formulasearchengine - User contributions [en]

Completeness of the real numbers

2013-12-27T10:04:06Z

71.246.72.13: /* Nested intervals theorem */

'''Multiple-prism grating laser oscillators''',<ref name="DLP">[[F. J. Duarte]], Narrow-linewidth pulsed dye laser oscillators, in ''Dye Laser Principles'' (Academic, New York, 1990) Chapter 4.</ref> or MPG laser oscillators, use [[beam expander|multiple-prism beam expansion]] to illuminate a diffraction grating mounted either in [[Littrow configuration]] or grazing-incidence configuration. Originally, these narrow-linewidth tunable dispersive oscillators were introduced as multiple-prism Littrow (MPL) grating oscillators,<ref>F. J. Duarte and J. A. Piper, A double-prism beam expander for pulsed dye lasers, ''Opt. Commun.'' '''35''', 100-104 (1980).</ref> or hybrid multiple-prism near-grazing-incidence (HMPGI) grating [[optical cavity|cavities]],<ref>F. J. Duarte and J. A. Piper, A prism preexpanded grazing incidence pulsed dye laser, ''Appl. Opt.'' '''20''', 2113-2116 (1981).</ref><ref>F. J. Duarte and J. A. Piper, Narrow linewidth high prf copper laser-pumped dye-laser oscillators, ''Appl. Opt.'' '''23''', 1391-1394 (1984).</ref> in organic [[dye laser]]s. However, these designs were quickly adopted for other types of lasers such as [[gas laser]]s,<ref>F. J. Duarte, Multiple-prism Littrow and grazing incidence pulsed CO2 lasers, ''Appl. Opt.'' '''24''', 1244-1245 (1985).</ref><ref>R. C. Sze and D. G. Harris, Tunable excimer lasers, in ''Tunable Lasers Handbook'', F. J. Duarte (Ed.) (Academic, New York, 1995) Chapter 3.</ref> [[diode laser]]s,<ref>P. Zorabedian, Characteristics of a grating-external-cavity semiconductor laser containing intracavity prism beam expanders, ''J. Lightwave Tech.'' '''10''', 330-335 (1992).</ref><ref>P. Zorabedian, Tunable external cavity semiconductor lasers, in ''Tunable Lasers Handbook'', F. J. Duarte (Ed.) (Academic, New York, 1995) Chapter 8.</ref> and more recently [[fiber laser]]s.<ref>T. M. Shay and F. J. Duarte, in ''Tunable Laser Applications'', 2nd Ed., F. J. Duarte (Ed.) (CRC, New York, 2009) Chapter 9.</ref>
[[Image:Duarte's multiple-prism grating laser oscillator.jpg|thumb|300px|Multiple-prism grating narrow-linewidth tunable laser oscillator.<ref>F. J. Duarte, T. S. Taylor, A. Costela, I. Garcia-Moreno, and R. Sastre, Long-pulse narrow-linewidth disperse solid-state dye laser oscillator, ''Appl. Opt.'' '''37''', 3987-3989 (1998).</ref> The grating in this particular oscillator is deployed in Littrow configuration.]]

==Excitation==

Multiple-prism grating laser oscillators can be excited either electrically, as in the case of gas lasers and semiconductor lasers,<ref name="TLO">F. J. Duarte, ''Tunable Laser Optics'' (Elsevier Academic, New York, 2003).</ref> or optically, as in the case of crystalline lasers and organic dye lasers.<ref name="DLP"/> In the case of optical excitation it is often necessary to match the polarization of the excitation laser to the polarization preference of the multiple-prism grating oscillator.<ref name="DLP"/> This can be done using a [[polarization rotator]] thus improving the laser conversion efficiency.<ref name="TLO"/>

==Linewidth performance==

The [[multiple-prism dispersion theory]] is applied to design these beam expanders either in additive configuration, thus adding or subtracting their dispersion to the dispersion of the grating, or in compensating configuration (yielding zero dispersion at a design wavelength) thus allowing the diffraction grating to control the tuning characteristics of the laser cavity.<ref name="TLO"/> Under those conditions, that is, zero dispersion from the multiple-prism beam expander, the single-pass [[laser linewidth]] is given by<ref name="DLP"/><ref name="TLO"/>

