OPAC 202 Optical Design and Inst.

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1 OPAC 202 Optical Design and Inst. Topic 9 Aberrations Department of Optical & Acustical Engineering Gaziantep University Apr 2018 Sayfa 1

2 Introduction The influences which cause different rays to converge to different points are called aberrations. Aberration leads to blurring of the image produced by an image-forming optical system. Optical Instrument-makers need to correct optical systems to compensate for aberration. Sayfa 2

3 First Order Optics The paraxial formulas developed earlier for image formation by spherical reflecting and refracting surfaces are, of course, only approximately correct. Rays that arrive small angles with respect to optical axis are known as paraxial rays. In paraxial optics, we use of the following approximate (1st order) Taylor expansions: sin tan cos 1 1/ 2 (1 x) 1 x / Paraxial theory is first studied by Gauss in Paraxial Optics = First Order Optics = Gaussian Optics 2 Sayfa 3

4 Third Order Optics Mathematically, the power expansions for the sine and cosine functions, given by: Taking first order terms in the expansion results in perfect imaging. Inclusion of higher-order terms in the derivations, predicts increasingly larger departures from perfect imaging with increasing angle. These departures are referred to as aberrations When the next term involving is included in the approximation for sin(x), a third-order aberration theory results. Sayfa 4

5 Third Order Optics The aberrations have been studied and classified by the German mathematician Ludwig von Seidel and are referred to as third-order or Seidel aberrations. For monochromatic light, there are five Seidel aberrations: spherical aberration coma astigmatism curvature of field distortion An additional aberration, chromatic aberration, results from the wavelength dependence of the imaging properties of an optical system. Sayfa 5

6 Third-Order Treatment of Refraction at a Spherical Interface Sayfa 6

7 Seidel Coefficients For the derivation of Seidel Coefficients see Chapter 20 3 rd Ed [Pedrotti]. Sayfa 7

8 Seidel Coefficients Using third order optics, aberrations at point Q is given by: These terms comprise the five monochromatic, or Seidel, aberrations, as follows: Sayfa 8

9 Ray and Wave Aberrations Consider two wavefronts are shown emerging from an optical system. W 1 = spherical wavefront representing paraxial approximation W 2 = the actual wavefront, an aspherical envelope whose shape represents an exact solution of the optical system LI = Longitudinal Aberration (LA) IS = Transverse Aberration (TA) Sayfa 9

10 Spherical Aberrations (SA) Figures given below illustrate spherical aberrations for parallel rays. Rays are converged to different points instead of a single focal point. Sayfa 10

11 Longitudinal & Transverse Aberrations in Lens For ray R: LA = Paraxial distance Trigonometric distance = OA - OB TA = LA tan(u) [U is the angle with the principle axis]. Sayfa 11

12 Longitudinal & Transverse Aberrations Sayfa 12

13 Longitudinal & Transverse Aberrations The spherical aberration of a system is usually represented graphically. Longitudinal spherical is plotted against the ray height. Transverse spherical is plotted against the final slope of the ray, Sayfa 13

14 Example* Plano-convex lens: R = 40.0 cm D = 12.0 cm h = 4.0 cm n = 1.5 (a) Find the effective focal length (paraxial focal length) (b) Find the locations of principle planes. (b) Find longitudinal and transverse spherical aberrations. (c) Plot h vs LA and h vs TA graphs. [Ans: f = 80 cm, LA = cm, TA = cm]. What if you change the orientation of the lens? (That is to say, curved face will look at the incoming ray) Repeat case (a), (b), (c) and (d). Sayfa 14

15 Example* (Plots) Sayfa 15

16 Longitudinal & Transverse Aberrations in Mirror For ray R: LA = Paraxial distance Trigonometric distance = OA - OB TA = LA tan(u) [U is the angle with the principle axis]. Sayfa 16

17 To overcome the spherical aberration in mirrors, parabolic reflecting surface rather than a spherical surface is used. Parallel light rays incident on a parabolic surface focus at a common point, regardless of their distance from the principal axis. Parabolic reflecting surfaces are used in many astronomical telescopes to enhance image quality. Sayfa 17

