The action of a perfect lens is to produce a spherical wave at the exit pupil under uniform illumination of monochromatic light. To assess aberrations in the lens, we measure phase deviations between the ideal and the aberrated wavefronts, and scientists have developed rigorous theories—mostly based on spherical surfaces—to address the effects of these imperfections. One commonly accepted theory is that aberrations can be evaluated in terms of Zernike polynomials (orthogonal circle polynomials).^{1}

Optical designers must reduce the effect of lens imperfection to improve resolution, and achieving this is costly, particularly for systems that require ultrahigh precision aberration suppression, such as lithographic projectors.^{2} It is now widely recognized that configurations using aspherical and freeform surfaces could obtain the same optical performance as spherical-based configurations, but with reduced complexity and at lower cost. Consequently, there have been great efforts to establish theories and methods to achieve such systems,^{3} yet the aberration theory still needs further exploration. Here, we consider how the so-called phase-space representation, which describes the local frequency spectrum of a wavefront, may provide the basis for an aspherical abberation theory.^{4}

A lens alters both the spatial and angular contents of a light field. Thus, we sought to observe evolution of both distributions in the lens simultaneously. This is the concept behind matrix optics,^{1} according to which the ray is represented by a matrix (*r*, *q*)^{T}, where *r* and *q* are the position and direction, respectively, and *T* means transpose. The extension of this matrix into wave optics yields the phase-space representation, which provides information about the local spatial frequency contents at every position, in a way analogous to a score of musical notes. The phase-space concept offers a powerful tool to characterize the behavior of waves propagating in the optical systems.^{5}

Recently, Herkommer drew a conceptual picture that the aberration theory of aspherical and freeform optics may be built on the basis of phase-space distributions, in particular, the Wigner distribution function (WDF): the Fourier transform of the coherence function of the wave.^{6} With this concept in mind, we examined how the aberrations of a spherical surface affect the WDF, to help predict the corresponding effects of aspherical and freeform optics.

To simplify our analysis, we shone a 1D uniform monochromatic wave of finite size onto the front surface of a 1D aberrated spherical lens, and examined the WDF at the back surface. In our experiment, the WDF has a well-known dog-bone shape, as shown in Figure 1(a). If we take an intersection of the WDF along the *q* axis at an arbitrary *r* (| *r* | < *T*/2), we can prove that it is actually a sinc function of *q*. The closer *r* gets to ±*T*/2, the wider the sinc function is. From this we can see how many frequency components the wave carries at each spatial position. If we make a projection along the *q* axis, all the spatial frequency components will add up together, producing the intensity distribution of the wave. In reverse, if we integrate with respect to *r*, we obtain the spectrum.

**Figure 1. **Theoretical results of phase-space representations: the Wigner distribution function (WDF) (a) of a 1D slit and (b) measured after the slit wave passes through a perfect lens. The results clearly show a shear along the q axis due to the chirp, and the shear serves as the reference to evaluate the aberration. The WDF is measured at the back surface of the lens when the lens is (c) tipped, (d) defocused, and (e) astigmatic. In (c) we see that the tip results in a shift of the WDF as a whole along the q axis, while (d) and (e) suggest that defocus and astigmatism result in additional shear of the WDF with respect to the effect of the perfect lens. q: Direction. r: Position.

When the wave passes through a perfect spherical lens, whose spatial frequency is a chirp function of *r* (where the frequency changes with time), it is expected that the WDF experiences a shear along the *q* axis, as depicted in Figure 1(b). Figure 1(c, d, e) shows the WDFs resulting from an aberration of the lens, specifically tipping, defocus, and astigmatism. Figure 1(c) suggests that the effect of tipping is to shift the WDF as a whole along the *q* axis, since the aberration is just a linear function of *r*. By contrast, defocus and astigmatism are quadratic functions of *r*, meaning that they make the WDF shear along the *q* axis, in a way similar to the effect of a perfect lens.

Our theoretical results suggest that by examining the shift, shearing, or other deformations of the WDF, it is possible to evaluate the aberration of the lens. We are currently working on a theory for higher-order aberrations, such as coma and trefoil (or even higher), and we expect these will induce other types of deformations in the WDF. In future work, we plan to extend our theory to the study of aberration in aspherical and possibly freeform optics.

Guowei Li, Guohai Situ

Shanghai Institute of Optics and Fine Mechanics

Shanghai, China

Guohai Situ's research interests include computational imaging, phase retrieval, phase space optics, and optical information processing. He has published 27 research papers in leading optics journals, and has delivered 10 invited presentations at international conferences. He is currently a topical editor with *Applied Optics*.

References:

1. M. Born, E. Wolf, *Principles of Optics*, Cambridge University Press, 2008.

2. H. J. Levinson,

*Principles of Lithography*, SPIE Press, 2010. doi:

10.1117/3.865363.

3. K. Thompson, Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry, *J. Opt. Soc. Am. A* 22, p. 1380-1401, 2005.

4. G. Li, G. Situ, Effects of lens aberrations in phase space,

*Proc. SPIE* *9276*, p. 927613, 2014. doi:

10.1117/12.2071394.

5. M. Testorf, B. Hennelly, J. Ojeda-Castañeda, *Phase-Space Optics: Fundamentals and Applications* , McGraw-Hill, 2009.

6. A. M. Herkommer, Phase space optics: an alternative approach to freeform optical systems,

*Opt. Eng.* 53, p. 031304, 2014. doi:

10.1117/1.OE.53.3.031304.