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New Separated Polynomial Solutions to the Zernike System on the Unit Disk and Interbasis Expansion

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New Separated Polynomial Solutions to the Zernike System on the Unit Disk and Interbasis Expansion 1844 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion 1,2,3 4, 1 GEORGE S. POGOSYAN, KURT BERNARDO WOLF, * AND ALEXANDER YAKHNO 1Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico 2Yerevan State University, Yerevan, Armenia 3Joint Institute for Nuclear Research, Dubna, Russia 4Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico *Corresponding author: [email protected] Received 23 May 2017; revised 7 August 2017; accepted 22 August 2017; posted 23 August 2017 (Doc. ID 296673); published 19 September 2017 The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solution bases involving Legendre and Gegenbauer polynomials in nonorthogonal coordinates, close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch–Gordan coefficients. © 2017 Optical Society of America OCIS codes: (000.3860) Mathematical methods in physics; (050.1220) Apertures; (050.5080) Phase shift; (080.1010) Aberrations (global); (350.7420) Waves. https://doi.org/10.1364/JOSAA.34.001844 1. INTRODUCTION: THE ZERNIKE SYSTEM to explain the symmetry hidden in the equal spacing of n familiar from the oscillator model. In 1934, Frits Zernike published a paper that gave rise to phase- ’ contrast microscopy [1]. That paper presented a differential In Refs. [8,9] we have interpreted Zernike s Eq. (1) as defin- ing a classical and a quantum system with a nonstandard equation of second degree to provide an orthogonal basis of “ ” − 1 Zˆ polynomial solutions on the unit disk to describe wavefront Hamiltonian 2 . This turns out to be interesting, because aberrations in circular pupils. This basis was also obtained in the classical system the trajectories turn out to be closed in Ref. [2] using the Schmidt orthogonalization process, as ellipses, and in the quantum system, this Hamiltonian partakes ’ in a cubic Higgs superintegrable algebra [10]. its authors noted that the reason to set up Zernike s differential “ ” equation had not been clearly justified. The two-dimensional The key to solve the system was to perform a vertical map D r x;y differential equation in r x;y that Zernike solved is from the disk in to a half-sphere in three-space ~r x;y;z, to be indicated as H ≔ fj~rj1;z ≥ 0g.On 2 2 ZbΨ r ≔ ∇ − r ∇ − 2r ∇ Ψ r −EΨ r ; H the issue of separability of solutions becomes clear: the · · (1) 1 orthogonal spherical coordinate system ϑ; φ, ϑ ∈ 0; 2 π, φ ∈ 2 on the unit disk D ≔ fjrj ≤ 1g and Ψ r ∈ L D (once two −π; π on H, projects on the polar coordinates r;ϕ of D. parameters had been fixed by the condition of self-adjointness). But as shown in Fig. 1, the half-sphere can also be covered with The solutions found by Zernike are separable in polar coordi- other orthogonal and separated coordinate systems (i.e., those nates r;ϕ, with Jacobi polynomials of degrees nr in the radius whose boundary coincides with one fixed coordinate): where r ≔ jrj times trigonometric functions eimϕ in the angle ϕ. The the coordinate poles are along the x axis and the range of spheri- 0 0 solutions are thus classified by nr ;m, which add up to non- cal angles is ϑ ∈ 0; π and φ ∈ 0; π. Since the poles of the negative integers n 2nr jmj, providing the quantized spherical coordinates can lie in any direction of the x;y plane eigenvalues En n n 2 for the operator in Eq. (1). and rotated around them, we take the x-axis orientation as rep- The spectrum nr ;m or n; m of the Zernike system is ex- resenting the whole class of new solutions, which we identify by actly that of the two-dimensional quantum harmonic oscillator. the label II, to distinguish them from Zernike’s polar-separated This evident analogy with the quantum oscillator spectrum has solutions, which will be labeled I. H been misleading, however. Two-term raising and lowering op- The coordinate system II is orthogonal on but projects erators do not exist; only three-term recurrence relations have on nonorthogonal ones on D; the new separated solutions con- been found [3–7]. Beyond the rotational symmetry that ex- sist of Legendre and