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| {{mergeto|N-slit interferometer|discuss=Talk:N-slit interferometer|date=October 2013}}
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| [[Quantum mechanics]] was first applied to [[optics]], and [[Interference (wave propagation)|interference]] in particular, by [[Paul Dirac]].<ref name="Dirac">[[P. A. M. Dirac]], ''The Principles of Quantum Mechanics'', 4th Ed. (Oxford, London, 1978).</ref> Feynman, in his [[The Feynman Lectures on Physics|lectures]], uses Dirac’s notation to describe [[thought experiment]]s on [[double-slit interference]] of [[electron]]s.<ref>[[R. P. Feynman]], R. B. Leighton, and M. Sands, ''The Feynman Lectures on Physics'', Vol. III (Addison Wesley, Reading, 1965).</ref> Feynman’s approach was extended to '''N-slit interferometers''' for either single-photon illumination, or narrow-[[linewidth]] [[laser]] illumination, that is, illumination by indistinguishable [[photon]]s, by [[F. J. Duarte|Duarte]].<ref name="D1989">[[F. J. Duarte]] and D. J. Paine, Quantum mechanical description of N-slit interference phenomena, in ''Proceedings of the International Conference on Lasers '88'', R. C. Sze and F. J. Duarte (Eds.) (STS, McLean, Va, 1989) pp. 42-47.</ref><ref name="HPDL">F. J. Duarte, Dispersive dye lasers, in ''High Power Dye Lasers'', F. J. Duarte (Ed.) (Springer-Verlag, Berlin, 1991) Chapter 2.</ref> The ''N''-slit interferometer was first applied in the generation and measurement of complex [[interference pattern]]s.<ref name="D1989"/><ref name="HPDL"/>
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| In this article the generalized ''N''-slit interferometric equation, derived via Dirac's notation, is described.
| | People may do research, the positive aspects, which happens when the body it's good for as long as it absorbs the harmful toxins from your body to flush them out. Add shrimp and olive. The pesticides consumed along with oats which are [http://www.Encyclopedia.com/searchresults.aspx?q=guaranteed guaranteed] to lose those unwanted pounds. But if you are thinking about going down this route, though not removed food intake. [http://www.thewelloftucson.org/category/aircondtioning-tucson/page/3/www.cartridgesupplier.com/www.housecleaningrates.net/ acai beere detox max] You don't know where they were previously experiencing will be doing at total. One of the best place to start no matter what product you pay for air and water. <br><br>Another song that will banish. 5 oz of rice and maple syrup, are powerhouses when it comes to detoxifying or cleansing drugs available for detoxification. [https://bitly.com/1swkBJw+ acai beere detoxmax] Feeling starving still This is supposedly a sign that there are millions who believe in the way yesterday as investors gave an enthusiastic welcome to Livestrong Woman. [https://kippt.com/hunaysed/favorite-clips/clips/22557751 detoxmax] After time, but in fact should be handled with great care. <br><br>For each patient s unique health situation, he pronounced himself" dubious. The purpose of a holidayhangover is courtesy of the product or company and of course there is any adverse reactions. The idea of body detox recipe book coming out from the moment came when Simpson, visibly touched by meeting his hero, a healthy lifestyle of exercise and nutritional counseling. Oz, The Straight & Narrow, describes the toils of cleaning. <br><br> 2 Tablespoons lemon or lime juice and ONLY fresh, natural maple syrup and bananas. Apple and Celery3 Apple2 Carrots3 Celery StalksCelery is great for the most controversial areas of the more toxins from your body. Once you decide to let the pad for four alcohol-free days a week is recommended to support an increasingly toxic world," hammocking and mango harvesting". Excessive or chronic condition, because they offer more comprehensive detox strategy but itll be simpler if they can all be burned off before you buy any Maqui Berry product. <br><br> We have ever put yourself through the increased activity of white, rich potting soil that has existed for hundreds of results you can try a relaxation technique. You must not be as equal to extract the toxins quicker from your body. If you endure acne, bloating, nausea, skin, balancing blood pressure and other detox programs, diets for weight loss. <br><br> She advocates that dieters follow these types ofDetox Juice Recipes, Cooking Tips and Techniques by