## Abstract

We have discovered an aplanatic design that contributes to a celebrated problem in classical optics in a novel way. In so doing new devices are envisioned with applications to illumination, concentration, and imaging.

© 2009 Optical Society of America

Perfect imaging has been a desired goal of optics from the days of Galileo, Newton, and Descartes. In geometrical optics this is approached by satisfying the aplanatic condition, a celebrated topic in optics. The most compact configuration is realized by two-mirror surfaces. The dilemma faced by previous investigators is that in many cases, when the two-mirror solution is fully developed (for example, to maximize light gathering power), the front surface blocks much of the incident light, defeating its original purpose. In this Letter the authors have avoided this dilemma by making the first surface a “one-way mirror” and in the process have extended aplantic designs to refractive media in a novel way. This opens a broad vista of applications to solar energy, imaging, light collection, and illumination.

It is remarkable that an on-axis condition that can be simply formulated ensures good off-axis performance. The condition is that rays parallel to the axis intersect rays converging to the focus on the surface of a sphere, called “Abbe Sphere” [1]. (For recent advances in general sine conditions, see [2].) Requiring both an on-axis image of infinity and the aplanatic condition clearly requires at least two surfaces (except for special configurations of high symmetry, like the Luneburg lens [3]). The method of designing a two-surface reflecting aplanat is illustrated by the classic monographs by Luneburg [3], Korsch [4], and Mertz [5]. Analytical solutions have been given by Schawarzchild [6], Puryayev and Gontcharov [7], and more recently by Bell [8]. Following Luneburg and assuming rotational symmetric around optical axis as shown in Fig. 1 , the entire two surfaces can be constructed by successive approximation starting from the optical axis. The key is to adjust the slope of both surfaces properly such that the aplanatic condition is satisfied. The starting positions of the two surfaces along the optical axis are free parameters. A family of two-mirror designs can be generated by changing these two parameters or their equivalent [8, 9].

The two-mirror aplanats are a “tour de force” of geometrical optics; they offer “no coma, no spherical aberration, and arbitrarily fast focal ratio” [9]. If we think of an image as pixels at the focal plane, an important property is the amount of light on each pixel. This is called the speed of the optical system and is related to the angular cone of light on the pixel; the larger the angle, the faster the system. The fully developed two-mirror solutions are intrinsically fast. However, many of these designs are impractical, because the front mirror blocks the incident light. An example can be seen from Fig. 1, where the last segment of the front surface blocks ray 0. It is a pity that such an elegant scheme cannot be implemented. Clearly, the problem cannot be solved unless a “one-way mirror” exists that allows the rays to be transmitted one way but reflected the other way.

Fortunately, nature does offer something close to a “one-way” mirror. It is well known that for light incident on an interface between two refractive media, light from the lower refractive index is transmitted (except for Fresnel reflection loss), while light from the higher refractive index at incident angles higher than the critical angle is totally reflected as if the interface is a mirror, an effect called total internal reflection (TIR). This suggests that the problem can be addressed by filling the space between the two mirror surfaces with a refractive medium and reflective coating the rear surface. Then, over portions of the internal surface where the incidence angle exceeds the critical angle, we indeed have a one-way mirror. So the design strategy will be to treat the refractive medium as a one-way mirror and discard the portions where TIR fails or simply coat with a reflective surface. There is, however, a more vexing obstacle to overcome, causality! The front surface transmits light but also introduces a disturbance in the horizontal ray direction by refraction. Unfortunately, this disturbance is indeterminate, because it depends on the slope, which is not determined until future steps in the iteration. We address this problem by a new prescription, which includes two keys: 1) a self-consistent initial configuration and 2) building the two surfaces from edge to center (optical axis) such that the slope of the front surface is determined before ray tracing is required at the location.

A self-consistent initial configuration is found by noticing the specialty of the ray striking the point where the front surface crosses the Abbe sphere, such as ray 0 in Fig. 2 . To satisfy the aplanatic condition, this ray must be reflected back on itself by the rear surface and then toward the center of the Abbe sphere by the front surface. Clearly in this case the refraction and reflection by the front surface happen at the same point. Using Snell’s law it is straightforward to get the slope of the front surface at the point to be $(({n}_{0}\u2215{n}_{1})+\mathrm{cos}\left(\theta \right))\u2215\mathrm{sin}\left(\theta \right)$, where ${n}_{1}$ and ${n}_{0}$ are the refractive indices of the material between the two surfaces and that of the ambient, respectively, and θ is the angle of the converging ray with respect to the optical axis. Obviously, the starting slope of the rear surface must be set such that the ray is incident normally. With this configuration as a starting point, the entire two surfaces can be constructed down to the optical axis by successive approximation. Causality is no longer an issue, because whenever the disturbance due to refraction needs to be evaluated, the slope has been determined by previous steps in the iteration. The final result also depends on the choice of two free parameters: the starting position of the front surface or angle θ, and the starting position of the rear surface or its projection on the optical axis.

Figure 3 shows a solution for an angular aperture of 120 deg (θ = 60 deg), which is a very fast system. The sharp focusing of both on-axial and off-axial (2 deg) rays indicates the aplanatic nature of the design. The central region of the front surface is coated with reflective material, but the obscuration of the input light is less than 4%. In other words, over most area of the front surface we have a “one-way” mirror. The aspect (height:diameter) of this aplanat is approximately 1:3. To our knowledge, this is a remarkably fast and compact system. We conceived of this as a high concentration device; therefore by the sine law of concentration [10] it has inherently small field of view. As a result, the higher-order Seidel aberrations [11], which depend on higher powers of field angle, are small, making it useful for imaging. For example, the size of the spot diagram at 2 deg off-axis is ∼0.001 of the entrance aperture. For comparison, this ratio would be ∼0.025 for a parabola. Chromatic aberration is certainly present. For nonimaging optical application such as solar concentration, we found the effect is to decrease the acceptance angle by about 4%, which is negligible in most cases. For imaging applications, this sets a limit on the width of the spectral band.

Compact, fast aplanats have a broad vista of applications to solar energy, imaging, light collection, and illumination. As imaging devices, they are a compact telescope with large light-gathering power. For illumination, with an LED at the focal plane, they are a bright, collimated flash light. For solar energy applications, they are a high-concentration, optimal acceptance angle device suitable for highly efficient but costly multijunction solar cells.

We are grateful to one of our reviewers for bringing additional references to our attention, and we thank project Dedalos for support of this research.

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