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Handmer CJ, Martijn de Sterke C, McPhedran RC, Botten LC, Steel MJ, Rahmani A. Blazing evanescent grating orders: a spectral approach to beating the Rayleigh limit. Opt Lett 2010; 35:2846-2848. [PMID: 20808344 DOI: 10.1364/ol.35.002846] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
We develop a way to enhance the amplitudes of the nonpropagating evanescent orders of resonant dielectric gratings. We use this blazing to design gratings with spectra tailored to generate steerable sub-Rayleigh field concentrations on a surface. We investigate the enhancement and customization of evanescent fields necessary to create a virtual and passive scanning probe with no moving parts. Spot size can be decreased 1 order of magnitude below the free-space Rayleigh limit.
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Affiliation(s)
- C J Handmer
- Institute for Photonics and Optical Science (IPOS) and Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
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2
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Mahmoodian S, Poulton CG, Dossou KB, McPhedran RC, Botten LC, de Sterke CM. Modes of shallow photonic crystal waveguides: semi-analytic treatment. Opt Express 2009; 17:19629-19643. [PMID: 19997183 DOI: 10.1364/oe.17.019629] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
We investigate the formation of photonic crystal waveguide (PCW) modes within the framework of perturbation theory. We derive a differential equation governing the envelope of PCW modes constructed from weak perturbations using an effective mass formulation based on the Luttinger-Kohn method from solid-state physics. The solution of this equation gives the frequency of the mode and its field. The differential equation lends itself to simple analytic approximations which reduce the problem to that of solving slab waveguide modes. By using this model, we demonstrate that the nature of the projected band structure and corresponding Bloch functions are central to the behaviour of PCW modes. With this understanding, we explain why the odd mode in a hexagonal PCW spans the entire Brillouin zone while the even mode is cut off.
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Affiliation(s)
- S Mahmoodian
- IPOS and CUDOS, School of Physics, University of Sydney, 2006, Australia.
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3
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Martijn de Sterke C, Dossou KB, White TP, Botten LC, McPhedran RC. Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes. Opt Express 2009; 17:17338-17343. [PMID: 19907519 DOI: 10.1364/oe.17.017338] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
We show that efficient coupling between fast and slow photonic crystal waveguide modes is possible, provided that there exist strong evanescent modes to match the waveguide fields across the interface. Evanescent modes are required when the propagating modes have substantially different modal fields, which occurs, for example, when coupling an index-guided mode and a gap-guided mode.
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4
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Campbell S, Botten LC, McPhedran RC, Martijn de Sterke C. Modal method for conical diffraction by slanted lamellar gratings. J Opt Soc Am A Opt Image Sci Vis 2009; 26:938-948. [PMID: 19340269 DOI: 10.1364/josaa.26.000938] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Slanted lamellar gratings made of dielectric materials are considered, used in conical diffraction mounts. We extend the modal method for slanted lamellar gratings from classical to conical incidence, develop fully generalized Fresnel matrices, and derive energy conservation relations for these matrices. Using the method, we verified a uniaxial crystal model for slanted lamellar gratings in a homogenization regime, examined the effects of grating symmetry on the maximum reflectance of Fano resonances, and showed that slanted lamellar gratings support Fano resonances despite the homogenization of their other optical properties.
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Affiliation(s)
- S Campbell
- Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS), School of Physics, University of Sydney, Sydney, NSW 2006, Australia
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5
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White TP, Botten LC, Martijn de Sterke C, Dossou KB, McPhedran RC. Efficient slow-light coupling in a photonic crystal waveguide without transition region. Opt Lett 2008; 33:2644-2646. [PMID: 19015695 DOI: 10.1364/ol.33.002644] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We consider the coupling into a slow mode that appears near an inflection point in the band structure of a photonic crystal waveguide. Remarkably, the coupling into this slow mode, which has a group index ng>1000, can be essentially perfect without any transition region. We show that this efficient coupling occurs thanks to an evanescent mode in the slow medium, which has appreciable amplitude and helps satisfy the boundary conditions but does not transport any energy.
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Affiliation(s)
- T P White
- School of Physics and Astronomy, University of St Andrews, Fife, UK.
