5. Aberration Theory

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1 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 theory Spherical aberrations Coma Astigmatism Curvature of Field Distortion

2 Aberrations Chromatic Monochromatic n (λ)( Unclear image Spherical Coma astigmatism Deformation of image Distortion Curvature A mathematical treatment can be developed by expanding the sine and tangent terms used in the paraxial approximation

3 Third-order (Seidel) aberrations Paraxial approximation

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

4 Aberrations: Chromatic 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

5 Aberrations: Spherical 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

Aberrations: 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

6 Aberrations: 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 f

7 Astigmatism and curvature of field Yields elliptically distorted images

8 Aberrations: Pincushion and Barrel Distortions 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 Ray and wave aberrations Wave aberration Ideal wavefront Actual wavefront LA Paraxial Image plane Paraxial Image point TA LA : longitudinal ray aberration TA : transverse (lateral) ray aberration

10 Longitudinal Ray Aberration n = 1.0 n L = 2.0 Modern Optics, R. Guenther, p R n sinθ = n sinθ sinθ = i i t i t sinθi 2

11 Longitudinal Aberration cont d n = 1.0 n L = 2.0 Modern Optics, R. Guenther, p Rsinθ t ( ) γ = π θ π θ = θ θ t i i t ( ) sin γ = sin θ θ = ( θ θ ) ( θ θ ) i t R R sinθt + Δz For paraxial rays : ρ R, sinθ θ sinθ θ sin i t i t i i t t ΔZ = ΔZ (θ i )

12 Longitudinal Aberration cont d n = 1.0 n L = 2.0 Modern Optics, R. Guenther, p ΔZ Δz/R ( ρ )

13 Third-order (Seidel) aberrations nn ( 1) + x = + nx + x + O x 2! n 2 3 (1 ) 1 ( ) Paraxial approximation

14 Third-order aberrations : On-axis imaging by a spherical interface Let s start the aberration calculation for a simple case. Q P O h I To the paraxial (first-order) ray approximation, PQI = POI according to Fermat s principle. Beyond a first approximation, PQI POI, thus the aberration at Q is ( ) = ( ) = ( + ) ( + ) a Q PQI POI n n ns n s opd

15 Refraction at a spherical interface cont d α l l β Q R P O s+r φ C β + R α cosφ = sinφ = = α + β s+ R s+ R ( β R) ( s+ R) α φ + φ = = 2 ( s+ R) = β + βr+ R + α sin cos ( ) ( ) s+ R = + 2βR+ R β = s+ R cosφ R Substituting and rearranging we obtain : 2 ( s R) R R( s R) = cosφ 2 2

16 Refraction at a spherical interface cont d O R φ l l ' C α φ s -R β I α β cosφ = sinφ = = ( R + α) + β s R s R α + β sin cos 1 ( s R) φ+ φ = = 2 ( s R) = β + α ( ) = R + 2αR+ α + β = R + 2αR+ s R ( s R) R ( s R) R( s R) α = cosφ = cosφ 2

17 Refraction at a spherical interface cont d Writing the cos φ term in terms of h we obtain : 2 1/ h h h cosφ = 1 sin φ = R 2R 8R where we have used the binomial expansion 2 1/2 x x Substituting into our expressions for and and rearranging ( x) = 1 + ( + ) 2 h R s Rs ( + ) 4 h R s 4Rs ( ) ( ) 2 4 h R s h R s = Rs 4R s Use the same binomial expansion and neglecting terms of 6 order h and higher we obtain + 1/2 1/2 ( + ) ( + ) ( + ) = Rs 8R s 8R s h R s h R s h R s h R s h R s h R s = Rs 8R s 8R s 2 ( ) ( ) ( ) 2

18 Refraction at a spherical interface cont d h h h h h h h = s s 2 Rs 8 R s 8 R s 8 s 4 Rs 8 R s h h h h h h h = s s 2 Rs 8 R s 8 R s 8 s 4 Rs 8 R s ( ) = ( + ) ( + ) a Q n n ns n s h n = s s R s' s' R 4 = ch n2 1 1 n n n n + = s s' R Imaging formula (first-order approx.) 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!

