Aberrations of Phase Space

1. 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

2. Polynomials and aberrations in one dimension In plane (D=1) optics, aberrations are generated by polynomials of phase space M = p q of rankk {1, 3/2, 2, …} weightm {-k,-k+1,…,k} and order A = 2k+1 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. k+m k--m k,m spherical coma astigmatism distorsion pocus aberration /curvature of field

3. Linear systems: Higher-order aberrations: Classical oscillator mechanics 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)

4. 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).

5. 3 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 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 ’ ’

6. 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 rankk {1, 2, 3, …}, symplectic spinj  {k, k-2, … 1 or 0} weightm {-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 (). k-j k,j,m j,j

7. 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). 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 the Wigner function above 

8. 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).

9. 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 comaglobal 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.

10. 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 dn, 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

11. 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)

12. 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:

13. 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

14. 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 effectin 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.