Modulational instability

<h2 id="initial-instability-and-gain">Initial instability and gain</h2>
Modulation instability only happens under certain circumstances. The most important condition is anomalous group velocity <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion</a>, whereby pulses with shorter <a href="/facts/Wavelength/kcJU0ymz">wavelengths</a> travel with higher <a href="/facts/Group_velocity/HPIMFwaV">group velocity</a> than pulses with longer wavelength.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> (This condition assumes a focusing <a href="/facts/Kerr_nonlinearity/IoJN6wlW">Kerr nonlinearity</a>, whereby refractive index increases with optical intensity.)<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, while at other frequencies, a perturbation will <a href="/facts/Exponential_growth/m4yEqKJ7">grow exponentially</a>. The overall <a href="/facts/Gain_(electronics)/CWbehg27">gain</a> spectrum can be derived <a href="/facts/Analytical_expression/y0xCXlSk">analytically</a>, 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.
The tendency of a perturbing signal to grow makes modulation instability a form of <a href="/facts/Amplifier/pnRgxncC">amplification</a>. By tuning an input signal to a peak of the gain spectrum, it is possible to create an <a href="/facts/Optical_amplifier/1Fj9Pfes">optical amplifier</a>.

<h3>Mathematical derivation of gain spectrum</h3>
The gain spectrum can be derived <a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> by starting with a model of modulation instability based upon the <a href="/facts/Nonlinear_Schr%C3%B6dinger_equation/gfGkRApW">nonlinear Schrödinger equation</a>

∂
              A
            
            
              ∂
              z
            
          
        
        +
        i
        
          β
          
            2
          
        
        
          
            
              
                ∂
                
                  2
                
              
              A
            
            
              ∂
              
                t
                
                  2
                
              
            
          
        
        =
        i
        γ
        
          |
        
        A
        
          
            |
          
          
            2
          
        
        A
        ,
      
    
    {\displaystyle {\frac {\partial A}{\partial z}}+i\beta _{2}{\frac {\partial ^{2}A}{\partial t^{2}}}=i\gamma |A|^{2}A,}

which describes the evolution of a <a href="/facts/Complex_number/2w7mqI7e">complex-valued</a> <a href="/facts/Slowly_varying_envelope_approximation/wh2emEog">slowly varying envelope</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 with time 
 
 
 
 t
 
 
 {\displaystyle t}
 
 and distance of propagation 
 
 
 
 z
 
 
 {\displaystyle z}
 
. The <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> 
 
 
 
 i
 
 
 {\displaystyle i}
 
 satisfies 
 
 
 
 
 i
 
 2
 
 
 =
 −
 1.
 
 
 {\displaystyle i^{2}=-1.}
 
 The model includes <a href="/facts/Group_velocity/HPIMFwaV">group velocity</a> dispersion described by the parameter 
 
 
 
 
 β
 
 2
 
 
 
 
 {\displaystyle \beta _{2}}
 
, and <a href="/facts/Kerr_effect/IoJN6wlW">Kerr nonlinearity</a> with magnitude 
 
 
 
 γ
 .
 
 
 {\displaystyle \gamma .}
 
 A <a href="/facts/Periodic_function/Sm8lJQsF">periodic</a> waveform of constant power 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is assumed. This is given by the solution

A
        =
        
          
            P
          
        
        
          e
          
            i
            γ
            P
            z
          
        
        ,
      
    
    {\displaystyle A={\sqrt {P}}e^{i\gamma Pz},}

where the oscillatory 
 
 
 
 
 e
 
 i
 γ
 P
 z
 
 
 
 
 {\displaystyle e^{i\gamma Pz}}
 
 <a href="/facts/Wave_phase/cpGtuVll">phase</a> factor accounts for the difference between the linear <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a>, and the modified <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a>, as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as

A
        =
        
          (
          
            
              
                P
              
            
            +
            ε
            (
            t
            ,
            z
            )
          
          )
        
        
          e
          
            i
            γ
            P
            z
          
        
        ,
      
    
    {\displaystyle A=\left({\sqrt {P}}+\varepsilon (t,z)\right)e^{i\gamma Pz},}

where 
 
 
 
 ε
 (
 t
 ,
 z
 )
 
