Vol. 2 (202 2 ), No. 4-222 JASCOME STABILITY OF ABERRATION RETRIEVAL METHOD USING SPOT IMAGES ) 2) 3) azuyohi OADA, enji AMAYA and Yuki ONISHI ) ( 52-8552 2-2--W8-36, E-mail: kokada@a.mei.titech.ac.jp) 2) ( 52-8552 2-2--W8-36, E-mail: kamaya@a.mei.titech.ac.jp) 3) ( 52-8552 2-2--W8-36, E-mail: 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) 202 0 2 202 5 Zernike 2. 2.. 3 Len A Len B Len C x-y z λ
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)
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 0 90 0 22 42 α i (i 0) 0 0 0 0 β β T (P (A T A) - P T ) - β<ε 2 β, β 2 0 0 β i (i, 2) Fig. 2 The baic concept of the hyperellipoid and it projection. 2.3. 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 3 7 3 7 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. 4..
= 2 5 =3 6 0.05 9 7 5 3 0.0 9 7 5 3 0.09.850 2.467 3.700 7.400.850 2.467 3.700 7.400 (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. 0.0 0.2 0.0 0.08 =.850 =2.467 =3.700 =7.400 3 5 7 9 =.850 =2.467 =3.700 =7.400 3 5 7 9 (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=5 0 5 20 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. 4.2. 650[nm] a 0.36 a 0.09 8 [µm] 0.858[µm] [µm] [µm] [µm] [µm] d d = ±(20000 500k) (k = 0,,, ) d = ±(20000 2500k) (k = 0,,, ) R[µm] R[µm]
3 = 0.05 3 =0. 0.5 0.2 0.25 2 0.07 0.08 0.09 2 0.3 0.35 0.4 0.45 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. 000 2 R Zernike M, N 3 4.3. 3 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 = 0.707 δβ σ β 9 Fig. 7 0.0 0.08 = =2 =3 0.5 0.4 0.3 0.2 0. = =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.
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