### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 3764-3773 |

Number of pages | 10 |

Journal | Applied Optics |

Volume | 54 |

Issue number | 12 |

DOIs | |

Publication status | Published - Apr 20 2015 |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics
- Engineering (miscellaneous)
- Electrical and Electronic Engineering

### Cite this

*Applied Optics*,

*54*(12), 3764-3773. https://doi.org/10.1364/AO.54.003764

**Skew ray tracing in a step-index optical fiber using geometric algebra.** / Ang, Angeleene S.; Sugon, Quirino Jr Mallorca; McNamara, Daniel J.

Research output: Contribution to journal › Article

*Applied Optics*, vol. 54, no. 12, pp. 3764-3773. https://doi.org/10.1364/AO.54.003764

}

TY - JOUR

T1 - Skew ray tracing in a step-index optical fiber using geometric algebra

AU - Ang, Angeleene S.

AU - Sugon, Quirino Jr Mallorca

AU - McNamara, Daniel J.

PY - 2015/4/20

Y1 - 2015/4/20

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84942371795&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84942371795&partnerID=8YFLogxK

U2 - 10.1364/AO.54.003764

DO - 10.1364/AO.54.003764

M3 - Article

VL - 54

SP - 3764

EP - 3773

JO - Applied Optics

JF - Applied Optics

SN - 1559-128X

IS - 12

ER -