Aberrationsof Phase Space

Kurt Bernardo Wolfin collaborations with

Sergey M. Chumakov, Ana Leonor Rivera, Natig M. Atakishiyev, S. Twareque Ali, George S.

Pogosyan, Miguel Angel Alonso, Luis Edgar Vicent

and Guillermo Krötzsch

Centro de Ciencias FísicasUniversidad Nacional Autónoma de México

Cuernavaca

Polynomials and aberrations in one dimensionIn plane (D=1) optics, aberrations aregenerated by polynomials of phase space

M = p q of rank k {1, 3/2, 2, …} weight m {-k,-k+1,…,k} and order A = 2k+1

k = 1 linear part Sp(2,R)k = 3/2 second order aberrationsk = 2 third order aberrationsk = 5/2 fourth orderk = 2 fifth order … ….

The 2k + 1 aberrations of rank k form amultiplet under linear Sp(2,R) systems.They form rank-k aberration algebras,and generate rank-k aberration groups.

They compose under concatenation,and aberrate phase space with terms upto order A (independently of the purpose –imaging or non-imaging— of the Apparatus --in the interaction frame.

k,mk+m k--m

spherical coma astigmatism distorsion pocusaberration /curvature of field

Classical oscillator mechanics

Metaxial régime Phase space, Hamiltonian systems,Lie algebras, Aberration Lie groups

An Sp(2,R)

Higher-orderaberrations:

Geometric paraxial optics

Linear Fourier optics

Quantum harmonic oscillator

Quantum optical field

Global (4) geometric opticsHelmholtz wave opticsFinite optics (signals in guides)

Relativistic coma

Finite Kerr medium

Linear systems:

Global systems:

Phase space in geometric optics

The manifold of oriented lines in spaceis four-dimensional.On the standard screen (2-dim position)Its momentum ranges on a sphere,i.e., two discs sown at their edges.

In flat optics, optical phase space is two-dimensional (and can be drawn).Hamilton equations are on the screen.

Free propagation deforms phase spaceSpherical aberration.Propagation along a guide rotates phase spacefractional Fourier transformation (paraxially).

Canonical transformations

Light is neither created nor destroyed,only transformed (pirated from Joseph Liouville)

In flat optics, this is all…In higher dimensions, the Hamilton equations must be preserved !Those transformations that preserve theHamiltonian structure are canonical.

Introduce Poisson brackets and operatorsand Lie exponential operators

Introduce one-parameter groups of:

Spherical aberration and pocus,Distorsion and coma, Fractional Fourier transformation

Introduce multiparameter Lie algebras and groupsof Hamiltonian flows of phase space

3

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Axis-symmetric aberrationsIn 3-dim optics (plane screens), phase space is 4-dim.Axis-symmetric optical systems produce axis-symmetricaberrations, characterized by their spot diagrams.They have a monomial basis (top) and aSymplectic basis Y (|p| , pq, |q| ) = (p q) Y (spherical harmonic) of rank k {1, 2, 3, …}, symplectic spin j {k, k-2, … 1 or 0} weight m {-j,-j+1,…,j} and order A = 2k+1

Classification of aberrations puts them in 1:1 correspondence with the states of the ordinary 3-dim quantum harmonic oscillator.

THEOREM: Under the paraxial subgroup Sp(4,R)only the Weyl-quantized operators are covariantwith their geometric (classical) generators. But under composition the aberrations differ by termsof powers of the wavenumber ().

k,j,mk-j

j,j

One aberration –astigmatismon a Gaussian ground stateEvolution under exp ( {², ²}Weyl )produces ‘quantum fluctuations’ in the Wigner function.

The classical Wigner probability distribution is conserved (simply follows phase space tfmns).

The ‘nonclasicality’ can be measured through the moments of the Wigner function W(p,x;t) : I k ( t ) ~ dp dq [W (p,x;t) ]

I1 = I2 = 1, while the higher moments Indicate fall from classicality.

k

k

Parameter values for theWigner function above

Aberrations of fractional Fourier transformers

Hamilton-Lie aberrations are in the Interaction frame of perturbation theory. As an application, we consider three fractional Fourier transformers:

a: Lens with polynomial faces between two screens.b: Elliptic-index-profile waveguide with warped face.c: Cat’s eye arrangement with warped back mirror.

Left: Uncorrected system:In the waveguide with flat face, we draw theaberration of phase space (interaction picture)for fractional Fourier angles every 15º (left).

Right: Partially corrected system:At each aberration order we can use one polynomial order of the lens face, and propose one or more correction tactics (right).

Relativistic coma aberration

The symmetries of vacuum are:translations, rotations,and Lorentz boost transformations.They are all canonical transformationsof optical phase space.

Optical phase space serves as homogeneous space for the Lorentz group.

Bradley’s `stellar aberration’ andBargmann’s deformation of the sphereare the momentum (ray direction) part;the image (position) part is therelativistic coma global aberration. SO(3,1) ASp(4,R)

A camera focused on a proximate object at rest begets comatic aberrrationswhen set in relative motion.

Wavefunctions of the finite oscillatorThe finite oscillator follows the dynamics of theordinary quantum harmonic oscillator: [, ] = -i , [, ] = i ,

but has the non-canonical commutator [,] = i 3 , 3 = – J – ½ ,so it is ruled by SU(2). It has 2J + 1 states. Its wavefunctions are the Wigner little-d functions

d n, q ( ½ ) The ground state is a binomial distribution function,the top state alternates its signs.

Figure: 33 points ( J = 16 ) and 33 stateslabeled by n = 0,1,2,…,32.n =

0

n = 1

n = 2

n = 16

n = 32

Wigner function for finite systemsGroup elements in polar coordinates.The Wigner operator is the ‘Fourier transform’of the group; an element of the group ring. Can be written as (—x).The Wigner function is the matrix element ofthe Wigner operator between the finite wavefunctions f. –Enter the Wigner matrix.

ContinuoussystemSp(2,R)

FinitesystemSU(2)

Have in commonThe fractionalFourier transform

Fractional Fourier-Kravchuk transformThe Wigner function for the finite SU(2) oscillator can be seen on the sphere. Ground state and top state,can be SU(2)-transformed to coherent states.

The time evolution of a coherent statecorresponds to the rotation of the sphere,and to fractional Fourier-Kravchuk transformation.

Rotations around Q and P axes in a harmonic guide:

Phase space of a q-oscillatorA q-oscillator is definedby the q-algebra suq(2).

Non-canonical commutatoris [,] = ½ i [2 3 ]q 3 = – J – ½ .

The Casimir operator yields a phase spacewhich is an ovoïd.This rotates around the 3 –axis,

The spectrum of (position of the sensors)is concentrated towardsthe center. The spectrum of is equally spaced.

Sensor positions (with q ) Energies

Kerr effectin ordinaryand finiteoscillator

Kerr effect on theordinary quantum oscillatorand its ‘classicality’ measures.--See the resonance times of the cat states.

Kerr effect on thefinite oscillator.--See the cat states.

The Kerr effect in geometric opticscorresponds to a guide with anelliptic index-profile n(q) = n0 – q²

h = – [n0² – (p² + q²)] = – n0 + H + H² + …