Slowly varying envelope approximation

<h2 id="example">Example</h2>
For example, consider the <a href="/facts/Electromagnetic_wave_equation/qUYSPdeN">electromagnetic wave equation</a>:

 
 
 
 
 ∇
 
 2
 
 
 E
 −
 
 
 1
 
 c
 
 2
 
 
 
 
 
 
 
 
 ∂
 
 2
 
 
 E
 
 
 ∂
 
 t
 
 2
 
 
 
 
 
 =
 0
 
 ,
 
 
 {\displaystyle \nabla ^{2}E-{\frac {1}{c^{2}}}{\frac {\partial ^{2}E}{\partial t^{2}}}=0\,,}

where 
 
 
 
 c
 =
 
 
 1
 
 
 μ
 
 0
 
 
 
 ε
 
 0
 
 
 
 
 
  
 .
 
 
 {\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}~.}

If k0 and ω0 are the <a href="/facts/Wave_number/cLaLAeW5">wave number</a> and <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> of the (characteristic) <a href="/facts/Carrier_wave/ae98pBlY">carrier wave</a> for the signal E(r,t), the following representation is useful:

 
 
 
 E
 (
 
 r
 
 ,
 t
 )
 =
 
 Re
 
 ⁡
 
 [
 
 
 E
 
 0
 
 
 (
 
 r
 
 ,
 t
 )
 
 
 e
 
 i
 (
 
 
 k
 
 
 0
 
 
 ⋅
 
 r
 
 −
 
 ω
 
 0
 
 
 t
 )
 
 
 
 ]
 
 ,
 
 
 {\displaystyle E(\mathbf {r} ,t)=\operatorname {\operatorname {Re} } \left[E_{0}(\mathbf {r} ,t)\,e^{i(\mathbf {k} _{0}\cdot \mathbf {r} -\omega _{0}t)}\right],}

where 
 
 
 
 Re
 ⁡
 [
 
 ⋅
 
 ]
 
 
 {\displaystyle \operatorname {Re} [\,\cdot \,]}
 
 denotes the <a href="/facts/Real_part/2w7mqI7e">real part</a> of the quantity between brackets, and 
 
 
 
 
 i
 
 2
 
 
 ≡
 −
 1.
 
 
 {\displaystyle i^{2}\equiv -1.}

In the slowly varying envelope approximation (SVEA) it is assumed that the <a href="/facts/Complex_amplitude/LyjSSUDo">complex amplitude</a> E0(r, t) only varies slowly with r and t. This inherently implies that E(r, t) represents waves propagating forward, predominantly in the k0 direction. As a result of the slow variation of E0(r, t), when taking derivatives, the highest-order derivatives may be neglected:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

|
          
            
              ∇
              
                2
              
            
            
              E
              
                0
              
            
          
          |
        
        ≪
        
          |
          
            
              
                k
              
              
                0
              
            
            ⋅
            ∇
            
              E
              
                0
              
            
          
          |
        
      
    
    {\displaystyle \left|\nabla ^{2}E_{0}\right|\ll \left|\mathbf {k} _{0}\cdot \nabla E_{0}\right|}
  
   and   
  
    
      
        
          |
          
            
              
                
                  ∂
                  
                    2
                  
                
                
                  E
                  
                    0
                  
                
              
              
                ∂
                
                  t
                  
                    2
                  
                
              
            
          
          |
        
        ≪
        
          |
          
            
              ω
              
                0
              
            
            
            
              
                
                  ∂
                  
                    E
                    
                      0
                    
                  
                
                
                  ∂
                  t
                
              
            
          
          |
        
        ,
      
    
    {\displaystyle \left|{\frac {\partial ^{2}E_{0}}{\partial t^{2}}}\right|\ll \left|\omega _{0}\,{\frac {\partial E_{0}}{\partial t}}\right|,}
  
   with   
  
    
      
        
          k
          
            0
          
        
        ≡
        
          |
          
            
              k
            
            
              0
            
          
          |
        
        .
      