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6
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Nicorovici NA, Milton GW, McPhedran RC, Botten LC. Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance. Opt Express 2007; 15:6314-6323. [PMID: 19546935 DOI: 10.1364/oe.15.006314] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
Discrete systems of infinitely long polarizable line dipoles are considered in the quasistatic limit, interacting with a two-dimensional cloaking system consisting of a hollow plasmonic cylindrical shell. A numerical procedure is described for accurately calculating electromagnetic fields arising in the quasistatic limit, for the case when the relative permittivity of the cloaking shell has a very small imaginary part. Animations are given which illustrate cloaking of discrete systems, both for the case of induced dipoles and induced quadrupoles on the interacting particles. The simulations clarify the physical mechanism for the cloaking.
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Affiliation(s)
- N A Nicorovici
- CUDOS, School of Physics, University of Sydney, NSW 2006, Australia
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7
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Dossou KB, McPhedran RC, Botten LC, Asatryan AA, de Sterke CM. Gap-edge Asymptotics of defect modes in 2D Photonic Crystals. Opt Express 2007; 15:4753-4762. [PMID: 19532721 DOI: 10.1364/oe.15.004753] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps.
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Affiliation(s)
- K B Dossou
- CUDOS, Department of Mathematical Sciences, University of Technology, Sydney, NSW 2007, Australia.
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8
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Asatryan AA, Botten LC, Byrne MA, McPhedran RC, de Sterke CM. Evidence of a mobility edge for photons in two dimensions. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 75:015601. [PMID: 17358216 DOI: 10.1103/physreve.75.015601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2006] [Indexed: 05/14/2023]
Abstract
A scaling analysis of conductance for photons in two dimensions is carried out and, contrary to widely held belief, we find strong evidence of a mobility edge. Such behavior is compatible with the existence of an Anderson transition for electronic systems under symplectic symmetry, and indeed we show that the transfer matrix in the photonic system we have modelled has such a symmetry. We verify single parameter scaling of the conductance and demonstrate the transition from the metallic phase to localization. Key parameters, including the critical disorder, the conductance, and the critical exponent of the localization length are calculated, and it is shown that the value of the critical exponent is similar to that for electronic systems with symplectic symmetry.
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Affiliation(s)
- A A Asatryan
- Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS) and Department of Mathematical Sciences, University of Technology, Sydney NSW 2007, Australia
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9
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Affiliation(s)
- D. R. Mckenzie
- a School of Physics, University of Sydney , New South Wales 2006, Australia
| | - R. C. Mcphedran
- a School of Physics, University of Sydney , New South Wales 2006, Australia
| | - N. Savvides
- a School of Physics, University of Sydney , New South Wales 2006, Australia
| | - L. C. Botten
- b School of Mathematical Sciences , New South Wales Institute of Technology , P.O. Box 123, Broadway , New South Wales 2007 , Australia
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10
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Botten LC, White TP, de Sterke CM, McPhedran RC. Wide-angle coupling into rod-type photonic crystals with ultralow reflectance. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 74:026603. [PMID: 17025553 DOI: 10.1103/physreve.74.026603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2005] [Indexed: 05/12/2023]
Abstract
We describe the surprising phenomenon of near-perfect coupling from free space into uniform two-dimensional rod-type photonic crystals over a wide range of incident angles. This behavior is shown to be a generic feature of many rod-type photonic crystal structures that is related to strong forward scattering resonances of the individual cylinders. We explain these results using both semianalytic analysis and two-dimensional numerical calculations and identify the conditions under which efficient, wide-angle coupling can occur. The results may lead to more efficient designs for in-band photonic crystal devices such as superprisms and self-collimation based photonic circuits.
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Affiliation(s)
- L C Botten
- Centre for Ultrahigh-Bandwidth Devices for Optical Systems (CUDOS) and Department of Mathematical Sciences, University of Technology, Sydney, Broadway NSW 2007, Australia.
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11
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Botten LC, Hansen RA, de Sterke CM. Supermodes in multiple coupled photonic crystal waveguides. Opt Express 2006; 14:387-396. [PMID: 19503352 DOI: 10.1364/opex.14.000387] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We analyze the supermodes in multiple coupled photonic crystal waveguides for long-wavelengths. In the tight-binding limit we obtain analytic results that agree with fully numerical calculations. We find that when the field flips sign after a single photonic crystal period, and there is an odd number of periods between adjacent waveguides, the supermode order is reversed, compared to that in conventional coupled waveguides, generalizing earlier results obtained for two coupled waveguides.
