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 . The nodal characteristic of coma field for a pupil decentered system is shown in Fig. 2, and the corresponding full field map is shown in Fig. 3.

Fig. 2. The nodal characteristic of coma field for a pupil decentered optical system: the node lies on the line along the vector direction of pupil decentration.

0.0500

-0.0000

-0.0500

-0.1000 Y Field Angle in Object Space - degrees

-0.1500

-0.1500 -0.1000 -0.0500 -0.0000 0.0500 0.1000 0.1500 X Field Angle in Object Space - degrees

Fig. 3. Full field display of a pupil decentered system for third-order coma.

3.3 Astigmatism The third group stands for astigmatism.  1 2 2 2 W= W222 H + W131 sH +2 W040 s  ⋅ ρ . (13)  2  It is shown that astigmatism, coma and spherical aberration in the rotationally symmetric system induce second-order, first-order and constant items of astigmatism in the new nonsymmetric system, respectively. A special condition is considered at first. When the rotationally symmetric system is astigmatism-free, then after decentering the pupil, linear or constant astigmatism appears in the new system. The concrete astigmatism field form depends on the magnitude of spherical aberration, coma and pupil decentration.

WhenW222 ≠ 0 , Eq. (13) leads to: 2 1  2 2W131 s 4 W040 s  2 WWHH=222  + + ⋅ ρ 2  WW222 222  2 (14)  W2 − 4 W W s 2  1 W131 s  ( 131 040 222 ) 2 =WH222  +  −  ⋅ ρ . 2 WW 2 222  222  Define W131 s aa ≡ − , (15) W222 and

2 WWW131− 4 040 222 ba ≡ 2 s , (16) W222 then third-order astigmatism in Eq. (14) is written as:

1 2 W= W H − a − b 2 ⋅ ρ 2 . (17) 2 222 ()a a 

#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 11656 Equation (17) reveals there are two nodal points for third-order astigmatism field H= aa ± b a . (18) From Eq. (16) and Eq. (18), the locations of the two nodal points depend on the sign of the 2 item (WWW131− 4 040 222 ) . They may be located on the line along the direction of vector s when 2 (WWW131 − 4040 222 ) > 0 , or at two points that are symmetric about the line when 2 (WWW131 − 4040 222 ) < 0 ,as shown in Fig. 4. The corresponding full field maps which illustrate these two conditions are shown in Fig. 5.

Fig. 4. Locations of the two nodal points of astigmatism field in a pupil decentered optical system. (a) On the line along the direction of pupil decentration vector. (b) Symmetric about the pupil vector direction line.

0.1500 . 0.1500 0.1000 0.1000

0.0500 0.0500 .. .

-0.0000 . -0.0000

-0.0500 -0.0500

-0.1000 -0.1000 Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees - Space Object in Angle Field Y

-0.1500 -0.1500

-0.1500 -0.1000 -0.0500 -0.0000 0.0500 0.1000 0.1500 -0.1500 -0.1000 -0.0500 -0.0000 0.0500 0.1000 0.1500 X Field Angle in Object Space - degrees X Field Angle in Object Space - degrees Fig. 5. Full field maps of two pupil decentered systems which correspond to the two different nodal positions as shown in Fig. 4 (a) and (b).

3.4 Defocus and field curvature The fourth group stands for defocus and field curvature.

#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 11657 W= W( H ⋅ H ) + 2 W s ⋅ H + W + 4 W s2  ()ρ⋅ ρ  220M 131 ( ) ( 020 040 )   2  2W (W020+ 4 W 040 s ) =W( H ⋅ H) +131 s ⋅ H +  ρ⋅ ρ (19) 220M () ()  WW   220M 220M  2 2 2  WW W020 W 220+ 4 W 040 W 220 s− W131 s  =W()() H +131 s ⋅ H +131 s + MM  ()ρ⋅ ρ . 220M WWW 2  220MM220 220M  This also brings the same conclusion in [3], i.e. there is a longitudinal focal shift apart from the decenter of the vertex of the quadratic medial surface. 3.5 Distortion The fifth group is the distortion item. 2 2 2 ∗ 2WsWH020+ 011 +4 Wss040 + 2 WsHWsH131 + 131  W =   ⋅ ρ . (20) +W H2 s∗ +2 W ( H ⋅ H ) s + W ( H ⋅ H ) H   222 220M 311  Since distortion doesn’t affect the imaging quality and can often be corrected via image procession, it is not discussed here. But from Eq. (20), it can inferred that distortion field has a three-node characteristic. 4. Conclusions We demonstrate the aberration fields of pupil decentered optical systems through third order in this paper. By describing the off axis factor with a pupil decentration vector in the aberrations expansion, the optical system excluding the pupil is manipulated as a whole. The aberration coefficients we get do not contain parameters of individual surfaces, and they are only functions of the pupil decentration vector and the system aberration coefficients of the rotationally symmetric system, so the aberration field of a pupil decentered optical system can be inferred from the original rotationally symmetric form. This work and its follow-up into higher-order optical aberrations are applicable to the design and analysis of off-axis systems that are formed by pupil decentration. Acknowledgments The work was supported by the Changjitu Special Cooperation Foundation of Jilin Province and Chinese Academy of Sciences (grants 2011CJT0004).