Equitable coloring: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>MathsPoetry
No edit summary
 
en>49ersBelongInSanFrancisco
 
Line 1: Line 1:
The first description of multiple-prism arrays, and multiple-prism dispersion, was given by [[Isaac Newton|Newton]] in his book ''[[Opticks]]''.<ref>I. Newton, ''Opticks'' (Royal Society, London, 1704).</ref>  Prism pair expanders were introduced by [[David Brewster|Brewster]] in 1813.<ref>D. Brewster, ''A Treatise on New Philosophical Instruments for Various Purposes in the Arts and Sciences with Experiments on Light and Colours'' (Murray and Blackwood, Edinburgh, 1813).</ref> A modern mathematical description of the single-prism dispersion was given by [[Max Born|Born]] and [[Emil Wolf|Wolf]] in 1959.<ref name=B1>M. Born and E. Wolf, ''Principles of Optics'', 7th Ed. (Cambridge University, Cambridge, 1999).</ref>  The generalized multiple-prism dispersion theory was introduced by [[F. J. Duarte|Duarte]] and Piper<ref name="DuarteOC">F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers", ''Opt. Commun.'' '''43''', 303–307 (1982).</ref><ref name="DuarteAJP">F. J. Duarte and J. A. Piper, "Generalized prism dispersion theory", ''Am. J. Phys.'' '''51''', 1132–1134 (1982).</ref> in 1982.


==Generalized multiple-prism dispersion equations==


The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. Here is more information about [http://kedahan.net/wordpress_backup_plugin_882917 backup plugin] take a look at the web page. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer. <br><br>But before choosing any one of these, let's compare between the two. To sum up, ensure that the tactics are aiming to increase the ranking and attracting the maximum intended traffic in the major search engines. For a much deeper understanding of simple wordpress themes", check out  Upon browsing such, you'll be able to know valuable facts. Provide the best and updated information to the web searchers and make use of these wonderful free themes and create beautiful websites. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Next you'll go by way of to your simple Word - Press site. So, we have to add our social media sharing buttons in website. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. The days of spending a lot of time and money to have a website built are long gone. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.
The generalized mathematical description of multiple-prism dispersion, as a function of the angle of incidence, prism geometry, prism refractive index, and number of prisms, was introduced as a design tool for [[multiple-prism grating laser oscillators]] by [[F. J. Duarte|Duarte]] and Piper,<ref name="DuarteOC"/><ref name="DuarteAJP"/> and is given by
 
:<math>(\partial\phi_{2,m}/\partial\lambda) = H_{2,m} (\partial n_m/\partial\lambda) + (k_{1,m}k_{2,m})^{-1}\bigg(H_{1,m}(\partial n_m/\partial\lambda) + (\partial\phi_{2,(m-1)}/\partial\lambda)\bigg)</math>
 
:<math>(\partial\phi_{2,m}/\partial\lambda) = H_{2,m} (\partial n_m/\partial\lambda) + (k_{1,m}k_{2,m})^{-1}\bigg(H_{1,m}(\partial n_m/\partial\lambda) - (\partial\phi_{2,(m-1)}/\partial\lambda)\bigg)</math>
[[Image:Duarte's multiple-prism grating laser oscillator.jpg|thumb|300px|Multiple-prism beam expander grating configuration as used in narrow-linewidth tunable laser oscillators<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>]]
 
where
 
:<math>\,k_{1,m}=cos\psi_{1,m}/cos\phi_{1,m}</math>
 
:<math>\,k_{2,m}=cos\phi_{2,m}/cos\psi_{2,m}</math>
 
:<math>\,H_{1,m}=(tan\phi_{1,m})/n_m</math>
 
:<math>\,H_{2,m}=(tan\phi_{2,m})/n_m</math>
 
Here, <math>\phi_{1,m}</math> is the angle of incidence, at the ''m''th prism, and <math>\psi_{1,m}</math> its corresponding angle of refraction. Similarly, <math>\phi_{2,m}</math> is the exit angle and <math>\psi_{2,m}</math> its corresponding angle of refraction.  The two main equations give the first order dispersion for an array of ''m'' prisms at the exit surface of the ''m''th prism. The plus sign in the second term in parentheses refers to a positive dispersive configuration while the minus sign refers to a compensating configuration.<ref name="DuarteOC"/><ref name="DuarteAJP"/>  The ''k'' factors are the corresponding beam expansions, and the ''H'' factors are additional geometrical quantities. It can also be seen that the dispersion of the ''m''th prism depends on the dispersion of the previous prism (''m'' - 1).
 