18 A parabolic solar dish The mirror of the Hubble Space Telescope Sayfa 18

$Law of reflection: and Law of refraction: 1 2 1 sin1 n2 sin can be derived from this principle.$

19 Optical Path Length Fermat s Principle of Least Time Light takes the path which requires the shortest time. Law of reflection: and Law of refraction: sin1 n2 sin can be derived from this principle. n 2 Sayfa 19

20 Optical Path Length Optical Path Length for a light beam is defined as follows: or OPL = (index of refraction) * (path travelled by light) OPL n s * If there are a number of mediums then OPL n 1 s 1 * If the medium consists of continues materials then: OPL nds n 2 s 2 n k s k k i1 n i s i Sayfa 20

21 Optical Path Length and Fermat s Principle Distance travelled by light in optical medium index of refraction n, is s v t Or time travelled by light in the same medium t s v s c / n ns c OPL c where c is the speed of light in vacuum which is a constant. Fermat s principle is related to optimum time. That is, Fermat s principle is equivalently related to optimum OPL. So, last form of the Fermat s principle is: Light travels in medium such that its total optical path length is optimum. Sayfa 21

22 Derivation of parabolic mirror Show that paralel rays can only be focused to a common point if one uses a parabolic surface. Sayfa 22

23 Derivation of Hyperbolic surface for a lens Determine the aspherical surface for a plano-hyperbolic collimator as follow: Sayfa 23

24 Surface Sag An important property of an optical surface is sag. Exact definition is: sag R R 2 y 2 When you measure y and sag, You can determine radius R. Note that if y<< R, one can show that sag 2 y 2R Sayfa 24

25 Aspheric Surfaces Consider a circle whose center is a origin of y-z plane. The equation of the circle is: z 2 y 2 R 2 If you translate the center to the right then z 2 2zR y 2 0 Sayfa 25

26 Aspheric Surfaces An equation that defines a conic asphere is: (1 k) z 2 2zR y 2 0 k is conic constant: k = 0 sphere k = -1 parabola k< -1 hyperbola k>0 or -1<k<0 ellipse % matlab m-file to plot % conic sections k = -1; R = 100; z = 0:0.1:R; y2 = 2*z*R - (1+k)*z.^2; plot(z, sqrt(y2)) hold on plot(z,-sqrt(y2)) grid on Sayfa 26

27 Sayfa 27

28 Aspheric Surfaces Most of the optical surfaces are spherical since they are easy to manufacture and measure (using spherometer). To reduce spherical aberration one way is to use a aspherical surface which is much harder to make and measure. Two examples are: 1. Usually primary mirrors of reflecting telescopes are parabolic. 2. Some camera lenses use injection-molded plastic glass elements which are aspheric. Sayfa 28

29 Example usage of aspheric surfaces in Zemax EX1: Single concave mirror with spherical mirror with parabolic mirror EX2: Cassegrain telescope with two sherical mirrors with primary mirror is prabolic and secondary is hyperbolic EX3: Single plano-convex lens with sherical refracting surface with hyperbolical refracting surface with elliptical refracting surface Sayfa 29

30 Reduction of S.A. in Zemax EX1: Single bi-convex lens Spherical aberration is proportional to 4 th power of aperture size. For R1 = R2 = 50 mm, ct = 5 mm, glass = BK7, λ = 550 nm. Aperture Size SPHA The smaller diameter The smaller spherical aberration Sayfa 30

31 Reduction of S.A. in Zemax EX2: Optimize lens radii * Start with R1 = R2 = 90 mm, ct=5mm, gla=bk7, λ=587nm, AS = 40 mm. * Target effl = 300 mm. * R1 and R2 are varible. Go to MFE: Click Optimize -> SPHA becomes 1% (initially 70%). Sayfa 31

32 Coddington shape factor Lens s maker formula for thin lens: 1 f 1 ( n 1)[ R 1 R Various choices of the radii of curvature, while not changing the focal length, may have a large effect on the degree of spherical aberration of the lens ] s o s i A measure of this bending is the Coddington shape factor: R R 2 2 R R 1 1 Bending of a single lens into various shapes having the same focal length. Sayfa 32

Coddington shape factor σ is determined by the physical shape of the lens. Also, Coddington position factor is sefined by: P s s s i = image distance to lens s o = onject distance to lens.