    
    {\displaystyle k_{0}\equiv \left|\mathbf {k} _{0}\right|.}

<h3>Full approximation</h3>
Consequently, the wave equation is approximated in the SVEA as:

 
 
 
 2
 i
 
 
 k
 
 
 0
 
 
 ⋅
 ∇
 
 E
 
 0
 
 
 +
 
 
 
 2
 i
 
 ω
 
 0
 
 
 
 
 c
 
 2
 
 
 
 
 
 
 
 ∂
 
 E
 
 0
 
 
 
 
 ∂
 t
 
 
 
 −
 
 (
 
 
 k
 
 0
 
 
 2
 
 
 −
 
 
 
 ω
 
 0
 
 
 2
 
 
 
 c
 
 2
 
 
 
 
 
 )
 
 
 E
 
 0
 
 
 =
 0
  
 .
 
 
 {\displaystyle 2i\mathbf {k} _{0}\cdot \nabla E_{0}+{\frac {2i\omega _{0}}{c^{2}}}{\frac {\partial E_{0}}{\partial t}}-\left(k_{0}^{2}-{\frac {\omega _{0}^{2}}{c^{2}}}\right)E_{0}=0~.}

It is convenient to choose k0 and ω0 such that they satisfy the <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relation</a>:

 
 
 
 
 k
 
 0
 
 
 2
 
 
 −
 
 
 
 ω
 
 0
 
 
 2
 
 
 
 c
 
 2
 
 
 
 
 =
 0
  
 .
 
 
 {\displaystyle k_{0}^{2}-{\frac {\omega _{0}^{2}}{c^{2}}}=0~.}

This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation:

 
 
 
 
 
 k
 
 
 0
 
 
 ⋅
 ∇
 
 E
 
 0
 
 
 +
 
 
 
 ω
 
 0
 
 
 
 c
 
 2
 
 
 
 
 
 
 
 
 ∂
 
 E
 
 0
 
 
 
 
 ∂
 t
 
 
 
 =
 0
  
 .
 
 
 {\displaystyle \mathbf {k} _{0}\cdot \nabla E_{0}+{\frac {\omega _{0}}{c^{2}}}\,{\frac {\partial E_{0}}{\partial t}}=0~.}

This is a <a href="/facts/Hyperbolic_partial_differential_equation/znXgGuvh">hyperbolic partial differential equation</a>, like the original wave equation, but now of first-order instead of second-order. It is valid for coherent forward-propagating waves in directions near the k0-direction. The space and time scales over which E0 varies are generally much longer than the spatial wavelength and temporal period of the carrier wave. A numerical solution of the envelope equation thus can use much larger space and time steps, resulting in significantly less computational effort.

<h3>Parabolic approximation</h3>
Assume wave propagation is dominantly in the z-direction, and k0 is taken in this direction. The SVEA is only applied to the second-order spatial derivatives in the z-direction and time. If 
 
 
 
 
 Δ
 
 ⊥
 
 
 ≡
 
 ∂
 
 2
 
 
 
 /
 
 ∂
 
 x
 
 2
 
 
 +
 
 ∂
 
 2
 
 
 
 /
 
 ∂
 
 y
 
 2
 
 
 
 
 {\displaystyle \Delta _{\perp }\equiv \partial ^{2}/\partial x^{2}+\partial ^{2}/\partial y^{2}}
 
 is the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a> in the x×y plane, the result is:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

 
 
 
 
 k
 
 0
 
 
 
 
 
 ∂
 
 E
 
 0
 
 
 
 
 ∂
 z
 
 
 
 +
 
 
 
 ω
 
 0
 
 
 
 c
 
 2
 
 
 
 
 
 
 
 ∂
 
 E
 
 0
 
 
 
 
 ∂
 t
 
 
 
 −
 
 
 1
 2
 
 
 
 i
 
 
 Δ
 
 ⊥
 
 
 
 E
 
 0
 
 
 =
 0
  
 .
 
