A geometric algebra reformulation of geometric optics

Sugon, Quirino M.; McNamara, Daniel J.

doi:10.1119/1.1621029

Cited by 9 publications

(6 citation statements)

References 6 publications

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“…which is the same expression for the law of reflection in Klein and Furtak. [15,24] 3. Law of Refraction…”

Section: Law Of Reflectionmentioning

confidence: 99%

“…The direct vector product form of the law of reflection [21][22][23] in Eq. (4a) and the exponential rotation form of the law of refraction [24] in Eq. (4b) were already known before.…”

Section: Introductionmentioning

confidence: 99%

“…If a light ray moves from its initial position r k at the k th interface by a distance s k+1 in the direction of the unit vector σ k+1 , then the ray's final position r k+1 at the (k + 1) th interface is [24] …”

Section: Law Of Propagationmentioning

confidence: 99%

“…(4a) and the exponential rotation form of the law of refraction [24] in Eq. (4b) were already known before.…”

Section: Introductionmentioning

confidence: 99%

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Skew ray tracing in a step-index optical fiber using geometric algebra

2015

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We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation, the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays.

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“…which is the same expression for the law of reflection in Klein and Furtak. [15,24] 3. Law of Refraction…”

Section: Law Of Reflectionmentioning

confidence: 99%

“…The direct vector product form of the law of reflection [21][22][23] in Eq. (4a) and the exponential rotation form of the law of refraction [24] in Eq. (4b) were already known before.…”

Section: Introductionmentioning

confidence: 99%

Section: Law Of Propagationmentioning

confidence: 99%

“…(4a) and the exponential rotation form of the law of refraction [24] in Eq. (4b) were already known before.…”

Section: Introductionmentioning

confidence: 99%

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Skew ray tracing in a step-index optical fiber using geometric algebra

2015

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“…( 11) and as we have shown can supersede to a large degree the various vector systems used today. New applications for Clifford algebra are steadily appearing in many fields of engineering and physics such as electromagnetism [29], optics [30], Fourier transforms [31], terahertz spectroscopy [14], satellite navigation [32], robotics [33], computer graphics and computer vision [25], [34], [35], quantum mechanics [36], [37], quantum computing [38], special and general relativity [39], [40].…”

Section: Discussionmentioning

confidence: 99%

The Vector Algebra War: A Historical Perspective

et al. 2016

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There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of nineteenth century science was to suitably describe vectors in three-dimensional space. This situation has also had the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford's multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science.

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Formulation of the Snell–Descartes Laws in Terms of Geometric Algebra

Fisanov

2019

Russ Phys J

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A geometric algebra reformulation of geometric optics

Cited by 9 publications

References 6 publications

Skew ray tracing in a step-index optical fiber using geometric algebra

Skew ray tracing in a step-index optical fiber using geometric algebra

The Vector Algebra War: A Historical Perspective

Formulation of the Snell–Descartes Laws in Terms of Geometric Algebra

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