Coupled mode theory

<h2 id="history">History</h2>
Coupled mode theory first arose in the 1950s in the works of Miller on microwave <a href="/facts/Transmission_lines/cYlqaPo3">transmission lines</a>,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> Pierce on <a href="/facts/Electron_beam/9jRXSquS">electron beams</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> and Gould on <a href="/facts/Backward_wave_oscillator/2r3xKPJr">backward wave oscillators</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> This put in place the mathematical foundations for the modern formulation expressed by <a href="/facts/H._A._Haus/gXsJYHDn">H. A. Haus</a> et al. for optical waveguides.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
In the late 1990s and early 2000s, the field of <a href="/facts/Nanophotonics/f2UAn4oH">nanophotonics</a> has revitalized interest in coupled mode theory. Coupled mode theory has been used to account for the <a href="/facts/Fano_resonance/7Ll5BejQ">Fano resonances</a> in photonic crystal slabs<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and has also been modified to account for optical resonators with non-orthogonal modes.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Since late 2000s, researchers have capitalized on coupled mode theory to explain the concept of magnetically coupled resonators.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="overview">Overview</h2>
The oscillatory systems to which coupled mode theory applies are described by second order partial differential equations. CMT allows the second order partial differential equation to be expressed as one or more coupled first order ordinary differential equations. The following assumptions are generally made with CMT:

<ul><li>Linearity</li>
<li>Time-reversal symmetry</li>
<li>Time-invariance</li>
<li>Weak mode coupling (small perturbation of uncoupled modes)</li>
<li>Energy conservation</li></ul>
<h2 id="formulation">Formulation</h2>
The formulation of the coupled mode theory is based on the development of the solution to an electromagnetic problem into modes. Most of the time it is eigenmodes which are taken in order to form a complete base. The choice of the basis and the adoption of certain hypothesis like parabolic approximation differs from formulation to formulation.
The classification proposed by <a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> of the different formulation is as follows:

<ol><li>The choice of starting differential equation. some of the coupled mode theories are derived directly from the <a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell differential equations</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> although others use simplifications in order to obtain a <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a>.</li>
<li>The choice of principle to derive the equations of the CMT. Either the reciprocity theorem <a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> or the <a href="/facts/Variational_principle/a5UkKEBj">variational principle</a> have been used.</li>
<li>The choice of orthogonality product used to establish the eigenmode base. Some references use the unconjugated form <a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> and others the complex-conjugated form.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li>
<li>Finally, the choice of the form of the equation, either vectorial <a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> or scalar.</li></ol>
When n modes of an <a href="/facts/Electromagnetic/oAkmALHT">electromagnetic</a> wave propagate through a media in the direction z without loss, the power transported by each mode is described by a modal power Pm. At a given frequency ω.

P
          
            ω
          
        
        (
        z
        )
        =
        
          ∑
          
            m
          
          
            n
          
        
        
          P
          
            m
          
          
            ω
          
        
        (
        z
        )
        =
        
          
            1
            4
          
        
        
          ∑
          
            m
          
          
            n
          
        
        
          N
          
            m
          
          
            ω
          
        
        
          
            |
            
              
                a
                
                  m
                
                
                  ω
                
              
              (
              z
              )
            
            |
          
          
            2
          
        
        
        
      
    
    {\displaystyle P^{\omega }(z)=\sum _{m}^{n}P_{m}^{\omega }(z)={\frac {1}{4}}\sum _{m}^{n}N_{m}^{\omega }\left|a_{m}^{\omega }(z)\right|^{2}\,\!}

where Nm is the norm of the mth mode and am is the modal amplitude.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Eigenmode_expansion/zy4qdfkv">Eigenmode expansion</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://wmm.computational-photonics.eu/cmt.html">WMM mode solver manual on couple mode theory</a></li></ul>

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

<ol>
<li id="fn:1">S.E.Miller,"Coupled wave theory and waveguide applications.", Bell System Technical Journal, 1954 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">J. R. Pierce, "Coupling of modes of propagations", Journal of Applied Physics, 25, 1954 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">R.W. Gould, "A coupled mode description of the backward-wave oscillator and the Kompfner dip condition" I.R.E. Trans. Electron Devices, vol. PGED-2, pp. 37–42, 1955. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Haus, H., et al. "Coupled-mode theory of optical waveguides." Journal of Lightwave Technology 5.1 (1987): 16-23. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">H. A. Haus, W. P. Huang. "Coupled Mode Theory."Proceedings of the IEEE, Vol 19, No 10, October 1991. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">S. Fan, W. Suh, J. Joannopoulos, "Temporal coupled-mode theory for the Fano resonance in optical resonators," JOSA A, vol. 20, no. 3, pp. 569–572, 2003. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," Quantum Electronics, IEEE Journal of, vol. 40, no. 10, pp. 1511–1518, 2004 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Elnaggar S.Y., Tervo, R.J., and Mattar, S.M., Energy Coupled Mode Theory for Electromagnetic Resonators, IEEE Transactions on Microwave Theory and Techniques, vol. 63, no. 7, pp. 2115-2123, July 2015, doi: 10.1109/TMTT.2015.2434377. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Kurs, A.; Karalis, A.; Moffatt, R.; Joannopoulos, J. D.; Fisher, P.; Soljacic, M. (6 July 2007). "Wireless Power Transfer via Strongly Coupled Magnetic Resonances". Science. 317 (5834): 83–86. Bibcode:2007Sci...317...83K. CiteSeerX 10.1.1.418.9645. doi:10.1126/science.1143254. PMID 17556549. S2CID 17105396. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Barybin and Dmitriev,"Modern Electrodynamics and Coupled-mode theory", 2002 <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Hardy and Streifer, "Coupled mode theory of parallel waveguides", Journal of Lightwave Technology,1985 <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">A. W. Snyder and J. D. Love, "Optical waveguide Theory", Chapman and Hall, 1983 <a href="/wiki/J._D._Love" target="_blank">/wiki/J._D._Love</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Hardy and Streifer, "Coupled mode theory of parallel waveguides", Journal of Lightwave Technology,1985 <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">A. W. Snyder and J. D. Love, "Optical waveguide Theory", Chapman and Hall, 1983 <a href="/wiki/J._D._Love" target="_blank">/wiki/J._D._Love</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Hardy and Streifer, "Coupled mode theory of parallel waveguides", Journal of Lightwave Technology,1985 <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">A. W. Snyder and J. D. Love, "Optical waveguide Theory", Chapman and Hall, 1983 <a href="/wiki/J._D._Love" target="_blank">/wiki/J._D._Love</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Hardy and Streifer, "Coupled mode theory of parallel waveguides", Journal of Lightwave Technology,1985 <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">A. W. Snyder and J. D. Love, "Optical waveguide Theory", Chapman and Hall, 1983 <a href="/wiki/J._D._Love" target="_blank">/wiki/J._D._Love</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
</ol>

Coupled mode theory open-in-new

Coupled mode theory