20. Aberration Theory

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1 0. Aberration Theory Wavefront aberrations ( 파면수차 ) Chromatic Aberration ( 색수차 ) Third-order (Seidel) aberration theory Spherical aberrations Coma Astigmatism Curvature of Field Distortion

2 Aberrations Chromatic Monochromatic n (λ)( Unclear image Deformation of image Spherical Coma astigmatism Distortion Field Curvature Five third-order (Seidel) aberrations

$Aberrations: Chromatic n (λ)( Because the focal length of a lens depends on the refractive index (n), and this in turn depends on the wavelength, n = n(λ), light of different$

3 Aberrations: Chromatic n (λ)( Because the focal length of a lens depends on the refractive index (n), and this in turn depends on the wavelength, n = n(λ), light of different colors emanating from an object will come to a focus at different points. A white object will therefore not give rise to a white image. It will be distorted and have rainbow edges

4 Five monochromatic Aberrations Unclear image Spherical Coma astigmatism Deformation of image Distortion Field curvature

5 Spherical aberration This effect is related to rays which make large angles relative to the optical axis of the system Mathematically, can be shown to arise from the fact that a lens has a spherical surface and not a parabolic one Rays making significantly large angles with respect to the optic axis are brought to different foci

brought to a focus at different points on the focal plane Found in lenses

6 Coma An off-axis effect which appears when a bundle of incident rays all make the same angle with respect to the optical axis (source at ) Rays are brought to a focus at different points on the focal plane Found in lenses with large spherical aberrations An off-axis object produces a comet-shaped image

7 Astigmatism and curvature of field Yields elliptically distorted images

8 Distortion: Pincushion, Barrel This effect results from the difference in lateral magnification of the lens. If f differs for different parts of the lens, si yi M T s y will differ also o o object M on axis less than off M on axis greater than axis (positive lens) off axis (negative lens) f i >0 f i <0 Pincushion image Barrel image

9 A mathematical treatment of the monochromatic aberrations can be developed by expanding the binomial series up to higher orders Third-order aberration theory Paraxial approximation

10 Now, let s derive the expression of the third-order aberrations (Seidel aberrations, Ludwig von Seidel)

11 0-1. Ray and wave aberrations Wave aberration Ideal wavefront Actual wavefront LA Paraxial Image plane Paraxial Image point TA LA : ray aberration - longitudinal TA : ray aberration - transverse (lateral)

12 Longitudinal Ray Aberration n = 1.0 n L =.0 Modern Optics, R. Guenther, p R R n sin n sin sin i i t t t sini

13 Longitudinal Aberration cont d n = 1.0 n L =.0 R Rsin t sin sin n sin n t i i t i sin i i t t t R R sint z Z = Z ( i )

14 Longitudinal Aberration cont d n = 1.0 n L =.0 sin n i t sin n R R sin i i t t sint z Z z/r ( ) Modern Optics, R. Guenther, p. 199.

15 0-. Third-order treatment of refraction at a spherical interface Let s start the aberration calculation for a simple case. Q h I P O Figure 0-3 To the paraxial (first-order) ray approximation, PQI = POI according to Fermat s principle. Beyond a first approximation, PQI (depends on the position of Q) POI. Thus we define the aberration at Q as a Q PQI POI n n ns n s opd 1 1

16 Refraction at a spherical interface cont d Let s describe l in terms of R, s,. l l Q R cos R sin sr sr s R R Rs R P R sr RR sin cos 1 s R sr RR sr cosr Substituting and rearranging we obtain : cos s O C

17 Refraction at a spherical interface cont d O R l l ' C s -R I R cos sin sr sr sin cos 1 s R s R R R R R sr s R R s R Rs R cos cos

18 Writing the cos term in terms of h we obtain : 1/ 4 h h h cos 1sin R R 8R where we have used the binomial expansion 1/ x x 1x 1 8 Substituting into our exp ressions for and and rearranging Refraction at a spherical interface cont d 6 order h and higher we obtain 4 h R s h R s s 1 3 Rs 4R s 1/ 4 h Rs h Rs s ' 1 3 Rs 4R s Use the same binomial exp ansion and neglecting terms of 4 4 h R s h R s h R s s Rs 8R s 8R s s ' 1 Rs 8R s 8R s 1/ 4 4 h R s h R s h R s 3 4 P Third-order angle effect O Q h I

19 Refraction at a spherical interface cont d h h h h h h h s s Rs R s R s s Rs R s h h h h h h h s1 s Rs 8 R s 8 R s 8 s 4 Rs 8 R s a Q n n ns n s 1 1 P O Q h a Q Imaging formula (first-order approx.) n n n n s s' R s s R s' s' R 4 h n11 1 n 1 1 ch 4 Wave aberration a Q Aberration for axial object points (on-axis imaging) : This aberration will be referred to as spherical aberration. The other aberrations will appear at off-axis imaging!

