Introduction to aberrations OPTI 518 Lecture 14

Transcription

1 Introduction to aberrations Lecture 14

2 Topics Structural aberration coefficients Examples

3 Structural coefficients Ж Requires a focal system Afocal systems can be treated with Seidel sums

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6 Structural stop shifting parameter ω = nu = Y 1 ϕ y /2 ( ) Using ω on we can express: S = y y P P 2Ж ϕ S ypypϕ ϕ s ϕ s' = = = 2Ж Y 1 s 2n Y + 1 ϕ s' 2 n' ( ) ϕ ( ) s is the distance from the front principal plane to entrance pupil s is the distance from the rear principal plane to exit pupil 2 ypδypϕ ypϕ δ S = = S 2Ж 2Ж

7 Structural aberration Coefficients for a System 3 4 k Φi ypi, I = i= 0 Φ yp Ii, 2 2 k Φi ypi, II = II k + i I i i= 0 Φ yp (, S,, ) L 2 k Φi ypi, = i= 0 Φ yp Li, Φ 2 (, 2S,, S,, ) k i III = III i + i II i + i I i i= 0 Φ k (, S,, ) = + T Ti i Li i= 0 Φ k i IV = i= 0 Φ IV, i 2 k y P V = V i + i IV i + III i + i II i + i I i i= 0 y Pi, 2 3 (, S, (, 3, ) 3S,, S,, ) S, i Φ y y = 2Ж i Pi, Pi,

8 Review of concepts Thin lens as the thickness tends to zero t φ = φ1+ φ2 φφ 1 2 n Shape of a lens and shape factor Conjugate factor to quantify how the lens is used. Related to transverse magnification Must know well first-order optics

9 Shape and Conjugate factors Y ω' + ω 1+ = = ω' ω 1 m m ω = nu X c + c R + R = = c c R R Lens bending concept

10 Shape X X=0 X=0 X=1 X=-1 X=1.7 X=-1.7 X=3.5 X=-3.5

11 Shape X=-3 X=-2 X=-1 X=0 X=3 X=2 X=1 X=0

12 Shape or bending factor X Quantifies lens shape Optical power of thin lens is maintained Not defined for zero power, R1=R2 X c + c R + R = = c c R R

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15 Example: Refracting surface free from spherical aberration Object at infinity Y=1 I n' + n n' n n' + n 1 2n 2 2n 1 = n' n n' n n' n = = 2 n' n n' n' n n' yp SI = ypφi + ( n) K = y 3 PφI + y 2 PφK 4 r 4 n' n ( ) ypφ n n 1 1 SI ypφ K = K 4 + = + n' n n' ( n' n) ( n' n) n' S I 2 n 2 = 0 = K = ε 2 n ' Parabola for reflection Ellipse for air to glass Hyperbola for glass to air

16 Note SI ypφi y 2 PφK ypφ = + = I + K 4 ( n' n) 4 n' n The contribution to the structural coefficient from the aspheric cap is 2 Icap = α = n' n 2 K For a reflecting surface is just the conic constant K

17 For the general case

18 Icap = α = K

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22 Spherical Mirror A spherical mirror can be treated as a convex/concave plano lens with n=-1. The plano surface acts as an unfolding flat surface contributing no aberration. X I II III IV V L T = ± 1 = Y 2 = Y = 1 = 1 = = = A = 4 B = 0 C = 1 1 D = 4 E = 0 F = 1 n=1 n=1 n=-1

