第三章單色差（monochromatic aberration）

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1 monochromatic aberration (The specification of ras).. (ra-intercept curves).. (comatic and astigmatic aberrations).. (defocusing)..5 (curvature of field and astigmatism)..6 (distortion). (aberration polnomials). (aberration tpes)..1 (spherical aberration).. (linear coma).. (linear astigmatism).. (distortion)..5 (oblique spherical aberration)..6 (elliptical coma)..7 (pupil aberration).5 (computation of aberration coefficients).6 Aldis (Aldis theorem).7 Zernike (Zernike analsis).1 OSLO (Seidel) (Buchdahl) -1

2 . (Smmetr properties of centered sstem) rotationall smmetrical surface optical ais (sharp) (undistorted) (conugate).1.1 O O O'O stigmatic O' O stigmatic image O O' meridional plane(.) -

3 . meridional plane O P field curvature defect aberration 1 distortion m O h,h stigmatic image pointo' h ',h ' O O' h ' = mh h ' = mh.1 m h h (ideal image) stigmatic free of field curvature free of distortion (paraial imager) 1 -

4 O O' h ',h ' O' h ',h '.1 displacement, = h = h ' mh ' mh. 0,..1 (The Specification of ras) (obect plane) Abbe (Abbe s Sine condition) Hopkins canonical coordinates OSLO aplantic ra aiming paraial ra aiming (obect distance) (finite) 0.5 direction cosine direction tangent, h,h, centered sstem h h =0.0 h,, -

5 θ ρ.. ρ, (pupil coordinates) = ρ cosθ = ρ sinθ. z =0 = meridional ra θ = π π θ = sagittal ra OSLO h OSLO (entrance beam radius)ebr (obect height) OBH fractional aperture world coordinates h EBR OBH( EBR OBH ).. ra-intercept curves stigmatic,,,h ra-intercept curves ra displacement centered doublet lens Doublet lens catalog lens Edmund Files> New Lens> (file tpes) catalog lens > Lens tpe (catalog)> doublet (Edmund) -5

6 . (on-ais) 70%(0.7 field) (full-field).edmund.5 Edmund doublet (EACH7).5 1. : (on-ais)70%field 0.7 deg (full-field)field 1 deg. meridional curve µm, µm, µm -6

7 z meridional ra (meridional ra-intercept curve) (nominal image plane) 0 - OSLO DY=0 FY cos =0 DX =0 FX sagittal fractional aperture coordinate = sin (meridional curve). paraial focus µm aial image 0. 1µm 0.7 DY v.s. FY coma astigmatism. sagittal ra =±π/ sagittal ra OSLO DX vs. FX (DY vs. FX 0,X, 0,-,- ( ) doublet lens -7

8 (off-ais) DX V.S. FX (on-ais) sagittal cross section. DX V.S. FX.. comatic and astigmatic aberration far from stigmatic O' O' distortion,,h ( ρ, θ, h) ( ρ, θ, h) ( ρ, θ h) = +. s c, [ ρ, θ, h) ] 1 s ( ρ, θ, h) = ( ρ, θ, h) (.5 1 c ( ρ, θ, h) = [ ( ρ, θ, h) + ( ρ, θ, h) ] s c ( ρ,θ, h) (astigmatic) (comatic). s c s s, -,- ρ,θ,h -ρ,θ,h ρ,θ,h smmetrical aberration c (,,h)(-,,h) ρ,θ,h asmmetrical aberration -8

9 (ra-intercept curves) (comatic) (astigmatic) 5.6 (comatic) (astigmatic) (DY) FY (astigmatic) (comatic) s.. defocusing stigmatic curvature of field (distortion) z h h ' ' = + v z z ' a = + v + ' a + ' ( m + zvb ) ' m + v ) h ( z b h.6.6 z v a 'v b ' (stigmatic) ~ ~ = + v = + v z z ' a ' a µm -9

10 h,h ' ( m + ) m~ = zv b ( ) ' '.7 z v, v defocusing term ' z z a z a.7 (ra-intercept curves) z v a ' (meridional aberration)..5 curvature of field and astigmatism (stigmatic).7, z = k.9 = k k.7 z k z =.10 v ' a k h plane obect (sag) z.10.9 (meridional) (sagittal) (.9) -10

11 = k = k.11 k k h.10 meridional and sagittal field surface k =k zt tangential focus zs sagittal focus field curve zt zs h h. (astigmatism) (curvature of field) h = = ' ( k + zva ) ρ cos ' ( k + v ) ρ sinθ s z a θ.1 (astigmatism) (curvature of field) z = zt tangential focal line z = zs sagittal focus line.7.6 tangential focal line sagittal focus line -11

12 ..6 distortion ( ) (stigmatic).1 h (=0) h % aperture stop pincushion distortion barrel distortion. (Aberration polnomials), h,h z,,h h h h + + h + h.1 h h =0 ξ = ρ ζ = h η = ρh cosθ = h = = h +.1 (overall wavefront) W W + W + +Λ.15 = W6 W = +.16 a1ρ + a ρh cosθ a h -1

