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On photonic band gaps in two-dimensional photonic crystal fibres. Analysis in the vicinity of the low-dielectric light line

Shane Cooper, Ilia Kamotski

TL;DR

The paper develops a rigorous operator-theoretic framework for Maxwell equations in two-dimensional photonic crystal fibres near the low-dielectric light line, showing that gaps in the line spectrum imply nearby photonic band gaps. It establishes uniform resolvent convergence from above the critical line to the line and demonstrates that ARROW fibres with small inclusions exhibit a low-frequency gap due to asymptotic eigenvalue scaling. The approach handles arbitrary dielectric contrasts and does not fix propagation constants along the fibre, enabling broad applicability to 1D and genuinely 2D PCFs. The results provide quantitative gap conditions and asymptotic expansions that connect line behavior to full Bloch spectra with potential for guiding PCF design and optimization.

Abstract

We consider `off-axis' electromagnetic wave propagation down the homogeneous direction of a low-loss two-dimensional periodic dielectric (or photonic crystal fibre) near the light line of the low-dielectric material constituent. Numerous physical and numerical experiments demonstrate the presence of photonic band gaps in the vicinity of this `critical' light line. We mathematically analyse the existence of photonic band gaps near the line and characterise them in terms of frequency gaps in the spectrum of the Maxwell equations restricted to the line. We apply the results to both one-dimensional photonic crystal fibres, and a genuinely two-dimensional photonic crystal fibres with `thin' inclusions, namely ``ARROW'' fibres. In the case of ARROW fibres, by an asymptotic analysis, in terms of the small inclusion parameter, we demonstrate the existence of low frequency photonic band gaps. It is important to note that our analysis does not assume any specific ratio between the dielectric contrasts; as a result, moderate or even low contrast models are fully encompassed. Similarly, no limiting assumptions are imposed on the wave propagation constant along the fibre.

On photonic band gaps in two-dimensional photonic crystal fibres. Analysis in the vicinity of the low-dielectric light line

TL;DR

The paper develops a rigorous operator-theoretic framework for Maxwell equations in two-dimensional photonic crystal fibres near the low-dielectric light line, showing that gaps in the line spectrum imply nearby photonic band gaps. It establishes uniform resolvent convergence from above the critical line to the line and demonstrates that ARROW fibres with small inclusions exhibit a low-frequency gap due to asymptotic eigenvalue scaling. The approach handles arbitrary dielectric contrasts and does not fix propagation constants along the fibre, enabling broad applicability to 1D and genuinely 2D PCFs. The results provide quantitative gap conditions and asymptotic expansions that connect line behavior to full Bloch spectra with potential for guiding PCF design and optimization.

Abstract

We consider `off-axis' electromagnetic wave propagation down the homogeneous direction of a low-loss two-dimensional periodic dielectric (or photonic crystal fibre) near the light line of the low-dielectric material constituent. Numerous physical and numerical experiments demonstrate the presence of photonic band gaps in the vicinity of this `critical' light line. We mathematically analyse the existence of photonic band gaps near the line and characterise them in terms of frequency gaps in the spectrum of the Maxwell equations restricted to the line. We apply the results to both one-dimensional photonic crystal fibres, and a genuinely two-dimensional photonic crystal fibres with `thin' inclusions, namely ``ARROW'' fibres. In the case of ARROW fibres, by an asymptotic analysis, in terms of the small inclusion parameter, we demonstrate the existence of low frequency photonic band gaps. It is important to note that our analysis does not assume any specific ratio between the dielectric contrasts; as a result, moderate or even low contrast models are fully encompassed. Similarly, no limiting assumptions are imposed on the wave propagation constant along the fibre.
Paper Structure (14 sections, 27 theorems, 247 equations, 1 figure)

This paper contains 14 sections, 27 theorems, 247 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The Maxwell system on the critical light line.
  4. The Maxwell system above the critical light line.
  5. Operator theoretic formulation of the spectral problems.
  6. Comparison of spectral problems in the vicinity of the critical line.
  7. Some properties of the reduced Maxwell operators $\mathcal{B}(\theta)$
  8. Photonic crystals which exhibit gaps in the Bloch spectrum on the critical light-line
  9. One-dimensional photonic crystal fibres
  10. ARROW fibres: two-dimensional photonic crystals with inclusions have vanishing volume faction
  11. Appendix
  12. Equivalence of \ref{['degp']} and the reduced Maxwell system \ref{['Max1']}-\ref{['Max2']} on the critical light line.
  13. Proof of Lemma \ref{['prethm.resolventasymp']}.
  14. Proof of Theorem \ref{['thmArrowAsymptotics']}

Key Result

Lemma 1.1

There exists a constant $C>0$ such that

Figures (1)

  • Figure 1: A two-dimensional photonic crystal fibre that is homogeneous in the $x_3$-direction with a cylindrical cross-section $R$ that is repeated periodically with respect to the reference cell $\square=[-\pi,\pi)$ in the $(x_1,x_2)$-plane.

Theorems & Definitions (46)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 36 more