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| In the fields of [[nonlinear optics]] and [[fluid dynamics]], '''modulational instability''' or '''sideband instability''' is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of [[Frequency spectrum|spectral]]-sidebands and the eventual breakup of the waveform into a train of [[wave packet|pulses]].<ref name="BenjaminFeir">{{cite journal
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| | doi = 10.1017/S002211206700045X
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| | volume = 27
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| | issue = 3
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| | pages = 417–430
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| | last1 = Benjamin
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| | first1 = T. Brooke
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| | author1-link = T. Brooke Benjamin
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| | first2 = J.E.
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| | last2 = Feir
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| | title = The disintegration of wave trains on deep water. Part 1. Theory
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| | journal = Journal of Fluid Mechanics
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| | year = 1967
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| |bibcode = 1967JFM....27..417B }}</ref><ref>{{Cite journal
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| | doi = 10.1098/rspa.1967.0123
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| | volume = 299
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| | issue = 1456
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| | pages = 59–76
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| | last = Benjamin
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| | first = T.B.
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| | author-link = T. Brooke Benjamin
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| | title = Instability of Periodic Wavetrains in Nonlinear Dispersive Systems
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| | journal = Proceedings of the Royal Society of London
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| | series = A. Mathematical and Physical Sciences
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| | year = 1967
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| |bibcode = 1967RSPSA.299...59B }} Concluded with a discussion by [[Klaus Hasselmann]].</ref><ref name="agrawal">{{cite book
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| | last = Agrawal
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| | first = Govind P.
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| | year = 1995
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| | title = Nonlinear fiber optics
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| | place = San Diego (California)
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| | publisher = Academic Press
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| | edition =2nd
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| | isbn = 0-12-045142-5
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| }}</ref>
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| The phenomenon was first discovered − and modelled − for periodic [[surface gravity wave]]s ([[Stokes wave]]s) on deep water by [[T. Brooke Benjamin]] and Jim E. Feir, in 1967.<ref>{{Cite journal
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| | doi = 10.1146/annurev.fl.12.010180.001511
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| | volume = 12
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| | pages = 303−334
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| | last1 = Yuen
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| | first1 = H.C.
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| | first2 = B.M.
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| | last2 = Lake
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| | title = Instabilities of waves on deep water
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| | journal = Annual Review of Fluid Mechanics
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| | year = 1980
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| |bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. It is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal
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| | doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
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| | volume = 33
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| | issue = 4
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| | pages = 863−884
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| | last = Janssen
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| | first = Peter A.E.M.
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| | title = Nonlinear four-wave interactions and freak waves
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| | journal = Journal of Physical Oceanography
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| | year = 2003
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| |bibcode = 2003JPO....33..863J }}</ref><ref>{{Cite journal
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| | doi = 10.1146/annurev.fluid.40.111406.102203
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| | volume = 40
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| | pages = 287−310
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| | last1 = Dysthe
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| | first1 = Kristian
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| | first2 = Harald E.
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| | last2 = Krogstad
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| | first3 = Peter
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| | last3 = Müller
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| | title = Oceanic rogue waves
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| | journal = Annual Review of Fluid Mechanics
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| | year = 2008
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| |bibcode = 2008AnRFM..40..287D }}</ref>
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| ==Initial instability and gain==
| | CMS provides the best platform to create websites that fulfill all the specifications of SEO. It is very easy to customize plugins according to the needs of a particular business. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Word - Press also provides protection against spamming, as security is a measure issue. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>The Internet is a vast open market where businesses and consumers congregate. If you are a positive thinker businessman then today you have to put your business online. This is the reason for the increased risk of Down Syndrome babies in women over age 35. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Word - Press makes it possible to successfully and manage your website. <br><br>The entrepreneurs can easily captivate their readers by using these versatile themes. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. 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. If you have any type of inquiries relating to where and how to use [http://kedahan.net/wordpress_backup_562132 backup plugin], you can call us at our own web-page. There are many advantages of hiring Wordpress developers for Wordpress project development:. If your site does well you can get paid professional designer to create a unique Word - Press theme. <br><br>This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. In fact portfolio Word - Press themes is a smooth and attractive but considerably flawed Word - Press theme in creating simpler to the photographers or designers to develop a specific internet site showcasing their most current perform since it appear modern-day and has fantastic typography and large photographs which would develop an attractive wanting portfolio internet site. Word - Press can also be quickly extended however improvement API is not as potent as Joomla's. Word - Press is the most popular personal publishing platform which was launched in 2003. Verify whether your company has a team of developers or programmers having hands-on experience and knowledge about all Word - Press concepts. |
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| Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focussing'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.) There is also a threshold power, below which no instability will be seen.<ref name="agrawal" />
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| The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, whilst at other frequencies, a perturbation will [[exponential growth|grow exponentially]]. The overall [[gain]] spectrum can be derived [[Analytical expression|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.
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| The tendency of a perturbing signal to grow makes modulation instability a form of [[amplifier|amplification]]. By tuning an input signal to a peak of the gain spectrum, it is possible to create an [[optical amplifier]].