 
 {\displaystyle k_{0}{\frac {\partial E_{0}}{\partial z}}+{\frac {\omega _{0}}{c^{2}}}{\frac {\partial E_{0}}{\partial t}}-{\frac {1}{2}}\,i\,\Delta _{\perp }E_{0}=0~.}
 
 
This is a <a href="/facts/Parabolic_partial_differential_equation/gJQfRUlr">parabolic partial differential equation</a>. This equation has enhanced validity as compared to the full SVEA: It represents waves propagating in directions significantly different from the z-direction.

<h3>Alternative limit of validity</h3>
In the one-dimensional case, another sufficient condition for the SVEA validity is

ℓ
          
            
              g
            
          
        
        ≫
        λ
      
    
    {\displaystyle \ell _{\mathsf {g}}\gg \lambda }
  
   and   
  
    
      
        
          ℓ
          
            
              p
            
          
        
        ≫
        λ
        
          (
          
            1
            −
            
              
                v
                c
              
            
          
          )
        
        
        ,
      
    
    {\displaystyle \ell _{\mathsf {p}}\gg \lambda \left(1-{\frac {v}{c}}\right)\,,}
  
   with   
  
    
      
        λ
        =
        
          
            
              2
              π
            
            
              k
              
                0
              
            
          
        
        
        ,
      
    
    {\displaystyle \lambda ={\frac {2\pi }{k_{0}}}\,,}

where 
 
 
 
 
 ℓ
 
 
 g
 
 
 
 
 
 {\displaystyle \ell _{\mathsf {g}}}
 
 is the length over which the radiation pulse is amplified, 
 
 
 
 
 ℓ
 
 
 p
 
 
 
 
 
 {\displaystyle \ell _{\mathsf {p}}}
 
 is the pulse width and 
 
 
 
 v
 
 
 {\displaystyle v}
 
 is the group velocity of the radiating system.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
These conditions are much less restrictive in the relativistic limit where 
 
 
 
 
 
 v
 c
 
 
 
 
 {\displaystyle {\frac {v}{c}}}
 
 is close to 1, as in a <a href="/facts/Free-electron_laser/ew2F96XV">free-electron laser</a>, compared to the usual conditions required for the SVEA validity.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ultrashort_pulse/V8joGdvF">Ultrashort pulse</a></li>
<li><a href="/facts/WKB_approximation/XyRnYvGh">WKB approximation</a></li></ul>

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

<ol>
<li id="fn:1">Arecchi, F.; Bonifacio, R. (1965). "Theory of optical maser amplifiers". IEEE Journal of Quantum Electronics. 1 (4): 169–178. Bibcode:1965IJQE....1..169A. doi:10.1109/JQE.1965.1072212. <a href="/wiki/IEEE_Journal_of_Quantum_Electronics" target="_blank">/wiki/IEEE_Journal_of_Quantum_Electronics</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Butcher, Paul N.; Cotter, David (1991). The Elements of Nonlinear Optics (reprint ed.). Cambridge University Press. p. 216. ISBN 0-521-42424-0. <a href="0-521-42424-0" target="_blank">0-521-42424-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Svelto, Orazio (1974). "Self-focussing, self-trapping, and self-phase modulation of laser beams". In Wolf, Emil (ed.). Progress in Optics. Vol. 12. North Holland. pp. 23–25. ISBN 0-444-10571-9. <a href="0-444-10571-9" target="_blank">0-444-10571-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Bonifacio, R.; Caloi, R.M.; Maroli, C. (1993). "The slowly varying envelope approximation revisited". Optics Communications. 101 (3–4): 185–187. Bibcode:1993OptCo.101..185B. doi:10.1016/0030-4018(93)90363-A. <a href="/wiki/Optics_Communications" target="_blank">/wiki/Optics_Communications</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Slowly varying envelope approximation open-in-new

Slowly varying envelope approximation