STABILITY OF ABERRATION RETRIEVAL METHOD USING SPOT IMAGES

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1 Vol. 2 (202 2 ), No JASCOME STABILITY OF ABERRATION RETRIEVAL METHOD USING SPOT IMAGES ) 2) 3) azuyohi OADA, enji AMAYA and Yuki ONISHI ) ( W8-36, kokada@a.mei.titech.ac.jp) 2) ( W8-36, kamaya@a.mei.titech.ac.jp) 3) ( W8-36, yonihi@a.mei.titech.ac.jp) We have developed the aberration retrieval method uing the pot image. In thi paper, we tudy the tability of the preent aberration retrieval method uing the pot image. At firt, the preent aberration retrieval method i explained. Secondly, the tability indicator for the preent aberration retrieval method i propoed. Finally, the tability indicator are calculated with changing the ome condition like the cell ize of the imaging device, the number of the pot image, the analytical region and the term number of the Zernike coefficient. ey Word : Invere Analyi, Optical Engineering, Aberration Retrieval, Spot Image. (, 2) Shack-Hartmann (3) Shack-Hartmann (4, 5, 6) Zernike Len A Len B Len C x-y z λ

2 a, a : Numerical aperture x, y Fringe Zernike Zernike i Table Relation between the Fringe Zernike index and the Len A a Len B Len C a O z aberration. Index i Aberration 0 Piton Imaging device X-Tilt Piezo actuator d 2 Y-Tilt 3 Defocu Fig. The et up of the optical ytem for the aberration meaurement and the definition of the coordinate ytem. Len A Len B a Len B d 0 z d Len C d (7, 8) g(ξ, η) h(ξ, η; d) f(x, y; d) (9) f(x, y; d) = ξ 2 e i2π λ a (xξ+yη) g(ξ, η)h(ξ, η; d) ξ 2 a2 (ξ 2 + η 2 ) dηdξ. a. h(ξ, η; d) (0). h(ξ, η; d) = exp a () [ i2π d ] a λ 2 (ξ 2 + η 2 ). (2) g(ξ, η) Zernike () (M + ) 0 N g(ξ, η) = A(ξ, η)e i2πφ(ξ,η), (3) A(ξ, η) = Φ(ξ, η) = M α i Z i (ξ, η), (4) i=0 N β i Z i (ξ, η). (5) i= Z i (ξ, η) Zernike α = {α i} β = {β i} Zernike Zernike 4 3rd Atigmaim 0 degree 5 3rd Atigmaim 45 degree 6 3rd Coma 0 degree 7 3rd Coma 90 degree 8 3rd Spherical Zernike α β h(x, y; d) 2.2. I(x, y; d) = f(x, y; d) 2, (6) I(x, y; d) ( I i,j (d) = I(x, y; d) (2) x i x, y j y t ) dxdy, (7) (2) (x, y) = (x) (y) (x) I i,j (d) = i,j ( x w(x, y) = (2) I i,j(d)w(x i xi + δx, y j yj + δy) + c, (8) ) (, y x x (2) t, y y t ) dx dy. (, t) (, t ) (x i, y j) (x i, y j ) w(x, y) (δ x, δ y) c Zernike α β δ x δ y c (9)

3 α, δ x, δ y, c T A T A<ε 2 Table 2 Range of the unknown vector for the tudy of the tability of the aberration retrieval method (L : Lowreolution, H : High-reolution). β Unknown quantity L H α α i (i 0) β β T (P (A T A) - P T ) - β<ε 2 β, β β i (i, 2) Fig. 2 The baic concept of the hyperellipoid and it projection d = d k I k I i,j (d k) I k J J() = k (I k () Īk) 2. (0) = {α T, β T, δ x, δ y, c} T M + N + 4 (0) β = arg min J(). () Gau-Newton Levenberg-Marquardt (2). 3. κ I = {I 0 T, I T, } T δi δ I I I A() = I/ κ = up lim up ɛ 0+ δi ɛ δ δi = up (A T A). (2) A T A T A < ɛ 2 2 (2) κ α δ x, δ y, c β Table 3 δ x, δ y c Lit of the condition for calculating the tability indicator (L : Low-reolution, H : High-reolution). Cell ize (L).85, 2.467, 3.7, 7.4 [µm] (H) 4.8, 29.6, 44.4, 59.2 Number of image (L), 3, 5, 7, 9, (H), 2, 3 Analytical reigion (L) 0, 5, 20 R [µm] (H) 97, 44, 404 Zernike term number (L) 8, 5 M = N (H) 8, 5 (M + N + 4) β N N β = P P R N (M+N+4) = up P (A T A) P T, (3) β 2- ( = up λ max P (AT A) P T). (4) λ max

