The $N$-achromat and beyond: a unified variational framework for optimal chromatic aberration correction

TL;DR

The paper addresses chromatic aberration in cemented lens systems and the limits of correcting a fixed set of wavelengths. It introduces a unified variational framework that minimizes residual chromatic aberration (RCA) over a spectral window using a constrained, KKT-based optimization, with dispersion stability aided by Chebyshev expansions and enhanced conditioning via a null-space approach. A pentachromat ($N=5$) is derived analytically and shown to outperform the classic superachromat, and the method generalizes to arbitrary $N$ with multi-window target capabilities, demonstrated through numerical results and a combinatorial search over glass catalogs. The approach balances theoretical performance with manufacturability by considering glass choices and curvature magnitudes, enabling practical, tailored optical designs with potential impact across photography, microscopy, lithography, and remote sensing.

Abstract

In this article, we present novel and effective methods for reducing chromatic aberrations in cemented lens systems. We derive an analytical solution coined the pentachromat, which corrects five distinct colors. This method can naturally be extended to accommodate an arbitrary number of lenses and to correct for a customized selection of spectral lines. Since correcting for specific rays rather than the entire residual spectrum can overconstrain the system, we introduce a variational formulation. This approach tames the residual spectrum by several orders of magnitude compared to conventional designs like the superachromat, while giving theoretical guarantees to reach the optimal solutions. Furthermore, this innovative methodology opens up previously uncharted design possibilities, such as multiple-focal-length achromatic systems. This allows for the selection of specific optical powers paired with desired bandwidths, enabling the design of highly specialized and tailored optical systems. Finally, we couple our variational framework with a combinatorial search, allowing to find the type of glasses and their geometry such that it reaches the best residual spectrum over an available catalogue.

Figures (4)

• Figure 1: Comparison between pentachromat and superchromat, solutions to \ref{['N_achromat']} for $N=5$ and $N=4$ respectively. The deviation with respect to target value is reduced by 2 orders of magnitude when a pentachromat solution is used.
• Figure 2: Optical power with respect to wavelength for the superachromat (four roots) and the optimal polynomials in either Chebyshev and Legendre basis. The optimal polynomial clearly outperforms the superachromat, as its $\mu$m$^{-2}$ MSE is $3.06e+1$$\mu$m$^{-2}$ versus the superachromat $2.12e+02$$\mu$m$^{-2}$.
• Figure 3: Optical power with respect to wavelength for the pentachromat (five roots) and the optimal polynomials in either Chebyshev and Legendre basis. The optimal polynomial clearly beats the pentachromat as its $\mu$m$^{-2}$ MSE is $2.6$$\mu$m$^{-2}$ versus the superachromat $7.73e1$$\mu$m$^{-2}$.
• Figure 4: Dispersion function of the optimal polynomial matching a target power function on the red windows, with $\phi_0 = 1$ m$^{-1}$ for 500 - 550 nm and $\phi_0 = 1.1$ m$^{-1}$ 650 - 700 nm. Root mean-squared error in m$^2$ is indicated for each windows.

Theorems & Definitions (6)

• Remark
• Remark
• Remark
• Remark
• Remark
• Remark