page_title: Lugiato–Lefever equation
section_title: Uniform stationary solutions. Connection with optical bistability. Four-wave mixing and pattern formation.
hierarchical_section_title: Lugiato–Lefever equation / Uniform stationary solutions. Connection with optical bistability. Four-wave mixing and pattern formation.
caption_attribution_description: English: optical bistability curve with theta=4 detuning
context_page_description: The model usually designated as Lugiato–Lefever equation was formulated in 1987 by Luigi Lugiato and Renè Lefever
  as a paradigm for spontaneous pattern formation in nonlinear optical systems. The patterns originate from the interaction of a coherent field, that is injected into a resonant optical cavity, with a Kerr medium that fills the cavity.
  The same equation governs two kinds of patterns: stationary patterns that arise in the planes orthogonal with respect to the direction of propagation of light and patterns that form in the longitudinal direction, travel along the cavity with the velocity of light in the medium and give rise to a sequence of pulses in the output of the cavity.
  The case of longitudinal patterns is intrinsically linked to the phenomenon of “Kerr frequency combs” in microresonators, discovered in 2007 by Tobias Kippenberg and collaborators, that has raised a very lively interest, especially because of the applicative avenue it has opened.
context_section_description: Let us focus on the case in which the envelope  is constant, i.e. on the stationary solutions that are independent of all spatial variables. By dropping all derivatives in Eqs.(1) and (2), and taking the squared modulus, one obtains the stationary equation
  If we plot the stationary curve of  as a function of , 
  when  we obtain a curve as that shown in Fig.3.
  The curve is -shaped and 
  there is an interval of values of  where one has three stationary states. However,the states that lie in the segment with negative slope are unstable, so that in the interval there are two coexisting stable stationary states: this phenomenon is called optical bistability. If the input intensity  is increased and then decreased, one covers a hysteresis cycle.
  If we refer to the modes of the empty cavity, in the case of the uniform stationary solutions described by Eq.(3) the electric field is singlemode, corresponding to the mode of frequency  quasi-resonant with the input frequency .
  In the transverse configuration of Eq.(1), in the case of these stationary solutions E corresponds to a singlemode plane wave  with , where  and  are the transverse components of the wave vector, exactly as the input field .
  The cubic Kerr nonlinearity of Eqs.(1) and (2) gives rise to four-wave mixing (FWM) , which can generate other modes, so that the envelope  displays a spatial pattern: in the transverse plane in the case of Eq.(1), along the cavity in the case of Eq.(2).
caption: Figure 3. Stationary curve of the normalized output intensity 
    
      
        
          
            |
          
          E
          
            
              |
            
            
              2
            
          
        
      
      {\displaystyle |E|^{2}}
    
   as a function of the normalized input intensity 
    
      
        
          
            |
          
          
            E
            
              in
            
          
          
            
              |
            
            
              2
            
          
        
      
      {\displaystyle |E_{\text{in}}|^{2}}
    
   for 
    
      
        
          θ
          =
          4
        
      
      {\displaystyle \theta =4}
    
  . The stationary states in the segment with negative slope are unstable. The arrows show the hysteresis cycle which is covered when 
    
      
        
          
            |
          
          
            E
            
              in
            
          
          
            
              |
            
            
              2
            
          
        
      
      {\displaystyle |E_{\text{in}}|^{2}}
    
   is increased and then decreased.
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