyze grating systems. These programs allow selected sets of parameter values governing the use and recording geometries to vary within prescribed limits. Optimal imaging is found by comparing the imaging properties for systems with different sets of parameters values.

Design techniques for grating systems that minimize aberrations may be classified into two groups: those that consider wavefront aberrations and those that consider ray deviations. The wavefront aberration theory of grating systems was developed by Beutler and Namioka, and was presented in Section 7.2. The latter group contains both the familiar raytrace techniques used in commercial optical design software and the Lie aberration theory developed by Dragt. The principles of optical raytrace techniques are widely known and taught in college courses, and are the basis of a number of commercially-available optical design software packages, so they will not be addressed here, but the concepts of Lie aberration theory are not widely known – for the interested reader they are summarized in Appendix B.

Design algorithms generally identify a merit function, an expression that returns a single value for any set of design parameter arguments; this allows two different sets of design parameter values to be compared quantitatively. Generally, merit functions are designed so that lower values correspond to better designs – that is, the ideal figure of merit is zero.

For grating system design, several merit functions may be defined. The MKS proprietary design software uses the function

M = w’ + ch’     (7-24)

where w’ and h’ are the width (in the dispersion plane) and height (perpendicular to the dispersion plane) of the image, and c is a constant weighting factor. Minimizing M therefore reduces both the width and the height of the diffracted image. Since image width (which affects spectral resolution) is almost always more important to reduce than image height, c is generally chosen to be much less than unity. If w’ is expressed not as a geometric width (say, in millimeters) but a spectral width (in nanometers), then M will have these units as well; since h’ is in millimeters (there being no dispersion in the direction in which h’ is measured), c will have the units of reciprocal linear dispersion (e.g., nm/mm) but it is not a measure of reciprocal linear dispersion – c is merely a weighting factor introduced in Eq. (7-24) to ensure that image width and image height are properly weighted in the optimization routine.

For optimization over a spectral range λ₁≤λ≤λ₂, Eq. (7-24) can be generalized to define the merit function as the maximum value of w’ + ch’ over all wavelengths:

M = sup{w(λ)+ ch'(λ)}     (7-25)

where the supremum function sup{} returns the maximum value of all of its arguments. Defining a merit function in the form of Eq. (7-25) minimizes the maximum value of w’ + ch’ over all wavelengths considered. [A more general form would allow the weighting factor to be wavelength-specific, i.e., c -> c(λ), and an even more general form would allow for wavelength-dependent weighting factors for w(λ) as well.]

Eqs. (7-24) and (7-25) consider the ray deviations in the image plane, determined either by direct ray tracing or by converting wavefront aberrations into ray deviations. An alternative merit function may be defined using Eqs. (7-19) and (7-20), the expressions for the tangential and sagittal focal distances. Following Schroeder, we define the quantity Δ(λ) as

Δ(λ) = |1/r'_t(λ) - 1/r'_s(λ)|     (7-26)

leading to the following merit function:

M = sup{Δ(λ)}     (7-27)

This version of M will consider second-order aberrations only (i.e., F₂₀ (defocus) and F₀₂ (astigmatism)) to minimize the distances between the tangential and sagittal focal curves for each wavelength in the spectrum.

Noda et al. have suggested using as the merit function the integral of the square of an aberration coefficient,

M = ∫dλ(F_ij(λ))²     (7-28)

where the integration is over the spectrum of interest (λ₁≤λ≤λ₂). Choosing defocus (F₂₀) as the aberration term would, however, not require the design routine to minimize astigmatism as well. A number N of aberrations may be considered, but this requires the simultaneous minimization of N merit functions of the form given by Eq. (7-28).

Two other merit functions have been used in the design of spectrometer systems are the Strehl ratio and the quality factor.