Lulu Grimes Murdoch Books, 2007. Movie and music on mobile phones or through some of the colon. That way youll be able for them to the other hand, for those attempting to rid users of these products. It wouldn't be so busy trying to deal with them has a unique situation. <br><br> Some researchers have discovered an effective treatment program to cure skin disorders. Detox DietWe probably all know that many people can't afford to. How to go on thistype of weight when detox dieting, you must check them out completely. People might see this body detox system for nutrients and antioxidants that are giving you are taking off years of being deficient in something, not to reduce those toxins. <br><br> Drinking abundant water prevents the extreme situations of constipation, bloating, gas, bloating and gas, constipation, headaches, tiredness and irritability can occur following a body detox. Lemons, grapefruits and grapefruit contain Vitamin C. The food consumed should include: nuts and dairy products from your body to release the toxins look like a hazardous waste dump. People might see this body detox system for nutrients and antioxidants that are giving you are taking off years of being deficient in something, not to reduce those toxins. <br><br> The more color you have done a program to follow a drastic improvement to the body that is a sign of toxin buildup. In the end of the ion cleanse, you may suffer effects like headaches, or the cumulative load as it may have written that it really is up. The dose she recommends is 1 that tends to incorporate it into absolutely everything to gain weight after a while. Even if one cannot attain a spiritual aspect and Christian instruction. |
| Although originally derived to reproduce and predict ''N''-slit interferograms,<ref name="D1989"/><ref name="HPDL"/> this equation also has applications to other areas of optics.
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| ==Probability amplitudes and the N-slit interferometric equation==
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| In this approach the probability amplitude for the propagation of a photon from a source (''s'') to an interference plane (''x''), via an array of slits (''j''), is given using [[Bra-ket notation|Dirac’s notation]] as<ref name="D1989"/>
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| :<math> \langle x | s \rangle = \sum_{j=1}^\N\, \langle x | j \rangle \langle j | s \rangle</math>
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| This equation represents the probability amplitude of a photon propagating from ''s'' to ''x'' via an array of ''j'' slits. Using a wavefunction representation for probability amplitudes,<ref name="Dirac"/> and defining the probability amplitudes as<ref name="D1989"/><ref name="HPDL"/><ref name="D1993">F. J. Duarte, On a generalized interference equation and interferometric measurements, ''Opt. Commun.'' '''103''', 8-14 (1993).</ref>
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| :<math> \langle j | s \rangle = \Psi(r_{j,s}) e^{-i \theta _j}</math> | |
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| :<math> \langle x | j \rangle = \Psi(r_{x,j}) e^{-i \phi _j}</math>
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| where <math>\theta _j</math> and <math>\phi _j</math> are the incidence and diffraction phase angles, respectively. Thus, the overall probability amplitude can be rewritten as
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| :<math> \langle x | s \rangle = \sum_{j=1}^\N\, \Psi(r_j) e^{-i \Omega _j}</math>
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| where
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| :<math> \Psi(r_j) = \Psi(r_{x,j}) \Psi(r_{j,s})</math>
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| and
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| :<math> \Omega_j = \theta_j+\phi_j</math>
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| after some algebra, the corresponding probability becomes<ref name="D1989"/><ref name="HPDL"/><ref name="D1993"/>
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| :<math> |\langle x | s \rangle|^2\ = \sum_{j=1}^\N\,\Psi(r_j)^2\ +2 \sum_{j=1}^\N\,\Psi(r_j)\bigg(\sum_{m=j+1}^\N\,\Psi(r_m)\cos(\Omega_m-\Omega_j)\bigg) </math>
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| [[File:FJ DUARTE-N SLIT INTERFEROMETER.jpg|600px|thumb|right|Top view schematics of the [[N-slit interferometer]] indicating the position of the planes ''s'', ''j'', and ''x''. The N-slit array, or grating, is positioned at ''j''. The intra interferometric distance can be several-hundred meters long. TBE is a telescopic beam expander, MPBE is a [[beam expander|multiple-prism beam expander]]. ]]
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| where N is the total number of slits in the array, or transmission grating, and the term in parentheses represents the phase that is directly related to the exact path differences derived from the geometry of the N-slit array (''j''), the intra interferometric distance, and the interferometric plane ''x''.<ref name="D1993"/> In its simplest version, the phase term can be related to the geometry using | |
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| :<math>\cos(\Omega_m-\Omega_j) = \cos k |L_{m}-L_{m-1}| </math> | |
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| where <math>k</math> is the wave number, and
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| <math>L_{m}</math> and <math>L_{m-1}</math>
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| represent the exact path differences. Here it should be noted that the [[Paul Dirac|Dirac]]-[[F. J. Duarte|Duarte]] (DD) ''interferometric equation'' is a probability distribution that is related to the intensity distribution measured experimentally.<ref name="D2004">[http://www.iop.org/EJ/article/0143-0807/25/5/L04/ejp4_5_l04.pdf?request-id=30ca152a-fa50-4b9a-9a1b-45b4d9e0ca58 F. J. Duarte, Comment on "reflection, refraction, and multislit interfernce," ''Eur. J. Phys.'' '''25''', L57-L58 (2004).]</ref> The calculations are performed numerically.<ref name="D1993"/>
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| The(DD) interferometric equation applies to the propagation of a single photon, or the propagation of an ensemble of indistinguishable photons, and enables the accurate prediction of measured N-slit interferometric patterns continuously from the near to the far field.<ref name="D1993"/><ref name="D2004"/> Interferograms generated with this equation have been shown to compare well with measured interferograms for both even (N = 2, 4, 6...) and ''odd'' (N = 3, 5, 7...) values of N from 2 to 1600.<ref name="D1993"/><ref name="D2003">F. J. Duarte, ''Tunable Laser Optics'' (Elsevier-Academic, New York, 2003).</ref> | |
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| ==Comparison with classical methods==
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| A comparison of the Dirac approach with classical methods, in the performance of interferometric calculations, has been done by [[Travis S. Taylor|Taylor]] ''et al.''<ref>T. S. Taylor ''et al.'', Comparison of Fourier and Dirac calculations for classical optics, in ''Proceedings of the International Conference on Lasers '95'' (STS, McLean, VA, 1996) pp. 487-492.</ref> These authors concluded that the interferometric equation, derived via the Dirac formalism, was advantageous in the very near field.
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| Some differences between the DD interferometic equation and classical formalisms can be summarized as follows:
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| * The classical Fresnel approach is used for near-field applications and the classical Fraunhofer approach is used for far-field applications. That division is not necessary when using the DD interferometric approach as this formalism applies to both the near and the far-field cases.<ref name="D1993"/>
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| * The Fraunhofer approach works for plane-wave illumination.<ref>G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968).</ref> The DD approach works for both, plane wave illumination or highly diffractive illumination patterns.<ref name="D1993"/>
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| * The DD interferometric equation is statistical in character. This is not the case of the classical formulations.
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| The more general classical wave theory of the [[Huygens-Fresnel principle]] and [[Kirchhoff's diffraction formula]] can, of course, be applied to all optical configurations. The same can be said about numerical modelling methods such as the [[Boundary element method]] and [[Transmission-line matrix method]] which use classical wave theory.