19 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 ( ) = ( ) a Q PQP PBP ( ) = ( ) a O POP PBP ( ) = ( ) ( ) a Q a Q a O opd opd a Q = a Q; r, θ, h' ( ) ( ) O B θ Q r ρ

20 Third-Order Aberration Theory After some very complicated analysis the third-order aberration equation is obtained: 4 ( ) = a Q C r 0 40 Spherical Aberration θ Q + C h r cosθ Coma O r ρ + C h r 2 22 cos θ Astigmatism B + C h r Curvature of Field + 3 3C11h r cosθ Distortion On-axis imaging 에서의 a(q) = ch 4 와일치

21 Spherical Aberration Transverse Spherical Aberration Optics, E. Hecht, p Longitudinal Spherical Aberration a Q = C r = ch ( )

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

23 Spherical Aberration For a thin lens with surfaces with radii of curvature R 1 and R 2, 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 σ R + R s s R R s + s 2 1 = p = h nl n L = σ ( nl + 1) pσ + ( 3nL + 2)( nl 1) p + s h s p 8 f nl( nl 1 ) nl 1 nl 1 Spherical aberration is minimized when : σ = 2 ( nl ) 2 1 n L + 2 p

24 Spherical Aberration Spherical aberration is minimized when : For an object at infinite ( p = -1, n L = 1.50 ), σ ~ 0.7 σ = σ = -1 σ = +1 2 ( nl ) 2 1 n + 2 L R + R σ = R R s s p = s + s p worse better Optics, E. Hecht, p. 222.

25 Spherical Aberration σ σ σ = p R R + R R = s s s + s

26 Coma ( ) 3 a Q = 1C31 h' r cos θ ( h' 0, cosθ 0)

27 Optics, E. Hecht, p Coma

28 Coma Least Coma Modern Optics, R. Guenther, p Most Coma

29 Astigmatism ( ) a Q = 2C22 h' r cos θ ( h' 0, cosθ 0) Optics, E. Hecht, p. 224.

30 Astigmatism Sagittal plane Tangential plane (Meridional plane)

31 Astigmatism Least Astig. Modern Optics, R. Guenther, p Most Astig.

32 Distortion ( ) 3 a Q = 3C11 h' rcos θ ( h' 0, cosθ 0)

33 Field Curvature ( ) 2 2 a Q = 2C20 h' r ( h' 0) Astigmatism when θ = 0. Optics, E. Hecht, p. 228.

34 Chromatic Aberration Optics, E. Hecht, p. 232.

35 Achromatic Doublet Optics, E. Hecht, p. 233.

36 1 1 1 P1D = = ( n1d 1) = ( n1d 1) K1 f1d r11 r P = = ( n 1) = ( n 1) K λ = nm = center of visible spectrum Achromatic Doublets 2D 2D 2D 2 f2d r21 r22 D For a lens separation of L : Chromatic aberration is eliminated when : P n1 n1 = K1 + K2 = λ λ λ In general, for materials with " normal " dispersion : n1 < 0 λ L = + P = P + P LP P f f f f f For a cemented doublet L = 0: = + = ( 1) + ( 1) K2 P P P n K n That means that to eliminate chromatic aberration, K and K must have opposite signs. 1 2 The partial derivative of refractive index with wavelength is approximated : n nf n C λ λ λ F C for an achromat in the visible region of the spectrum. In the above equation, λ = nm and λ = nm. F C

37 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 λ λf λc n1d 1 ( λf λc) V1 K n = K n n n 1 = P 2D 2F 2C 2D 2D 2 2 λ λf λc n2d 1 ( λf λc) V2 V and V are functions only of the material 1 2 properties of the two lenses. P 1D 2D ( λ λ ) V ( λ λ ) + P 0 V2 P1D V1P2D 0 V = + = F C 1 F C 2 Achromatic condition for doublets

38 Achromatic Doublets We can solve for the power of each lens in terms of the desired power of the doublet : V V P = P + P P = P P = P K 1 2 D 1D 2D 1D D 2D D V2 V1 V2 V1 P = K = 1D 2D 1 2 1D 2D ( n 1) ( n 1) P From the values of K and K the radii of curvature for the two surfaces 1 2 of the lenses can be determined. If lens1 is bi - convex with equal curvature for each surface : r r r r r = = = 1 K 2 r 12 r

39 Achromatic Doublets

20. Aberration Theory

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