13 (curvature of the wavefront) (tilt) piston error ξ, η, ζ, ξη, ξζ, ηζ.17 ζ W = b ρ + b ρ h cosθ + b ρ h cos θ + b ρ h + b ρ cosθ h.18 Seidel S I -S V W 1 = SΙρ (S Π + S 1 S ) ρ 1 ρ h cos θ + Sρ h 1 h + Sρ h cos θ Π cos θ +.19 W W α W α.0.19 σ = σ ρ cosθ + σ 1 = σ ρ sinθ + σ 1 ( + cos θ ) ρ h + ( σ + σ ) ρh ρ h sin θ + ( σ + σ ) ρh sinθ cosθ + σ h 5.1 ρ h W -1

14 5 5 5 = µ ρ cosθ ( µ + µ cos θ ) + µ ρh ( µ + µ cos θ ) ρ h + ( µ + µ cos θ ) ρ h + µ 5 = µ ρ sinθ + µ ρ h sin θ ρh cosθ + µ h ( µ + µ cos θ ) ρ h ρ h cosθ sinθ + µ ρ h sin θ sinθ σ µ aberration coefficient 1 9 µ = µ µ = µ + µ µ = µ + µ 9..8 H=Lagrange n'= v' a = v' b = S = Seidel 1 (Spherical Aberration) (Coma) S II / (Sagittal Coma) (Tangential Coma) (Astigmatism) S III / T-S (T-S distance) Petzval (Petzval) Petzval (Image Plane to Petzval surface) Petzval (Curvature of S I /8 S I /(n u a ) -S I /(n u a ) S IV / S II /(n u a ) S II /(n u a ) S III /(n u a ) -S IV /(n u a ) - n S IV /H -1

15 Petzval surface) (Sagittal Field of curvature) (Tangential Field of Curvature) (S III + S IV )/(n u a ) -( S III +S IV )/(n u a ) (S III + S IV )/(n u a ) (Distortion) S V / S V /(n u a ) (Fractional distortion) (Entrance Pupil Spherical aberration) S V /H S I /(n u b ).8 -( S III +S IV )/(n u a ) -S I /(n u b ). (Aberration tpes) ρ h ρ ρ ρ h ρ n-s h s n-s n s n-s n n-s ρ h ρ h 5 (cubic astigmatism).9 (Spherical aberration) n n n (Coma or linear coma) -15

16 (Astigmatism) n (distortion) (oblique spherical aberration) (elliptical coma) n n n (n-) n (n-).9.9 (ra-intercept curves) annular aperture curve (60 )..1 Spherical aberration 5 7 = σ ρ + µ ρ + τ +Λ ρ n n n.10-16

17 .10 (meridional ra-intercept curves) (Linear coma) h comatic = σ ( + cos θ ) ρ h + ( µ + µ cos θ ) = σ ρ h sin θ + µ ρ h sin θ ρ h.5-17

18 0 annular aperture curve ρ(.1).1 σ σ.1.1 ρ (sagittal ra) -18

19 5.. (Linear astigmatism).1.1. = = ( σ + σ ) ρh cosθ + µ 10 ρh cos ( σ + σ ) ρh sinθ + µ ρh sinθ 11 θ.6 σ σ tangential sagittal (field angle).1.1 h

20 σ Petzval Petzval (sag) zp σ = v ' h.7 a σ Petzval field surface Petzval Petzval σ Petzval.6 Petzval tangential field surface Petzval sagittal field surface 1.5 Petzval Petzval Petzval Petzval curvature of field Petzval.. distortion n n = σ h = µ h σ µ (pincushion distortion) (barrel distortion) -0

21 Oblique spherical aberration 6 8 (oblique spherical aberration) = = ( µ + µ 6 cos θ) ρ h cosθ ( µ + µ cos θ) ρ h sin θ µ5 µ µ 6.9 µ =µ 5 +µ µ 5 =µ 6-1

22 .17 (µ 5 µ 6 ).17 µ 5 /µ 6 oval shaped curve µ 5 /µ µ 5 /µ 6 =0 µ 5 /µ 6 0 µ 5 /µ 6-1 µ 5 /µ 6-1 (.17m5/m6 µ 5 /µ 6 ) Elliptical coma ρ h n n- ( µ + µ cos θ ) = µ ρ h = sin θ ρ h.0 -

23 ( ) µ 7 =µ 8 +µ µ 8 /µ 9 (.19m8/m9 µ 8 /µ 9 ) µ 8 /µ 9.19 µ 7 µ 8 /µ 9 µ 8 /µ 9 (eccentricit) µ 8 = Pupil aberration -

24 OSLO PSAPCMPAS PDS PTZ 1 eepiece. (Computation of aberration coefficients) OSLO calculate>>aberration Analsis sei unconverted transverse angular transverse aberration angular form Seidel Si.7.1 OSLO Smith Modern Lens Design OSLO Smith SA TSC CMA CC AST TAC Pertzval PTZ TPC DIS DC.0 OSLO Smith Modern Lens Design OSLO fif buc Buchdabl MU1,, MU1. µ 1,, µ 1.1 fif OSLO Buchdahl SA5,MU1 CMA5,MU AST5,(MU10-MU11)/ -

