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10.1117/2.1200607.0041

Designing aplanatic thick lensesJorge Castro-Ramos and Sergio Vazquez-Montiel

An iterative method can be used to design thick refractive lenses thatare free from spherical and coma aberrations.

The lens is the most frequently used part of an optical system,and any problems with image formation are often caused duringthe concentration ofmonochromatic light into a singlepoint. Thisfailure to produce exact point-to-point correspondence betweenan object and its image is referred to as an aberration. Two of themost common in an optical system are spherical aberration (thevariation of the image position within the aperture) and comaaberration (the variation of magnification with aperture). If anoptical system is free from both of these errors, we refer to it asaplanatic, a term originated by Ernest Abbe.1

There has been much research on how to avoid opticalaberrations and achieve aplanatic lenses. For example, twocenturies ago, Kepler, Descartes, and Huygens tried todetermine the shape of the surfaces that make image formationon a point possible.1 Later, Conrady2 and Kingslake3 explainedthat there are three well-analyzed cases where a sphericalsurface is free of spherical aberration: when the object is onthe vertex of the surface, when the object is at the centerof curvature, and when the object is at the aplanatic points(given by the relations between the object distance L and thecorresponding image distance L : see Figure 1). In his analysis,Mahajan4 showed that the spherical aberration of a thin lenscannot be zero when both the object and its image are real, butthe coma of a thin lens is zero for some shapes and positionsof an object. Therefore, he concluded that there were only twoaplanatic points for a thin lens.

Our research derived an iterative method to design thickrefractive lenses that are free from spherical and comaaberrations. The resulting lenses can be used in convergingor diverging beams and for near and far objects withmonochromatic light, regardless of the objects positions. Inthis technique, we begin with a spherical lens and obtain thefollowing Gaussian properties: positions of the object l, imagel, thin lens power, magnification m and the radii of curvaturer1 and r2 . Then, we use the third-order design to calculatethe bending factor B of the corrective lens to obtain the leastspherical aberration.4

Figure 1. By using these key parameters, it is possible to design a thicklens free from spherical and coma aberration.

In order to accommodate a thick lens,we add an axial thicknessd1 . However, when we insert the axial thickness value, the focallength changes. This can be addressed by using a techniquedescribedbyKingslake.3 Byusing an iterative processto calculatethe radii of curvature of the surfaces, the focal length and the bestshape factor can bemaintained.Toeliminatesphericalaberration,we use the conic constant of the first surface as a variable. To befree from spherical aberrations, the optical path length of an axialray and the optical path length traveled by a marginal ray mustbe equal (see Figure 1). This relationship is expressed as:

D0 + n D1 + D2 = d0 + n d1 + d2 , (1)

If we realize an exact marginal ray trace3,4 and propose cosinedirectors (L0 , M0 ,N0 ), then it is possible to obtain the incidenceand refraction angles at the first surface. By employing therefraction law in vector form and by knowing the initialcoordinates (x1 , y1, z1) of the incident ray S1 , we can determinethe coordinates (x2 , y2 , z2) of the refracted ray S2 . Hence, we canobtain the direction cosines (L1 , M1 ,N1) of the refracted ray afterit passes through the first surface of the lens.

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After we substitute the (x2 , y2), coordinates of the incident rayon the second surface into Equation 1, it is possible to obtainan equation that depends on the conic constant. By solvingthe equation for the sag of the first surface, we can determineits conic constant and obtain a lens that is free from sphericalaberration. (Additionally, this methodology could be employedto correct the height of incidence of the marginal ray.)

Next we can eliminate the coma aberration. According toKingslake,3 a spherically corrected lens is free from comanear the center of the field if the marginal M and paraxialmagnifications m are equal. For a very distant object, the sinecondition takes a different form. In the equation below, fp isthe distance from the second principal plane to the focal pointmeasurement along any paraxial ray, and fm is the focal lengthfor any marginal ray. In order to obtain a thick lens free fromcoma aberration, the Abbe sine condition should be satisfiedwhile the lens is free from spherical aberration. This means that:

fp = Fm (2)

From Figure 1 we can determine a relationship between themagnificationm and the last angle of the marginal ray. Using theparaxial curvatures of the first and second surfaces, respectively,the separation between surfaces, and the back focal length, it ispossible to calculate the effective focal length.Next, to determinethe conic constants of the first and second surfaces respectively,we solved equations one and two for z1 by using z2 = z1 d1 N1D1 .As a result, we can compute the sag for the second surface andthe conic constants of both surfaces.

Our research resulted in a general design method to enhancethe 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 sphericalsurfaces, this methodology allowed us to obtain an analytic andexact expression to determine the conic constant in order tocorrect for marginal spherical and coma aberrations.

Author Information

Jorge Castro-RamosNational Institute of Astrophysics, Optics, and ElectronicsTonantzintla, Puebla, Mexico

Jorge Castro-Ramos works at the National Institute ofAstrophysics, Optics, and Electronics in the instrumentationgroup. He concentrates on optical design and test. In addition,he is a member of SPIE and has presented twelve papers at SPIEmeetings held during the last two years.

Sergio Vazquez-MontielNational Institute of Astrophysics, Optics, and ElectronicsTonantzintla, Puebla, Mexico

References

1. F. A. Jenkins and H. E. White, Fundamentalsof Optics, McGraw-Hill Inc., NewYork, USA, 4th ed., 1981.2. A. E. Conrady, Applied Optics and Optical Design, Part I and II, Dover, NewYork, 1960.3. R. Kingslake, LensDesign Fundamentals,AcademicPress Inc., NewYork, USA,1978.4. W. T. Welford, Aberrations of Optical Systems, Adam Hilger, Bristol, GreatBritain, 1991.

c 2006 SPIEThe International Society for Optical Engineering