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Aberrations
of Phase Space
Kurt Bernardo Wolf
in 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 Físicas
Universidad Nacional Autónoma de México
Cuernavaca
Polynomials and aberrations in one dimension
In plane (D=1) optics, aberrations are
generated by polynomials of phase space
Mk,m = p k+m q k--m
k=1
k = 3/2
k=2
k = 5/2
k=2
…
of rank k  {1, 3/2, 2, …}
weight m  {-k,-k+1,…,k}
and order A = 2k+1
linear part Sp(2,R)
second order aberrations
third order aberrations
fourth order
fifth order
….
The 2k + 1 aberrations of rank k form a
multiplet under linear Sp(2,R) systems.
They form rank-k aberration algebras,
and generate rank-k aberration groups.
spherical coma
aberration
astigmatism
/curvature of field
distorsion pocus
They compose under concatenation,
and aberrate phase space with terms up
to order A (independently of the purpose
–imaging or non-imaging— of the
Apparatus --in the interaction frame.
Linear systems:
Classical oscillator mechanics
Geometric paraxial optics
Linear Fourier optics
Higher-order
aberrations:
Quantum harmonic oscillator
Quantum optical field
Metaxial régime
Phase space, Hamiltonian systems,
Lie algebras, Aberration Lie groups
An Sp(2,R)
Global systems:
Global (4) geometric optics
Helmholtz wave optics
Finite optics (signals in guides)
Relativistic coma
Finite Kerr medium
Phase space in geometric optics
The manifold of oriented lines in space
is 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 space
Spherical aberration.
Propagation along a guide rotates phase space
fractional Fourier transformation (paraxially).
Canonical transformations
3
’
’
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 the
Hamiltonian structure are canonical.
Introduce Poisson brackets and operators
and Lie exponential operators
Introduce one-parameter groups of:
Spherical aberration and pocus,
Distorsion and coma,
Fractional Fourier transformation
Introduce multiparameter Lie algebras and groups
of Hamiltonian flows of phase space
Axis-symmetric aberrations
In 3-dim optics (plane screens), phase space is 4-dim.
Axis-symmetric optical systems produce axis-symmetric
aberrations, characterized by their spot diagrams.
They have a monomial basis (top) and a
Symplectic basis
k-j
Yk,j,m (|p| , pq, |q| ) = (p  q) j,jY (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 covariant
with their geometric (classical) generators.
But under composition the aberrations differ by terms
of powers of the wavenumber ().
One aberration –astigmatism
on a Gaussian ground state
Evolution under exp (  {², ²}Weyl )
produces ‘quantum fluctuations’
in the Wigner function.
The classical Wigner probability distribution
is conserved
(simply follows phase space tfmns).
k
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)k ]
Parameter values for the
Wigner function above
I1 = I2 = 1, while the higher moments

Indicate fall from classicality.
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 the
aberration 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 transformations
of optical phase space.
Optical phase space serves as
homogeneous space for the Lorentz group.
Bradley’s `stellar aberration’ and
Bargmann’s deformation of the sphere
are the momentum (ray direction) part;
the image (position) part is the
relativistic coma global aberration.
SO(3,1)  ASp(4,R)
A camera focused on a proximate
object at rest begets comatic aberrrations
when set in relative motion.
n = 32
Wavefunctions of the finite oscillator
The finite oscillator follows
the dynamics of the
ordinary quantum harmonic oscillator:
[, ] = -i , [, ] = i ,
n = 16
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
n=2
d n, q ( ½  )
n=1
The ground state is a
binomial distribution function,
the top state alternates its signs.
n=0
Figure: 33 points ( J = 16 ) and 33 states
labeled by n = 0,1,2,…,32.
Wigner function for finite systems
Group 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 of
the Wigner operator between the finite
wavefunctions f. –Enter the Wigner matrix.
Continuous
system
Sp(2,R)
Have in common
The fractional
Fourier transform
Finite
system
SU(2)
Fractional Fourier-Kravchuk transform
The 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 state
corresponds 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-oscillator
A q-oscillator is defined
by the q-algebra
suq(2).
Non-canonical commutator
is [,] = ½ i [2 3 ]q
3 =  – J – ½ .
Sensor positions (with q )
Energies
The Casimir operator
yields a phase space
which is an ovoïd.
This rotates around
the 3 –axis,
The spectrum of 
(position of the sensors)
is concentrated towards
the center.
The spectrum of 
is equally spaced.
Kerr effect
in ordinary
and finite
oscillator
Kerr effect on the
ordinary quantum oscillator
and its ‘classicality’ measures.
--See the resonance times
of the cat states.
Kerr effect
on the
finite oscillator.
--See the cat states.
The Kerr effect in geometric optics
corresponds to a guide with an
elliptic index-profile n(q) = n0 – q²
h = – [n0² – (p² + q²)]
= – n0 + H + H² + …