134 Aberration Theory
Conversion to wave aberrations requires that the |
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2nkuk |
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s1 ¼ |
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Spherical aberration |
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i¼1 |
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s2 ¼ |
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Coma |
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k |
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s3 ¼ |
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Astigmatism |
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ðyn 1u 1 |
yn 1u 1Þ2 |
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Petzval |
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s4 ¼ |
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4l |
is4 |
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i¼1 |
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s5 ¼ |
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Distortion |
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where l is the wavelength and the wave aberrations are measured at the edge of the exit pupil in units of wavelength.
ENDNOTES
1H. A. Buchdahl, Optical Aberration Coefficients, Dover Publications, New York (1968).
2Historically, lens designers used a left-hand Cartesian coordinate system with positive slopes of rays bending downwards. This was done for computational convenience and error mitigation when doing manual computations. Most current optical design and analysis software packages use the right-hand Cartesian coordinate system.
3It should be understood that astigmatic means not stigmatic. This should not be confused with astigmatism or more specific astigmatic aberrations, which will be discussed later. In a like manner, the term anastigmatic lens means a highly corrected lens having sensibly perfect imagery in contrast to meaning a stigmatic lens (not not stigmatic).
4The entrance pupil is the image of the aperture stop formed by all of the optical elements preceding the aperture stop. The exit pupil is the image of the aperture stop formed by all of the optical elements following the aperture stop.
5Note that changing the sign of r is the same as changing the signs of both X and Y, or the angle y by p.
6The value of r can have values of either sign. Consequently, a ray having entrance pupil coordinates of ( r,0o) is equivalent to having entrance pupil coordinates of (r,180o).
7The number of independent aberration coefficients for the nth-order is given by
ðn þ 3Þðn þ 5Þ 1 8
For n ¼ 1, or the first-order, there are two independent coefficients, namely magnification and defocus.
Chapter 4 |
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8G. C. Steward, The Symmetrical Optical System, Cambridge University Press (1928).
9A. E. Conrady, Applied Optics and Optical Design, Dover Publications, New York; Part I (1957), Part II (1960).
10Andrew Rakich and Raymond Wilson, “Evidence supporting the primacy of Joseph Petzval in the discovery of aberration coefficients and their application to lens design,” SPIE Proc. 6668:66680B (2007).
11A. E. Conrady, p. 289–290.
12R. Barry Johnson, “A Historical perspective on the understanding optical aberrations,” SPIE Proc., CR41:18–29 (1992).
13A negative singlet lens has overcorrected or positive spherical aberration.
14If the primary coefficients negligible, then the identity m9 ¼ m7 m8 is reasonably valid if the system is well corrected. See previous Buchdahl Eq. (31.8).
15Extrinsic contributions are also referred to as transfer contributions.
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Chapter 5
Chromatic Aberration
5.1 INTRODUCTION
In 1661, Huygens created the two-lens compound negative eyepiece which generally corrected lateral color, that is, yielding an image of a white object which subtends the same angle for all colors (see Chapter 16). This was a remarkable achievement and won him acclaim at the scientific conferences, since other eyepieces of the day yielded poor performance and contained often 5, 8, and even 19 lenses. An interesting point is that Huygens had no concept of achromatizing his eyepieces or any other kind of optical system for that matter; nevertheless, it worked better than other eyepieces of the day. The reason for Huygens’ lack of understanding was that no one understood the dispersive properties of glass.
About two years later, Newton began to study the dispersion of glass, in part, to understand why, the Huygens compound eyepiece was corrected for lateral color. Newton was the first, it should be noted, to develop the concept of the dispersion of glass. Remarkably, Newton failed to recognize one important property of glass—different glasses have different dispersions. In contrast, he put forth the concept that all glasses have the same dispersion; consequently, he asserted that one could not achieve an achromatic system. Newton was also the first to differentiate between the aberrations of spherical and color by assigning the spherical aberration to the surface and the color to the materials. He was the first to explain that spherical aberration varied with the cube of the aperture, and published the results in his book OPTICKS.1 Also, Newton presented a detailed description of chromatic aberrations.
After the work by Newton, there was a lull in the development of optical aberrations of about 60 years. Then in 1729, Chester Hall discovered, rather accidentally it is noted, that achromatic lenses could be constructed by cementing positive and negative lenses together when the lenses were made of different glasses. By achromatic, it was meant only in the context that the chromatic
Copyright # 2010, Elsevier Inc. All rights reserved. |
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DOI: 10.1016/B978-0-12-374301-5.00009-7
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Chromatic Aberration |
aberrations of the lenses were not corrected, but notably reduced. Hall’s discovery gave rise to renewed research into understanding optical materials and recognizing that the dispersion of glasses can vary from type to type. John Dolland, a London optician, in 1757 began to design and fabricate a variety of achromatic lenses after he empirically determined by experimenting with a variety of positive and negative lens combinations that longitudinal chromatic aberration could be mitigated by combining a convex crown-glass lens with a weaker concave flint-glass lens. According to Conrady, John Dolland produced the first achromatic telescope objective and was the first person to patent the achromatic doublet.2
The Swedish mathematician Klingenstierna, in 1760, was the first to develop a mathematical theory of achromatic lenses and, what was called at that time, the aplanatic lens. Part of Klingenstierna’s work was based on John Dolland’s initial understanding of achromatic lenses. The next year, Clairaut was the first to explain the concept of secondary spectrum (see Section 5.5) and he also observed that certain crown and flint glasses had different partial dispersions. He further deduced theorems for pairing glasses, not unlike those found in modern books. Also that same year, John Dolland made an effort to correct secondary spectrum by the use of a third glass. In 1764, D’Alembert described a triple glass objective in which he also distinguished between longitudinal and transverse features of spherical aberration and chromatic aberration.3
A discussion was presented in Chapter 2 about the refractive index of glass and other optical materials changing with wavelength. From this behavior of optical materials, it follows that every property of a lens depending on its refractive index will also change with wavelength. This includes the focal length, the back focus, the spherical aberration, field curvature, and all of the other aberrations. In this chapter, we explore field-independent chromatic aberrations4 while field-dependent chromatic aberrations (including lateral color) are discussed in Chapter 11.
Figure 5.1 depicts the chromatic aberration of a single positive lens having “white” light incident upon the lens. As will be mentioned in Section 5.9.1, it is common to select F (blue), d (yellow), and C (red) spectral lines for design and analysis of visual systems.5 As seen in the figure, the focus for F light is
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F (blue) |
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chromatic |
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aberration |
Longitudinal axial chromatic aberration
Figure 5.1 Undercorrected chromatic aberration of a simple lens.