 
 {\displaystyle \varepsilon (t,z)}
 
 is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as 
 
 
 
 A
 
 
 {\displaystyle A}
 
). Substituting this back into the nonlinear Schrödinger equation gives a <a href="/facts/Perturbation_theory/JWrNgbVm">perturbation equation</a> of the form

∂
              ε
            
            
              ∂
              z
            
          
        
        +
        i
        
          β
          
            2
          
        
        
          
            
              
                ∂
                
                  2
                
              
              ε
            
            
              ∂
              
                t
                
                  2
                
              
            
          
        
        =
        i
        γ
        P
        
          (
          
            ε
            +
            
              ε
              
                ∗
              
            
          
          )
        
        ,
      
    
    {\displaystyle {\frac {\partial \varepsilon }{\partial z}}+i\beta _{2}{\frac {\partial ^{2}\varepsilon }{\partial t^{2}}}=i\gamma P\left(\varepsilon +\varepsilon ^{*}\right),}

where the perturbation has been assumed to be small, such that 
 
 
 
 
 |
 
 ε
 
 
 |
 
 
 2
 
 
 ≪
 P
 .
 
 
 {\displaystyle |\varepsilon |^{2}\ll P.}
 
 The <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 is denoted as 
 
 
 
 
 ε
 
 ∗
 
 
 .
 
 
 {\displaystyle \varepsilon ^{*}.}
 
 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

ε
        =
        
          c
          
            1
          
        
        
          e
          
            i
            
              k
              
                m
              
            
            z
            −
            i
            
              ω
              
                m
              
            
            t
          
        
        +
        
          c
          
            2
          
        
        
          e
          
            −
            i
            
              k
              
                m
              
              
                ∗
              
            
            z
            +
            i
            
              ω
              
                m
              
            
            t
          
        
        ,
      
    
    {\displaystyle \varepsilon =c_{1}e^{ik_{m}z-i\omega _{m}t}+c_{2}e^{-ik_{m}^{*}z+i\omega _{m}t},}

where 
 
 
 
 
 k
 
 m
 
 
 
 
 {\displaystyle k_{m}}
 
 and 
 
 
 
 
 ω
 
 m
 
 
 
 
 {\displaystyle \omega _{m}}
 
 are the <a href="/facts/Wavenumber/cLaLAeW5">wavenumber</a> and (real-valued) <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> of a perturbation, and 
 
 
 
 
 c
 
 1
 
 
 
 
 {\displaystyle c_{1}}
 
 and 
 
 
 
 
 c
 
 2
 
 
 
 
 {\displaystyle c_{2}}
 
 are constants. The nonlinear Schrödinger equation is constructed by removing the <a href="/facts/Carrier_wave/ae98pBlY">carrier wave</a> of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore, 
 
 
 
 
 ω
 
 m
 
 
 
 
 {\displaystyle \omega _{m}}
 
 and 
 
 
 
 
 k
 
 m
 
 
 
 
 {\displaystyle k_{m}}
 
 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, provided 
 
 
 
 
 c
 
 2
 
 
 =
 
 c
 
 1
 
 
 ∗
 
 
 
 
 {\displaystyle c_{2}=c_{1}^{*}}
 
 and subject to the condition

k
          
            m
          
        
        =
        ±
        
          
            
              β
              
                2
              
              
                2
              
            
            
              ω
              
                m
              
              
                4
              
            
            +
            2
            γ
            P
            
              β
              
                2
              
            
            
              ω
              
                m
              
              
                2
              
            
          
        
        .
      
    
    {\displaystyle k_{m}=\pm {\sqrt {\beta _{2}^{2}\omega _{m}^{4}+2\gamma P\beta _{2}\omega _{m}^{2}}}.}

This dispersion relation is vitally dependent on the sign of the term within the square root, as if positive, the wavenumber will be <a href="/facts/Real_number/R02gw5Pb">real</a>, corresponding to mere <a href="/facts/Oscillation/6vSrm84a">oscillations</a> around the unperturbed solution, whilst if negative, the wavenumber will become <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary</a>, corresponding to exponential growth and thus instability. Therefore, instability will occur when

β
 
 2
 
 
 2
 
 
 
 ω
 
 m
 
 
 2
 
 
 +
 2
 γ
 P
 
 β
 
 2
 
 
 <
 0
 ,
 
 
 {\displaystyle \beta _{2}^{2}\omega _{m}^{2}+2\gamma P\beta _{2}<0,}
 
   that is for   
 
 
 
 
 ω
 
 m
 
 
 2
 
 
 <
 −
 2
 
 
 
 γ
 P
 
 
 β
 
 2
 
 
 
 
 .
 
