1. Chaper 6 Image Quality of Optical System
outline
1). Overview and Definitions 2). The Axial Monochromatic Aberration-Spherical Aberration(轴向单色像差-球差) 3). Off-axial Monochromatic Aberrations( comatic aberration, astigmatism, field curvature and distortion) （离轴单色像差-彗差、像散、场曲、畸变） 4). Color Dispersion and Chromatic Aberrations（色散、像差）

2. 1. Overview of Image Quality
Two requirements for the imaging properties
Imaging properties
paraxial region
(近轴区域成像特性)
Image quality
the focal length
the object and image distances
the magnifications
the entrance pupil position and distance
The image should be
similar to the object
clear enough
distort as little as possible
The job of optical design is to make sure structure parameters according to imaging properties and image quality.

3. 2 What is aberrations?
The ideal optical system theory assumes that rays make small angles with the principal axis and that the lenses are thin.
A precise analysis of image formation requires tracing each ray, using Snell's law and reflection law. This procedure shows that the rays from a point object do not focus at a single point.
The departures of actual images from the ideal predicted by theory are called aberrations.

4. 2.1 Aberration types Monochromatic Aberration Aberrations Chromatic
the axial aberration
spherical aberration
Monochromatic
Aberration
comatic aberration
astigmatism
field curvature
distortion
off-axial aberrations
Aberrations
axial chromatic aberration
Chromatic
Aberrations
off-axial chromatic aberrations

5. 3 The Axial Monochromatic Aberration -Spherical Aberration
The definition of spherical aberration
The measurement of spherical aberration
Spherical aberration Curve(球面像差曲线)
The effect of spherical aberration
The correction of spherical aberration

6. 3.1What is spherical aberration?
When the object distance L of the object point on the axis is determined, the image distance L' is the function of aperture angle U (or height h). The distance between real image point and ideal image point is axial spherical aberration. namely:δL'=L'-l', Lateral spherical aberration: δT'=δL'tgU'.
The sign convention of the spherical aberration δL' is that δL' will be positive
if the ray intersection point is on the right of the paraxial image point A0', and
will be negative if on the left of point A0'.

7. Spherical aberration

8. Spherical aberration 图

9. definition 返回

10. Spherical aberration curve（1）corrected

11. 3.2The measurement of spherical aberration
Although the rays in pairs with the same heights will meet at one point, the rays with different heights will not intersect　at one point,but result in
an imperfect image. The rays which have the largest aperture intesect at point A'1.0; and the rays of 0.85 aperture intersect at A'0.85, and so on.

12. 3.3 Spherical Aberration Curve

13. 3.4The effect of spherical aberration
When there are spherical aberration, it will produce circular defocused spots on the imaging plane.
The infinite object point images at different imaging plane

14. perfect imaging system
real imaging system
The finite object point on axis

15. 3.5 The correction of spherical aberration
Three locations that don't produce spherical aberration by a single spherical surface
Aplanatic lens(消球差透镜)
Single lens
The combination of positive and negative lens
Parallel glass block

16. Object point is on the centre of sphere
When object point is on the centre of sphere, namely L=r, all emited light
from object point is along normal direction and have no deflection through
the sphere.

17. Object point is on the peak of sphere
When object point is on the peak of sphere, object point and image point
are the same point.

18. Object distance
When object distance , image distance

19. Aplanatic lens（消球差透镜）
positive aplanatic lens
nagetive aplanatic lens

20. The spherical aberration of a single lens
For a single thin lens, focal power is
Focal power is relative with optical material and curvature. When Focal power is fixed, optical material varies with curvature difference. so there are two methods to decrease spherical aberration:
(1) Increase refraction index of optical material.
(2) Optical material and curvature difference are fixed, change the whole lens shape.

21. The spherical aberration curve varies with positive lens shape
The spherical aberration of positive lens is always negative

22. The spherical aberration curve varies with negative lens shape
conclusion:
(1) The spherical aberration can not be zero in a single lens.
(2) For a negative lens or a positive lens, there is a optimal shape, which has minimal spherical aberration.
The spherical aberration of negative
lens is always positive.