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Wilcox S, Botten LC, McPhedran RC, Poulton CG, de Sterke CM. Modeling of defect modes in photonic crystals using the fictitious source superposition method. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:056606. [PMID: 16089667 DOI: 10.1103/physreve.71.056606] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2004] [Indexed: 05/03/2023]
Abstract
We present an exact theory for modeling defect modes in two-dimensional photonic crystals having an infinite cladding. The method is based on three key concepts, namely, the use of fictitious sources to modify response fields that allow defects to be introduced, the representation of the defect mode field as a superposition of solutions of quasiperiodic field problems, and the simplification of the two-dimensional superposition to a more efficient, one-dimensional average using Bloch mode methods. We demonstrate the accuracy and efficiency of the method, comparing results obtained using alternative techniques, and then concentrate on its strengths, particularly in handling difficult problems, such as where a mode is highly extended near cutoff, that cannot be dealt with in other ways.
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Affiliation(s)
- S Wilcox
- Centre for Ultrahigh-bandwidth Devices for Optical Systems, School of Physics, University of Sydney, New South Wales 2006, Australia
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13
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Asatryan AA, Botten LC, Byrne MA, Langtry TN, Nicorovici NA, McPhedran RC, de Sterke CM, Robinson PA. Conductance of photons in disordered photonic crystals. Phys Rev E Stat Nonlin Soft Matter Phys 2005; 71:036623. [PMID: 15903621 DOI: 10.1103/physreve.71.036623] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2004] [Revised: 11/12/2004] [Indexed: 05/02/2023]
Abstract
The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H(||) (TE) polarization for weak and intermediate disorder while, for E(||) (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar.
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Affiliation(s)
- A A Asatryan
- Department of Mathematical Sciences, University of Technology, Sydney, New South Wales, Australia
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14
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White TP, Botten LC, de Sterke CM, McPhedran RC, Asatryan AA, Langtry TN. Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications. Phys Rev E Stat Nonlin Soft Matter Phys 2004; 70:056607. [PMID: 15600779 DOI: 10.1103/physreve.70.056607] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2004] [Indexed: 05/24/2023]
Abstract
The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.
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Affiliation(s)
- T P White
- Centre for Ultrahigh-Bandwidth Devices for Optical Systems (CUDOS) and School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia.
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15
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Botten LC, White TP, Asatryan AA, Langtry TN, de Sterke CM, McPhedran RC. Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory. Phys Rev E Stat Nonlin Soft Matter Phys 2004; 70:056606. [PMID: 15600778 DOI: 10.1103/physreve.70.056606] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2004] [Indexed: 05/24/2023]
Abstract
We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.
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Affiliation(s)
- L C Botten
- Centre for Ultrahigh-Bandwidth Devices for Optical Systems (CUDOS) and Department of Mathematical Sciences, University of Technology, Sydney, Broadway New South Wales 2007, Australia.
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16
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Abstract
We consider the modes of coupled photonic crystal waveguides. We find that the fundamental modes of these structures can be either even or odd, in contrast with the behavior in coupled conventional waveguides, in which the fundamental mode is always even. We explain this finding using an asymptotic model that is valid for long wavelengths.
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Affiliation(s)
- C Martijn de Sterke
- Centre for Ultrahigh-Bandwidth Devices for Optical Systems and School of Physics, University of Sydney, New South Wales 2006, Australia.
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17
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McPhedran RC, Botten LC, McOrist J, Asatryan AA, De Sterke CM, Nicorovici NA. Density of states functions for photonic crystals. Phys Rev E Stat Nonlin Soft Matter Phys 2004; 69:016609. [PMID: 14995738 DOI: 10.1103/physreve.69.016609] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2003] [Indexed: 05/24/2023]
Abstract
We discuss density of states functions for photonic crystals, in the context of the two-dimensional problem for arrays of cylinders of arbitrary cross section. We introduce the mutual density of states (MDOS), and show that this function can be used to calculate both the local density of states (LDOS), which gives position information for emission of radiation from photonic crystals, and the spectral density of states (SDOS), which gives angular information. We establish the connection between MDOS, LDOS, SDOS and the conventional density of states, which depends only on frequency. We relate all four functions to the band structure and propagating states within the crystal, and give numerical examples of the relation between band structure and density of states functions.
