Generation of spherical aberration with axially translating phase plates via extrinsic aberration

We show that spherical aberration of all orders can be generated as an extrinsic aberration in a system of axially translating plates. Some practical examples are provided. In particular for two phase plates that are 10 mm in diameter it is possible to generate from −10 to 10 waves of fourthorder spherical aberration with an axial displacement of +/− 0.65 mm. We also apply the phenomenon of extrinsic aberration for the generation of a conical wavefront and other non-axially symmetric wavefronts, in other words we propose what can be called a generalized zoom plate. ©201 Optical Society of America OCIS codes: (220.0220) Optical design and fabrication; (220.1000) Aberration compensation. References and links 1. L. W. Alvarez and W. E. Humphrey, “Variable power lens and system,” U.S. Patent 3,507,565 (1970). 2. L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. Patent 3,305,294 (1967). 3. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9(7), 1669–1671 (1970). 4. N. López-Gil, H. C. Howland, B. Howland, N. Charman, and R. Applegate, “Generation of third order spherical and coma aberrations by use of radially symmetrical fourth order lenses,” J. Opt. Soc. Am. A 15(9), 2563–2571 (1998). 5. I. A. Palusinski, J. M. Sasián, and J. E. Greivenkamp, “Lateral-shift variable aberration generators,” Appl. Opt. 38(1), 86–90 (1999). 6. T. Hellmuth, A. Bich, R. Börret, A. Holschbach, and A. Kelm, “Variable phase plates for focus invariant optical systems,” Proc. SPIE 5962, 596215 (2005). 7. E. Acosta and S. Bará, “Variable aberration generators using rotated Zernike plates,” J. Opt. Soc. Am. A 22(9), 1993–1996 (2005). 8. B. M. Pixton and J. E. Greivenkamp, “Spherical aberration gauge for human vision,” Appl. Opt. 49(30), 5906– 5913 (2010). 9. E. Acosta and J. Sasián, “Phase plates for generation of variable amounts of primary spherical aberration,” Opt. Express 19(14), 13171–13178 (2011). 10. M. T. Chang and J. Sasian, “Variable spherical aberration generators” Proc. SPIE 3129, 217 (1997). 11. J. P. Mills, T. A. Mitchell, K. S. Ellis, and P. K. Manhart, “Conformal dome aberration correction with counterrotating phase plates,” Proc. SPIE 3705, 201 (1999). 12. J. Sasian, “Extrinsic aberrations in optical imaging systems,” Adv. Opt. Technol. 2(1), 75–80 (2013). 13. P. S. Tsai, B. Migliori, K. Campbell, T. N. Kim, Z. Kam, A. Groisman, and D. Kleinfeld, “Spherical aberration correction in nonlinear microscopy and optical ablation using a transparent deformable membrane,” Appl. Phys. Lett. 91(19), 191102 (2007). 14. E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004). 15. A. Kołodziejczyk and Z. Jaroszewicz, “Diffractive elements of variable optical power and high diffraction efficiency,” Appl. Opt. 32(23), 4317–4322 (1993). 16. Anees Ahmad, “High resonance adjustable mirror mount,” U.S. Patent 4,726,671, (1988). 17. K. Kato, A. Ono, W. Inami, and Y. Kawata, “Plasmonic nanofocusing using a metal-coated axicon prism,” Opt. Express 18(13), 13580–13585 (2010). 18. K. Petelczyc, S. Bará, A. C. Lopez, Z. Jaroszewicz, K. Kakarenko, A. Kolodziejczyk, and M. Sypek, “Imaging properties of the light sword optical element used as a contact lens in a presbyopic eye model,” Opt. Express 19(25), 25602–25616 (2011). 19. B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283(9), 1678–1683 (2010). 20. F. M. Dickey and J. D. Conner, “Annular ring zoom system using two positive axicons,” Proc. Soc. Photo Opt. Instrum. Eng. 8130, 81300B (2011). #198683 $15.00 USD Received 1 Oct 2013; revised 28 Nov 2013; accepted 2 Dec 2013; published 2 Jan 2014 (C) 2014 OSA 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000289 | OPTICS EXPRESS 289


Introduction
After the seminal work of Alvarez [1,2] a variety of papers have been published about the generation of aberration by the relative translation, or rotation of two complementary phase plates [3][4][5][6][7][8][9][10][11]. The basic theory is that the subtraction of the wavefront deformation introduced by one plate from the wavefront introduced by the other plate results in a particular type of aberration.
While the phenomenon of extrinsic (or induced) aberration is discussed in higher order aberration theory [12], it has not been fully exploited. Rather that translating or rotating two phase plates for generating aberration, it also possible to generate aberration by the axial displacement of two phase plates due to extrinsic aberration. Extrinsic aberration results when there is previous aberration incoming to an element that contributes aberration. Extrinsic aberration can be considered as a cross term, or synergy, or the interaction of an aberrated beam with an element that contributes aberration.
We first discuss extrinsic aberrations and then present some application for generating aberration. In particular we highlight the generation of fourth-order spherical aberration which is currently a subject of interest. This paper follows our previous work on phase plates [9]. The system of two phase plates presented here is quite a simple and can be used in ophthalmology, optical testing, microscopy, optical alignment, and other applications [13,14]. While the concept is implemented using refractive optics, it can be also implemented with diffractive optics as well [15].