23. The combination of positive and negative lens
Doublet lens: two kind of materials should match well .
material 2
material 1
material 1

24. The combination of positive and negative lens
Air-spaced doublet: it can correct spherical aberration and another aberration.
material 2
material 1 d

25. Three types of spherical aberration
Undercorrection
Overcorrection
Correction

26. 3.Off-axial Monochromatic Aberrations
(1) Comatic aberration
(2) Astigmatism
(3) Field curvature
(4) Distortion

27. Pre-knowledge
Principal ray: the oblique ray through the center of the aperture of a lens system is called the principal, or chief ray.
First paraxial ray emitted from axial object point A and passes through aperture margin.
Second paraxial ray emitted from off-axial object point B and passes through aperture center.
Meridional plane is determined by the principal ray and the axis.
Sagittal plane passes through the chief ray and is perpendicular to the meridional plane.
Upper ray emitted from off-axial object point and passes through aperture upper margin
Lower ray emitted from off-axial object point and passes through aperture lower margin.
Front ray emitted from off-axial object point and passes through aperture upper margin.
Rear ray emitted from off-axial object point and passes through aperture rear margin.

28. What is Meridional plane and Sagittal plane？ (子午面和弧矢面）
Spherical surface

29. Meridional Aberrations（子午像差）
Entrance pupil
Gaussian image plane
Vertical distance KT' from the principal ray to intersect point BT' of the rays perpendicular to the pricipal ray is called meridional coma.
Meridional coma KT' denotes the variation of magnification with aperture, or the dissymmetry of the meridional rays to the pricipal ray.

30. Meridional Aberrations（子午像差）
If we want to know the convergences of the total rays, we can calculate different couples of rays of different aperture and find out their corresponding KT'. Usually we calculate rays with (±1，±0.85，±0.7，±0.5，±0.3) hm，and here hm represents the largest aperture height.
In order to know the image quality of the whole image plane, we must know the aberrations of different image heights, which means we must calculate different fields of view,usually (1,0.85,0.7,0.5,0.3)ym, where ym represents the largest field of view.

31. Sagittal aberrations（弧矢像差）
Entrance pupil
Gaussian image plane
The vertical distance KS' from intersect point BS' of the rays perpendicular to the principal ray is called saggital coma.

32. Sagittal aberrations（弧矢像差）
Compared with meridional aberrations, sagittal aberrations change smoothly with apeture, so we can calculate less saggital rays than meridional rays.
In order to know the image quality of the whole image plane, we should calculae different fields of view, usually (1,0.85,0.7,0.5,0.3)ym, where ym represents the largest field of view.

33. Sine aberration
For an optical system whose field angle of view is small, since the image height is very small, the value of aberration coma will be much smaller, which is unsuitable to denote the property of the comet by its aboslute value. In this situation, we usually use the ratio of coma to the image height to denote the coma aberration. this is called sine aberration,represented by SC'
The calculation equation of SC' is

34. 4 The sketch rays for the coma aberration（彗差）
For a bundle of oblique rays there are always some tangential and sagittal comas, and the sagittal coma is approximately one third as large as the tangential coma. That means, with any one of the two kinds of comas we can estimate the system's coma.

35. 4.1 Correction of comatic aberration
There is no comatic aberration when entrance pupil is on the spherical center.
Comatic aberration is changed when the position of entrance pupil is changed.
There is no comatic aberration when structure is symmetry.

36. 5 Astigmatism（像散）
If only astigmatism exists in a system, the tangential and sagittal rays will intersect at two points on the principal ray. The two intersect points may not coincide. The astigmatism of an optical system is usually denoted by the aberration curves t, s in the following picture.

38. The effect of astigmatism

39. The correction of astigmatism
There is no astigmatism when entrance pupil is on the spherical center.
Astigmatism is changed when the position of entrance pupil is changed.
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40. 6 Field curvature
If only the field curvature exists in an optical system, all of the rays will intersect at one point. However, this point may not coincide with the paraxial image point.
Meridional field curvature
Sagittal field curvature

41. Field curvature cueves

42. The effect of field curvature

43. Spot Diagrams
The first line is caused by axial object point. The second line is caused by off-axial object point. The third line is the position of image.

