Chebyshev center: Difference between revisions

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Relaxed Chebyshev center: ... The constraint that \Delta \le x x^T is not convex. It has been changed to \Delta \ge x x^T. The claims at the end of the page are still valid as per the assumptions.
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'''Electromagnetically induced grating''' (EIG) is an optical [[Interference (wave propagation)|interference]] phenomenon where an [[interference pattern]] is used to build a dynamic spatial [[diffraction grating]] in matter. EIGs are dynamically created by [[light]] interference on optically resonant materials and rely on [[population inversion]] and/or optical [[coherence (physics)|coherence]] properties of the material. They were first demonstrated with population gratings on atoms.<ref>Mitsunaga et al.,  Phys. Rev. A 59, 4773 (1999)</ref>  EIGs can be used for purposes of atomic/molecular velocimetry,<ref>{{cite journal |doi=10.1016/S0030-4018(99)00228-X |title=Transient Bragg diffraction by a transferred population grating: application for cold atoms velocimetry |year=1999 |author=Tabosa, J |journal=Optics Communications |volume=165 |pages=59}}</ref> to probe the material optical properties such as coherence and population life-times,<ref>G.C.Cardoso, Phys. Rev. A 65, 033803 B (2002).</ref> and switching and routing of light.<ref>Brown AW, Xiao, M, Opt. Lett. 30, 699  2005 ; Su XM, Ham BS, Dynamic control of the photonic band gap using quantum coherence
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Phys. Rev. A 71, 013821 (2005)</ref> Related but different effects are thermally induced gratings and  [[photolithographic|photolithography]] gratings.


== Writing, Reading and Phase matching conditions for EIG diffraction ==
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Figure 1 shows a possible beam configuration to write and read an EIG. The period of the grating is controlled by the angle <math>\theta</math>. The writing and reading frequencies are not necessarily the same. E<sub>_B</sub> is referred as the "backward" reading beam and E<sub>R</sub> is the signal obtained by diffraction on the grating.
 
[[File:Write-Read EIG color.jpg|left|thumb|260px|'''Figure 1''': Writing and reading an EIG]]
 
The phase-matching conditions for the EIG for the plane-wave approximation is given by the simple geometric relation:
 
<math>\sin\beta=n \Bigl(\frac{\omega_1}{\omega_2}\Bigr) \sin(\theta/2)</math>,
 
where the angles are given according to Fig. 2, <math>\omega_1</math> and <math>\omega_2</math> are the frequencies of the writing (W, W') and reading beam (R), respectively, and n is the effective index of refraction of the medium.
 
[[File:Bragg-condition.jpg|center|thumb|350px|'''Figure 2''': Phase matching conditions for EIG diffraction]]
 
== Types of EIG ==
 
[[File:Grating-types.jpg|right|thumb|300px|'''Figure 3''': Difference between a "matter grating" a "population grating". The [[smiley]]s :-( and :-) represent ground and optically excited atoms, respectively.]]
 
'''Matter Gratings''': The writing lasers form a grating by modulating density of matter or by localizing matter (trapping) on the regions of maxima (or minima) of the writing interference fields. A thermal grating is an example. Matter gratings have slow dynamics (milliseconds) compared to population and phase gratings (potentially nanoseconds and faster).
 
'''Population Gratings''': The writing lasers are resonant with optical transitions in the matter and the grating is formed by [[optical pumping]]. (See Fig. 3)
 
'''Coherence Gratings''': A grating where the writing lasers form a coherent matter pattern. An example is a pattern of [[electromagnetically induced transparency]]
 
== Applications ==
 
Usually two [[laser]]s at an angle are used to build an EIG. The EIG is used to diffract a third laser, to  monitor the behavior of the underlying substrate where the EIG was written or to serve as a switch for one of the lasers involved in the process.
 
==See also==
*[[Atomic coherence]]
*[[Electromagnetically induced transparency]]
*[[Bragg's law]]
*[[Optical lattice]]
 
==Notes==
{{reflist}}
 
==References==
*Laser-induced thermal grating effects in flames, S Williams,La Rahn, Ph Paul, JW Forsman, RN Zare Opt. Lett. 19:2121, 1681–1683, 1994
*{{cite journal |doi=10.1103/PhysRevA.65.033803 |title=Electromagnetically induced gratings in a degenerate open two-level system |year=2002 |author=Cardoso, G. C. |journal=Physical Review A |volume=65 |pages=033803 |last2=Tabosa |first2=J. |issue=3}}
 
[[Category:Wave mechanics]]
[[Category:Quantum mechanics]]
[[Category:Optics]]

Revision as of 15:09, 7 February 2014

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