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Aberration Theory |
4.3ABERRATION DETERMINATION USING RAY TRACE DATA
The elements comprising the total ray aberration can be computed directly from specific ray trace data. How this is done and the relationship to the aberration coefficients are presented in this section. It should be noted that the method discussed decouples the defocus element from the astigmatic elements thereby enhancing the utility of these elements in the optical design process. Each of the following aberrations is briefly introduced and will be discussed in detail in subsequent chapters.
4.3.1 Defocus
Defocus can be used as a first-order aberration that is measured from the paraxial image plane. It depends only on entrance pupil coordinates, not on the object height or field angle. Defocus impacts imagery uniformly over the entire field of view. Often defocus can be used to balance or improve symmetric (astigmatic) aberrations, while having no effect on asymmetric (comatic) aberrations. Figure 4.6 shows the upper and lower marginal rays exiting the optical system, focusing at the paraxial image plane, and forming a blur at the image plane located a longitudinal distance x from the paraxial image plane. Defocus can be expressed as
DFðr; xÞ ¼ x tan ua0 |
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¼ |
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f |
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where u 0a is the angle of the marginal paraxial ray in image space and f is the focal length.
Image plane
Upper marginal ray |
v ′ |
a |
x |
Lower marginal ray |
Paraxial image plane
Figure 4.6 Defocus.
4.3 Aberration Determination Using Ray Trace Data |
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For finite conjugate systems, f should be replaced by the paraxial image distance. The intersection height of the marginal ray with the image plane is the transverse defocus aberration, and the defocus blur is
2xr
f
The ray fan plot of ex or ey versus r is simply a straight line. A plot of~e versus y for a fixed value of r shows a circle because of the radial symmetry about the optical axis. As will be discussed in subsequent chapters defocus can be used as a means to improve the image quality when astigmatic errors are present; however, defocus has no effect on comatic aberrations.
PROBLEM: Show that DFðr; xÞ is independent of the image height and is therefore a field-independent aberration.
4.3.2 Spherical Aberration
Spherical aberration can be defined as a variation with aperture of the image distance or focal length in the case of infinite conjugates. Figure 4.7 shows a positive lens that suffers undercorrected or negative spherical aberration, which is typical of such lenses.13 A close-up view of the image region of Figure 4.7a is shown in Figure 4.7b. The paraxial rays come to a focus at the paraxial focal plane while, in this case, meridional rays farther from the optical axis progressively intersect this axis farther from the paraxial image plane and closer to the lens. This is referred to as longitudinal spherical aberration and is referenced to the marginal ray intercept as shown in the figure. In a similar manner, these rays intercept the paraxial image plane below the optical axis and are referred to as transverse spherical aberration.
Figure 4.8a presents the meridional ray fan plot, which more clearly presents the transverse ray error ey as a function of entrance pupil radius r. Figure 4.8b shows the longitudinal spherical aberration as a plot of the axial intercept location as a function of the entrance pupil radius r. An alternative presentation of the ray error is the spherical aberration contribution to the wavefront error as a function of the entrance pupil radius r as illustrated in Figure 4.8c. As will be explained, the longitudinal, transverse, and wave presentations of spherical aberration are related to each by simple multiplicative factors. Each form of spherical aberration has utility and none has general superiority.
The transverse spherical aberration at the paraxial image plane is given by the displacement of a ray having coordinates ðr; 00; 0; 0Þ from the optical axis, which can be expressed as
SPHðr; 00; 0Þ ¼ Yðr; 00; 0; 0Þ
(4-5)
¼ s1r3 þ m1r5 þ t1r7 þ . . .
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Aberration Theory |
Paraxial image plane
(a)
Longitudinal spherical aberration
Transverse spherical aberration
Paraxial image plane
(b)
Figure 4.7 (a) Positive lens that suffers undercorrected or negative spherical aberration.
(b) Close-up view of the image region.
where Yðr; 00; 0; 0Þ is the real ray value in the polynomial expansion as also shown.
Figure 4.9 illustrates the general behavior of the third-, fifthand seventhorder spherical aberration terms. In this particular case, s1; m1; and t1 are all given a value of unity. It should be noted that the higher the order of the terms, the flatter the plots are until progressively larger values of r are reached, at which point the curves increase rapidly. The distance from the paraxial image plane to the intersection point of the ray with the optical axis is called longitudinal spherical aberration. Assuming that the ray slope is negative, then the longitudinal spherical aberration is considered positive, or overcorrected, if the intersection point is beyond the paraxial image plane; and is considered negative, or undercorrected, if the intersection point precedes the paraxial image plane.
4.3 Aberration Determination Using Ray Trace Data |
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ey |
r |
r
0 +Z
(a) |
(b) |
Waves
r
(c)
Figure 4.8 (a) Transverse spherical aberration. (b) Longitudinal spherical aberration.
(c) Spherical aberration contribution to the wavefront error.
4.3.3 Tangential and Sagittal Astigmatism
Field-dependent astigmatism and curvature of field are inherently related to displace the image from the paraxial image plane. As is illustrated in Figure 4.10, the meridional rays come to a focus some distance from the paraxial image plane, forming a line lying in the sagittal plane whose length is determined by the width of the sagittal fan of rays at that point. In a like manner, the sagittal focus is determined by where the sagittal fan focuses in the tangential plane and has a length determined by the width of the tangential fan at that point. The tangential astigmatism for a given value of r and H can be determined exactly by tracing three rays, namely the corresponding upper and lower off-axis rays,
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Aberration Theory |
ey (normalized)
1.0
Third order
0.5
Seventh order
0.0
–0.5
Fifth order
–1.0 
–1.0 –0.5 0.0 0.5 1.0
r
Figure 4.9 Orders of spherical aberration.
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H ′ |
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y |
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Meridional |
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focus |
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Sagittal |
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focus |
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H ′ |
Z |
Y |
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x |
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Image |
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plane |
X |
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Exit |
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pupil |
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Figure 4.10 Tangential and sagittal astigmatism.
4.3 Aberration Determination Using Ray Trace Data |
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and the marginal ray. Consequently, the tangential astigmatism is computed using the ray data in the following equation.
TASTðr; HÞ ¼ Yðr; 00; H; xÞ Yðr; 1800; H; xÞ 2Yðr; 00; 0; xÞ |
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¼ 2ASTyðr; 00; HÞ |
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¼ 2½ð3s3 þ s4ÞH2 þ m10H4 þ t18H6 þ . . .&r |
(4-6) |
þ 2½ðm4 þ m6ÞH2 þ ðt11 þ t12ÞH4 þ . . .&r3 |
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þ 2½ðt4 þ t6ÞH2 þ . . .&r5 þ . . . |
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In addition, the resulting aberration coefficients are also shown. Notice that the polynomial expansion is expanded in odd orders of r. The importance of this will be explained presently. The purpose of including the marginal ray in the above calculation is to remove the field-independent components from the upper and lower rays, that is, defocus and spherical aberration. The portion of the expansion that is linear with r is known as linear tangential astigmatism and has a ray fan plot similar to the plot for defocus. Figure 4.11 illustrates the behavior of tangential astigmatism for r; r3; and r5 for a particular value of H. Notice that these plots have the same form as defocus, and thirdand fifth-order spherical aberration; however, TASTðr; HÞ varies with H.
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0.5 |
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ey |
0 |
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–0.5 |
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Field angle = H |
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–1 |
–0.5 |
0 |
0.5 |
1 |
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–1 |
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r
Figure 4.11 Tangential ray fan plot.