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  • American Institute of Physics (AIP)  (11)
  • 1
    Electronic Resource
    Electronic Resource
    The Euclidean root of Snell's law I. Geometric polarization optics (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 2390-2408 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Coset spaces of the Euclidean group of rigid motions in three-space are used as a model of the rays in geometric polarization optics. The Haar measure invariance leads to canonicity of the phase space transformations. The phase space of optical rays undergoes a canonical map also under the effect of smooth refracting or reflecting surfaces between two optical media that is governed by Snell's law. This is a conservation law between two irreducible representations of the Euclidean group labeled by the refractive indices n, n' of the media. These surface transformations are canonical and, furthermore, factorize into two root transformations that are also canonical. This factorization applies equally to the transformation of the polarization vector. The root transformation permits the computation of an aberration expansion of the polarization field over the object screen. The generators of infinitesimal surface transformations are found, i.e., those that transform between media n and n+dn. It is shown that, together with displacements, they yield the Hamilton equations of optics for inhomogeneous media.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529608
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  • 2
    Electronic Resource
    Electronic Resource
    Hidden symmetry and potential group of the Maxwell fish-eye (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2757-2768 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Maxwell fish-eye is an exceptional optical system that shares with the Kepler problem and the point rotor (mass point on a sphere) a hidden, higher rotation symmetry. The Hamiltonian is proportional to the Casimir invariant. The well-known stereographic map is extended to canonical transformations between of the phase spaces of the constrained rotor and the fish-eye. Their dynamical group is a pseudoorthogonal one that permits a succint "4π'' wavization of the constrained system. The fish-eye exhibits, unavoidably, chromatic dispersion. Further, a larger conformal dynamical group contains the potential group, that relates the closed, inhomogeneous fish-eye system to similar, scaled ones. Asymptotically, it is related to free propagation in homogenous media.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528979
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  • 3
    Electronic Resource
    Electronic Resource
    The relativistic coma aberration. I. Geometrical optics (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 2457-2462 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown, in the framework of Hamilton–Lie geometrical optics, that the image on a moving screen undergoes comatic aberration as the conjugate sphere of ray directions distorts under Lorentz boosts.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528524
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  • 4
    Electronic Resource
    Electronic Resource
    Lie algebras for systems with mixed spectra. I. The scattering Pöschl–Teller potential (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 973-983 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Starting from an N-body quantum space, we consider the Lie-algebraic framework where the Pöschl–Teller Hamiltonian, − 1/2 ∂2χ +c sech2 χ+s csch2 χ, is the single sp(2,R) Casimir operator. The spectrum of this system is mixed: it contains a finite number of negative-energy bound states and a positive-energy continuum of free states; it is identified with the Clebsch–Gordan series of the D+×D− representation coupling. The wave functions are the sp(2,R) Clebsch–Gordan coefficients of that coupling in the parabolic basis. Using only Lie-algebraic techniques, we find the asymptotic behavior of these wave functions; for the special pure-trough potential (s=0) we derive thus the transmission and reflection amplitudes of the scattering matrix.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526559
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  • 5
    Electronic Resource
    Electronic Resource
    The metaplectic group within the Heisenberg–Weyl ring (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 29-36 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Heisenberg–Weyl ring contains the metaplectic group of canonical transforms acting unitarily on L 2(R). These ring elements are characterized through (i) the integral transform kernels, (ii) coset distributions, and (iii) classical functions under any quantization scheme. The isomorphism under group composition leads to several new relations involving twisted products and quantization of Gaussian classical functions. The Wigner inversion operator is a special central group element. It is shown that the only quantization scheme invariant under metaplectic transformations is the Weyl scheme. The structure studied here appears to be relevant to the study of wave optics with aberration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527333
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  • 6
    Electronic Resource
    Electronic Resource
    The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 2498-2507 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A Lie-theoretical classification of the aberrations of systems modeled by asymmetric optical devices is given. The classification is done on the basis of aberration order and axial symmetry of the first-order part. This leads to finite-dimensional (nonunitary) representations of sp(4,R) reduced with respect to its sp (2,R) subalgebra, with helicity and "symplectic spin'' labels. Based on pure-magnifier systems, a weight label reproduces and completes Seidel's traditional classification of axis-symmetric aberrations. Based on other first-order systems such as optical fibers, other classification schemes are indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527739
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  • 7
    Electronic Resource
    Electronic Resource
    The group-theoretical treatment of aberrating systems. II. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1458-1465 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The group-theoretical description of geometrical optical systems whose elements are, in general, inhomogeneous but axis symmetric, in third aberration order, is presented. Closed algebraic expressions are given for fiber elements, i.e., media homogeneous under translations along the optical axis, and refracting interfaces between them.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527104
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  • 8
    Electronic Resource
    Electronic Resource
    The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1449-1457 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The constituents of a lens system, i.e., slabs of homogeneous transparent material and the refracting surfaces between them, considered to third aberration order, are associated to elements of a nine-parameter aberration group. Three parameters correspond to Gaussian systems and six to group-classified aberrations. The group multiplication operation (through matrix-cum-vector algebra) corresponds to their concatenation, and the linear group action on an eight-dimensional homogeneous space corresponds to the nonlinear action of the system on the optical phase space. This leads to economical computation algorithms that may be extended to aberrating systems of higher order.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527103
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  • 9
    Electronic Resource
    Electronic Resource
    A simple difference realization of the Heisenberg q-algebra (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 3253-3260 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A realization of the Heisenberg q-algebra whose generators are first-order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q-oscillator Hamiltonian are given explicitly in terms of the q−1-Hermite polynomials. The nonuniqueness of the measure for these q-oscillator states is also studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530464
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  • 10
    Electronic Resource
    Electronic Resource
    Wigner distribution function for finite systems (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 6247-6261 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We construct a Wigner distribution function for finite data sets. It is based on a finite optical system; a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry. The dynamical group for this model is SU(2) and the wave functions are sets of N=2l+1 data points. The Wigner distribution function assigns classical c-numbers to the operators of position, momentum, and wave guide mode. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532636
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