 
 {\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}}.}

This condition describes the requirement for anomalous dispersion (such that 
 
 
 
 γ
 
 β
 
 2
 
 
 
 
 {\displaystyle \gamma \beta _{2}}
 
 is negative). The gain spectrum can be described by defining a gain parameter as 
 
 
 
 g
 ≡
 2
 
 |
 
 ℑ
 {
 
 k
 
 m
 
 
 }
 
 |
 
 ,
 
 
 {\displaystyle g\equiv 2|\Im \{k_{m}\}|,}
 
 so that the power of a perturbing signal grows with distance as 
 
 
 
 P
 
 
 e
 
 g
 z
 
 
 .
 
 
 {\displaystyle P\,e^{gz}.}
 
 The gain is therefore given by

g
 =
 
 
 {
 
 
 
 2
 
 
 −
 
 β
 
 2
 
 
 2
 
 
 
 ω
 
 m
 
 
 4
 
 
 −
 2
 γ
 P
 
 β
 
 2
 
 
 
 ω
 
 m
 
 
 2
 
 
 
 
 ,
 
 
 
 for 
 
 
 
 ω
 
 m
 
 
 2
 
 
 <
 −
 2
 
 
 
 γ
 P
 
 
 β
 
 2
 
 
 
 
 ,
 
 
 
 
 
 0
 ,
 
 
 
 for 
 
 
 
 ω
 
 m
 
 
 2
 
 
 ≥
 −
 2
 
 
 
 γ
 P
 
 
 β
 
 2
 
 
 
 
 ,
 
 
 
 
 
 
 
 
 
 {\displaystyle g={\begin{cases}2{\sqrt {-\beta _{2}^{2}\omega _{m}^{4}-2\gamma P\beta _{2}\omega _{m}^{2}}},&{\text{for }}\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}},\\[2ex]0,&{\text{for }}\displaystyle \omega _{m}^{2}\geq -2{\frac {\gamma P}{\beta _{2}}},\end{cases}}}

where as noted above, 
 
 
 
 
 ω
 
 m
 
 
 
 
 {\displaystyle \omega _{m}}
 
 is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for 
 
 
 
 
 ω
 
 2
 
 
 =
 −
 γ
 P
 
 /
 
 
 β
 
 2
 
 
 .
 
