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Optical Design & Engineering

Designing aplanatic thick lenses

An iterative method can be used to design thick refractive lenses that are free from spherical and coma aberrations.
7 August 2006, SPIE Newsroom. DOI: 10.1117/2.1200607.0041

The lens is the most frequently used part of an optical system, and any problems with image formation are often caused during the concentration of monochromatic light into a single point. This failure to produce exact point-to-point correspondence between an object and its image is referred to as an aberration. Two of the most common in an optical system are spherical aberration (the variation of the image position within the aperture) and coma aberration (the variation of magnification with aperture). If an optical system is free from both of these errors, we refer to it as aplanatic, a term originated by Ernest Abbe.1

There has been much research on how to avoid optical aberrations and achieve aplanatic lenses. For example, two centuries ago, Kepler, Descartes, and Huygens tried to determine the shape of the surfaces that make image formation on a point possible.1 Later, Conrady2 and Kingslake3 explained that there are three well-analyzed cases where a spherical surface is free of spherical aberration: when the object is on the vertex of the surface, when the object is at the center of curvature, and when the object is at the aplanatic points (given by the relations between the object distance L and the corresponding image distance L': see Figure 1). In his analysis, Mahajan4 showed that the spherical aberration of a thin lens cannot be zero when both the object and its image are real, but the coma of a thin lens is zero for some shapes and positions of an object. Therefore, he concluded that there were only two aplanatic points for a thin lens.

Our research derived an iterative method to design thick refractive lenses that are free from spherical and coma aberrations. The resulting lenses can be used in converging or diverging beams and for near and far objects with monochromatic light, regardless of the objects' positions. In this technique, we begin with a spherical lens and obtain the following Gaussian properties: positions of the object l, image l', thin lens power, magnification m and the radii of curvature r1 and r2. Then, we use the third-order design to calculate the bending factor B of the corrective lens to obtain the least spherical aberration.4

In order to accommodate a thick lens, we add an axial thickness d1. However, when we insert the axial thickness value, the focal length changes. This can be addressed by using a technique described by Kingslake.3 By using an iterative process to calculate the radii of curvature of the surfaces, the focal length and the best shape factor can be maintained. To eliminate spherical aberration, we use the conic constant of the first surface as a variable. To be free from spherical aberrations, the optical path length of an axial ray and the optical path length traveled by a marginal ray must be equal (see Figure 1). This relationship is expressed as:

If we realize an exact marginal ray trace3,4 and propose cosine directors (L0,M0,N0), then it is possible to obtain the incidence and refraction angles at the first surface. By employing the refraction law in vector form and by knowing the initial coordinates (x1,y1,z1) of the incident ray Ŝ1, we can determine the coordinates (x2,y2,z2) of the refracted ray Ŝ2. Hence, we can obtain the direction cosines (L1,M1,N1) of the refracted ray after it passes through the first surface of the lens.

After we substitute the (x2,y2), coordinates of the incident ray on the second surface into Equation 1, it is possible to obtain an equation that depends on the conic constant. By solving the equation for the sag of the first surface, we can determine its conic constant and obtain a lens that is free from spherical aberration. (Additionally, this methodology could be employed to correct the height of incidence of the marginal ray.)

Next we can eliminate the coma aberration. According to Kingslake,3 a spherically corrected lens is free from coma near the center of the field if the marginal M and paraxial magnifications m are equal. For a very distant object, the sine condition takes a different form. In the equation below, fp is the distance from the second principal plane to the focal point measurement along any paraxial ray, and fm is the focal length for any marginal ray. In order to obtain a thick lens free from coma aberration, the Abbe sine condition should be satisfied while the lens is free from spherical aberration. This means that:

From Figure 1 we can determine a relationship between the magnification m and the last angle of the marginal ray. Using the paraxial curvatures of the first and second surfaces, respectively, the separation between surfaces, and the back focal length, it is possible to calculate the effective focal length. Next, to determine the conic constants of the first and second surfaces respectively, we solved equations one and two for z1 by using z2 = z1d1N1D1. As a result, we can compute the sag for the second surface and the conic constants of both surfaces.

Figure 1. By using these key parameters, it is possible to design a thick lens free from spherical and coma aberration. (Click to enlarge image.) 

Our research resulted in a general design method to enhance the performance of thick lenses for monochromatic light, regardless of whether the object is near or far from the lens. Overcoming the current problems for lenses with spherical surfaces, this methodology allowed us to obtain an analytic and exact expression to determine the conic constant in order to correct for marginal spherical and coma aberrations.

Jorge Castro-Ramos and Sergio Vazquez-Montiel  
National Institute of Astrophysics, Optics, and Electronics
Tonantzintla, Puebla
Jorge Castro-Ramos works at the National Institute of Astrophysics, Optics, and Electronics in the instrumentation group. He concentrates on optical design and test. In addition, he is a member of SPIE and has presented twelve papers at SPIE meetings held during the last two years.

1. F. A. Jenkins, H. E. White,
Fundamentals of Optics,
4th Edition, McGraw Hill Inc., New York, 1981.
2. A. E. Conrady,
Applied Optics and Optical Design, Part I and II,
Part I and II, Dover, New York, 1960.
3. R. Kingslake,
Lens Design Fundamentals,
Academic Press, Inc., New York, 1978.
4. W. T. Welford,
Aberrations of Optical Systems,
Adam Hilger, Bristol, UK, 1991.