Light intensity distribution in conical refraction

Uhlmann, Günther

doi:10.1002/cpa.3160350104

Cited by 17 publications

(6 citation statements)

References 4 publications

(2 reference statements)

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“…The theorem generaUzes various results of the authors (see [CI,C2,C3,U]) and is closely related to a recent result of Seeley [S], obtained independently. The proof is very classical in spirit and uses fundamentally the Mellin transform and its properties.…”

mentioning

confidence: 86%

Singular asymptotics approach to partial differential equations with isolated singularities in the coefficients

Callias¹,

Uhlmann²

1984

Bull. Amer. Math. Soc.

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1. Introduction. In this note we announce some results on the heat equation ( §3) and scattering theory ( §4) associated with some differential operators with isolated singularities in the coefficients. The results are based on recent joint work by the authors [CU] and on independent work by the first author [CI, C2, C3]. We study the small time asymptotics of the heat kernel and the large frequency asymptotics of the scattering amplitude. These questions are classical but need reinterpretation because of the singularities. Our general approach is to solve the differential equation in question away from the singularity and then try to extend it across. In all of our problems, we found that we can handle the situation by using a singular asymptotics lemma in P'(R) ( §2) for functions of the form f(s/x,x) as s -• 0 with certain quite general conditions on ƒ. We believe that the latter result is of general interest and wide applicability [CI, C2, C3, CM, CU, U].

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confidence: 86%

Singular asymptotics approach to partial differential equations with isolated singularities in the coefficients

Callias¹,

Uhlmann²

1984

Bull. Amer. Math. Soc.

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“…Singularities along the double characteristics (optical axis) propagate along a cone; the cone of conical refraction (see [19]). A more detailed study of the intensity of light at the optical axis was carried out in [21], leading to an explanation of the "double ring phenomenon". The propagation of polarization has been studied in [8].…”

Section: Historical Background To the Problemmentioning

confidence: 99%

“…Although propagation of singularities gives some information about the inverse problem, in many cases one needs more quantitative information about the solution. For instance, in [17] the results of [16] and [21] are used to solve an inverse problem for Maxwell's equations in a biaxial crystal when the bicharacteristic sheets are involutive. The results in [17] show how to estimate the singularities (e.g.…”

Section: Historical Background To the Problemmentioning

confidence: 99%

Parametrices for symmetric systems with multiplicity

Nolan

Uhlmann

2007

Wave Motion

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“…In this sense the set of bicharacteristics that is to be excluded is small. An analysis of conical refraction has been carried out by Melrose and Uhlmann [74] and Uhlmann [104]. When the elastic tensor (for n = 3) has symmetries it is determined by less than 21 coefficients.…”

Section: Propagation Of Elastic Waves In Smoothly Varying Mediamentioning

confidence: 99%

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Stolk¹,

Hoop²

2001

Comm Pure Appl Math

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Abstract. We review applications of microlocal analysis (MA) to reflection seismology. In this inverse method one attempts to estimate the index of refraction of waves in the earth from seismic data measured at the Earth's surface. Seismic imaging creates images of the Earth's upper crust using seismic waves generated by artificial sources and recorded into extensive arrays of sensors (geophones or hydrophones). The technology is based on a complex, and rapidly evolving, mathematical theory that employs advanced solutions to a wave equation as tools to solve approximately the general seismic inverse problem, with complications introduced by the heterogeneity and anisotropy of the Earth's crust. We describe several important developments using MA to generate these wave-solutions by manipulating the wavefields directly on their phase space. We also consider some recent applications of MA to global seismology.

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Light intensity distribution in conical refraction

Cited by 17 publications

References 4 publications

Singular asymptotics approach to partial differential equations with isolated singularities in the coefficients

Singular asymptotics approach to partial differential equations with isolated singularities in the coefficients

Parametrices for symmetric systems with multiplicity

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

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