Gegenbauer polynomials [9]. Of course, plains the multiplets in fmg, no Lie algebra has been shown the spectrum fEng in Eq. (1) is the same as in the coordinate 1084-7529/17/101844-05 Journal © 2017 Optical Society of America Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A 1845 system I. Recall also that coordinates that separate a differential equation lead to extra commuting operators and constants of the motion. In this paper, we proceed to find the I–II interbasis expansions between the original Zernike and the newly found solution bases; its compact expression in terms of su(2) Clebsch–Gordan coefficients certainly indicates that some kind of deeper symmetry is at work. The solutions of the Zernike system [1] in the new coordi- I II I nate system, which we indicate by ϒ and ϒ on H, and Ψ and ΨII on D, are succinctly derived and written out in Section 2. In Section 3, we find the overlap between them, add some remarks in the concluding Section 4, and reserve for Appendix A some special-function developments. 2. TWO COORDINATE SYSTEMS, TWO FUNCTION BASES The Zernike differential Eq. (1)inr x;y on the disk D can be “elevated” to a differential equation on the half-sphere H in Fig. 1. Top row: orthogonal coordinate systems that separate on the ~ H D Fig. 1 through first defining the coordinates ξ ξ1; ξ2; ξ3 by half-sphere . Bottom row: their vertical projection on the disk . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Left: spherical coordinates with their pole at the z axis; separated 2 2 ξ1 ≔ x;ξ2 ≔ y; ξ3 ≔ 1 − x − y ; (2) solutions will be marked by I. Right: spherical coordinates with their pole along the x axis, whose solutions are identified by II. The latter H D then relating the measures of and through maps on nonorthogonal coordinates on the disk that also separate dξ dξ dxdy d2r solutions of the Zernike equation. d2S ξ~ 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 (3) ξ3 1 − x − y 1 − jrj 2 2 V W r ≔ − r ∕8 1 − r , which has the form of a repulsive and the partial derivatives by ∂x ∂ξ − ξ1∕ξ3∂ξ 1 3 oscillator constrained to −1; 1, whose rather delicate and ∂y ∂ξ − ξ2∕ξ3∂ξ . 2 3 boundary conditions were addressed in Ref. [9]. H D ξ~ A. Map Between and Operators The coordinates can be now expressed in terms of the two mutually orthogonal systems of coordinates on the sphere [11], Due to the change in measure (3), the Zernike operator on D, b as shown in Fig. 1: Z in Eq. (1), must be subject to a similarity transformation by 2 ~ 2 the root of the factor between d S ξ and d r; thus we define System I∶ ξ1 sin ϑ cos φ; ξ2 sin ϑ sin φ; the Zernike operator on the half-sphere H and its solutions as π∕2 π ξ3 cos ϑ; ϑj0 ; φj−π; (8) Wb ≔ 1−21∕4Zb 1 − jrj2−1∕4; 0 0 0 System II∶ ξ1 cos ϑ ; ξ2 sin ϑ cos φ ; ϒ ξ~ ≔ 1 − jrj21∕4Ψ r: (4) 0 0 0 π 0 π ξ3 sin ϑ sin φ ; ϑ j0; φ j0: (9) In this way the inner products required for functions on the disk and on the sphere are related by In the following, we succinctly give the normalized solutions Z for the differential Eq. (6) in terms of the angles for H in 0 2 0 Ψ; Ψ D ≔ d rΨ r Ψ r the coordinate systems I and II, and their projection as wave- D ZD fronts on the disk of the optical pupil. The spectrum of n; m 2 ~ ~ 0 ~ 0 quantum numbers that classify each eigenbasis, and d S ξϒ ξ ϒ ξ ≕ ϒ; ϒ H : (5) n ;n H 1 2, will indeed be formally identical with that of the two-dimensional quantum harmonic oscillator in polar and Perhaps rather surprisingly, the Zernike operator Wb in Eq. (4) ~ Cartesian coordinates, respectively. on ξ ∈ H has the structure of (−2 times) a Schrödinger Hamiltonian, B. Solutions in System I [Eq. (8)] ξ2 ξ2 ’ b ~ 1 2 ~ ~ Zernike s differential Eq. (1) is clearly invariant under rotations W ϒ ξ ΔLB 2 1 ϒ ξ−Eϒ ξ; (6) Wb 4ξ3 around the center of the disk, corresponding to rotations of in Eq. (6) around the ξ3 axis of the coordinate system (8) on the – which is a sum of the Laplace Beltrami operator sphere. Written out in those coordinates, it has the form of a Δ Lˆ 2 Lˆ 2 Lˆ 2 LB 1 2 3, where Schrödinger equation, Lˆ ≔ ξ ∂ − ξ ∂ ; Lˆ ≔ ξ ∂ − ξ ∂ ; Lˆ ≔ ξ ∂ − ξ ∂ I 2 I 1 3 ξ2 2 ξ3 2 1 ξ3 3 ξ1 3 2 ξ1 1 ξ1 1 ∂ ∂ϒ ϑ; φ 1 ∂ ϒ ϑ; φ sin ϑ (7) sin ϑ ∂ϑ ∂ϑ sin2θ ∂φ2 1 are the generators of a formal so(3) Lie algebra.
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