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| ===Mathematical derivation of gain spectrum===
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| The gain spectrum can be derived <ref name="agrawal" /> by starting with a model of modulation instability based upon the [[Nonlinear Schrödinger equation]]
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| :<math>\frac{\partial A}{\partial z}+i\beta_2\frac{\partial^2A}{\partial t^2}=i\gamma|A|^2A</math>
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| which describes the evolution of a [[Slowly varying envelope approximation|slowly varying envelope]] <math>A</math> with time <math>t</math> and distance of propagation <math>z</math>. The model includes group velocity dispersion described by the parameter <math>\beta_2</math>, and Kerr nonlinearity with magnitude <math>\gamma</math>. A waveform of constant power <math>P</math> is assumed. This is given by the solution
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| :<math>A = \sqrt{P}e^{i\gamma Pz}</math>
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| where the oscillatory <math>e^{i\gamma Pz}</math> phase factor accounts for the difference between the linear [[refractive index]], and the modified refractive index, as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as
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| :<math>A = \left(\sqrt{P}+\epsilon\left(t,z\right)\right)e^{i\gamma Pz}</math>
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| where <math>\epsilon\left(t,z\right)</math> is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as <math>A</math>). Substituting this back into the Nonlinear Schrödinger equation gives a [[perturbation theory|perturbation equation]] of the form
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| :<math>\frac{\partial \epsilon}{\partial z}+i\beta_2\frac{\partial^2\epsilon}{\partial t^2}=i\gamma P \left(\epsilon+\epsilon^*\right)</math>
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| where the perturbation has been assumed to be small, such that <math>\epsilon^2\approx 0</math>. Instability can now be discovered by searching for solutions of the perturbation equation which grow exponentially. This can be done using a trial function of the general form
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| :<math>\epsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m z + i \omega_m t}</math>
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| where <math>\omega_m</math> and <math>k_m</math> are the [[frequency]] and [[wavenumber]] of a perturbation, and <math>c_1</math> and <math>c_2</math> are constants. The Nonlinear Schrödinger equation is constructed by removing the [[carrier wave]] of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore <math>\omega_m</math> and <math>k_m</math> don't represent absolute frequencies and wavenumbers, but the ''difference'' between these and those of the initial beam of light. It can be shown that the trial function is valid, subject to the condition | |
| :<math>k_m = \pm\sqrt{\beta_2^2\omega_m^4 + 2 \gamma P \beta_2 \omega_m^2}</math>
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| This dispersion relation is vitally dependent on the sign of the term within the square root, as if positive, the wavenumber will be [[real number|real]], corresponding to mere [[oscillation]]s around the unperturbed solution, whilst if negative, the wavenumber will become [[imaginary number|imaginary]], corresponding to exponential growth and thus instability. Therefore, instability will occur when | |
| :<math>\beta_2^2\omega_m^2 + 2 \gamma P \beta_2 < 0</math>
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| This condition describes both the requirement for anomalous dispersion (such that <math>\beta_2</math> is negative) and the requirement that a threshold power be exceeded. The gain spectrum can be described by defining a gain parameter as <math>g</math> <math>\equiv</math> <math>\Im\left[2|k_m|\right]</math>, so that the power of a perturbing signal grows with distance as <math>P</math> <math>\propto</math> <math>e^{g z}</math>. The gain is therefore given by
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| :<math>g = \begin{cases} 2\sqrt{-\beta_2^2\omega_m^4-2\gamma P \beta_2\omega_m^2} &;\, -\beta_2^2\omega_m^2 - 2 \gamma P \beta_2 > 0 \\ 0 &;\, -\beta_2^2\omega_m^2 - 2 \gamma P \beta_2 \leq 0\end{cases} </math>
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| where as noted above, <math>\omega_m</math> is the difference between the frequency of the perturbation and the frequency of the initial light.
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| ==Breakup==
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| The waveform will eventually break up into a train of pulses.<ref name="agrawal" />
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * {{Cite journal
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| | doi = 10.1016/j.physd.2008.12.002
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| | volume = 238
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| | issue = 5
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| | pages = 540–548
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| | last1 = Zakharov
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| | first1 = V.E.
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| | author1-link = Vladimir E. Zakharov
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| | first2 = L.A.
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| | last2 = Ostrovsky
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| | title = Modulation instability: The beginning
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| | journal = Physica D: Nonlinear Phenomena
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| | year = 2009
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| | url = http://www.math.umass.edu/~kevrekid/math697/sdarticle_ZO.pdf
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| |bibcode = 2009PhyD..238..540Z }}
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| {{physical oceanography}}
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| {{DEFAULTSORT:Modulational Instability}}
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| [[Category:Nonlinear optics]]
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| [[Category:Photonics]]
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| [[Category:Water waves]]
| |
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The Internet is a vast open market where businesses and consumers congregate. If you are a positive thinker businessman then today you have to put your business online. This is the reason for the increased risk of Down Syndrome babies in women over age 35. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Word - Press makes it possible to successfully and manage your website.
The entrepreneurs can easily captivate their readers by using these versatile themes. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors.
The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. 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. If you have any type of inquiries relating to where and how to use backup plugin, you can call us at our own web-page. There are many advantages of hiring Wordpress developers for Wordpress project development:. If your site does well you can get paid professional designer to create a unique Word - Press theme.
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