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Affiliation(s)
- R C McPhedran
- CUDOS & School of Physics, University of Sydney, New South Wales 2006, Australia
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18
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White TP, Botten LC, McPhedran RC, de Sterke CM. Ultracompact resonant filters in photonic crystals. Opt Lett 2003; 28:2452-2454. [PMID: 14690112 DOI: 10.1364/ol.28.002452] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
A novel design for an ultracompact, high-Q notch-rejection filter is presented, and an analytic expression for the transmission properties is derived. This folded directional coupler shares the properties of a Fabry-Perot resonator and a directional coupler. We compare and contrast the device to high-Q Fabry-Perot cavities in photonic crystal waveguides.
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Affiliation(s)
- T P White
- Centre for Ultrahigh-Bandwidth Devices for Optical Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia.
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Langtry TN, Asatryan AA, Botten LC, de Sterke CM, McPhedran RC, Robinson PA. Effects of disorder in two-dimensional photonic crystal waveguides. Phys Rev E Stat Nonlin Soft Matter Phys 2003; 68:026611. [PMID: 14525137 DOI: 10.1103/physreve.68.026611] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2002] [Indexed: 05/24/2023]
Abstract
The effects of randomness on the guiding properties of waveguides embedded in disordered two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length are investigated for TM-polarized radiation. Different degrees of disorder in the radius, filling fraction, refractive index, and position are considered for both straight and 90 degrees bent guides. The crystals exhibit similar sensitivity to refractive index and radius disorder, with a degree of disorder from 15%-20% yielding little substantial change in the guiding properties. A smaller range of position disorder is also considered. For strong disorder in radius and refractive index, the guide effectively closes. These results were obtained by a Monte Carlo simulation method, and the performance of this method is analyzed. The method requires at least ten realizations in some cases for convergence to commence; substantially more realizations are required for moderate and strong disorder to achieve accurate results.
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Affiliation(s)
- T N Langtry
- CUDOS and Department of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia
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Botten LC, Asatryan AA, Langtry TN, White TP, Martijn de Sterke C, McPhedran RC. Semianalytic treatment for propagation in finite photonic crystal waveguides. Opt Lett 2003; 28:854-856. [PMID: 12779169 DOI: 10.1364/ol.28.000854] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry-Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.
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Affiliation(s)
- L C Botten
- Center for Ultrahigh-bandwidth Devices for Optical Systems, Department of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway NSW 2007, Australia.
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Smith GH, Botten LC, McPhedran RC, Nicorovici NA. Cylinder gratings in conical incidence with applications to woodpile structures. Phys Rev E Stat Nonlin Soft Matter Phys 2003; 67:056620. [PMID: 12786309 DOI: 10.1103/physreve.67.056620] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2003] [Indexed: 05/24/2023]
Abstract
We use our previous formulation for cylinder gratings in conical incidence to discuss the photonic band gap properties of woodpile structures. We study scattering matrices and Bloch modes of the woodpile, and use these to investigate the dependence of the optical properties on the number of layers. We give data on reflectance, transmittance and absorptance of metallic woodpiles as a function of wavelength and number of layers, using both the measured optical constants of tungsten and using a perfect conductivity idealization to characterize the metal. For semi-infinite metallic woodpiles, we show that polarization of the incident field is important, highlighting the role played by surface effects as opposed to lattice effects.
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Affiliation(s)
- G H Smith
- CUDOS ARC Centre of Excellence and Department of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia
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Asatryan AA, Robinson PA, McPhedran RC, Botten LC, Martijn de Sterke C, Langtry TL, Nicorovici NA. Diffusion and anomalous diffusion of light in two-dimensional photonic crystals. Phys Rev E Stat Nonlin Soft Matter Phys 2003; 67:036605. [PMID: 12689177 DOI: 10.1103/physreve.67.036605] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2002] [Indexed: 05/24/2023]
Abstract
The transport properties of electromagnetic waves in disordered, finite, two-dimensional photonic crystals composed of circular cylinders are considered. Transport parameters such as the transport and scattering mean free paths and the transport velocity are calculated, for the case where the electromagnetic radiation has its electric field along the cylinder axes. The range of the parameters in which the diffusion process can take place is specified. It is shown that the transport velocity upsilon(E) can be as much as 10(8) times less than its free space value, while just outside the cluster upsilon(E) can be 0.3c. The effects of weak and strong disorders on the transport velocity are investigated. Different regimes of the wave transport-ordered propagation, diffusion, and anomalous diffusion-are demonstrated, and it is inferred that Anderson localization is incipient in the latter regime. Exact numerical calculations from the Helmholtz equation are shown to be in good agreement with the diffusion approximation.