33 Coddington shape factor σ is determined by the physical shape of the lens. Also, Coddington position factor is sefined by: P s s s i = image distance to lens s o = onject distance to lens. i i When s o = inf then s i = f and P = -1, incident light is parallel. Note that, spherical Aberration is minimum when* s s o o 2( n 1) n 2 2 P To find the best form of the lens having minimum S.A. R 1 2 f ( n 1) 1 R 2 f ( n 1) 1 * Proof of this equation can be found in some text books. 2 Sayfa 33

34 Example Determine the radii of curvatures of a lens of f = +100 mm, n = 1.5, which for parallel incident light has minimum spherical aberration Solution: Position factor: P s s i i s s o o Optimum shape factor 2 2( n 1) n 2 P 2 2(1.5 1) ( 1) Radii of curvatures: R R 2 f ( n 1) f ( n 1) mm mm Sayfa 34

35 Coma Coma is an off-axial aberration. Coma increases rapidly as the third power of the lens aperture. Sayfa 35

36 Coma Coma can be reduced by placing an aperture stop after the second surface. Sayfa 36

37 Example Consider a bi-convex lens, R1 = R2 = 100 mm, ct=5mm, glass=bk7, λ=632.8 nm and Field angle is 5 o. Using Zemax, (a) Plot comatic aberration vs aperture stop size for a fixed aperture position. (b) Plot comatic aberration vs aperture stop position for a fixed aperture size. Sayfa 37

Astigmatism is one where rays that propagate in two perpendicular planes

38 Other Types of Aberrations Distortion is a deviation from rectilinear projection. Coma refers to aberration due to imperfection in the lens. Astigmatism is one where rays that propagate in two perpendicular planes have different focal lengths. This is due to manufacturing the lens. Sayfa 38

39 Monochromatic Aberration vs stop shift Sayfa 39

40 Zemax: Seidel Coefficients Analysis > Aberration Coefficients > Seidel Coefficients Displays Seidel (unconverted, transverse, and longitudinal), and Wavefront aberration coefficients. The Seidel coefficients are listed surface by surface, spherical aberration (SPHA, S1), coma (COMA, S2), astigmatism (ASTI, S3), field curvature (FCUR, S4), distortion (DIST, S5), longitudinal color (CLA, CL), and transverse color (CTR, CT). The units are always the same as the system lens units. Sayfa 40

41 Zemax: OPD Fan Analysis > Fans > Optical Path OPD Fan is a plot of the optical path difference as a function of pupil coordinate. In a perfect optical system, the optical path of the wavefront will be identical to that of an aberration-free spherical wavefront in the exit pupil. Try this for paraxial and non paraxial lens in Zemax. Sayfa 41

42 Zemax: Ray Fan Plots Analysis > Fans > Ray Aberrations The Ray Fan plots ray aberrations as a function of pupil coordinate. Generally, a given ray which passes through the optical system an onto the image surface, its point of intersection falls on some small but nonzero distance away from the chief ray. Once again, in a perfect optical system, the ray aberrations should be zero across the pupil. chief ray, or principal ray, is the central ray in this bundle of rays. Sayfa 42

43 Zemax: Ray Fan Plots Sayfa 43

44 Zemax: Ray Fan Plots Sayfa 44

45 Zemax: Ray Fan Plots Sayfa 45

46 Zemax: Ray Fan Plots Sayfa 46

47 Zemax: Ray Fan Plots Sayfa 47

48 Zemax: Ray Fan Plots Sayfa 48

49 Zemax: Ray Fan Plots Sayfa 49

50 Zemax: Ray Fan Plots Sayfa 50

$CA occurs because lenses have a different refractive index$

51 Chromatic Aberrations (CA) A lens will not focus different colors in exactly the same place. CA occurs because lenses have a different refractive index for different wavelengths of light (the dispersion of the lens). v c n( ) violet red Sayfa 51

52 Sayfa 52

53 In Zemax [Wav] -> F,d,C (Visible) [Gen] -> Entrance pupil = 20 mm Tools -> Miscellaneous -> Quick Focus Sayfa 53

54 Example Figure shows a equiconvex lens made up of a BK7. Determine the distance (x) between paraxial focal lengths for the Fraunhofer F and C lines. Use Zemax for R = 100 mm, ct = 8 mm and D = 20 mm. Sayfa 54