These equations can also be used to quantify the angular dispersion in prism arrays, as described in [[Isaac Newton]]'s book ''[[Opticks]]'', and as deployed in dispersive instrumentation such as multiple-prism spectrometers.  A comprehensive review on practical multiple-prism [[beam expander]]s and multiple-prism angular dispersion theory, including explicit and ready to apply equations (engineering style), is given by Duarte.<ref name="TLO"/>
 
More recently, the generalized multiple-prism dispersion theory has been extended to include positive and [[negative refraction]].<ref>F. J. Duarte, Multiple-prism dispersion equations for positive and negative refraction, ''Appl. Phys. B'' '''82''', 35-38 (2006).</ref> Also, higher order phase derivatives have been derived using a Newtonian iterative approach.<ref>[http://www.springerlink.com/content/98gr542030t720h7/?p=521f833aeb0c4542a661a42336f49994&pi=28 F. J. Duarte, Generalized multiple-prism dispersion theory for laser pulse compression: higher order phase derivatives, ''Appl. Phys. B'' '''96''', 809-814 (2009)].</ref> This extension of the theory enables the evaluation of the Nth higher derivative via an elegant mathematical framework.  Applications include further refinements in the design of prism pulse compressors and nonlinear optics.
 
===Single prism dispersion===
 
For a single generalized prism (''m'' = 1), the Duarte-Piper equation simplifies to<ref name=B1/><ref name="DLP">F. J. Duarte, Narrow-linewidth pulsed dye laser oscillators, in ''Dye Laser Principles'' (Academic, New York, 1990) Chapter 4.</ref>
 
:<math>(\partial\phi_{2,1}/\partial\lambda) =  (sin\psi_{2,1}/cos\phi_{2,1})(\partial n_1/\partial\lambda) + (cos\psi_{2,1}/cos\phi_{2,1})tan\psi_{1,1}(\partial n_1/\partial\lambda)</math>
 
If the single prism is a right-angled prism with the beam exiting normal to the output face, that is <math>\phi_{2,m}</math> equal to zero, this equation reduces to<ref name="TLO">F. J. Duarte, ''Tunable Laser Optics'' (Elsevier Academic, New York, 2003) Chapter 4.</ref>
 
:<math>(\partial\phi_{2,1}/\partial\lambda) = tan\psi_{1,1} (\partial n_1/\partial\lambda)</math>
 
[[Image:prism-compressor.svg|thumb|300px|A two-prism pulse compressor as deployed in some femtosecond laser configurations.]]
[[Image:Multiple prisms used for tuning a dye laser.JPG|thumb|300px|This multiple-prism arrangement is used with a [[diffraction grating]] to provide tuning in a dye laser.]]
 
==Intracavity dispersion and laser linewidth==
 
The first application of this theory was to evaluate the [[laser linewidth]] in multiple-prism grating  laser oscillators.<ref name="DuarteOC"/>  The total intracavity angular dispersion plays an important role in the [[laser linewidth|linewidth narrowing]] of pulsed tunable lasers through the equation<ref name="DuarteOC"/><ref name="TLO"/>
 
:<math> \Delta\lambda \approx \Delta \theta \left({\partial\Theta\over\partial\lambda}\right)^{-1}</math>
 
where <math>\Delta \theta</math> is the beam divergence and the ''overall intracavity angular dispersion'' is the quantity in parentheses (elevated to –1).  Although originally classical in origin, in 1992 it was shown that this laser cavity linewidth equation can also be derived from [[N-slit interferometric equation|interferometric quantum principles]].<ref>F. J. Duarte, Cavity dispersion equation: a note on its origin, ''Appl. Opt.'' '''31''', 6979-6982 (1992).</ref>
 
