Figures 3-6. Alternative approaches to asphere placement in Double Gauss lens assembly.
Placing one asphere near an aperture stop or a pupil will correct spherical aberration, which is a variation in focus location with aperture height. This is a rotationally symmetric aberration and is constant across the field of view. The aperture stop is the ideal location for asphere placement since that location affects all fields of view simultaneously.

A second asphere placed at an intermediate image will correct field aberrations, such as coma and astigmatism. These are non-rotationally symmetric aberrations. Intuitively, having an asphere at an intermediate image allows the field to be directly mapped to the optical surface.

These ideal placements are not always practical in a real optical system. For instance, many commercial optical systems allow control of the aperture setting by using an iris, which is a moving part requiring space for mechanical actuation. In these cases it’s not feasible to place an asphere directly on the aperture stop. Instead, it must be placed as close to the aperture stop as possible.

Similarly, it is not advisable to place an optical surface at an intermediate image, because dust on that surface or surface imperfections will be imaged to the detector plane. So in practice an asphere is placed close to an intermediate image to correct field aberrations. This is easily done if the optical system is a re-imager, meaning there is a focus inside the optical system. Some system requirements have tight packaging or cost specifications, however, which do not allow for an intermediate focus. In these cases an asphere is placed as close as possible to the image plane.

Dueling Aspheres

Without the ideal locations available for aspheres, the lens designer has to be careful not to design in “dueling aspheres” that can inadvertently drive up system cost. The cost of an asphere roughly corresponds to its fabrication difficulty, and practitioners of asphere manufacturing use aspheric departure as a metric for how difficult an asphere is to make. This departure is the maximum difference from a best fit sphere and is specified in microns or waves. Dueling aspheres increase each other’s aspheric departure requirements.

As an example, let’s imagine that the entrance pupil (image of the aperture stop in object space) is on the front surface of the optical system and
the exit pupil (image of the stop in image space) is at the aft optical surface. These two surfaces are conjugate to one another. This is a fancy way of saying that placing a scratch on the front surface will image onto the back surface. If aspheres are placed on both of these surfaces they may duel one another. For instance, one could end up with a design where the front surface has an asphere with +53 waves of departure, while the back could have -50 waves. The all spherical equivalent could simply have -3 waves of spherical aberration and it could be corrected by placing one asphere of +3 waves on either the front or the back surface.

Lessons Learned

A literature and patent search on camera objectives that utilized aspheres shows mostly telephoto systems designed in Japan during the 80’s and 90’s. Most of these camera lenses are derivatives of the Double Gauss lens, which doesn’t have an intermediate image and usually contains an iris. It is interesting to note that the various locations of aspheres in these designs did not conform to the “theoretical optimum locations”. Several classes of asphere location emerged and they are re-plotted as they would appear on an ideal Double Gauss lens in Figure 3 through Figure 6.

Figure 5 is an interesting case since the form breaks the Double Gauss symmetry about the aperture stop. The increase in airspace between the iris and the aft optics allows the second asphere to reduce the field aberrations due to the beam wander over the aperture with changing field, while the asphere near the stop minimizes spherical aberration.

Figure 6 is appealing because of a cost savings due to the aspheres being located on a single element. If the element is glass molded this approach can yield tremendous cost reductions compared to two aspheres.

One could argue many of these cases set up dueling aspheres. Perhaps instead of “dueling” aspheres these are “split” aspheres, where the aspheric departure is distributed between two surfaces.

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