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Affiliation(s)
- A A Asatryan
- School of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia
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Asatryan AA, Busch K, McPhedran RC, Botten LC, de Sterke CM, Nicorovici NA. Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0959-7174/13/1/302] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Botten LC, McPhedran RC, Nicorovici NA, Asatryan AA, de Sterke CM, Robinson PA, Busch K, Smith GH, Langtry TN. Rayleigh Multipole Methods for Photonic Crystal Calculations. ACTA ACUST UNITED AC 2003. [DOI: 10.2528/pier02010802] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Smith GH, Botten LC, McPhedran RC, Nicorovici NA. Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 66:056604. [PMID: 12513620 DOI: 10.1103/physreve.66.056604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2002] [Indexed: 05/24/2023]
Abstract
We develop a formulation for cylinder gratings in conical incidence, using a multipole method. The theory, and its numerical implementation, is applied to two-dimensional photonic crystals consisting of a stack of one-dimensional gratings, each characterized by its plane wave scattering matrix. These matrices are used in combination with Bloch's theorem to determine the band structure of the photonic crystal from the solution of an eigenvalue problem. We show that the theory is well adapted to the difficult task of locating the complete band gaps needed to support air-guided modes in microstructured optical fibers, that is, optical fibers in which the confinement of light in a central air hole is achieved by photonic band-gap effects in a periodic cladding comprising a lattice of air holes in a glass matrix.
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Affiliation(s)
- G H Smith
- School of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia
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White TP, McPhedran RC, de Sterke CM, Botten LC, Steel MJ. Confinement losses in microstructured optical fibers. Opt Lett 2001; 26:1660-1662. [PMID: 18049691 DOI: 10.1364/ol.26.001660] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
We describe a multipole formulation that can be used for high-accuracy calculations of the full complex propagation constant of a microstructured optical fiber with a finite number of holes. We show how the imaginary part of the microstructure, which describes confinement losses not associated with absorption, varies with hole size, the number of rings of holes, and wavelength, and give the minimum number of rings of holes required for a specific loss for given parameters.
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Botten LC, Nicorovici NA, McPhedran RC, Sterke CM, Asatryan AA. Photonic band structure calculations using scattering matrices. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:046603. [PMID: 11690166 DOI: 10.1103/physreve.64.046603] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2001] [Indexed: 05/23/2023]
Abstract
We consider band structure calculations of two-dimensional photonic crystals treated as stacks of one-dimensional gratings. The gratings are characterized by their plane wave scattering matrices, the calculation of which is well established. These matrices are then used in combination with Bloch's theorem to determine the band structure of a photonic crystal from the solution of an eigenvalue problem. Computationally beneficial simplifications of the eigenproblem for symmetric lattices are derived, the structure of eigenvalue spectrum is classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed form expressions for the reflection and transmission scattering matrices of finite stacks of gratings are established. A new, fundamental quantity, the reflection scattering matrix, in the limit in which the stack fills a half space, is derived and is used to deduce the effective dielectric constant of the crystal in the long wavelength limit.
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Affiliation(s)
- L C Botten
- School of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia
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Steel MJ, White TP, Martijn de Sterke C, McPhedran RC, Botten LC. Symmetry and degeneracy in microstructured optical fibers. Opt Lett 2001; 26:488-90. [PMID: 18040361 DOI: 10.1364/ol.26.000488] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
The symmetry of an optical waveguide determines its modal degeneracies. A fiber with rotational symmetry of order higher than 2 has modes that either are nondegenerate and support the complete fiber symmetry or are twofold degenerate pairs of lower symmetry. The latter case applies to the fundamental modes of perfect microstructured optical fibers, guaranteeing that such fibers are not birefringent. We explore two numerical methods and demonstrate their agreement with these symmetry constraints.
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Asatryan AA, Busch K, McPhedran RC, Botten LC, de Sterke CM, Nicorovici NA. Two-dimensional Green's function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 63:046612. [PMID: 11308973 DOI: 10.1103/physreve.63.046612] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2000] [Revised: 10/24/2000] [Indexed: 05/23/2023]
Abstract
Using the exact theory of multipole expansions, we construct the two-dimensional Green's function for photonic crystals, consisting of a finite number of circular cylinders of infinite length. From this Green's function, we compute the local density of states (LDOS), showing how the photonic crystal affects the radiation properties of an infinite fluorescent line source embedded in it. For frequencies within the photonic band gap of the infinite crystal, the LDOS decreases exponentially inside the crystal; within the bands, we find "hot" and "cold" spots. Our method can be extended to three dimensions as well as to treating disorder and represents an important and efficient tool for the design of photonic crystal devices.