For the special case of zero dispersion from the multiple-prism beam expander, the single-pass [[laser linewidth]] is given by<ref name="TLO"/><ref name="DLP"/>
 
:<math> \Delta\lambda \approx \Delta \theta \left(M {\partial\theta\over\partial\lambda}\right)^{-1}</math>
 
where ''M'' is the beam magnification provided by the beam expander that multiplies the angular dispersion provided by the diffraction grating.  In practice, ''M'' can be as high as 100-200.<ref name="TLO"/><ref name="DLP"/>
 
When the dispersion of the multiple-prism expander is not equal to zero, then the single-pass linewidth is given by<ref name="DuarteOC"/><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 dispersion from the multiple-prism beam expander (given in the section above).<ref name="TLO"/><ref name="DLP"/>
 
==Further applications==
 
In 1987 the multiple-prism angular dispersion theory was extended to provide explicit second order equations directly applicable to the design of [[prism compressor|prismatic pulse compressors]].<ref>F. J. Duarte, "Generalized multiple-prism dispersion theory for pulse compression in ultrafast dye lasers", ''Opt. Quantum Electron.'' '''19''', 223–229 (1987)</ref>
The generalized multiple-prism dispersion theory is applicable to:
 
* [[Amici prism]]s<ref>F. J. Duarte, Tunable organic dye lasers: physics and technology of high-performance liquid and solid-state narrow-linewidth oscillators, ''Progress in Quantum Electronics'' '''36''', 29-50 (2012).</ref><ref>F. J. Duarte, Tunable laser optics: applications to optics and quantum optics, ''Progress in Quantum Electronics'' '''37''', 326-347 (2013). </ref>
* laser [[microscopy]],<ref>B. A. Nechay, U. Siegner, M. Achermann, H. Bielefeldt, and U. Keller, Femtosecond pump-probe near-field optical microscopy, ''Rev. Sci. Instrum.'' '''70''', 2758-2764 (1999).</ref><ref>U. Siegner, M. Achermann, and U. Keller, Spatially resolved femtosecond spectroscopy beyond the diffraction limit, ''Meas. Sci. Technol.'' '''12''', 1847-1857 (2001).</ref>
* narrow-linewidth [[tunable laser]] design,<ref>[http://www.opticsjournal.com/tlo.htm F. J. Duarte, ''Tunable Laser Optics'' (Elsevier Academic, New York, 2003) Chapter 7.]</ref>
* [[beam expander|prismatic beam expanders]]<ref name="DuarteOC"/><ref name="DuarteAJP"/>
* [[prism compressor]]s for [[femtosecond pulse]] lasers.<ref>L. Y. Pang, J. G. Fujimoto, and E. S. Kintzer, Ultrashort-pulse generation from high-power diode arrays by using intracavity optical nonlinearities, ''Opt. Lett.'' '''17''', 1599-1601 (1992).</ref><ref>K. Osvay, A. P. Kovács, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser, ''Opt. Commun.'' '''248''', 201-209 (2005).</ref><ref>J. C. Diels and W. Rudolph, ''Ultrashort Laser Pulse Phenomena'', 2nd Ed. (Elsevier Academic, New York, 2006).</ref>
 
==See also==
 
* [[Beam expander]]
* [[Laser linewidth]]
* [[Multiple-prism grating laser oscillator]]
 
==References==
{{reflist}}
 
==External links==
*[http://www.opticsjournal.com/multiple-prismtheory.htm References on multiple-prism dispersion theory]
*[http://www.opticsjournal.com/prismpulsecompression.htm Prism and Multiple-Prism Pulse Compression: Tutorial ]
 
[[Category:Optics]]
[[Category:Nonlinear optics]]
[[Category:Prisms]]

Latest revision as of 10:16, 8 January 2014

The first description of multiple-prism arrays, and multiple-prism dispersion, was given by Newton in his book Opticks.[1] Prism pair expanders were introduced by Brewster in 1813.[2] A modern mathematical description of the single-prism dispersion was given by Born and Wolf in 1959.[3] The generalized multiple-prism dispersion theory was introduced by Duarte and Piper[4][5] in 1982.