4 = 2 5 = (b) M = N =5. Fig. 3 The tability indicator uing the low-reolution image in the cae of R = 5 with repect to the cell ize and the number of the image =.850 =2.467 =3.700 = =.850 =2.467 =3.700 = (b) M = N =5. Fig. 4 The tability indicator uing the low-reolution image in the cae of R = 5 with repect to the number of the image. 0.0 M=N=8 M=N= Analyticalregion[µm] R Fig. 5 The tability indicator uing the low-reolution image in the cae of = 7.4, = with repect to the analytical reigion R [nm] a 0.36 a [µm] 0.858[µm] [µm] [µm] [µm] [µm] d d = ±( k) (k = 0,,, ) d = ±( k) (k = 0,,, ) R[µm] R[µm]

5 3 = = Fig. 6 (b) M = N =5. The tability indicator uing the high-reolution image in the cae of R = 44 with repect to the cell ize and the number of the image R Zernike M, N R = 5 4 R = 5 5 = 7.4 = R 6 R = 44 7 R = 44 8 = 29.6 = R R M = N 4.4. = 7.4 R = 5 M = N = 8 δi σ I L δi σ I L δβ σ β σ β σ I L/N δi σ I = δβ σ β 9 Fig = =2 = = =2 =3 Cellize[µm] 0.0 Cellize[µm] (b) M = N =5. The tability indicator uing the high-reolution image in the cae of R = 44 with repect to the cell ize.

6 M=N=8 M=N=5 () :,, 2(200), pp (2) G. Harber, P. J. unt, and G. W. R. Leibbrant: Analyi of lateral hearing interferogram by ue of Zernike polynomial, APPLIED OPTICS, 35(996), pp Fig Analyticalregion[µm] R The tability indicator uing the high-reolution image in the cae of = 29.6, = with repect to the analytical reigion R. σ β [λ] Fig The tandard deviation σ β uing the low-reolution image in cae of = 7.4, R = 5, M = N = 8 with repect to the number of the image. 9 [λ] = 0.05[λ] > 9 5. (3) W.H.Southwell: Wavefront etimation from wavefront lope meaurement, Journal of Optical Society of America, 70(980), pp (4) Maahi UESHIMA, enji AMAYA, and oei ATAOA: Development of Practical Performance Aberration Retrieval Method from Spot Intenity Image uing Invere Analyi, Optical Review, 6(2009), pp (5) azuyohi OADA, enji AMAYA, Maahi UESHIMA, and Yuki ONISHI: Development of Off-Axi Aberration Retrieval Method uing Spot Image, 7th International Conference on Optic-photonic Deign & Fabrication, (200). (6),, :, 23, (200), pp (7) :, CAE, (992),. (8) Deepu Rajan and Subhai Chaudhuri: Simultaneou Etimation of Super-reolved Intenity and Depth Map from Low Reolution Defocued Obervation of a Scene, Proceeding Eighth IEEE International Conference on Computer Viion ICCV 200, (200), pp (9),, :, 6, (202). (0) E. G. William: Fourier Acoutic: Sound Radiation and Nearfield Acoutical Holography, (2005), Springer- Verlag. () M. Born and E. Wolf: Principle of Optic, (974), Pergamon Pre. (2).Maden, H.B.Nielen, and O.Tingleff: METHODS FOR NON-LINEAR LEAST SQUARES PROBLEMS, (2004), Informatic and Mathematical Modelling, Technical Univerity of Denmark.

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