:<math> \Delta\lambda \approx \Delta \theta \left(M {\partial\theta\over\partial\lambda}\right)^{-1}</math>

where <math>\Delta \theta</math> is the beam divergence and ''M'' is the beam magnification provided by the beam expander that multiplies the angular dispersion provided by the diffraction grating. In the case of multiple-prism beam expanders this factor can be as high as 100-200.<ref name="DLP"/><ref name="TLO"/>

When the dispersion of the multiple-prism expander is not equal to zero, then the single-pass linewidth is given by<ref name="DLP"/><ref name="TLO"/>

:<math> \Delta\lambda \approx \Delta \theta \left(M {\partial\theta\over\partial\lambda} + {\partial\phi_{2,m}\over\partial\lambda} \right)^{-1}</math>

where the first differential refers to the angular dispersion from the grating and the second differential refers to the overall [[multiple-prism dispersion theory|dispersion from the multiple-prism beam expander]].<ref name="DLP"/><ref name="TLO"/>

Optimized solid-state multiple-prism grating laser oscillators have been shown, by [[F. J. Duarte|Duarte]], to generate pulsed single-longitudinal-mode emission limited only by [[Heisenberg's uncertainty principle]].<ref name=FJD1999>F. J. Duarte, Multiple-prism grating solid-state dye laser oscillator: optimized architecture, ''Appl. Opt.'' '''38''', 6347-6349 (1999).</ref> The [[laser linewidth]] in these experiments is reported as <math>\Delta \nu</math> ≈ 350 MHz (or <math>\Delta \lambda</math> ≈ 0.0004 nm at 590 nm) in pulses ~ 3 ns wide, at power levels in the kW regime.<ref name=FJD1999/>

==Applications==
Applications of these tunable narrow-linewidth lasers include:

* [[Coherent anti-Stokes Raman spectroscopy]] and combustion diagnostics<ref>R. J. Hall and A. C. Eckbreth, [[Coherent anti-Stokes Raman spectroscopy]]: applications to combustion diagnostics, in ''Laser Applications'' (Academic, New York, 1984) pp. 213-309.</ref>
* [[LIDAR]]<ref>W. B. Grant, Lidar for atmospheric and hydrospheric studies, in ''Tunable Laser Applications'', 1st Ed. (Marcel-Dekker, New York, 1995) Chapter 7.</ref>
* Laser [[spectroscopy]]<ref>[[W. Demtröder]], ''Laserspektroscopie: Grundlagen und Techniken'', 5th Ed. (Springer, Berlin, 2007).</ref><ref>W. Demtröder, ''Laser Spectroscopy: Basic Principles'', 4th Ed. (Springer, Berlin, 2008).</ref>
* [[Atomic vapor laser isotope separation]]<ref>S. Singh, K. Dasgupta, S. Kumar, K. G. Manohar, L. G. Nair, U. K. Chatterjee, High-power high-repetition-rate capper-vapor-pumped dye laser, ''Opt. Eng.'' '''33''', 1894-1904 (1994).</ref><ref>A. Sugiyama, T. Nakayama, M. Kato, Y. Maruyama, T. Arisawa, Characteristics of a pressure-tuned single-mode dye laser oscillator pumped by a copper vapor oscillator, ''Opt. Eng.'' '''35''', 1093-1097 (1996).</ref>

==See also==
*[[Dye lasers]]
*[[Solid state dye lasers]]
*[[optical cavity|Laser cavity]]
*[[Laser linewidth]]
*[[Multiple-prism dispersion theory]]
*[[Polarization rotator]]
*[[Tunable laser]]s

==References==
{{reflist}}

==External links==
*[http://www.opticsjournal.com/laseroscillators.htm Diagrams of MPG laser oscillators]
*[http://www.opticsjournal.com/tutorial.htm MPG tunable laser oscillators: Tutorial ]

[[Category:Optics]]
[[Category:Prisms]]
[[Category:Laser types]]

Sakuma–Hattori equation

2011-09-07T16:16:07Z

71.246.31.124: Added the units for wavelength and temperature that correspond to the value of the Second Radiation Constant used in the equation

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2010-02-02T20:47:44Z

71.246.85.162:

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