We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The trajectories turn out to be closed ellipses. We show that this is due to the existence of higher-order invariants that close into a cubic Higgs algebra. The Zernike classical system thus belongs to the class of superintegrable systems. Its Hamilton-Jacobi action separates into three vertical projections of polar coordinates of sphere, polar, and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.
REFERENCES
1.
Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects
, edited by Bagarello
, F.
, Gazeau
, J.-P.
, Szafraniec
,F. H.
, and Znojil
, M.
(John Wiley & Sons
, Hoboken
, 2015
).2.
Bender
, C. M.
, Brody
, D. C.
, and Müller
, M. P.
, “Hamiltonian for the zeros of the Riemann zeta function
,” Phys. Rev. Lett.
118
, 130201
(2017
).3.
Bertrand
, J.
, “Théorème relatif au mouvement d’un point attiré vers un centre fixe
,” C. R. Acad. Sci.
77
, 849
–853
(1873
).4.
Bhatia
, A. B.
and Wolf
, E.
, “On the circle polynomials of Zernike and related orthogonal sets
,” Math. Proc. Cambridge Philos. Soc.
50
, 40
–48
(1954
).5.
Born
, M.
and Wolf
, E.
, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light
, 7th ed. (Cambridge University Press
, 1999
), p. 986
.6.
Cariñena
, J. F.
, Rañada
, M. F.
, and Santander
, M.
, “Two important examples of nonlinear oscillators
,” e-print arXiv:math-ph/0505028.7.
Cariñena
, J. F.
, Perelomov
, A. M.
, and Rañada
, M. F.
, “Isochronous classical systems and quantum systems with equally spaced spectra
,” J. Phys.: Conf. Ser.
87
, 012007
(2007
).8.
Gradshteyn
, I. S.
and Ryzhik
, I. M.
, Table of Integrals, Series, and Products
, 6th ed. (Academic Press
, 2000
).9.
Higgs
, P. W.
, “Dynamical symmetries in a spherical geometry
,” J. Phys. A: Math. Gen.
12
, 309
–323
(1979
).10.
Ismail
, M. E. H.
and Zhang
, R.
, “Classes of bivariate orthogonal polynomials
,” SIGMA
12
, 021
(2016
); e-print arXiv:1502.07256.11.
Kintner
, E. C.
, “On the mathematical properties of the Zernike polynomials
,” Opt. Acta
23
, 679
–680
(1976
).12.
Lukac
, I.
and Smorodinskiĭ
, Ya. A.
, “Wave functions for the asymmetric top
,” Sov. Phys. JETP
30
, 728
–730
(1970
).13.
Lukach
, I.
, “A complete set of the quantum-mechanical observables on a two-dimensional sphere
,” Theor. Math. Phys.
14
, 271
–281
(1973
).14.
Miller
, Jr., W.
, Post
, S.
, and Winternitz
, P.
, “Classical and quantum superintegrability with applications
,” J. Phys. A: Math. Theor.
47
, 423001
(2014
).15.
Myrick
, D. R.
, “A generalization of the radial polynomials of F. Zernike
,” SIAM J. Appl. Math.
14
, 476
–489
(1966
).16.
Olevskiĭ
, M. N.
, “Triorthogonal systems in spaces of constant curvature in which the equation allows a complete separation of variables
,” Mat. Sbornik
27
, 379
–426
(1950
).17.
Pogosyan
, G. S.
, Sissakian
, A. N.
, and Winternitz
, P.
, “Separation of variables and Lie algebra contractions. Applications to special functions
,” Phys. Part. Nuclei
33
(Suppl. 1
), S123
–S144
(2002
).18.
Pogosyan
, G. S.
and Yakhno
, A.
, “Lie algebra contractions and separation of variables on two-dimensional hyperboloidal coordinate systems
,” e-print arXiv 1510.03785 V1 (2015
).19.
Rodrigo
, J. A.
, Alieva
, T.
, and Bastiaans
, T. J.
, “Phase space rotators and their applications in optics
,” in Optical and Digital Image Processing: Fundamentals and Applications
, edited by Cristóbal
, G.
, Schelkens
, P.
, and Thienpont
, H.
(Wiley-VCH Verlag
, 2011
), pp. 251
–271
.20.
Shakibaei
, B. H.
and Paramesran
, R.
, “Recursive formula to compute Zernike radial polynomials
,” Opt. Lett.
38
, 2487
–2489
(2013
).21.
Simon
, R.
and Wolf
, K. B.
, “Structure of the set of paraxial optical systems
,” J. Opt. Soc. Am. A
17
, 342
–355
(2000
).22.
Tango
, W. J.
, “The circle polynomials of Zernike and their application in optics
,” Appl. Phys.
13
, 327
–332
(1977
).23.
Winternitz
, P.
, Lukac
, P.
, and Smorodinskiĭ
, Ya. A.
, “Quantum numbers in the little groups of the Poincaré group
,” Sov. J. Nucl. Phys.
7
, 139
–145
(1968
).24.
Winternitz
, P.
, Smorodinsky
, J.
, and Sheftel
, M.
, “Poincaré and Lorentz invariant expansions of relativistic amplitudes
,” Sov. J. Nucl. Phys.
7
, 785
–792
(1968
).25.
26.
Wünsche
, A.
, “Generalized Zernike or disc polynomials
,” J. Comput. Appl. Math.
174
, 135
–163
(2005
).27.
Zernike
, F.
, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form der phasenkontrastmethode
,” Physica
1
, 689
–704
(1934
).© 2017 Author(s).
2017
Author(s)
You do not currently have access to this content.