55 One way to minimize this aberration is to use glasses of different dispersion in a doublet or other combination. The use of a strong positive lens made from a low dispersion glass like crown glass (n<1.6) coupled with a weaker high dispersion glass like flint glass (n>1.6) can correct the chromatic aberration for two colors, e.g., red and blue. Such doublets are often cemented together and called achromat. An achromatic lens (or achromat) is designed to limit the effects of chromatic and spherical aberration. Sayfa 55

56 Sayfa 56

57 Example Figure shows an achromatic lens consisting of a bi-convex crown glass (BK7) and a bi-concave flint glass (SF5) where radius of curvature is R = 10 cm. Find the distance between paraxial focal lengths for blue and red rays with and without flint glass. Ans: Δf (BK7) = cm Δf (BK7 + SF5) = cm Glass Refractive Index 400 nm 700 nm BK7 : SF5 : Sayfa 57

Optimum Doublet Consider achromatic doublet in contact. If the focal lengths of the two (thin) lenses for light at the yellow Fraunhofer D-line (587.

58 Optimum Doublet Consider achromatic doublet in contact. If the focal lengths of the two (thin) lenses for light at the yellow Fraunhofer D-line (587.6 nm) are f 1 and f 2, then best correction occurs for the condition: f 1 V1 f2 V2 where V 1 and V 2 are the Abbe numbers of the materials of the first and second lenses, respectively. Since Abbe numbers are positive, one of the focal lengths must be negative. 0 Note that if f is the focal length of the for system D-line, then: 1 f V1 V V f 1 V2 f2 V2 V [Prove these relations in the class] 1 1 f Sayfa 58

59 Optimum Doublet Alternatively, one can also show that (for D-lines): P V P 1 2 P2 V1 1 V1 P V V Geometric factors: ( Pi 1/ fi ) P 2 V2 P V V 2 1 K 1 P1 n 1 1 K 2 P2 n 1 2 For simplicity one can select r 11 = r 12. Then radii of curvature: r 12 r 11 r21 r12 r 22 r12 1 K r 2 12 [Matlab code can be written] Sayfa 59

60 Example (Pedrotti 3 rd Ed.) Consider 520/636 crown glass and 617/366 flint glass are used in designing an achromat of focal length 15 cm. Equations lead to lenses with radii of curvature given by: r 11 = cm r 12 = cm r 21 = cm r 22 = cm Sayfa 60

61 Example (Pedrotti 3 rd Ed.) Consider 520/636 crown glass and 617/366 flint glass are used in designing an achromat of focal length 15 cm. Equations lead to lenses with radii of curvature given by: r 11 = cm r 12 = cm r 21 = cm r 22 = cm Sayfa 61

62 Optimum Doublet Design using Matlab % achromat lens design clear; clc; F = 500; % target system focal length P = 1/F; % Lens 1 nd1 = ; nf1 = ; nc1 = ; V1 = (nd1-1)/(nf1-nc1); t1 = 4; n1 = nd1; % Lens 2 nd2 = ; nf2 = ; nc2 = ; V2 = (nd2-1)/(nf2-nc2); t2 = 3; n2 = nd2; % powers P1 = -P*V1/(V2-V1); P2 = +P*V2/(V2-V1); K1 = P1/(n1-1); K2 = P2/(n2-1); % radii of curvatures r11 = F/2 % usually F/2 (suggested) Output r11 = 250 r12 = -250 r21 = -250 r22 = Goodness = r12 = -r11 r21 = r12 r22 = r12/(1-k2*r12) % Pers performance Goodness = P1*V1 + P2*V2 % You can also print focal lengths for F, D and C lines separately % like in the table on page 56 of these slides. Sayfa 62

63 Exercise ZEMAX Analysis: Kidger Triplet at: Sayfa 63

Waves & Oscillations

Physics 42200 Waves & Oscillations Lecture 33 Geometric Optics Spring 2013 Semester Matthew Jones Aberrations We have continued to make approximations: Paraxial rays Spherical lenses Index of refraction