Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration.

We describe a way of selecting pairs of glasses for both thin cemented achromatic doublets and thin aplanatic achromatic doublets with a reduced secondary spectrum. By taking one pair of glasses at a time, we can compute and display the secondary spectrum in increasing value. The number of solutions based on the magnitude of the secondary spectrum alone is huge: 40,804 pairs. Some tests are applied at different stages of the design procedure to reduce the number of acceptable solutions. Aberrations that cannot be corrected, namely, spherochromatism and fifth-order spherical aberration, are further calculated to reduce drastically the number of acceptable solutions. To do this, we establish tolerance conditions based on the relationship between the Strehl intensity ratio and the rms wave-aberration error so that the rms wave error is minimized in the presence of the secondary spectrum, spherochromatism, and the fifth-order spherical aberration.

Selection of Glasses for Achromatic Doublets with Reduced Secondary Spectrum. II. Application of the Method for Selecting Pairs of Glasses with Reduced Secondary Spectrum.

The method for selecting pairs of glasses for thin aplanatic achromatic doublets and cemented achromatic doublets with a reduced secondary spectrum, presented in Part I, is applied to the design of two optical systems. The first is a Lister-type 10x microscope objective with a numerical aperture of 0.25 working in the visible band. The second is a camera f/7.5 working in the near-IR spectral band, 0.8521 mum < lambda < 2.3254 mum, of a spectrograph for the San Pedro Martir Observatory in Ensenada, México. Improvement in the performance of both optical systems is shown.

Differential equation for normal glass dispersion and evaluation of the secondary spectrum.

Some of the questions concerning secondary chromatic aberration at both sides of the visible band of the spectrum are the following: (1) What is the bandwidth at different wavelengths, given the permissible chromatic aberration circle and the lens aperture? (2) What is the size of the chromatic aberration circle, given the wavelength, the bandwidth, and the lens aperture? The answers to these and other questions may be found with the new definitions of V-number and relative partial dispersion P based on infinitesimal bandwidths that we propose. In addition, an alignment chart for the secondary color of a normal glass doublet is presented, so fast answers to the questions posed above and to other questions concerned with secondary color can be found. In addition, a continual challenge in computer-aided lens design is the use of optical glasses as design parameters in simultaneous optimization of lens systems over various regions of the spectrum. This problem could be solved if we could find an ideal glass family, not too different from real glasses, such that, given the refractive index n and the V-number at any wavelength, the indices at all wavelengths could be determined. Therefore we derive a differential equation for normal glass dispersion and present a recursive solution.

Least-squares fitting of orthogonal polynomials to the wave-aberration function.

The wave-aberration function of systems with circular and square apertures can be expanded in terms of Zernike and Legendre polynomials. The polynomial terms form orthogonal sets; therefore each coefficient independently determined by an integral satisfies the principle of least squares. To evaluate the integral the pupil is divided into small areas where the wave-aberration function is approximated by the first three terms of a Taylor series expansion: the optical path difference and components of geometric aberration. In final form the coefficients are expressed by the sum of three bilinear terms by combining three matrices and six vectors. The former depend on the construction parameters and the latter on the ray pattern.