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

[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).

[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).

[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).

[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).

Euclidean group ISO(3)geometric model

wave model

Euclidean group ISO(3)

Heisenberg-Weyl group

geometric model

wave model

paraxial models

contraction

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

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

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

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

In the Beginning there was Symmetry…

In the Beginning there was Symmetry…

The perfect symmetry

In the Beginning there was Symmetry…

The perfect symmetry

In the Beginning there was Symmetry…

The perfect symmetry

of empty space…

In the Beginning there was Symmetry…

The perfect symmetry

of empty space…

And it was seen that symmetries formed a group

And it was seen that symmetries formed a group

And it was seen that symmetries formed a group

and It was called

The fundamental objects of the world

are determined by their symmetries

The fundamental objects of the world

are determined by their symmetries

the z axis

The fundamental objects of the world

are determined by their symmetries

the z axis

the x – y plane

The rays in geometric optics:

manifold of rays (cosets) symmetry

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

Postulates of geometry + dynamics

↔ Conserved quantities under linear canonical tfmns

Position (continuity)

Momentum (refract law)

Geometric optics on phase space –results

● Hamilton equations● canonical transforms● factorization of refraction

into 2 canonical root tmns

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)…

The relativistic coma aberration

The relativistic coma aberrationgeometric:

The relativistic coma aberrationgeometric:

The relativistic coma aberrationgeometric:

The relativistic coma aberration

The relativistic coma aberrationHelmholtz wavefield

The relativistic coma aberrationHelmholtz wavefield

Generator of relativistic boosts

The relativistic coma aberrationHelmholtz wavefield

Generator of relativistic boosts

`reduced’ problem: exponentiate 1+2+3 order diff ops

The relativistic coma aberrationHelmholtz wavefield

Thesis work, Cristina Salto-Alegre

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:

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

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!

Contraction: Euclidean → Heisenberg-Weyl

let ε→0 and rename

to have the Heisenberg-Weyl algebra + rotations

Quadratic extension of the Heisenberg-Weyl algebra

generators:

when with matrix

we obtain the six symplectic conditions:

that include:

The u(2)F Fourier subalgebra:

4D rotations:

The u(2)F Fourier subalgebra:

4D rotations:

anisotropic Fourier:

isotropic Fouirer:

gyrations :

rotations:

The Sp(2D,R) of integral

linear canonical transforms:

The Sp(2D,R) of integral

linear canonical transforms:

phase & norm:

integral kernel:

The Sp(2D,R) of integral

linear canonical transforms:

phase & norm:

integral kernel:

¡ Ahh ! …and the covering metaplectic Mp(2D,R) sign

The algebra and group of

aberrations

The algebra and group of

aberrations

of rank weight

nonlinear action:

group composition to a rank

The Euclidean mother group ISO(3)

The Euclidean mother group ISO(3)is itself a contraction

The Euclidean mother group ISO(3)is itself a contraction of itsgrand-mother group: SO(4) of 4D rotations

The Euclidean mother group ISO(3)is itself a contraction of itsgrand-mother group: SO(4) of 4D rotations

In the 1D finite oscillator:

position: momentum:mode:

Kravchuk functionorthogonal & complete on 2j+1 points

And there occurs an accident:

And there occurs an accident:

And there occurs an accident:

Not only that, but:

And there occurs an accident:

Not only that, but:

Cartesian basis polar basis

Hermite-G-Kravchuk

Laguerre-Grad-Kravchuk

Clebsch-Gordan

Hermite-G-Kravchuk

Laguerre-Grad-Kravchuk

Clebsch-Gordan

← UNITARY →

On 2D pixellated

RECTANGULARscreens

The Cartesian basis…

The Cartesian basisis orthogonaland complete…

We can define the

“Laguerre-Kravchuk” modeson rectangular screens

MSc Thesis of

Alejandro R. Urzúa

JOSAA 33,643 (2016)

The complete mode-angularmomentumbasis

rectangular

In 2D we thus have the : Fourier group on pixellated screens,

Fourier transformations (domestic)

gyrations (imported)

rotations (imported)

Rotations:

These ensue fromthe rotations ofall modes

Virtue:no info is lostSin:it is the

slowest !

These ensue fromthe rotations ofall modes

Gyrations ofall modes:

Pending matters…

Pending matters…

Pixellations distinctfrom Cartesian or polarwith their own modes…

Pending matters…

Pixellations distinctfrom Cartesian or polarwith their own modes…

Unitary aberrations in 2 dimensions

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…

Muchas Gracias

desde Cuernavaca…