44. the correction of field curvature
The combination of positive lens and negative lens
Thick lens
If astigmatism is zero, field curvature exists or not?
Answer: When astigmatism is zero, although Meridional image coincides with sagittal image, this point does not coincide with the paraxial image point.

45. 7 Distortion（畸变）
1. Lateral magnification of perfect optical system is constant. So
the object is similar to the image.
2. Lateral magnification of real optical system is not a constant. It
changes along with the field.
3. The amount of distortion is the displacement of the image
from the paraxial potion, and can be either direct or as a
percentage of the ideal image height.

46. Distortion is the function of chief ray
1. If aperture is on the a, 1 is chief ray and y1’ is image height.
2. If aperture is on the b, 2 is chief ray and y2’ is image height.
3. If aperture is on the c, 3 is chief ray and y3’ is image height.
β1 ≠ β2 ≠ β3

47. Relative distortion（相对畸变）
In optical design, using relative distortion q’

48. Distortion types
(a). If the images are displaced outward from the correct position, the distortion is called overcorrected or barrel
distortion.
(b). If the distortion is of the opposite type, and the corners of the square are pulled inward more than the sides, the distortion is called pincushion distrotion

49. The correction of distortion
1. When aperture is on the spherical center, there is no distortion.
2. For a single lens or a single lens group, when aperture is closely coincide with it, there is no distortion.
3. A symmetry system has no distortion.

50. 8 Color Dispersion
So, the index of the red light will be the smallest and that of purple light will be the largest.
For a certain medium the difference of two indexes for two different wavelengths is called the dispersion.

51. The relationship between wavelength and focal length
For single thin lens, the focal length is
Because of the fact that the index of refraction varies as a function of the wavelength of light, the focal length f ’ also varies with wavelength. Thus for a lens the focal length of the red light will be the largest, and that of the purple light will be the smallest.

52. 8 Chromatic aberrations
C light: λ=656.28nm
F light: λ=486.13nm
Usually, the distance along the axis between two image points of wavelength F and wavelength C is called the axis chromatic aberration.（轴向色差）
Usually, the axis chromatic aberration is defined as:
In the paraxial region:

53. Chromatic aberrations

54. Chromatic aberrations
Spherochromatic aberration
Secondary spectrum

55. Chromatic aberration curve
It can be seen:
Monochromatic spherical aberration varies with aperture height.
The axial chromatic aberration varies with aperture height.
Spherical varies with wavelength.
Secondary spectrum

56. Spot Diagrams It can be seen:
Axial point: polychromatic light and monochromatic light
It can be seen:
The axial chromatic aberration and spherical aberration are axial point aberration.
The axial chromatic aberration and spherical aberration both produce circle spot diagrams.
spherical aberration produces monochromatic spot diagram and axial chromatic aberration produces colorful circle spot diagrams.

57. Axial chromatic aberration in thin lens system:
For glue lens, the heights of light in lens is same. So achromatic condition:

58. Derivations of formulas
The differential from Gauss formula:
The differential from focal length of thin lens:
According to:
 Abbe constant
So :
dl’=0

59. Correction of chromatic aberration for doublet lens

60. Axial chromatic aberration in parallel glass block

61. lateral chromatic aberrations （倍率位置色差）
The image height y’ can be calculated according to the following equation (n’=n=1)

62. Lateral chromatic aberration 倍率色差
lateral chromatic aberration makes image edge colorful, and it will affect image resolution, particularly in large field,
so we must correct it.

63. Primary lateral chromatic aberration of thin lens:
（薄透镜的主要横向色差）
For glue thin lens:

64. Correction
For glue thin lens, the lateral chromatic aberration and axial chromatic aberration are corrected at the same time.
When aperture coincides with glue thin lens, the lateral chromatic aberration will not be produced.
For double air-space lens, if using the same material and the gap meet needs, the lateral chromatic aberration will be corrected.

65. The relationship between primary chromatic aberration and aperture, field
From the formula:
We can know: