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Optical models and symmetries
Kurt Bernardo Wolf
Instituto de Ciencias Físicas
Universidad Nacional Autónoma de México
Cuernavaca
Selected Topics in
Mathematical Physics
in honour of Natig M. Atakishiyev
Instituto de Matemáticas, UNAM/CuernavacaNovember 30, 2016
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[63] N.M. Atakishiyev and K.B. Wolf, Generalized coherent states for a relativistic
model of the linear oscillator in a homogeneous external field. Reports in
Mathematical Physics 28, 21–27 (1990).
[64] N.M. Atakishiyev, W. Lassner and K.B. Wolf, The relativistic coma aberration.
I. Geometrical optics. Journal of Mathematical Physics 30, 2457–2462 (1989).
[65] N.M. Atakishiyev W. Lassner and K.B. Wolf, The relativistic coma aberration.
II. Helmholtz wave optics. Journal of Mathematical Physics 30, 2463–2468
(1989).
[78] N.M. Atakishiyev, A. Frank and K.B. Wolf, A simple difference realization of
the Heisenberg q-algebra. Journal of Mathematical Physics 35, 3253–3260
(1994).
[79] N.M. Atakishiyev and K.B. Wolf, Approximation on a finite set of points
through Kravchuk functions. Revista Mexicana de F´ısica 40, 366–377 (1994).
[85] N.M. Atakishiyev, Sh.M. Nagiyev and K.B. Wolf, Realization of Sp(2,R) by
finite difference operators: the relativistic oscillator in an external field. Journal of
Group Theory and Its Applications 3, 61–70 (1995).
[86] N.M. Atakishiyev, A. Ronveaux and K.B. Wolf, Difference equation for the
associated polynomials on the linear lattice. Teoreticheskaya i Matematicheskaya
Fizika 106, 76–83 (1996).
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[88] N.M. Atakishiyev, S.M. Chumakov, A.L. Rivera and K.B. Wolf, On the phase
space description of quantum nonlinear dynamics. Physics Letters A 215, 128–
134 (1996).
[91] N.M. Atakishiyev and K.B. Wolf, Fractional Fourier-Kravchuk transform.
Journal of the Optical Society of America A 14, 1467–1477 (1997).
[92] A.L. Rivera, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Evolution under
polynomial Hamiltonians in quantum and optical phase spaces. Physical Review A
55, 876–889 (1997).
[93] N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Wigner distribution function
for finite systems. Journal of Mathematical Physics 39, 6247–6261 (1998).
[98] N.M. Atakishiyev, Sh.M. Nagiyev and K.B. Wolf, On the Wigner distribution
function for a relativistic oscillator, Theoretical and Mathematical Physics 114,
322–334 (1998).
[100] L.M. Nieto, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Wigner
distribution function for Euclidean systems. Journal of Physics A 31, 3875–3895
(1998).
[101] K.B. Wolf N.M. Atakishiyev, S.M. Chumakov, and L.M. Nieto, Wigner
operator and function for various optical systems, Yadernaya Fizika 61, 1828–
1835 (1998); Traducci´on: Physics of Atomic Nuclei 61, 1713–1721 (1998).
[102] N.M. Atakishiyev, E.I. Jafarov, Shakir M. Nagiyev and K.B. Wolf, Meixner
oscillators. Revista Mexicana de F´ısica 44, 235–244 (1998).
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[106] N.M. Atakishiyev, L.E. Vicent and K.B. Wolf, Continuous vs. discrete
fractional Fourier transforms. Journal of Computational and Applied Mathematics
107, 73–95 (1999).
[108] M. Arık, N.M. Atakishiyev and K.B. Wolf, Quantum algebraic structures
compatible with the harmonic oscillator Newton equation, Journal of Physics A 32,
L371–L376 (1999).
[109] S.T. Ali, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, The Wigner
function for general Lie groups and the wavelet transform, Annales Henri Poincaré
1, 685–714 (2000).
[114] N.M. Atakishiyev, Sh.M. Nagiyev, L.E. Vicent and K.B. Wolf, Covariant
discretization of axis-symmetric linear optical systems. Journal of the Optical
Society of America A 17, 2301–2314 (2000).
[117] N.M. Atakishiyev, G.S. Pogosyan, L.E. Vicent and K.B. Wolf, Finite two-
dimensional oscillator. I: The Cartesian model. Journal of Physics A 34, 9381–
9398 (2001).
[118] N.M. Atakishiyev, G.S. Pogosyan, L.E. Vicent and K.B. Wolf, Finite two-
dimensional oscillator. II: The radial model. Journal of Physics A 34, 9399–9415
(2001).
[123] N.M. Atakishiyev, G.S. Pogosyan and K.B. Wolf, Contraction of the finite
one-dimensional oscillator, International Journal of Modern Physics A 18, 317–327
(2003).
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[124] N.M. Atakishiyev, G.S. Pogosyan and K.B. Wolf, Contraction of the finite
radial oscillator, International Journal of Modern Physics A 18, 329–341 (2003).
[128] N.M. Atakishiyev, A.U. Klimyk and K.B. Wolf, Finite q-oscillator, Journal of
Physics A 37, 5569–5587 (2004).
[129] N.M. Atakishiyev, G.S. Pogosyan and K.B. Wolf, Finite models of the
oscillator, Physics of Particles and Nuclei (Fizika Elementarnikh Chastits i
Atomnogo Yadra) Suppl. 3 36, 521–555 (2005).
[140] N.M. Atakishiyev, J. Rueda-Paz and K.B. Wolf, On q-extended eigenvectors
of the integral and finite Fourier transforms, Journal of Physics A 40, 12701–
12707 (2007).
[142] N.M. Atakishiyev, A.U. Klimyk, and K.B. Wolf, Discrete quantum model of
the harmonic oscillator, Journal of Physics A 41, art. 085201, 14p. (2008). Article
selected by IOP Select.
[151] N.M. Atakishiyev, M.R. Kibler and K.B. Wolf, SU(2) and SU(1,1) approaches
to phase operators and temporally stable phase states: Application to mutually
unbiased bases and discrete Fourier transform, Symmetry 2, 1461–1482 (2010),
doi: 10.3390/sym 2031461.
[159] M.K. Atakishiyeva, N.M. Atakishiyev and K.B. Wolf, Kravchuk oscillator
revisited, J. Phys. Conf. Series 512, art. 012031 (2014).
[161] I. Area, N. Atakishiyev, E. Godoy and K.B. Wolf, Bivariate raising and
lowering differential operators for eigenvectors of a 2D Fourier transform, Journal
of Physics A 48, art. 075201 (12 p.) (2015).
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Euclidean group ISO(3)geometric model
wave model
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Euclidean group ISO(3)
Heisenberg-Weyl group
geometric model
wave model
paraxial models
contraction
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Euclidean group ISO(3)
linear symplectic group Sp(2,R)
Heisenberg-Weyl group
geometric model
wave model
paraxial models
thin lenses,small angles;canonical transforms
quadratic extension
contraction
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Euclidean group ISO(3)
linear symplectic group Sp(2,R)
Heisenberg-Weyl group
Fourier group U(2)
geometric model
wave model
paraxial models
thin lenses,small angles;canonical transforms
rotations,gyrations,Fourier transforms
compact subgroup
quadratic extension
contraction
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Euclidean group ISO(3)
linear symplectic group Sp(2,R)
Heisenberg-Weyl group
nonlinear aberration group
Fourier group U(2)
geometric model
wave model
paraxial models
thin lenses,small angles;canonical transforms
rotations,gyrations,Fourier transforms
classification,compositionaberrationless (?)
Fourier transforms
compact subgroup
quadratic extension
contraction
covering algebra
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Euclidean group ISO(3)
4D rotation group SO(4)
linear symplectic group Sp(2,R)
Heisenberg-Weyl group
nonlinear aberration group
Fourier group U(2)
geometric model
wave model
paraxial models
thin lenses,small angles;canonical transforms
rotations,gyrations,Fourier transforms
classification,compositionaberrationless (?)
Fourier transforms
finite data pointspixelated screensrotation, gyration, etc.
grand-mother group
mother groupcontraction
compact subgroup
quadratic extension
contraction
covering algebra
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In the Beginning there was Symmetry…
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In the Beginning there was Symmetry…
The perfect symmetry
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In the Beginning there was Symmetry…
The perfect symmetry
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In the Beginning there was Symmetry…
The perfect symmetry
of empty space…
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In the Beginning there was Symmetry…
The perfect symmetry
of empty space…
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And it was seen that symmetries formed a group
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And it was seen that symmetries formed a group
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And it was seen that symmetries formed a group
and It was called
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The fundamental objects of the world
are determined by their symmetries
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The fundamental objects of the world
are determined by their symmetries
the z axis
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The fundamental objects of the world
are determined by their symmetries
the z axis
the x – y plane
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The rays in geometric optics:
manifold of rays (cosets) symmetry
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The rays in geometric optics:
The δ-planes of polychromatic wave optics
manifold of rays (cosets)
manifold of planes (cosets)
symmetry
symmetry
+ Hilbert spaces+ Fourier transform
Helmholtz equation
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Postulates of geometry + dynamics
↔ Conserved quantities under linear canonical tfmns
Position (continuity)
Momentum (refract law)
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Geometric optics on phase space –results
● Hamilton equations● canonical transforms● factorization of refraction
into 2 canonical root tmns
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Results in Helmholtz (monochromatic) optics
plane ↔ sphere
Wave transform with value and normal derivative on a screen:
For example,:
Finding the
Helmholtz wavefield
of minimal energy
that passes through
a finite number of
data points.
Hilbert space with non-local measure:
Relativistic transformations (with Atakishiyev and Lassner 1994)…
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The relativistic coma aberration
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The relativistic coma aberrationgeometric:
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The relativistic coma aberrationgeometric:
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The relativistic coma aberrationgeometric:
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The relativistic coma aberration
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The relativistic coma aberrationHelmholtz wavefield
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The relativistic coma aberrationHelmholtz wavefield
Generator of relativistic boosts
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The relativistic coma aberrationHelmholtz wavefield
Generator of relativistic boosts
`reduced’ problem: exponentiate 1+2+3 order diff ops
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The relativistic coma aberrationHelmholtz wavefield
Thesis work, Cristina Salto-Alegre
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The relativistic coma aberrationHelmholtz wavefield
Thesis work, Cristina Salto-Alegre
Helmholtz wavefields on screen ↔ function on the sphere
narrowest image on the screen ↔ Bessel function J0(kr)
Translations multiply by phases:
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The relativistic coma aberrationHelmholtz wavefield
Helmholtz wavefields on screen ↔ function on the sphere
narrowest image on the screen ↔ Bessel function J0(kr)
Boosts deform the sphere with a multiplier:
Thesis work, Cristina Salto-Alegre
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The relativistic coma aberrationHelmholtz wavefield
Helmholtz wavefields on screen ↔ function on the sphere
narrowest image on the screen ↔ Bessel function J0(kr)
Boosts deform the sphere with a multiplier:
Thesis work, Cristina Salto-Alegre
integrate!
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Contraction: Euclidean → Heisenberg-Weyl
let ε→0 and rename
to have the Heisenberg-Weyl algebra + rotations
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Quadratic extension of the Heisenberg-Weyl algebra
generators:
when with matrix
we obtain the six symplectic conditions:
that include:
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The u(2)F Fourier subalgebra:
4D rotations:
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The u(2)F Fourier subalgebra:
4D rotations:
anisotropic Fourier:
isotropic Fouirer:
gyrations :
rotations:
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The Sp(2D,R) of integral
linear canonical transforms:
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The Sp(2D,R) of integral
linear canonical transforms:
phase & norm:
integral kernel:
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The Sp(2D,R) of integral
linear canonical transforms:
phase & norm:
integral kernel:
¡ Ahh ! …and the covering metaplectic Mp(2D,R) sign
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The algebra and group of
aberrations
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The algebra and group of
aberrations
of rank weight
nonlinear action:
group composition to a rank
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The Euclidean mother group ISO(3)
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The Euclidean mother group ISO(3)is itself a contraction
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The Euclidean mother group ISO(3)is itself a contraction of itsgrand-mother group: SO(4) of 4D rotations
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The Euclidean mother group ISO(3)is itself a contraction of itsgrand-mother group: SO(4) of 4D rotations
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In the 1D finite oscillator:
position: momentum:mode:
Kravchuk functionorthogonal & complete on 2j+1 points
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And there occurs an accident:
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And there occurs an accident:
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And there occurs an accident:
Not only that, but:
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And there occurs an accident:
Not only that, but:
Cartesian basis polar basis
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Hermite-G-Kravchuk
Laguerre-Grad-Kravchuk
Clebsch-Gordan
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Hermite-G-Kravchuk
Laguerre-Grad-Kravchuk
Clebsch-Gordan
← UNITARY →
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On 2D pixellated
RECTANGULARscreens
The Cartesian basis…
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The Cartesian basisis orthogonaland complete…
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We can define the
“Laguerre-Kravchuk” modeson rectangular screens
MSc Thesis of
Alejandro R. Urzúa
JOSAA 33,643 (2016)
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The complete mode-angularmomentumbasis
rectangular
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In 2D we thus have the : Fourier group on pixellated screens,
Fourier transformations (domestic)
gyrations (imported)
rotations (imported)
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These ensue fromthe rotations ofall modes
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Virtue:no info is lostSin:it is the
slowest !
These ensue fromthe rotations ofall modes
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Gyrations ofall modes:
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Pending matters…
Pixellations distinctfrom Cartesian or polarwith their own modes…
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Pending matters…
Pixellations distinctfrom Cartesian or polarwith their own modes…
Unitary aberrations in 2 dimensions
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Pending matters…
Pixellations distinctfrom Cartesian or polarwith their own modes…
Unitary aberrations in 2 dimensions
Zernikes, superintegrability !!!
criptographycoloured screensentangled pixelated states…and other problems found on the road…
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Muchas Gracias
desde Cuernavaca…