Aberrations of Phase Space

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Aberrationsof 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

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Polynomials and aberrations in one dimension
In plane (D1) optics, aberrations are generated
by polynomials of phase space M p q
of rank k ? 1, 3/2, 2,
weight m ? -k,-k1,,k
and order A 2k1 k
1 linear part Sp(2,R) k 3/2 second order
aberrations k 2 third order aberrations k
5/2 fourth order k 2 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. 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.
km
k--m
k,m
spherical coma astigmatism
distorsion pocus aberration
/curvature of field
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Classical oscillator mechanics
Linear systems
Higher-order aberrations
Geometric paraxial optics
Linear Fourier optics
Quantum harmonic oscillator
Quantum optical field
Metaxial régime
Phase space, Hamiltonian systems, Lie algebras,
Aberration Lie groups An Sp(2,R)
Global systems
Relativistic coma Finite Kerr medium
Global (4?) geometric optics
Helmholtz wave optics
Finite optics (signals in guides)
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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. Propagat
ion along a guide rotates phase space fractional
Fourier transformation (paraxially).
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Canonical transformations
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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


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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 Y (p ,
p?q, q ) (p ? q) Y (spherical harmonic)
of rank k ? 1, 2,
3, , symplectic
spin j ? k, k-2, 1 or 0
weight m ? -j,-j1,,j
and order A 2k1
Classification of aberrations puts them in 11
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 (?).
k-j
k,j,m
j,j
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One aberration astigmatism on a Gaussian ground
state Evolution under exp ( ? ?², ?²Weyl
) 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,xt)
I k ( t ) ? dp dq W (p,xt) I1 I2
1, while the higher moments Indicate fall from
classicality.
k
k
Parameter values for the Wigner function above
?
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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 Cats 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).
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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. Bradleys stellar aberration
and Bargmanns 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.

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Wavefunctions of the finite oscillator
n 32
The finite oscillator follows the dynamics of
the ordinary 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
states labeled by n 0,1,2,,32.
n 16
n 2
n 1
n 0
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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)
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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
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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 . 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.
Sensor positions (with q ) Energies
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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.
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
Kerr effect on the finite oscillator. --See the
cat states.