Submicron Structure of “Amorphous” Opal

Greer, Raymond T.

doi:10.1038/2241199a0

Cited by 14 publications

(14 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Plugging in the expressions of u (1) (X, ξ ) and u (2) (X, ξ ) in (2.4), we get at the zeroth order,…”

Section: We Rewrite the Equation In The Formmentioning

confidence: 99%

“…and f (1) 0 (X) and f (2) 0 (X) are the modulating functions whose governing equation we seek to find. It is then clear that U (1) 0 (ξ ) and U (2) 0 (ξ ) have the form…”

Section: We Rewrite the Equation In The Formmentioning

confidence: 99%

“…Opals are another example, which consist of tiny spherical particles of silica arranged in a face-centred cubic array, which act like a diffraction grating to create the beautiful colours we see [1,2]. A sea mouse has a wonderful iridescence which is caused by a hexagonal array of voids in a matrix of chitin [3].…”

Section: Introductionmentioning

confidence: 99%

“…0 (x/ ) e −i(k·(x/ )−ω 1 (t/ )) and V (2) 0 (x/ ) e −i(m·(x/ )−ω 2 (t/ )) are the Bloch solutions at the wavevectors k/ and m/ , (in which V (1) 0 (x/ ) and V (2) 0 (x/ ) have the same unit cell of periodicity as the periodic material we are considering) and the modulating functions f and ω/ = g(k/ ) is the dispersion relation. This effective equation admits as solutions, the expected travelling waves high-frequency homogenization are those of Birman & Suslina [23], which provides a rigorous justification of high-frequency homogenization.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

High-frequency homogenization for travelling waves in periodic media

Harutyunyan

Milton

2016

Proc. R. Soc. A.

View full text Add to dashboard Cite

We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 2. We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1=ω 2 and [Formula: see text] where Λ=(λ1λ2…λ d ) is the periodicity cell of the medium and for any two vectors [Formula: see text] the product a⊙b is defined to be the vector (a 1 b 1,a 2 b 2,…,a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue

show abstract

“…Plugging in the expressions of u (1) (X, ξ ) and u (2) (X, ξ ) in (2.4), we get at the zeroth order,…”

Section: We Rewrite the Equation In The Formmentioning

confidence: 99%

“…and f (1) 0 (X) and f (2) 0 (X) are the modulating functions whose governing equation we seek to find. It is then clear that U (1) 0 (ξ ) and U (2) 0 (ξ ) have the form…”

Section: We Rewrite the Equation In The Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

High-frequency homogenization for travelling waves in periodic media

Harutyunyan

Milton

2016

Proc. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…The mesoscale can be probed by imaging-or diffraction-related methods. The first family of techniques was exclusively limited to surface imaging, namely to scanning electron microscopy (SEM) and a replica-technique variety of transmission electron microscopy (TEM) [1][2][3]. Diffraction methodsvisible light diffraction and small-angle x-ray scattering-provide invaluable information on the volume-specific structure, but mainly on the average one [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Towards High-Resolution X-ray Microscopy on Mesoscopic Structures

Bosak¹,

Snigirev²,

McNulty³

et al. 2011

AIP Conference Proceedings

View full text Add to dashboard Cite

Abstract. We introduce a high-resolution x-ray microscopy (HRXRM) based on parabolic compound refractive lenses for the volume-specific studies of mesoscopic photonic crystals. The use of coherent properties of x-ray beams along with the tunable refractive optics allows retrieval of the high-resolution diffraction pattern and real-space images in the same experimental setup. Methodologically the proposed approach is similar to the studies of "traditional" crystals by high-resolution transmission electron microscopy (HRTEM).

show abstract