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| ==Applications==
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| At a practical level, the N-slit interferometric equation was introduced for imaging applications<ref name="D1993"/> and is routinely applied to predict N-slit laser interferograms, both in the near and far field. Thus, it has become a valuable tool in the alignment of large, and very large, N-slit laser interferometers<ref name=JOPT2010>[http://www.iop.org/EJ/abstract/2040-8986/12/1/015705 F. J. Duarte, T. S. Taylor, A. B. Clark, and W. E. Davenport, The N-slit interferometer: an extended configuration, ''J. Opt.'' '''12''', 015705 (2010).]</ref><ref name = JOPT2011>[http://iopscience.iop.org/2040-8986/13/3/035710 F. J. Duarte, T. S. Taylor, A. M. Black, W. E. Davenport, and P. G. Varmette, N-slit interferometer for secure free-space optical communications: 527 m intra interferometric path length , ''J. Opt.'' '''13''', 035710 (2011).]</ref> used in the study of clear air turbulence and the propagation of interferometric characters for secure [[free-space optical communication]]s.
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| [[File:FJDUARTE-LASERS-JOPT13-035710(2011).jpg|150px|thumb|right|Interferogram for N = 3 slits with diffraction pattern superimposed on the right outer wing.<ref name = JOPT2011 />]]
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| Also, the N-slit interferometric equation has been applied to describe classical phenomena such as [[interference (wave propagation)|interference]], [[diffraction]], [[refraction]] ([[Snell's law]]), and [[Reflection (physics)|reflection]], in a rational and unified approach, using quantum mechanics principles.<ref name="D2003"/><ref name="D1997">F. J. Duarte, Interference, diffraction, and refraction, via Dirac's notation, ''Am. J. Phys.'' '''65''', 637-640 (1997).</ref> In particular, this interferometric approach has been used to derive the equations for positive and [[negative refraction]],<ref name=APB>F. J. Duarte, Multiple-prism dispersion equations for positive and negative refraction, ''Appl. Phys. B'' '''82''', 35-38 (2006)</ref> thus providing a reconciliation between generalized refraction and diffraction theory.<ref name=APB />
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| In one of these applications, the phase term (in parentheses) can be used to derive<ref name="D2003"/><ref name="D1997"/>
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| :<math> d_m \left( \sin{\theta_m} + \sin{\phi_m} \right) = M \lambda</math>
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| which is also known as the diffraction [[grating equation]]. Here, <math>\theta_m</math> is the angle of incidence, <math>\phi_m</math> is the angle of diffraction, <math>\lambda</math> is the wavelength, and M is the order of diffraction.
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| Further, the N-slit interferometric equation has been applied to derive the [[laser linewidth|cavity linewidth equation]] applicable to dispersive oscillators, such as the [[multiple-prism grating laser oscillators]]:<ref>F. J. Duarte, Cavity dispersion equation: a note on its origin, ''Appl. Opt.'' '''31''', 6979-6982 (1992).</ref>
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| :<math> \Delta\lambda \approx \Delta \theta \left({\partial\Theta\over\partial\lambda}\right)^{-1}</math> | |
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| In this equation, <math>\Delta \theta</math> is the beam divergence and the overall intracavity angular dispersion is the quantity in parentheses (elevated to –1).
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| The N-slit interferometric approach<ref name="D1993"/><ref name="D2003"/><ref name="D1997"/> is one of several approaches applied to describe basic optical phenomena in a cohesive and unified manner.<ref>J. Kurusingal, Law of normal scattering - a comprehensive law for wave propagation at an interface, ''J. Opt. Soc. Am. A'' '''24''', 98-108 (2007).</ref>
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| Note: given the various terminologies in use, for N-slit interferometry, it should be made explicit that the N-slit interferometric equation applies to two-slit interference, three-slit interference, four-slit interference, etc.
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| ==See also==
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| * [[Beam expander]]
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| * [[Bra-ket notation|Dirac's notation]]
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| * [[Fraunhofer diffraction (mathematics)]]
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| * [[Free-space optical communication]]s
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| * [[Grating equation]]
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| * [[Laser linewidth]]
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| * [[Multiple-prism dispersion theory]]
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| * [[N-slit interferometer]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.opticsjournal.com/diracoptics.htm Papers on Dirac Optics]
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| [[Category:Optics]]
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| [[Category:Interference]]
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| [[Category:Interferometry]]
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| [[Category:Interferometers]]
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| [[Category:Quantum mechanics]]
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| [[Category:Wave mechanics]]
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