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Affiliation(s)
- A A Asatryan
- School of Physics, University of Sydney, Sydney, NSW 2006, Australia
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Abstract
The most intense colours displayed in nature result from either multilayer reflectors or linear diffraction gratings. Here we investigate the spectacular iridescence of a spine (notoseta) from the sea mouse Aphrodita sp. (Polychaeta: Aphroditidae). The spine normally appears to be deep red in colour, but when light is incident perpendicular to the axis of the spine, different colours are seen as stripes running parallel to the axis of the spine; over a range of smaller incident angles, the complete visible spectrum is reflected with a reflectivity of 100% to the human eye. The simple structure responsible for this effect is a remarkable example of photonic engineering by a living organism.
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Affiliation(s)
- A R Parker
- Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
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Botten LC, Nicorovici NA, Asatryan AA, McPhedran RC, de Sterke CM, Robinson PA. Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method. J Opt Soc Am A Opt Image Sci Vis 2000; 17:2165-2176. [PMID: 11140475 DOI: 10.1364/josaa.17.002165] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.
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Affiliation(s)
- L C Botten
- School of Mathematical Sciences, University of Technology, Sydney, NSW, Australia.
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Botten LC, Nicorovici NA, Asatryan AA, McPhedran RC, de Sterke CM, Robinson PA. Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation. J Opt Soc Am A Opt Image Sci Vis 2000; 17:2177-2190. [PMID: 11140476 DOI: 10.1364/josaa.17.002177] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.
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Affiliation(s)
- L C Botten
- School of Mathematical Sciences, University of Technology, Sydney, NSW, Australia
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McKenzie DR, Savvides N, McPhedran RC, Botten LC, Netterfield RP. Optical properties of a-Si and a-Si:H prepared by DC magnetron techniques. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/16/24/022] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Asatryan AA, Robinson PA, Botten LC, McPhedran RC, Nicorovici NA. Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000; 62:5711-5720. [PMID: 11089130 DOI: 10.1103/physreve.62.5711] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2000] [Indexed: 05/23/2023]
Abstract
The effects of disorder in the geometry and refractive index on the transmittance of two-dimensional photonic crystals composed of dielectric circular cylinders are considered, including randomness of radii, positions of the cylinder centers, and thickness of each layer of the photonic crystal. The effects of combinations of different types of strong disorder are also considered. The localization and homogenization properties of disordered photonic crystals are investigated. Analytical expressions for the two-dimensional localization length in the form of integrals are presented for both polarizations. It is shown numerically that the slope of the exponential divergence of the localization length in two dimensions is proportional to the inverse of the square of randomness for strong disorder and proportional to the inverse of the randomness for weak disorder. The effective dielectric constants for both polarizations in the case of strong disorder are also found. The transition from localization to homogenization is discussed and the terms responsible for this transition are identified and investigated.
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Affiliation(s)
- AA Asatryan
- School of Physics, The University of Sydney, New South Wales 2006, Australia
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McPhedran RC, Botten LC, Asatryan AA, Nicorovici NA, Robinson PA, de Sterke CM. Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1999; 60:7614-7. [PMID: 11970724 DOI: 10.1103/physreve.60.7614] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/1999] [Indexed: 04/18/2023]
Abstract
A method is developed to calculate electromagnetic properties of arrays of metallic and dielectric cylinders. It incorporates and exploits cylindrical boundary conditions and Rayleigh identities for efficient, high-accuracy calculation of scattering off individual layers that are stacked into arrays using scattering matrices. The method enables absorption, dispersion, and randomness to be incorporated efficiently, and reproduces known results with vastly improved speed and accuracy. It is used to demonstrate existence of states introduced into photonic band gaps of a dielectric array by disorder, and anomalous absorption behavior in arrays of aluminum cylinders.