Generalized multiple-prism dispersion equations

The generalized mathematical description of multiple-prism dispersion, as a function of the angle of incidence, prism geometry, prism refractive index, and number of prisms, was introduced as a design tool for multiple-prism grating laser oscillators by Duarte and Piper,[4][5] and is given by

(∂ϕ2,m/∂λ)=H2,m(∂nm/∂λ)+(k1,mk2,m)−1(H1,m(∂nm/∂λ)+(∂ϕ2,(m−1)/∂λ))
(∂ϕ2,m/∂λ)=H2,m(∂nm/∂λ)+(k1,mk2,m)−1(H1,m(∂nm/∂λ)−(∂ϕ2,(m−1)/∂λ))
File:Duarte's multiple-prism grating laser oscillator.jpg
Multiple-prism beam expander grating configuration as used in narrow-linewidth tunable laser oscillators[6]

where

k1,m=cosψ1,m/cosϕ1,m
k2,m=cosϕ2,m/cosψ2,m
H1,m=(tanϕ1,m)/nm
H2,m=(tanϕ2,m)/nm

Here, ϕ1,m is the angle of incidence, at the mth prism, and ψ1,m its corresponding angle of refraction. Similarly, ϕ2,m is the exit angle and ψ2,m its corresponding angle of refraction. The two main equations give the first order dispersion for an array of m prisms at the exit surface of the mth prism. The plus sign in the second term in parentheses refers to a positive dispersive configuration while the minus sign refers to a compensating configuration.[4][5] The k factors are the corresponding beam expansions, and the H factors are additional geometrical quantities. It can also be seen that the dispersion of the mth prism depends on the dispersion of the previous prism (m - 1).

These equations can also be used to quantify the angular dispersion in prism arrays, as described in Isaac Newton's book Opticks, and as deployed in dispersive instrumentation such as multiple-prism spectrometers. A comprehensive review on practical multiple-prism beam expanders and multiple-prism angular dispersion theory, including explicit and ready to apply equations (engineering style), is given by Duarte.[7]

More recently, the generalized multiple-prism dispersion theory has been extended to include positive and negative refraction.[8] Also, higher order phase derivatives have been derived using a Newtonian iterative approach.[9] This extension of the theory enables the evaluation of the Nth higher derivative via an elegant mathematical framework. Applications include further refinements in the design of prism pulse compressors and nonlinear optics.

Single prism dispersion

For a single generalized prism (m = 1), the Duarte-Piper equation simplifies to[3][10]

(∂ϕ2,1/∂λ)=(sinψ2,1/cosϕ2,1)(∂n1/∂λ)+(cosψ2,1/cosϕ2,1)tanψ1,1(∂n1/∂λ)

If the single prism is a right-angled prism with the beam exiting normal to the output face, that is ϕ2,m equal to zero, this equation reduces to[7]

(∂ϕ2,1/∂λ)=tanψ1,1(∂n1/∂λ)
A two-prism pulse compressor as deployed in some femtosecond laser configurations.
This multiple-prism arrangement is used with a diffraction grating to provide tuning in a dye laser.

Intracavity dispersion and laser linewidth

The first application of this theory was to evaluate the laser linewidth in multiple-prism grating laser oscillators.[4] The total intracavity angular dispersion plays an important role in the linewidth narrowing of pulsed tunable lasers through the equation[4][7]

Δλ≈Δθ(∂Θ∂λ)−1

where Δθ is the beam divergence and the overall intracavity angular dispersion is the quantity in parentheses (elevated to –1). Although originally classical in origin, in 1992 it was shown that this laser cavity linewidth equation can also be derived from interferometric quantum principles.[11]

