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Title: What is the Brillouin zone of an anisotropic photonic crystal?

Abstract

The concept of the Brillouin zone (BZ) in relation to a photonic crystal fabricated in an optically anisotropic material is explored both experimentally and theoretically. In experiment we used femtosecond laser pulses to excite THz polaritons and image their propagation in lithium niobate and lithium tantalate photonic crystal (PhC) slabs. We directly measured the dispersion relation inside PhCs and observed that the lowest band gap expected to form at the BZ boundary forms inside the BZ in the anisotropic lithium niobate PhC. Our analysis shows that in an anisotropic material the BZ—defined as the Wigner-Seitz cell in the reciprocal lattice—is no longer bounded by Bragg planes and thus does not conform to the original definition of the BZ by Brillouin. We construct an alternative Brillouin zone defined by Bragg planes and show its utility in identifying features of the dispersion bands. We show that for an anisotropic two-dimensional PhC without dispersion, the Bragg plane BZ can be constructed by applying the Wigner-Seitz method to a stretched or compressed reciprocal lattice. We also show that in the presence of the dispersion in the underlying material or in a slab waveguide, the Bragg planes are generally represented by curved surfaces rather thanmore » planes. In conclusion, the concept of constructing a BZ with Bragg planes should prove useful in understanding the formation of dispersion bands in anisotropic PhCs and in selectively tailoring their optical properties.« less

Authors:
 [1];  [1];  [1];  [1]
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  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Publication Date:
Research Org.:
Energy Frontier Research Centers (EFRC) (United States). Solid-State Solar-Thermal Energy Conversion Center (S3TEC)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES); Natural Sciences and Engineering Research Council of Canada (NSERC); National Science Foundation (NSF)
OSTI Identifier:
1470240
Alternate Identifier(s):
OSTI ID: 1237502
Grant/Contract Number:  
SC0001299; FG02-09ER46577; CHE-1111557
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 93; Journal Issue: 5; Related Information: S3TEC partners with Massachusetts Institute of Technology (lead); Boston College; Oak Ridge National Laboratory; Rensselaer Polytechnic Institute; Journal ID: ISSN 2469-9950
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Sivarajah, P., Maznev, A. A., Ofori-Okai, B. K., and Nelson, K. A. What is the Brillouin zone of an anisotropic photonic crystal?. United States: N. p., 2016. Web. doi:10.1103/PhysRevB.93.054204.
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Sivarajah, P., Maznev, A. A., Ofori-Okai, B. K., & Nelson, K. A. What is the Brillouin zone of an anisotropic photonic crystal?. United States. https://doi.org/10.1103/PhysRevB.93.054204
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Sivarajah, P., Maznev, A. A., Ofori-Okai, B. K., and Nelson, K. A. Tue . "What is the Brillouin zone of an anisotropic photonic crystal?". United States. https://doi.org/10.1103/PhysRevB.93.054204. https://www.osti.gov/servlets/purl/1470240.
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@article{osti_1470240,
title = {What is the Brillouin zone of an anisotropic photonic crystal?},
author = {Sivarajah, P. and Maznev, A. A. and Ofori-Okai, B. K. and Nelson, K. A.},
abstractNote = {The concept of the Brillouin zone (BZ) in relation to a photonic crystal fabricated in an optically anisotropic material is explored both experimentally and theoretically. In experiment we used femtosecond laser pulses to excite THz polaritons and image their propagation in lithium niobate and lithium tantalate photonic crystal (PhC) slabs. We directly measured the dispersion relation inside PhCs and observed that the lowest band gap expected to form at the BZ boundary forms inside the BZ in the anisotropic lithium niobate PhC. Our analysis shows that in an anisotropic material the BZ—defined as the Wigner-Seitz cell in the reciprocal lattice—is no longer bounded by Bragg planes and thus does not conform to the original definition of the BZ by Brillouin. We construct an alternative Brillouin zone defined by Bragg planes and show its utility in identifying features of the dispersion bands. We show that for an anisotropic two-dimensional PhC without dispersion, the Bragg plane BZ can be constructed by applying the Wigner-Seitz method to a stretched or compressed reciprocal lattice. We also show that in the presence of the dispersion in the underlying material or in a slab waveguide, the Bragg planes are generally represented by curved surfaces rather than planes. In conclusion, the concept of constructing a BZ with Bragg planes should prove useful in understanding the formation of dispersion bands in anisotropic PhCs and in selectively tailoring their optical properties.},
doi = {10.1103/PhysRevB.93.054204},
journal = {Physical Review B},
number = 5,
volume = 93,
place = {United States},
year = {Tue Feb 09 00:00:00 EST 2016},
month = {Tue Feb 09 00:00:00 EST 2016}
}
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Journal Article:
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Publisher's Version of Record
https://doi.org/10.1103/PhysRevB.93.054204
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Cited by: 2 works
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Figures / Tables:

