Figure 1 shows the Luneburg lens from 1944 [1], a gradient index ball with a radially symmetric index gradient. It forms a perfect monochromatic image of collimated light on the back surface of the ball. The light rays are curved inside the ball, and the perfect focus occurs for any field angle incident on the ball. The index of refraction is 1.414 (square root of 2) at the center and 1.0 at the rim of the ball. This is not physically realizable but can be approximated with structures in the microwave region. Figure 2 shows a related design, the Maxwell fisheye lens from 1854 [2]. It is also a perfect system, monochromatically, and focuses any point on the surface of the spherical ball onto the opposite side of the ball at unit magnification. The ray curves inside the ball are arcs of circles. The refractive index in the center is 3.0, and at the rim of the ball, it is 1.5.

Figure 1 Luneburg gradient index lens.

Figure 2 Maxwell Fisheye gradient index lens.

It is hard to imagine that there could be a simple two-element design that is nearly perfect and with no index gradient, but Figure 3 shows just such a design [3].

Figure 3 A nearly perfect 0.99-NA monocentric design.

It is a monocentric catadioptric system, and all four surfaces, one of them reflecting, have the same center of curvature. There are only four design variables: two-element thicknesses and an airspace on either side of the common center of curvature. With these four variables, it is possible to correct for the focal length and third-, fifth-, and seventh-order spherical aberration. Higher orders beyond that are extremely small. The glass can be any typical optical glass.

If the aperture stop is at the common center of curvature of this monocentric design, then, there are no field aberrations. The image is curved and also has the same center of curvature as all the surfaces. Now, here is a remarkable feature of this design. The higher-order spherical aberration is so small that this design works very well at 0.99 NA. When optimized for the wavefront, a 0.99-NA design with a 20-mm focal length has an r.m.s. OPD of 0.0035 wave at 0.55 μ, assuming a glass index of about 1.6. The performance goes to about 0.006 waves r.m.s. When the design is reoptimized for the lower glass index of n=1.5. If the NA is extended to 0.999 NA, the wavefront hardly changes. Because of that, this design has the very unusual feature of not needing an aperture stop. Total internal reflection at the second surface as the design NA approaches 1.0 will effectively define an aperture stop, and reflection losses will apodize the pupil near its rim so that there is very little energy beyond 0.99 NA.

The Figure 3 picture shows the design with a 20° field. The obscuration due to the image surface is very small, about 15% diameter for a 20° full field 0.99 NA design. The performance is the same for any field angle. This is a monochromatic design, but it could be achromatized by adding two ‘buried surfaces’ to the design. The main practical limitations of this remarkable system are that its size is large compared to the pupil diameter and its curved image surface. The variation of the orders of spherical aberration with these very few monocentric design parameters is extremely nonlinear, and it is quite difficult to fully optimize this design.

There is a new perfect optical design [4], which is the ultimate in simplicity – a single optical element – and also the ultimate in image quality, with no monochromatic aberrations. It is shown in Figure 4, and it has perfect image quality at NA=1.0, in air, and at NA=3.0 with an immersion focal surface. This design is simply a gradient index Maxwell’s fisheye lens that has been cut in half and then given a reflective outer surface. By means of a very simple geometrical construction, it is possible to prove that this new design, just ½ of Maxwell’s design and with a reflective outer surface, is a perfect design with no geometrical aberrations of any order. It images perfectly, to a point, any point on the flat surface to another point on that same surface. The radial gradient index goes from 3.0 at the center to 1.5 at the outer surface.

Figure 4 Perfect 1.0X gradient index catadioptric design.

Prior to this new design, there were only four known perfect optical systems. The Luneburg gradient index lens and the Maxwell fisheye lens both form real images on a curved surface. Another perfect system is the single surface aplanatic surface, a spherical surface between two different index of refraction materials, with an object and image conjugate ratio that is the same as the ratio of the refractive index on opposite sides of the aplanatic surface. This gives perfect imagery, but both the object and image are curved, and one conjugate is virtual. The last of the well-known perfect optical systems is a flat mirror. This is the only one where both object and image can be flat, with perfect imagery, but one has to be virtual.

This new design here is the first new perfect optical system in many decades and has both the object and image flat and real, with no virtual conjugate. Of course, there is no practical way right now to make this large change in the index gradient, but the design shows that gradient index is a very powerful design tool, and more complicated designs may not require such large index differences. The design is only perfect monochromatically, but there is a partial equivalent to this system in the Wynne-Dyson design [5], which does have color correction and does not use gradient index.

Figure 5 shows how two of these elements can be joined so that the input and output are in opposite directions.

Figure 5 Opposite input and output directions.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.