### Abstract

In periodic optical media, the group velocity is defined as the gradient with respect to wave-vector of the corresponding Bloch mode frequency dispersion curve, forming the photonic band structure. Instead of deducing it from the numerically computed photonic crystal band structure, the group velocity can be calculated directly from the integral of the Poynting vector over the crystal unit cell, the physical meaning of which is immediately perceivable. The related formula, which can be regarded as the application of Hellmann–Feynman theorem to electromagnetism, has been reported previously though without proof. We provide hereafter a full derivation of that formula starting from Maxwell’s equations and we discuss its usefulness in photonics.

Original language | English |
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Pages (from-to) | 213-220 |

Number of pages | 8 |

Journal | Journal of Modern Optics |

Volume | 65 |

Issue number | 2 |

DOIs | |

Publication status | Published - 19 Jan 2018 |

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### Keywords

- Photonic crystals
- group velocity
- slow light

### Cite this

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**Alternative expression of the Bloch wave group velocity in loss-less periodic media using the electromagnetic field energy.** / Deparis, Olivier; Lambin, Philippe.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Alternative expression of the Bloch wave group velocity in loss-less periodic media using the electromagnetic field energy

AU - Deparis, Olivier

AU - Lambin, Philippe

PY - 2018/1/19

Y1 - 2018/1/19

N2 - In periodic optical media, the group velocity is defined as the gradient with respect to wave-vector of the corresponding Bloch mode frequency dispersion curve, forming the photonic band structure. Instead of deducing it from the numerically computed photonic crystal band structure, the group velocity can be calculated directly from the integral of the Poynting vector over the crystal unit cell, the physical meaning of which is immediately perceivable. The related formula, which can be regarded as the application of Hellmann–Feynman theorem to electromagnetism, has been reported previously though without proof. We provide hereafter a full derivation of that formula starting from Maxwell’s equations and we discuss its usefulness in photonics.

AB - In periodic optical media, the group velocity is defined as the gradient with respect to wave-vector of the corresponding Bloch mode frequency dispersion curve, forming the photonic band structure. Instead of deducing it from the numerically computed photonic crystal band structure, the group velocity can be calculated directly from the integral of the Poynting vector over the crystal unit cell, the physical meaning of which is immediately perceivable. The related formula, which can be regarded as the application of Hellmann–Feynman theorem to electromagnetism, has been reported previously though without proof. We provide hereafter a full derivation of that formula starting from Maxwell’s equations and we discuss its usefulness in photonics.

KW - Photonic crystals

KW - group velocity

KW - slow light

UR - http://www.scopus.com/inward/record.url?scp=85031400857&partnerID=8YFLogxK

U2 - 10.1080/09500340.2017.1384513

DO - 10.1080/09500340.2017.1384513

M3 - Article

VL - 65

SP - 213

EP - 220

JO - Journal of Modern Optics

JF - Journal of Modern Optics

SN - 0950-0340

IS - 2

ER -