23 Field curves

24 Field curves Rt~f/3.66 Rm~f/2.66 Rs~f/1.66 Rp~f/0.66

25 Application of Coddington Equations

26 Application of Coddington Equations

27 Thin lens Spherical aberration and coma I II Fourth-order

28 Spherical aberration of a F/4 lens Asymmetry For high index 4 th order predicts well the aberration

29 Thin lens spherical aberration n=1.517 Y I X

30 Thin lens coma aberration n=1.517 Y II X

31 This lens Aplanatic solutions Y ω' + ω 1+ m = = ω' ω 1 m X=4 Y=5 n=1.5

32 N=1.5 N=2 N=2.5 N=3 N=3.5 N=4 Thin lens Spherical and Y=0 I II Strong index dependence

33 Thin lens special cases stop at lens

34 Thin lens special cases stop at lens

35 Thin lens special cases stop at lens

36 Thin lens special cases stop at lens

37 Thin lens special cases stop at lens

38 Achromatic doublet Two thin lenses in contact The stop is at the doublet φ = φ + φ y = y φ1 ρ1 = φ φ2 ρ2 = φ ρ + ρ = 1 Y Y Y = ρ2 ρ 1 Y ρ1 = ρ 2

39 Achromatic doublet Correction for chromatic change of focus L 2 k Φi ypi, = i= 0 Φ yp Li, L ρ1 ρ2 = + ν ν 1 2 ν ρ = ( 1 ν ) ν1 ν2 ν ρ = ( 1 ν ) ν1 ν2 L L For an achromatic doublet: = L 0

40 Achromatic doublet Correction for spherical aberration I 3 4 k Φi ypi, = i= 0 Φ yp Ii, ( ) ( ) I = ρ AX BXY+ CY + D + ρ AX BXY + CY + D For a given conjugate factor Y, spherical aberration is a function Of the shape factors X 1 and X 2. For a constant value of spherical aberration we obtain a hyperbola as a function of X1 and X2.

41 Achromatic doublet Correction for coma aberration 2 2 k Φi ypi, II = II i + i I i i= 0 Φ yp (, S, ) ( ) ( ) II = ρ E X FY + ρ E X FY For a given conjugate factor Y and a constant amount of coma the graph of X1 and X2 is a straight line.

42 Achromatic doublet Astigmatism aberration Φ k i III = III i + i II i + i I i i= 0 Φ ( 2 ), 2S, S, = ρ 1 + ρ 1 III ( ) ( ) 1 2 Astigmatism is independent of the relative lens powers, shape factors, or conjugate factors.

43 Achromatic doublet Field curvature aberration IV k i = i= 0 Φ Φ IV, i ρ IV = + n ρ n For an achromatic doublet there is no field curvature n ν n = ν IV = 0 if

44 Achromatic doublet Distortion and chromatic change of magnification 2 k y P V = V i + i IV i + III i + i II i + k I i i= 0 y Pi, 2 3 (, S (, 3, ) 3S, S, ) k T = Ti, + S i Li, i= 0 ( ) V T = = 0 0

45 Cemented achromatic doublet c ( X ) ( n ) ( X ) ( n ) ( X ) ( X ) ( n ) ( X ) ( n ) X 1 φ + 1 φ = = c = 1 ρ + 1 ρ = = α α = α = ( n 1) ( n 1) ( n 1) ( n 1) ρ ρ ν ν For achromat For a cemented achromatic lens the graph of X1 and X2 is a straight line.

46 Crown in front: BK7 and F8

47 Flint in front: BK7 and F8

48 Cemented achromatic doublet

49 Cemented doublet solutions Crown in front Flint in front

50 Two mirror afocal system

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54 Merssene afocal system Anastigmatic Confocal paraboloids

55 Paul-Baker system Anastigmatic-Flat field Anastigmatic Parabolic primary Spherical secondary and tertiary Curved field Tertiary CC at secondary Anastigmatic, Flat field Parabolic primary Elliptical secondary Spherical tertiary Tertiary CC at secondary

56 Meinel s two stage optics concept (1985) Large Deployable Reflector Second stage corrects for errors of first stage; fourth mirror is at the exit pupil.

