CAPÍTULO 1: MARCO TEÓRICO
1.1. Estado del Arte
1.1.2. Mypes
1.1.2.3. Mypes en el Perú
The theory we have studied in Chapter 6 is useful to design optical systems with a relatively small aperture and a relatively small field of view. If the aperture of field is not small, the image is strongly affected by the high-order aberrations. In this sec-tion, we will study in more detail these high-order aberrations. Some expressions for transverse aberration polynomials were proposed by Buchdahl (1954, 1956, 1958a, 1958b, 1958c, 1959, 1960a, 1960b, 1960c, 1961, 1962a, 1962b, 1965, 1970) and by Cruickshank and Hills (1960), using Seidel coefficients for the primary aberrations and his coefficients for high-order aberrations. They are now commonly used in commercial lens design and evaluation software. These expressions are as follows:
TA Sy( , , )θh =σ1S3cosθ σ+ ( +cos θ)hS +( σ +σ )h Sc
and
where σi are the coefficients for the five primary aberrations and μi are the Buchdahl coefficients for the nine fifth-order aberrations. We have 12 Buchdahl fifth-order coefficients, but only nine independent aberrations. It can be shown (Cruickshank and Hills 1960) that in a system where the third-order aberrations are corrected, the following relations between the coefficients exist:
µ µ
We can use these relations between the coefficients to reduce its number to nine, but they are retained for a simpler interpretation. The image structure and shape for these aberrations have been described in some detail by Steward (1928). These fifth-order aberrations are as follows:
Spherical aberration
From Equations 7.28 and 7.29, the primary and fifth-order transverse spherical aberrations are represented by
TAy (S,θ,h) = σ1S3 cosθ + μ1S5 cosθ (7.31) and
TAx (S,θ,h) = σ1S3 sinθ + μ1S5 sinθ. (7.32) These expressions do not depend on the image height h. Because of the
rotational symmetry of the system, TAx and TAy are identical. More details on this aberration can be found in Chapter 5.
Coma
The primary and fifth-order transverse coma aberration, sometimes called linear coma, is given by
TAy (S,θ,h) = σ2(2 + cos2θ)hS2 + (μ2 + μ3 cos2θ)hS4 (7.33) and
170 Handbook of Optical Design
TAx (S,θ,h) = σ2hS2 sin2θ + μ3hS4 sin2θ. (7.34) The primary coma has been studied in detail in Chapter 6. The structure
of the primary and fifth-order coma is quite similar, with small differences in the dimensions, as illustrated in Figure 7.3. We have a fixed relation between μ2 and μ3 given by Equation 7.30.
Astigmatism
The primary and fifth-order transverse aberrations for astigmatism, also called linear astigmatism, are
TAy (S,θ,h) = (3σ3 + σ4)h2S cosθ + μ10h4S cosθ (7.35) and
TAx (S,θ,h) = (σ3 + σ4)h2S sinθ + μ11h4S sinθ. (7.36) The Seidel component (first term) is the primary astigmatism at the Petzval
surface (σ3) with the proper defocusing (σ4) to shift the image to the Gaussian image plane. The primary astigmatism grows with the square of the image height while the fifth-order astigmatism grows with the fourth power of the image height. As shown in Figure 7.4, the images for primary astigmatism as well as the images for high-order astigmatism are ellipses at the Petzval sur-face. The major and minor semiaxes at this surface have the values μ10 and μ11. Distortion
For primary and fifth-order distortion, the transverse aberrations are TAy (S,θ,h) = σ5h3 + μ12h5 (7.37)
32
Third-order linear coma Fifth-order linear coma σ2ρ2h
2σ2ρ2h µ3ρ4h
µ3ρ4h
FIGURE 7.3 Image structures for third- and fifth-order linear coma.
and
TAx (S,θ,h) = 0. (7.38)
These aberrations move the image in the y direction. The image distor-tion appears because this shift is not linear with the image height h.
Oblique spherical aberration
This aberration is commonly known as oblique spherical aberration;
according to Cruickshank and Hills (1960), it is more appropriate to con-sider it as a cubic astigmatism, following the original Hopkins proposal.
The transverse aberrations for oblique spherical aberration are given by TAy (S,θ,h) = (μ4 + μ6 cos2θ)h2y3cosθ (7.39) and by
TAx (S,θ,h) = (μ5 + μ6 cos2θ)h2y3sinθ. (7.40) The image shape and structure for several ratio values of μ5/μ6 are
illustrated in Figure 7.5. According to Cruickshank and Hills (1960), this aberration is so important that it may be the limiting factor in the off-axis resolution of medium or large field objectives.
Elliptical coma
Finally, the transverse aberrations for elliptical coma are represented by TAy (S,θ,h) = (μ7 + μ8 cos 2θ)h3y2 (7.41)
σ4ρh2
µ10ρh4
µ11ρh4 σ3ρh2
σ4ρh2
3σ3ρh2
Third-order astigmatism Fifth-order astigmatism
FIGURE 7.4 Image structures for third- and fifth-order astigmatism.
172 Handbook of Optical Design
and by
TAx (S,θ,h) = μ9 S2h3 sin 2θ. (7.42) The image structure of this aberration is similar to that of the primary
coma, as illustrated in Figure 7.6. The main difference is that the elliptical coma grows with the cube of the image height while the primary coma grows linearly with the image height. The other difference is that the light rays passing through concentric circles in the exit pupil are in ellipses. They become circles only when μ8/μ9 = 1. The three parameters that define this image are not linearly independent; they are related by Equation 7.30.
The computer analysis of an optical system by evaluation of the Buchdahl transverse aberrations by tangential and sagittal ray plots will be described in the next chapter.
µ8 = 1
µ9 µ8 = 0.5
µ9 µ8 = 0
µ9 µ8 = –0.5
µ9 µ9ρ2h3
µ8ρ2h3
µ8ρ2h3 µ8ρ2h3
µ7ρ2h3 µ7ρ2h3
µ7ρ2h3 µ7ρ2h3
µ9ρ2h3 µ9ρ2h3 µ9ρ2h3
Elliptical coma
FIGURE 7.6 Image structures for elliptical coma.
= 1 µ5ρ3h2
µ5
µ6 µ5 = 0.5
µ6 µ5 = 0
µ6 µ5 = –1
µ6
µ5ρ3h2 µ5ρ3h2 µ5ρ3h2
(µ4 + µ6)ρ3h2
Oblique spherical aberration
FIGURE 7.5 Image structures for oblique spherical aberration.
7.3.5 waVefront aberration Polynomialsfor
noncenteredand asymmetric systems
Frequently, more general expressions without assuming any symmetry for the opti-cal system are useful. This need arises from the fact that real-life systems are not perfect. They may have lens decentrations and tilts and also surface imperfections.
Up to degree n, a set of (n + 2)(n + 1)/2 linearly independent polynomials must be obtained. In other words, no polynomial could be expressed as a linear combination of some of the others.
We may define an index m, beginning at 0 and with the last term equal to n, for all polynomials with the same power n. Thus, the whole set of polynomials may be ordered with a single index r defined by
r n n
= ( + + +1) m
2 1. (7.43)
The inverse procedure, to find the values of n and m if the value of r is known, can be performed with the expressions
n r
m r n
=next integer greater than− + +3 1 8 / ; = − 2
[ ]1 2 (nn+ −1
2 ) 1. (7.44) TABLE 7.2
Aberration Polynomials up to the Fourth Degree with the Center of Curvature of the Reference Sphere at the Parabasal Focus Position
n m r
Cartesian Coordinates
Polar
Coordinates Name
0 0 1 1 1 Piston
1 0 2 x S sin θ Tilt about y axis
1 3 y S cos θ Tilt about x axis
2 0 4 y2 + x2 S2 Defocusing
1 5 yx S2 sin θ cos θ Astigmatism, axis at ±45°
2 6 y2 – x2 S2 cos 2θ Astigmatism, axis at 0° or 90°
3 0 7 x(y2 + x2) S3 sin θ Coma, along x axis 1 8 y(y2 + x2) S3sin θ Coma, along y axis
2 9 x(y2 – x2) S3 sin θ cos 2θ Triangular astigmatism, 30°, 150°, 270°
3 10 y(y2 – x2) S3 cos θ cos 2θ Triangular astigmatism, 0°, 120°, 240°
4 0 11 (x2 + y2)2 S4 Spherical aberration
1 12 xy(y2 + x2) 0.5 S4 sin 2θ Fifth-order astigmatism, axis at 45°
2 13 y4 – x4 S4 cos 2θ Fifth-order astigmatism, axis at 0° or 90°
3 14 y2 – x2 S4 sin2 θ cos2θ Ashtray at ±45°
4 15 xy(y2 – x2) 0.25S4 sin 4θ Ashtray at 22.5°
The angle θ is measured from the y axis.
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Piston Tilt about y axisTilt about x axis DefocusingAstigmatism ±45ºAstigmatism 0º or 90º Triangular astigmatismTriangular astigmatism ΔComa along y axisComa along x axis Spherical aberrationFifth-order astigmatism ±45
ºFifth-order astigmatism 0
º or 90
ºAshtray ±45ºAshtray 22.5º
yx yxyx yxyxyx yxyxyxyx yxyxyxyxyx
Δ
FIGURE 7.7Isometric plots for the aberration polynomials in Table 7.2
If we take the medium focus as the center of curvature for the reference sphere, the polynomials in Table 7.2 are obtained. The wavefront polynomial is a linear combination of these polynomials, as follows:
W x y A A x A y A y x A y x A xy
Examples of some of these polynomials are illustrated in the isometric plots in Figure 7.7 including some high-order aberrations.
It is interesting to notice that any aberration calculated with the coefficients de fined in Chapter 5 and with the wavefront polynomial aberrations has opposite sign conventions.