25 PTZ5,(5MU11-MU10)/ DIS5,MU1 MU+MU6 MU5 MU7+MU8 MU9.OSLO Buchdahl OSLO Buchdahl.5 Aldis (Aldis theorem) Aldis Seidel M. L. Aldis A. Co A Sstem of Optical Design Aldis W. T. Welford Aberration of Optical Sstem ' paraial field Lagrange H n u i A =n i =n' i' J,,z K,L,M Welford, k 1 A ( = = A Z K + K + L ).1 n' k u' k M ' k 1 M ' + M k 1 A H ( = = A Z L + K + L ). n' k u' k M ' k 1 M ' + M Co Welford (aspheric) Aodis surface-b-surface Seidel -5

26 OSLO SCP *aldis total ra aberration ad, ae, af, ag asr *aldis fb f f fb fractional obect heightf f User main menu General Analsis pull-right Seidel Aldis Seidel DELTA Y DELTA OSLO DY DX SUM Seidel DYref DELTA Y SUM DELTA Y DYref SUM.6 Zernike Zernike interferometer optical testing Zernike complete set Zernike Zernike Wand Creath (Appl. Optics and Optical Engineering Vol.XI,ed.R.R.Shannon and J.C. Want,199) ρ θzernike Zernike Want Creath Zernike Zernike -6

27 Zernike 1Zρ,θ ρ Rρ θ Gθ ( ρ θ ) R( ρ) G( θ ) Z, =. G π Q Q+α ( θ + α) = G( θ) G( α) G. G(θ) general solutions comple eponential ± imθ ( θ) = e G (.5 m 0 R(ρ)n ρ ρ m m R(ρ) m R(ρ) R m n (ρ) Jacobi 1 m ( ρ) R ( ρ) m R ρdρ = δ n n' nn' ( n + 1) δ nn' kronecker dalta n=n' δ nn' =1 nn' δ nn' =0 R = 1.7 m n ( ρ) = ( ρ) ρ m m R m n m Q n.8 (n-m) Q m n (ρ) Q m n n m s= 0 ( ρ) = ( ) s ( n m s)! ( n s)(! n m s) 1 ρ s!! ( n m s ).9 W(ρ, θ)zernike -7

28 n 0 m m W ( ρ, θ) = W + A nq n ( ρ) + Q n ( ρ) ρ ( Bnm cos mθ + C nm sin mθ).0 n= 1 m= 1 W A n B nm C nm 1 Zernike W A 0 0 m m W ( ρ, θ) = A + A Q ( ρ) + Q ( ρ) ρ ( B cos mθ + C sin θ) =.1 n 0 n n nm nm m n= 1 m 1 n z θ Zernike σ n An 1 Bnm + C nm σ = +. n= 1 n + 1 m= 1 n + 1 m rms OPD σ σ ((.) Zernike Zernike (. ) Zernike Zernike Zernike Zernike OSLO Zernike 7 n m No. Zernike ρcos θ ρsin θ 0 ρ 1 1 ρ cos θ 5 ρ sin θ 6 (ρ ) ρcosθ 7 (ρ ) ρsin θ 0 8 6ρ 6ρ + 1-8

29 9 ρ cosθ 10 ρ sin θ 11 (ρ ) ρ cos θ 1 (ρ ) ρ sin θ 1 1 (10ρ 1ρ + ) ρcosθ 1 (10ρ 1ρ + ) ρsin θ ρ 0ρ + 1ρ 1 16 ρ cos θ 17 ρ sin θ 18 (5ρ ) ρ cosθ 19 (5ρ ) ρ sin θ 0 (15ρ 0ρ + 6) ρ cos θ 1 (15ρ 0ρ + 6) ρ sin θ 1 6 (5ρ 60ρ + 0ρ ) ρcosθ 6 (5ρ 60ρ + 0ρ ) ρsin θ ρ 10ρ + 90ρ 0ρ ρ cos5θ 6 ρ sin 5θ 7 (6ρ 5) ρ cos θ 8 (6ρ 5) ρ sin θ 9 (1ρ 0ρ + 10) ρ cosθ 0 (1ρ 0ρ + 10) ρ sin θ 1 6 (56ρ 105ρ + 60ρ 10) ρ cos θ 6 (56ρ 105ρ + 60ρ 10) ρ sin θ (16ρ 80ρ + 10ρ 60ρ + 5) ρcosθ 8 6 (16ρ 80ρ + 10ρ 60ρ + 5) ρsin θ ρ 60ρ + 560ρ 10ρ + 0ρ ρ 77ρ + 150ρ 1680ρ + 0ρ ρ + 1.Zernike. Zernike 0 Zernike 0 0 OPD

30 1. OSLO Optical Reference version 5 (Sinclair Optics, 1996) Chapter. OSLO Optical References Analsis of the aberrations of a smmetric sstem, note prepared b P. J. Sands, CSIRO, Australia. -0

31 1. Matlab. OSLO catalog lens OSLO -1

第１０章　繞射光學（Diffractive Optics）

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第三章 單色差（monochromatic aberration）

第三章單色差（monochromatic aberration）