Extrinsic aberrations
We assume a system of two phase plates in air where a light beam is transmitted and aberrated by the first plate by 1 where 12 ( ) W ρ  is the extrinsic aberration and ρ  is the normalized aperture vector that specifies where a ray intersects the first plate in the system.
The ray error ρ Δ  at the second plate is given by where d is the distance between the plates and a is the physical radius of the plates. A given ray that intersects the first plate at the point defined by aρ  will intersect the second plate at the point defined by ( ) a ρ ρ + Δ   . We assume that the incoming beam is collimated.
The aberration introduced by the second plate will change due to the change of ray position ρ ρ + Δ   . Then we will have 2 ( )  for the aberration introduced by the second plate. The total aberration introduced by both plates is then Let us first analyze the case of having the plates contributing an axially symmetric wavefront deformation in the form Table 1 presents the total aberration when only on term is present at a time. Note that all aberration generated is positive in sign.
d W a ρ Noteworthy is that it is possible to generate a conical wavefront deformation similar to the aberration introduced by an axicon, focus, fourth, sixth, and eight-order spherical aberration as well. The amount of aberration is proportional to the axial displacement d between the plates.
Let us second analyze the case of having the plates contributing a non-axially symmetric wavefront deformation in the form 1 2 Where i  is a unit vector which defines a particular direction to define the aberration symmetry. Table 2 presents the total aberration when only on term is present at a time. The second order terms, focus and astigmatism, contribute also focus and astigmatism respectively as an extrinsic aberration. Coma and line coma produce fourth-order terms. Other aberration forms can be developed using Cartesian coordinates rather than polar coordinates.
The assembly of two phase plates as indicated above could be named as a generalized zoom plate due to the fact that axial displacements provide a variable amount of both axial and non-axial wavefront aberrations.

Spherical aberration
The case of fourth order spherical aberration is of practical interest as there are applications in ophthalmology, optical testing, and optical alignment [8,13,14] where having the ability to introduce a desired amount of spherical aberration would be of value. Here we optimize two phase plates as shown in Fig. 1 for generating fourth-order spherical aberration. One plate is plano concave and the other is plano convex. Both are optically strong. The design of the plates involves also specifying not only cubic, but fourth, and fifth order terms as to obtain as much as possible pure fourth-order spherical aberration, and for having no aberration for a spacing of 0.75 mm. We use a wavelength of 500 nm. The relationship between the wavefront deformation 1 ( ) W ρ  and the surface asphericity is where n is the index of refraction, Cartesian coordinates, and S stands for the aspheric surface coefficients which are presented in Table 3.
The axial displacement range is from 0.1 mm for −10 waves, 0.75 mm for zero waves, and 1.35 mm for + 10 waves. The maximum error in producing fourth-order spherical aberration is less than 0.05 waves at a displacement of 1.35 mm. The material used for the design of the plates is acrylic plastic 1.49 n ≈ .  This follows the relationship for the coma, coma W , generated by an element that introduces spherical aberration, spherical W , when there is a lateral displacement y Δ :

Conical wavefront
The generation of a wavefront deformation in the shape of a cone is of interest. Axicons are optical elements shaped as cones and produce a conical wavefronts. They can be used in a variety of applications from optical alignment to the generation of special beams [17,18].
Recently the control of the light distribution in the focal region has been proposed by using two [19] and three [20] axicons. In this work, the extrinsic aberration would be a linear term proportional to displacement if we generate terms of the form 3/ 2 3/ 2 W ρ according to theory. Figure 2 (Left) shows a system of two plates that generate a conical wavefront as shown in Fig. 2 (Center). The aperture of these plates is 10mm, the surface coefficient is , and the amount of conical wavefront generated is about 1.23 waves per millimeter of axial displacement. Fig. 2. Left, phase plates for generating a conical wavefront; center, wave fan for an axial displacement of 1 mm corresponding to 1.24 waves of wavefront amplitude; Right, point spread function when the beam is focused by a perfect lens. Note the strong plate asphericity. The beam diameter is 10 mm.