Poisson commutator-anticommutator brackets for ray tracing and longitudinal imaging via geometric algebra

Quirino M. Sugon Jr. and Daniel J. McNamara

(Submitted on 16 Dec 2008)

Abstract: We use the vector wedge product in geometric algebra to show that Poisson commutator brackets measure preservation of phase space areas. We also use the vector dot product to define the Poisson anticommutator bracket that measures the preservation of phase space angles. We apply these brackets to the paraxial meridional complex height-angle ray vectors that transform via a 2×2 matrix, and we show that this transformation preserves areas but not angles in phase space. The Poisson brackets here are expressed in terms of the coefficients of the ABCD matrix. We also apply these brackets to the distance-height ray vectors measured from the input and output sides of the optical system. We show that these vectors obey a partial Moebius transformation, and that this transformation preserves neither areas nor angles. The Poisson brackets here are expressed in terms of the transverse and longitudinal magnifications.

Comments: 10 pages, 9 figures
Subjects:      Mathematical Physics (math-ph)
Cite as:          arXiv:0812.2979v1 [math-ph]

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Revisiting 2×2 matrix optics: Complex vectors, Fermion combinatorics, and Lagrange invariants

Quirino M. Sugon Jr. and Daniel J. McNamara

(Submitted on 3 Dec 2008)

Abstract: We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray’s 2×2 matrix operators should be right-acting, so that the matrix product succession would go with light’s left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants.

Comments: 10 pages, 6 figures
Subjects: Optics (physics.optics)
Cite as: arXiv:0812.0664v1 [physics.optics]

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Paraxial meridional ray tracing equations from the unified reflection-refraction law via geometric algebra

Authors: Quirino M. Sugon Jr., Daniel J. McNamara

(Submitted on 29 Oct 2008)

Abstract: We derive the paraxial meridional ray tracing equations from the unified reflection-refraction law using geometric algebra. This unified law states that the normal vector to the interface is a rotation of the incident ray or of the refracted ray or of the reflected ray by an angle equal to the angle of incidence or of refraction. We obtain the finite meridional ray tracing equations by simply equating the arguments of the exponential rotation operators. We then derive the paraxial limits of these equations with the help of sign function identities. We show that by embedding the sign functions in the ray tracing equations, we explicitly declare our chosen sign conventions in symbols and not in prose.

Comments: 5 pages, 2 figures
Subjects: Optics (physics.optics)
Cite as: arXiv:0810.5224v1 [physics.optics]