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Third-Order Aberration Fields of Pupil Decentered Optical Systems

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Third-Order Aberration Fields of Pupil Decentered Optical Systems Third-order aberration fields of pupil decentered optical systems Jian Wang,1,2,* Banghui Guo,1,2 Qiang Sun,1 and Zhenwu Lu1 1Opto-electronic Technology Center, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin, 130033, China 2Graduate School of the Chinese Academy of Sciences, Beijing, 100039, China *[email protected] Abstract: By introducing a transformed pupil vector into the aberration expansions of an axially symmetric optical system, the aberration coefficients through third order of a pupil decentered off-axis optical system are obtained. Nodal aberration characteristics are revealed only by means of the pupil decentration vector and the aberration coefficients of the axially symmetric system, which shows great convenience since parameters of individual surface such as radius of curvature, decenter as well as the shifted center of the aberration field are not used in the analysis. ©2012 Optical Society of America OCIS codes: (080.1005) Aberration expansions; (080.2740) Geometric optical design. References and links 1. R. V. Shack, “Aberration theory, OPTI 514 course notes,” College of Optical Sciences, University of Arizona, Tucson, Arizona. 2. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980). 3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). 4. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976). 5. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950). 6. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express 16(20), 15655–15670 (2008). 1. Introduction The theory about aberration fields of nonsymmetric optical systems was developed by Shack [1] and Thompson [2,3] in 1970s, based on the work of Buchroeder [4]. In a nonsymmetric optical system, third-order aberrations are still the sum of the individual surface contributions just like in an axially symmetric system [4]. Each surface has its shifted center of the aberration field from the perturbed Gaussian image center, which is often denoted by a vector σ j . The total field of each aberration type is a function of the vector σ j and the aberration coefficient of individual surfaces. This theory is applicable to all types of off-axis optical systems. In reflective optical systems, a principal application of off-axis formation is to avoid the obstruction of the primary aperture by the secondary optical element. One of the off axis methods is to decenter the pupil. Although aberrations of such systems can undoubtedly be analyzed with the available vectorial aberration theory, it relies on the detailed parameters of all the optical elements. In this paper we describe the third-order aberration fields of pupil decentered optical systems by considering these optical elements as a whole excluding the pupil (equally the aperture stop). The aberration fields are thus independent of the parameters of individual elements. 2. Vector-form aberration expansion of pupil decentered systems The wave aberration expansion of surface j in a vector form can be written as [3]: #165319 - $15.00 USD Received 23 Mar 2012; revised 19 Apr 2012; accepted 23 Apr 2012; published 7 May 2012 (C) 2012 OSA 21 May 2012 / Vol. 20, No. 11 / OPTICS EXPRESS 11652 ∞ ∞ ∞ p n m Wj = ∑∑∑()()()(),WHHHklm j ⋅ρ ⋅ ρ ⋅ ρ p n m (1) k=2 p + m , l = 2 n + m , where H is the normalized field vector and ρ is the normalized aperture vector. The indices p , m , n are integer numbers, k and l are the power of H and ρ , respectively. Fig. 1. Pupil vector relation before and after pupil decentration. When it comes to the problem in this paper, only the pupil is decentered while the other elements hold their common axis. As shown in Fig. 1, the relation between the new pupil coordinate and the old one is ρ′ = ρ + s , (2) where s is the normalized pupil decentration vector. For simplicity but without loss of generality, we make s in the y axis direction. The vector-form aberration expansion for a pupil decentered system can be modified as: ∞ ∞ ∞ m W W H Hp s sn H s j= ∑∑∑()()()()(). klm j ⋅[ ρ + ⋅ ρ +] ⋅ ρ + (3) p n m The total aberration field is simply the sum of individual surfaces: WW= ∑ j . (4) j According to Eq. (3) and Eq. (4), the aberration function of a pupil decentered system through third order is given as: 2 W=∑ W020 j ()()ρ + s ⋅ ρ + s +∑ W111 j H ⋅()()()ρ + s +∑ W040 j ρ+ s ⋅ ρ + s j j j 1 2 +W H ⋅ρ + s ρ + s ⋅ ρ + s + W H2 ⋅ρ + s (5) ∑ 131 j ()()() ∑ 222 j () j 2 j + W( H⋅ H) ()()ρ + s ⋅ ρ + s + W() H ⋅ H H ⋅()ρ + s , ∑ 220M j ∑ 311 j j j in which W , the medial astigmatic component proposed by Hopkins [5], is used: 220M 1 WWW= + . (6) 220M 2202 222 #165319 - $15.00 USD Received 23 Mar 2012; revised 19 Apr 2012; accepted 23 Apr 2012; published 7 May 2012 (C) 2012 OSA 21 May 2012 / Vol. 20, No. 11 / OPTICS EXPRESS 11653 Because the vector s is independent of surface indices, the aberration coefficients expansion in Eq. (5) can be further arranged. The aberrations of individual surfaces can be added together directly, which is the same state as in an axially symmetric system. Expand Eq. (5) by using the vector multiplication operation [3], and the aberration function changes to: 2 WW=020(ρ ⋅ ρ) +2W 020( ρ ⋅s) + W 020 s +W111()() H ⋅ρ + W 111 H ⋅ s W()ρ⋅ ρ2 + W s4 +2 W s 2 ⋅ ρ 2 + 4 W s2 ()ρ ⋅ ρ + 040 040 040 040 2 +4W040 ()()() s ⋅ρ ρ ⋅ ρ + 4W040 s s ⋅ ρ 2 2 WH131 ()⋅ρ() ρ ⋅ ρ +2W131 s()() H ⋅ρ + W131 sH ⋅ ρ + (7) +2W s ⋅ Hρ ⋅ ρ + W s2 s ⋅ H +W s2 H ∗ ⋅ ρ 131 ()() 131 () 131 () 1 2 21 2 2 2 ∗ +W222 ()H ⋅ρ + W222 ()() H ⋅ s + W222 H s ⋅ ρ 2 2 2 +W220 () HH ⋅()ρ ⋅ ρ + WsHHW220 ( ⋅ ) + 2220 ( HHs ⋅ )( ⋅ ρ ) M M M +W311 ( H ⋅ H )( H ⋅ρ ) + W311 ( H ⋅ H )( H ⋅ s ) , where s2 = s ⋅ s is a constant scalar caused by pupil decentration, s2 = ss is a square vector manipulated by vector multiplication operation, WW= is the aberration coefficient klm ∑( klm ) j j of the optical system with axial symmetry. Each aberration group listed in Eq. (7) is induced by one type of optical aberration, in sequence as defocus, tilt, spherical, coma, astigmatism, field curvature and distortion. 3. Aberration function and field characteristics The aberration coefficients can be grouped according to their dependence on the power of the aperture vector as in [6]. In this way, the aberration expansion in Eq. (7) for a pupil decentered system can be grouped as: 2 WW=040 ()ρ ⋅ ρ + (4W040s+ W 131 H ) ⋅ρ() ρ ⋅ ρ 1 2 2 2 + W222 H + W131 sH +2 W040 s ⋅ ρ 2 +W( H ⋅ H) + 2 W s ⋅ H + W + 4 W s2 ()ρ⋅ ρ 220M 131 () ( 020 040 ) (8) 2 2 2 ∗ 2WsWH020+ 111 +4 Wss040 + 2 WsHWsH131 + 131 + ⋅ ρ +W H2 s∗ +2 W ( H ⋅ H ) s + W ( H ⋅ HH) 222 220M 311 W s2 + W H ⋅ s + W s4 + W s2 s ⋅ H 020 111 () 040 131 () + . 1 2 2 2 +W222 Hs ⋅ + W220 sHHWHHHs( ⋅ ) +311 ( ⋅ )( ⋅ ) 2 () M A similar result was also given by Shack in his course notes [1]. Because piston is not a true optical aberration and it’s often neglected in aberration analysis, attention will be paid to the five monochromatic optical aberrations, especially to coma and astigmatism. #165319 - $15.00 USD Received 23 Mar 2012; revised 19 Apr 2012; accepted 23 Apr 2012; published 7 May 2012 (C) 2012 OSA 21 May 2012 / Vol. 20, No. 11 / OPTICS EXPRESS 11654 3.1 Spherical aberration The first item in Eq. (8) is third-order spherical aberration. It can be seen when an optical system becomes off axis by decentering the pupil, spherical aberration doesn’t change. This can also be concluded from the available theory by Thompson [3]. 3.2 Coma The second group is third-order coma. W=(4 W040 s + W 131 H ) ⋅ρ() ρ ⋅ ρ . (9) From Eq. (9) we can find an interesting property about third-order coma. When a symmetric system does not have spherical aberration, coma will not change with the decentration of the pupil. This is just like the coma property of a spherical-aberration-free system when the aperture stop is shifted axially. When the symmetric system is coma-free but has residual spherical aberration, the system with pupil decentration will demonstrate a constant coma, and the coma is linearly dependent on the pupil decentration magnitude. When neither W040 nor W131 equals 0, Eq. (9) can be described as: 4W040 W= W131 H + s ⋅ρ() ρ ⋅ ρ . (10) W131 Define a vector 4W040 ac ≡ − s , (11) W131 and third-order coma can be shown as W= W131 ( H − ac ) ⋅ρ() ρ ⋅ ρ . (12) Equation (12) gives the usual characteristic field behavior of coma in a pupil decentered optical system. A node exists away from the center of the Gaussian image, and the displacement relates to the ratio of spherical aberration and coma of the original symmetric system, as well as the pupil decentration magnitude. According to Eq. (11), the coma node lies on the line along the vector direction of s .
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