FIG. 1 FIG. 1: Crossing points of the empty lattice dispersion curves identify the Bragg planes. (a) TE dispersion curves for propagation along $Γ$-M-$Γ$ in a 2D empty square lattice with an isotropic refractive index n = 5 and lattice periodicity a. The first two crossing points are labeled as kB-1 andmore » kB-2. (b) Corresponding dispersion curves with a finite perturbation from air holes of radius r = 0.15a. (c) The first several BZ boundaries in reciprocal space constructed using the WS geometrical method. The propagation direction (red arrow), irreducible Brillouin zone (shaded blue), and reciprocal lattice vectors b1 and b2 are also indicated.« less
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Works referenced in this record:

Chemically assisted femtosecond laser machining for applications in LiNbO3 and LiTaO3
journal, July 2013

  • Sivarajah, Prasahnt; Werley, Christopher A.; Ofori-Okai, Benjamin K.
  • Applied Physics A, Vol. 112, Issue 3
  • DOI: 10.1007/s00339-013-7833-x

Spontaneous Emission Enhancement of Quantum Dots in a Photonic Crystal Wire
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High-Resolution, Low-Noise Imaging in THz Polaritonics
journal, May 2013

  • Werley, C. A.; Teo, S. M.; Ofori-Okai, B. K.
  • IEEE Transactions on Terahertz Science and Technology, Vol. 3, Issue 3
  • DOI: 10.1109/TTHZ.2013.2250580

Direct experimental visualization of waves and band structure in 2D photonic crystal slabs
journal, May 2014

  • Ofori-Okai, Benjamin K.; Sivarajah, Prasahnt; Werley, Christopher A.
  • New Journal of Physics, Vol. 16, Issue 5
  • DOI: 10.1088/1367-2630/16/5/053003

Large Absolute Band Gap in 2D Anisotropic Photonic Crystals
journal, September 1998

Anisotropic photonic crystals: Generalized plane wave method and dispersion symmetry properties
journal, November 2008

Spatiotemporal Coherent Control of Lattice Vibrational Waves
journal, January 2003

Spontaneous emission from a two-level atom in two-band anisotropic photonic crystals
journal, October 2003

Guided modes in photonic crystal slabs
journal, August 1999

  • Johnson, Steven G.; Fan, Shanhui; Villeneuve, Pierre R.
  • Physical Review B, Vol. 60, Issue 8
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Strong localization of photons in certain disordered dielectric superlattices
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Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal
journal, August 1999

  • Yoshino, Katsumi; Shimoda, Yuki; Kawagishi, Yoshiaki
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Inhibited Spontaneous Emission in Solid-State Physics and Electronics
journal, May 1987

Terahertz polariton propagation in patterned materials
journal, September 2002

  • Stoyanov, Nikolay S.; Ward, David W.; Feurer, Thomas
  • Nature Materials, Vol. 1, Issue 2
  • DOI: 10.1038/nmat725

Terahertz Polaritonics
journal, August 2007

Symmetry properties of two-dimensional anisotropic photonic crystals
journal, January 2006

  • Alagappan, G.; Sun, X. W.; Shum, P.
  • Journal of the Optical Society of America A, Vol. 23, Issue 8
  • DOI: 10.1364/JOSAA.23.002002

Group theoretical investigations of photonic band structures
journal, June 2003

Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide
journal, May 2007

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    Works referencing / citing this record:

    Formation of Bragg Band Gaps in Anisotropic Phononic Crystals Analyzed With the Empty Lattice Model
    journal, May 2016

    • Wang, Yan-Feng; Maznev, Alexei; Laude, Vincent
    • Crystals, Vol. 6, Issue 5
    • DOI: 10.3390/cryst6050052
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      Figures / Tables found in this record:

      FIG. 1 (p. 3)figure
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      FIG. 6. (p. 7)figure
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