57 Aplanatic, Anastigmatic, Flat-field, Orthoscopic (free from distortion, rectilinear, JS 1987) Spherical primary telescope. The quaternary mirror is near the exit pupil. Spherical aberration and Coma are then corrected with a single aspheric surface. The Petzval sum is zero. If more aspheric surfaces are allowed then more aberrations can be corrected.

58 Lister objective

59 Lister objective Two achromatic doublets that are spaced Telecentric in image space Normalized system Ж = 1 ya = 1 ua = 1 IA = 0 = 0 IB

60 Lister objective The aperture stop is at the first lens. The system is telecentric φ B = 1 φ y A B = 1 = 1 y B

61 Lister objective

62 Condition for zero coma 2 2 k Φi ypi, II = II i + i I i i= 0 Φ yp = φ y + φ y II A A IIA B B IIB IB = 0 ( y ) 2 2 II B IIA yb IIB = 1 + = 0 IIA y 2 B ( 1 y ) B 2 IIB (, S,, ) =

63 Condition for zero astigmatism Φ k i III = III i + i II i + i I i i= 0 Φ ( 2 ), 2S,, S,, ( y ) = φ + φ 1+ = 0 III A B B IIB ( y ) ( y ) = = 0 III B B IIB S, i = Φ y i Pi, Pi, 2Ж y 2 yb + yb IIB = 0 2 yb = 1 IIB 2 yb IIB = y IIA = y B B( 2 yb) 2 ( 1 y ) B

64 Lister Objective Choose: IIB = IIA 2 y y B = y B ( 1 y ) B 2 2 yb 1 2y + y = y y ϕ B A = IIA IIB B( 2 yb) 2 ( 1 y ) 2 2 B B B 1 = 2 1 = 2 = 3 = 3 B

65 Petzval objective R N-crown=1.517; N-flint=1.576 T / F=100 mm

66 Petzval objective Artificially flattening the field by adding negative astigmatism

67 Variance criterion = W W W W W H W W H W H ( W040 ) + ( W131H) + ( W222H ) Minimize the variance with astigmatism 2 dw = W220 + W222 H + W222H = dw W222 = W220 = W220P + W W222 = W220P 3 3 W = W Flat medial surface P 0

68 Zero medial field curve curvature Φ k i IV = IV, i = ( 1 yb ) IVA + IVB i= 0 Φ 2 2 y ( 1 y ) ( 1 y ) III B B IIB IV B IV IV IIB IIA = + + = 2 III (( ) ( )) = 1 y + 1+ y = 2+ y y + 2= y 1 IV = 2 B + 2 ( 1 ) = = y B IIB ( ) 2 2 y B B( 2 yb) 2 ( 1 y ) y B B B IIB B B IIB IV IIB

69 Grating linear phase change mλ nsin ( I' ) nsin ( I) = d mλ nsin ( I' ) y ynsin ( I) = y d

70 Ray diffractive law (1D)

71 Diffractive optics high-index model 1/d

72 Diffractive optics high-index model 1/d

73 Diffractive lens (n very X=0) = + I II 2 3Y 1 = 2Y = 1 III A=0 B=0 C=3 D=1 E=0 F=2 = IV = V L = ν = T diffractive 0

74 MIRRORS

75 Spherical Mirror

76 Aspheric mirrors

77 Aspheric mirrors

78 Spherical mirror stop at mirror

79 Spherical mirror with stop shift

80 Stop shifting S ypypϕ ϕ s ϕ s' = = = 2Ж Y 1 s 2n Y + 1 ϕ s' 2 n' ( ) ϕ ( )

81 Spherical Mirror Object at infinity Y=1

82 Aspheric mirror

83 With stop shift

84 Paraboloid with object at infinity

85 Aspheric mirror with corrector plate Shift stop from mirror and insert corrector at stop

86 Special aplanatic cases

87 Wright and Schmidt systems

88 Wright and Schmidt systems

89 Summary Structural coefficients Basic treatment Use of Y-Ybar diagram Analysis of simple systems See R. Shack notes