Absolute astigmatism correction for flat field spectrographs Christopher Palmer

Milton Roy Analytical Products Division, 820 Linden Ave­nue, Rochester, New York 14625. Received 18 January 1989. Sponsored by W. R. Hunter, Sachs/Freeman Associates. 0003-6935/89/091605-03$02.00/0. © 1989 Optical Society of America. Absolute astigmatism correction for flat field spectro­

graphs is possible over an arbitrary wavelength range, while absolute defocus correction is not possible.

Although the Rowland circle mount has long been known to provide exceptional image resolution, only the two lowest-order horizontal aberrations (defocus and coma) are elimi­nated when a classical grating is used, their aberration terms vanishing identically at all wavelengths in this mount. No corresponding analytic mount is known for the primary ver­tical aberration (astigmatism); thus spectrometer designers generally accept mediocre vertical correction or employ holo­graphic gratings whose designs usually defy algebraic analy­sis and require numerical optimization. When spectro­meters with flat field detectors are used, the ideal focal curves are no longer circles tangent to the grating center but straight lines. The analysis below shows that this condition allows ideal astigmatism correction over an arbitrarily large wavelength range.

In Fig. 1 a typical flat field spectrometer is shown; the system lies in the x-y (principal) plane. The entrance slit A is fixed a distance r from the grating center and oriented an angle a from its normal (the Ox axis), while the diffracted wavelengths in the range λs ≤ λ ≤ λL are imaged along the line BsBL, the intersection of the planar detector and the principal plane. An ideal flat field spectrometer would be

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curves,

Fig. 1. Flat field spectrometer geometry. Light from point A(r,a) is diffracted by the grating centered at 0; the ideal image points of the ends of the spectrum (λS and λL are located at BS and BL, the ends of the detector, respectively. An intermediate wavelength λ is diffracted to an ideal image point between BS and BL located at point

where

one in which the focal curves for all aberrations lie along this line, given by

χ' and y' being the Cartesian coordinates of the image: χ' = r' cosβ, y' = r' sinβ, where r', the distance from the grating center to the center of the diffracted image, depends on λ. Converting the equation of this line into plane polar coordi­nates yields

Comparison of Eq. (6) with Eq. (2) shows that the tangential focal curve can never be of the form of Eq. (2) except near β = 0, about which point the variation in cos2β vanishes. On the other hand, Eqs. (7) and (2) can be made to agree if we set

Since b = r'(0), the first condition requires the sagittal focal curve to pass through the center of curvature of the grating: r'(0) = R (unrationalized). The remaining condi­tions may be achieved by setting

where the λ dependence of r' is shown explicitly as a depen­dence on the diffraction angle β. This replacement is possi­ble since the incidence angle a is constant in the grating equation

For an aberration to be completely absent from the diffract­ed images of all wavelengths, the corresponding aberration term must have as its solution a focal curve which may be written in the form of Eq. (2). The constants a and b are the slope of the detector line and its intercept with the grating normal: b = r'(0).

The lowest-order aberration coefficients for a flat field spectrograph are found to be1

which leaves the slope of the detector as a free parameter,

which is true for spherical substrates,2 and

which relates the entrance slit distance r with the incidence angle a and the slope of the detector α.

As an example, consider a flat field detector in which the detector is oriented normally to the grating normal. In this case α = 0, from which Eq. (1) gives

corresponding to defocus and astigmatism, respectively. The recording wavelength is λ0, and H20 and H02 are holo­graphic coefficients which are nonzero unless the grating grooves are equally spaced and vertical (to lowest order), respectively. The constants α20 and α02 are the expansion coefficients corresponding to y2 and z2, respectively, in the MacLaurin expansion of the grating surface. The distances r and r' have been rationalized by the major radius R of the grating blank for convenience: the entrance slit and image distances are actually rR and r'R. The focal curves are defined by F20 = 0 and F02 = 0; solving Eqs. (4) and (5) for r'(β) yields equations for the tangential and sagittal focal

since r'(0) = 1. For the sagittal focal curve F02 = 0 to be coincident with the detector, Eq. (15) imposes the restriction H02 = 0, implying that a holographic grating used in this mount must provide no holographic correction to the astig­matism of the diffracted images (i.e., it must be classical as far as astigmatism is concerned). Equation (16) is satisfied for spherical gratings, and Eq. (17) yields

Equations (18) and (19) are the astigmatic analogs of the rationalized Rowland circle constraints

which provide F20(λ) = 0 identically at all diffracted wave­lengths provided H20 = 0.

A more general example involves a flat field detector with a

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holographic grating. The example above shows that the detector cannot be oriented normally to the grating normal, given H02 ≠ 0, if perfect astigmatism correction is to be achieved. For a specified detector slope a and laser record­ing wavelength λ0, the holographic coefficient H02 is deter­mined by Eq. (15), leaving either r or a as a free parameter in Eq. (17).

Given the ideal astigmatic focal curve, it would be desir­able to render the defocus curve F20 = 0 coincident with it, thereby eliminating both second-order aberrations from the diffracted images of every wavelength in the spectrum. As seen from Eq. (6), the presence of the cos2β factor in the numerator makes this generally impossible, but we may min­imize the deviation between the tangential and ideal focal curves:

Differentiation of the difference between Eqs. (6) and (7) yields a complicated expression in β which is best solved numerically. Solving this equation for β provides regions in which the tangential and sagittal focal curves are roughly coincident over wavelength ranges centered on those wave­lengths λi corresponding to the roots βi of Eq. (21), where

Equation (21) may be written

since D, E, and F are determined when the sagittal curve is placed, while B = 1 for a spherical grating; finally, A and C include only H20 as a free parameter. In spectrometer de­sign, the tangential focal curve is generally determined first since resolution is crucial, after which the sagittal curve is matched as closely as possible. The method described above chooses the values of the free parameters in minimizing astigmatism, after which only the holographic defocus coeffi­cient H20 and either r or a are left to adjust the location of the tangential focal curve.

It has been shown that absolute astigmatism correction for flat field spectrographs is possible over an arbitrary wave­length range, while absolute defocus correction is not possi­ble. Linear astigmatic focal curves have been derived previ­ously for constant-deviation monochromators,2 but only in those designs for which H20 = 0.

The author is also with the Physics Department of Bryn Mawr College.

References 1. T. Namioka, "Theory of the Concave Grating," J. Opt. Soc. Am.

49, 446 (1959). 2. W. R. McKinney and C. Palmer, "Numerical Design Method for

Aberration-Reduced Concave Grating Spectrometers," Appl. Opt. 26, 3108 (1987).

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