For the special case of zero dispersion from the multiple-prism beam expander, the single-pass laser linewidth is given by[7][10]

Δλ≈Δθ(M∂θ∂λ)−1

where M is the beam magnification provided by the beam expander that multiplies the angular dispersion provided by the diffraction grating. In practice, M can be as high as 100-200.[7][10]

When the dispersion of the multiple-prism expander is not equal to zero, then the single-pass linewidth is given by[4][7]

Δλ≈Δθ(M∂θ∂λ+∂ϕ2,m∂λ)−1

where the first differential refers to the angular dispersion from the grating and the second differential refers to the overall dispersion from the multiple-prism beam expander (given in the section above).[7][10]

Further applications

In 1987 the multiple-prism angular dispersion theory was extended to provide explicit second order equations directly applicable to the design of prismatic pulse compressors.[12] The generalized multiple-prism dispersion theory is applicable to:

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. ↑ I. Newton, Opticks (Royal Society, London, 1704).
  2. ↑ D. Brewster, A Treatise on New Philosophical Instruments for Various Purposes in the Arts and Sciences with Experiments on Light and Colours (Murray and Blackwood, Edinburgh, 1813).
  3. ↑ 3.0 3.1 M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University, Cambridge, 1999).
  4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers", Opt. Commun. 43, 303–307 (1982).
  5. ↑ 5.0 5.1 5.2 5.3 F. J. Duarte and J. A. Piper, "Generalized prism dispersion theory", Am. J. Phys. 51, 1132–1134 (1982).
  6. ↑ 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).
  7. ↑ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 F. J. Duarte, Tunable Laser Optics (Elsevier Academic, New York, 2003) Chapter 4.
  8. ↑ F. J. Duarte, Multiple-prism dispersion equations for positive and negative refraction, Appl. Phys. B 82, 35-38 (2006).
  9. ↑ F. J. Duarte, Generalized multiple-prism dispersion theory for laser pulse compression: higher order phase derivatives, Appl. Phys. B 96, 809-814 (2009).
  10. ↑ 10.0 10.1 10.2 10.3 F. J. Duarte, Narrow-linewidth pulsed dye laser oscillators, in Dye Laser Principles (Academic, New York, 1990) Chapter 4.
  11. ↑ F. J. Duarte, Cavity dispersion equation: a note on its origin, Appl. Opt. 31, 6979-6982 (1992).
  12. ↑ F. J. Duarte, "Generalized multiple-prism dispersion theory for pulse compression in ultrafast dye lasers", Opt. Quantum Electron. 19, 223–229 (1987)
  13. ↑ F. J. Duarte, Tunable organic dye lasers: physics and technology of high-performance liquid and solid-state narrow-linewidth oscillators, Progress in Quantum Electronics 36, 29-50 (2012).
  14. ↑ F. J. Duarte, Tunable laser optics: applications to optics and quantum optics, Progress in Quantum Electronics 37, 326-347 (2013).
  15. ↑ B. A. Nechay, U. Siegner, M. Achermann, H. Bielefeldt, and U. Keller, Femtosecond pump-probe near-field optical microscopy, Rev. Sci. Instrum. 70, 2758-2764 (1999).
  16. ↑ U. Siegner, M. Achermann, and U. Keller, Spatially resolved femtosecond spectroscopy beyond the diffraction limit, Meas. Sci. Technol. 12, 1847-1857 (2001).
  17. ↑ F. J. Duarte, Tunable Laser Optics (Elsevier Academic, New York, 2003) Chapter 7.
  18. ↑ L. Y. Pang, J. G. Fujimoto, and E. S. Kintzer, Ultrashort-pulse generation from high-power diode arrays by using intracavity optical nonlinearities, Opt. Lett. 17, 1599-1601 (1992).
  19. ↑ K. Osvay, A. P. Kovács, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser, Opt. Commun. 248, 201-209 (2005).
  20. ↑ J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd Ed. (Elsevier Academic, New York, 2006).
Retrieved from "https://en.formulasearchengine.com/index.php?title=Equitable_coloring&oldid=23784"