 
 {\displaystyle \omega ^{2}=-\gamma P/\beta _{2}.}

<h2 id="modulation-instability-in-soft-systems">Modulation instability in soft systems</h2>
Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> Modulation instability of spatially and temporally incoherent light is possible owing to the non-instantaneous response of photoreactive systems, which consequently responds to the time-average intensity of light, in which the femto-second fluctuations cancel out.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Vladimir_E._Zakharov/xm8HLW1C">Zakharov, V.E.</a>; Ostrovsky, L.A. (2009). <a href="http://www.math.umass.edu/~kevrekid/math697/sdarticle_ZO.pdf">"Modulation instability: The beginning"</a> (PDF). Physica D: Nonlinear Phenomena. 238 (5): 540–548. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2009PhyD..238..540Z">2009PhyD..238..540Z</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.physd.2008.12.002">10.1016/j.physd.2008.12.002</a>.[permanent dead link‍]</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Benjamin, T. Brooke; Feir, J.E. (1967). "The disintegration of wave trains on deep water. Part 1. Theory". Journal of Fluid Mechanics. 27 (3): 417–430. Bibcode:1967JFM....27..417B. doi:10.1017/S002211206700045X. S2CID 121996479. <a href="/wiki/T._Brooke_Benjamin" target="_blank">/wiki/T._Brooke_Benjamin</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Benjamin, T.B. (1967). "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 299 (1456): 59–76. Bibcode:1967RSPSA.299...59B. doi:10.1098/rspa.1967.0123. S2CID 121661209. Concluded with a discussion by Klaus Hasselmann. <a href="/wiki/T._Brooke_Benjamin" target="_blank">/wiki/T._Brooke_Benjamin</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. ISBN 978-0-12-045142-5. <a href="978-0-12-045142-5" target="_blank">978-0-12-045142-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Yuen, H.C.; Lake, B.M. (1980). "Instabilities of waves on deep water". Annual Review of Fluid Mechanics. 12: 303–334. Bibcode:1980AnRFM..12..303Y. doi:10.1146/annurev.fl.12.010180.001511. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Piliptetskii, N. F.; Rustamov, A. R. (31 May 1965). "Observation of Self-focusing of Light in Liquids". JETP Letters. 2 (2): 55–56. <a href="http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml" target="_blank">http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bespalov, V. I.; Talanov, V. I. (15 June 1966). "Filamentary Structure of Light Beams in Nonlinear Liquids". ZhETF Pisma Redaktsiiu. 3 (11): 471–476. Bibcode:1966ZhPmR...3..471B. Archived from the original on 31 July 2020. Retrieved 17 February 2021. <a href="https://web.archive.org/web/20200731112029/http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml" target="_blank">https://web.archive.org/web/20200731112029/http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Janssen, Peter A.E.M. (2003). "Nonlinear four-wave interactions and freak waves". Journal of Physical Oceanography. 33 (4): 863–884. Bibcode:2003JPO....33..863J. doi:10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2. <a href="https://doi.org/10.1175%2F1520-0485%282003%2933%3C863%3ANFIAFW%3E2.0.CO%3B2" target="_blank">https://doi.org/10.1175%2F1520-0485%282003%2933%3C863%3ANFIAFW%3E2.0.CO%3B2</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Dysthe, Kristian; Krogstad, Harald E.; Müller, Peter (2008). "Oceanic rogue waves". Annual Review of Fluid Mechanics. 40 (1): 287–310. Bibcode:2008AnRFM..40..287D. doi:10.1146/annurev.fluid.40.111406.102203. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. ISBN 978-0-12-045142-5. <a href="978-0-12-045142-5" target="_blank">978-0-12-045142-5</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. ISBN 978-0-12-045142-5. <a href="978-0-12-045142-5" target="_blank">978-0-12-045142-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. ISBN 978-0-12-045142-5. <a href="978-0-12-045142-5" target="_blank">978-0-12-045142-5</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Burgess, Ian B.; Shimmell, Whitney E.; Saravanamuttu, Kalaichelvi (2007-04-01). "Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium". Journal of the American Chemical Society. 129 (15): 4738–4746. doi:10.1021/ja068967b. ISSN 0002-7863. PMID 17378567. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Basker, Dinesh K.; Brook, Michael A.; Saravanamuttu, Kalaichelvi (2015). "Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides". The Journal of Physical Chemistry C. 119 (35): 20606–20617. doi:10.1021/acs.jpcc.5b07117. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Biria, Saeid; Malley, Philip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-03-03). "Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization". The Journal of Physical Chemistry C. 120 (8): 4517–4528. doi:10.1021/acs.jpcc.5b11377. ISSN 1932-7447. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Biria, Saeid; Malley, Phillip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-11-15). "Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring". ACS Macro Letters. 5 (11): 1237–1241. doi:10.1021/acsmacrolett.6b00659. PMID 35614732. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Kewitsch, Anthony S.; Yariv, Amnon (1996-01-01). "Self-focusing and self-trapping of optical beams upon photopolymerization" (PDF). Optics Letters. 21 (1): 24–6. Bibcode:1996OptL...21...24K. doi:10.1364/ol.21.000024. ISSN 1539-4794. PMID 19865292. <a href="https://authors.library.caltech.edu/2845/1/KEWol96.pdf" target="_blank">https://authors.library.caltech.edu/2845/1/KEWol96.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Spatial Solitons | Stefano Trillo | Springer. <a href="https://www.springer.com/us/book/9783540416531" target="_blank">https://www.springer.com/us/book/9783540416531</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Modulational instability open-in-new

Modulational instability