20 0-3. Spherical Aberration 4 a Q ch h Optics, E. Hecht, p.. n Transverse Spherical Aberration b y Longitudinal Spherical Aberration b z b y 4 cs ' n h 3 b z 4 cs ' n h

21 Spherical Aberration Least SA Modern Optics, R. Guenther, p Most SA

22 Spherical Aberration For a thin lens with surfaces with radii of curvature R 1 and R, refractive index n L, object distance s, image distance s', the difference between the paraxial image distance s' p and image distance s' h is given by h nl n L 4 3 nl 1 p 3nL nl 1 p s h s p 8 f nlnl 1 nl 1 nl 1 where, R R s s R R s s 1 ( shape factor), p 1 Spherical aberration is minimized when : nl 1 n L p

23 Spherical Aberration Spherical aberration is minimized when : For an object at infinite ( p = -1, n L = 1.50 ), ~ 0.7 nl 1 n L p R R s s p R R s s 1, 1 = -1 = +1 worse better Optics, E. Hecht, p..

24 Spherical Aberration ~ 0.7 p R R R R 1 1 s s s s

25 Third-Order Aberration : Off-axis imaging by a spherical interface Now, let s calculate the third-order aberrations in a general case. Q h Q O B ' r b rbcos, aq; r,, h ' 4 4 ( ' ) a Q a Q a O c b a Q a Q PQPPBP c( BQ) c ' b h' opd 4 4 a O POPPBP c( BO) cb opd 4 4 Q O b r B On the lens surface

26 Third-Order Aberration Theory After some very complicated analysis the third-order aberration equation is obtained: C h' r 4 a Q C r 0 40 Spherical Aberration Q C hr cos Coma O r C h r cos Astigmatism B C h r 0 Curvature of Field 3 3C11h r cos Distortion On-axis imaging 에서의 a(q) = ch 4 와일치

27 0-4. Coma 3 a Q 1C31 h' r cos ( h' 0, cos 0) O Q r B

28 Optics, E. Hecht, p. 4. Coma

29 Coma Least Coma Modern Optics, R. Guenther, p. 05. Most Coma

30 Coma

31 0-5. Astigmatism a Q C h' r cos ( h' 0, cos 0) Optics, E. Hecht, p. 4.

32 Astigmatism Sagittal plane Tangential plane (Meridional plane)

33 Astigmatism Coma

34 Astigmatism Least Astig. Modern Optics, R. Guenther, p. 07. Most Astig.

35 Field Curvature a Q C0 h' r ( h' 0) Astigmatism when = 0. O B Q r A flat object normal to the optical axis cannot be brought into focus on a flat image plane. This is less of a problem when the imaging surface is spherical, as in the human eye.

36 Field Curvature a Q C0 h' r ( h' 0) Astigmatism when = 0. If no astigmatism is present, the sagittal and tangential image surfaces coincide on the Petzval surface. The best image plane, Petzval surface, is actually not planar, but spherical. This aberration is called field curvature.

37 0-6. Distortion 3 a Q 3C11 h' rcos ( h' 0, cos 0)

38 0-7. Chromatic Aberration Optics, E. Hecht, p. 3.

39 Achromatic Doublet Figure 0-13 (1) () 0-13 Power of two lenses, P 1D and P D are different at the Fraunhofer wavelength, D = nm. * Note: What is the definition of the Fraunhofer wavelengths (lines)?

* Note: Fraunhofer wavelengths (lines) Spectrum of a

at around 3 or 4 pm on a clear day. D = 587.

3 nm (red) The dark lines in the solar spectrum were

40 * Note: Fraunhofer wavelengths (lines) Spectrum of a blue sky somewhat close to the horizon pointing east at around 3 or 4 pm on a clear day. D = nm (yellow) F = nm (blue) C = nm (red) The dark lines in the solar spectrum were caused by absorption by those elements in the upper layers of the Sun.

41 Achromatic Doublets Chromatic aberration is eliminated when : P n 1 n 1K 1D 1D 1D 1 f1d r11 r P n 1 n 1K nm center of visible spectrum D D D fd r1 r D For a lens separation of L : Total power is L P P P LP P f f f f f 1 1 For a cemented doublet L 0: 1 1K P P P n K n P n1 n K1 K 0 In general, for materials with " normal " dispersion : n 0 That means that to eliminate chromatic aberration, K and K must have opposite signs. 1 The partial derivative of refractive index with wavelength is approximated : n nf n C F for an achromat in the visible region of the spectrum. In the above equation, nm and nm. F C C

42 Achromatic Doublets Defining the dispersive constant V : V nd 1 n n F C we can write K n n K n n 1 P 1D 1F 1C 1D 1D 1 1 F C n1d 1 F CV1 K n K n n n 1 P D F C D D F C nd 1 F CV V and V are functions only of the material 1 properties of the two lenses. P 1D D V P 0 V P1D V1PD 0 V F C 1 F C Achromatic condition for doublets

43 D 1D D Achromatic Doublets We can solve for the power of each lens in terms of the desired power of the doublet : P P P V VP VP 0 P P P P K 1 1D 1 D 1D D D D V V1 V V1 1 P1D PD, K n 1 n 1 1D D where P n 1 K n 1 K V D 1D 1 D From the values of K and K, the 4 radii of curvature for the two surfaces 1 of the lenses can be determined. If lens1 is bi - convex with equal curvature for each surface : r r r r r where, K 1 Kr1 r 1 1 r r 1

44 Table 0-1 Achromatic Doublets

5. Aberration Theory

5. Aberration Theory Last lecture Matrix methods in paraxial optics matrix for a two-lens system, principal planes This lecture Wavefront aberrations Chromatic Aberration Third-order (Seidel) aberration