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Affiliation(s)
- R C McPhedran
- School of Physics, University of Sydney, NSW 2006, Australia
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Yardley JG, McPhedran RC, Nicorovici NA, Botten LC. Addition formulas and the Rayleigh identity for arrays of elliptical cylinders. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1999; 60:6068-80. [PMID: 11970512 DOI: 10.1103/physreve.60.6068] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/1998] [Revised: 05/25/1999] [Indexed: 04/18/2023]
Abstract
We apply the Rayleigh method to solve the problem where a uniform electrostatic field is imposed upon a rectangular array of elliptical cylinders embedded in a matrix of unit dielectric constant. This new formulation overcomes geometric restrictions inherent in previous methods and is shown in principle and in various examples to converge for all possible geometries of the array and inclusion. Also presented are forms of both the interior and exterior addition formulas for harmonic functions in elliptical coordinates that possess optimal regions of convergence.
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Affiliation(s)
- J G Yardley
- Department of Theoretical Physics, School of Physics, University of Sydney, New South Wales 2006, Australia
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Asatryan AA, Robinson PA, Botten LC, McPhedran RC, Nicorovici NA, Martijn de Sterke C. Effects of disorder on wave propagation in two-dimensional photonic crystals. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1999; 60:6118-27. [PMID: 11970517 DOI: 10.1103/physreve.60.6118] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/1999] [Indexed: 04/18/2023]
Abstract
The electromagnetic transmittance of disordered two-dimensional photonic crystals composed of circular cylinders is investigated as a function of wavelength and polarization. At short wavelengths, the transmittance shows a band structure similar to that found in the optical absorption spectrum of amorphous semiconductors, with impurity states increasingly appearing on the long wavelength side of the band gaps as the degree of disorder is increased. In the long-wavelength limit, Anderson localization of waves is found, provided that the wavelength is not so large that the random photonic crystal can be viewed as homogeneous. The localization properties in this regime are studied and an analytic expression for the dependence of the localization length on wavelength is derived. In the limit of extremely long wavelengths, the system homogenizes and can be replaced by an equivalent one with uniform effective refractive index, whose form is derived for both polarizations. Analysis of the crossover between localization and homogenization is also presented.
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Affiliation(s)
- A A Asatryan
- School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
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Poulton CG, Botten LC, McPhedran RC, Movchan AB. Source-neutral Green'functions for periodic problems in electrostatics, and their equivalents in electromagnetism. Proc Math Phys Eng Sci 1999. [DOI: 10.1098/rspa.1999.0351] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- C. G. Poulton
- School of Physics, University of Sydney, NSW 2006, Australia
| | - L. C. Botten
- School of Mathematical Sciences, University of Technology Sydney, NSW 2007, Australia
| | - R. C. McPhedran
- School of Physics, University of Sydney, NSW 2006, Australia
| | - A. B. Movchan
- School of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
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McPhedran RC, Botten LC, Asatryan AA, Nicorovici NA, Sterke CMD, Robinson PA. Ordered and Disordered Photonic Band Gap Materials. ACTA ACUST UNITED AC 1999. [DOI: 10.1071/ph98110] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
Abstract
We discuss a formulation and computer implementation of a new method that can
be used to determine the electromagnetic properties of ordered and disordered
dielectric and metallic cylinders, using periodic boundary conditions in one
direction. We show results which exhibit strong parallels with the behaviour
of electrons in disordered semiconductors, but also illustrate some
characteristics which clearly differentiate between photonic and electronic
behaviour. Among these are strong polarisation sensitivity and effects due to
metallic absorption.
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Nicorovici NA, McPhedran RC, Botten LC. Photonic band gaps for arrays of perfectly conducting cylinders. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1995; 52:1135-1145. [PMID: 9963518 DOI: 10.1103/physreve.52.1135] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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McPhedran RC, Botten LC, McKenzie DR, Netterfield RP. Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements. Appl Opt 1984; 23:1197. [PMID: 18204704 DOI: 10.1364/ao.23.001197] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
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Mills DR, Botten LC. Lower emissivity limits indicated for high temperature selective surfaces. Appl Opt 1983; 22:3182. [PMID: 18200176 DOI: 10.1364/ao.22.003182] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
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45
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McKenzie DR, McPhedran RC, Botten LC, Savvides N, Netterfield RP. Hydrogenated carbon films produced by sputtering in argon-hydrogen mixtures. Appl Opt 1982; 21:3615-3617. [PMID: 20396283 DOI: 10.1364/ao.21.003615] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
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Abstract
We describe a technique for determining the groove depth of gratings having sinusoidal grooves. Absolute efficiencies measured using a laser are compared with numerical results to give an indication of groove depth.
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