page_title: Fabry–Pérot interferometer
section_title: Other Airy distributions
hierarchical_section_title: Fabry–Pérot interferometer / Theory / Other Airy distributions
caption_attribution_description: English: Airy distribution A t r a n s &#x2032; {\displaystyle A'_{\rm {trans}}} (solid lines), corresponding to light transmitted through a Fabry-Pérot resonator, calculated for different values of the reflectivities R 1 = R 2 {\displaystyle R_{1}=R_{2}} , and comparison with a single Lorentzian line (dashed lines) calculated for the same R 1 = R 2 {\displaystyle R_{1}=R_{2}} .&#91;1&#93; At half maximum (black line), with decreasing reflectivities the FWHM linewidth &#x0394; &#x03BD; A i r y {\displaystyle \Delta \nu _{Airy}} of the Airy distribution broadens compared to the FWHM linewidth &#x0394; &#x03BD; c {\displaystyle \Delta \nu _{c}} of its corresponding Lorentzian line: R 1 = R 2 = 0.9 , 0.6 , 0.32 , 0.172 {\displaystyle R_{1}=R_{2}=0.9,0.6,0.32,0.172} results in &#x0394; &#x03BD; A i r y / &#x0394; &#x03BD; c = 1.001 , 1.022 , 1.132 , 1.717 {\displaystyle \Delta \nu _{Airy}/\Delta \nu _{c}=1.001,1.022,1.132,1.717} , respectively.
context_page_description: In optics, a Fabry–Pérot interferometer or etalon is an optical cavity made from two parallel reflecting surfaces. Optical waves can pass through the optical cavity only when they are in resonance with it. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899. Etalon is from the French étalon, meaning "measuring gauge" or "standard".
  Etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. Recent advances in fabrication technique allow the creation of very precise tunable Fabry–Pérot interferometers. The device is called an interferometer when the distance between the two surfaces can be changed, and an etalon when the distance is fixed.
context_section_description: Once the internal resonance enhancement, the generic Airy distribution, is established, all other Airy distributions can be deduced by simple scaling factors. Since the intensity launched into the resonator equals the transmitted fraction of the intensity incident upon mirror 1,
  and the intensities transmitted through mirror 2, reflected at mirror 2, and transmitted through mirror 1 are the transmitted and reflected/transmitted fractions of the intensity circulating inside the resonator,
  respectively, the other Airy distributions  with respect to launched intensity  and  with respect to incident intensity  are
  The index "emit" denotes Airy distributions that consider the sum of intensities emitted on both sides of the resonator.
  The back-transmitted intensity  cannot be measured, because also the initially back-reflected light adds to the backward-propagating signal. The measurable case of the intensity resulting from the interference of both backward-propagating electric fields results in the Airy distribution
  It can be easily shown that in a Fabry-Pérot resonator, despite the occurrence of constructive and destructive interference, energy is conserved at all frequencies:
  The external resonance enhancement factor (see figure "Resonance enhancement in a Fabry-Pérot resonator") is
  At the resonance frequencies , where  equals zero, the external resonance enhancement factor is
  Usually light is transmitted through a Fabry-Pérot resonator. Therefore, an often applied Airy distribution is
  It describes the fraction  of the intensity  of a light source incident upon mirror 1 that is transmitted through mirror 2 (see figure "Airy distribution "). Its peak value at the resonance frequencies  is
  For  the peak value equals unity, i.e., all light incident upon the resonator is transmitted; consequently, no light is reflected, , as a result of destructive interference between the fields  and .
   has been derived in the circulating-field approach by considering an additional phase shift of  during each transmission through a mirror,
  resulting in
  Alternatively,  can be obtained via the round-trip-decay approach by tracing the infinite number of round trips that the incident electric field  exhibits after entering the resonator and accumulating the electric field  transmitted in all round trips. The field transmitted after the first propagation and the smaller and smaller fields transmitted after each consecutive propagation through the resonator are
  respectively. Exploiting
  results in the same  as above, therefore the same Airy distribution  derives. However, this approach is physically misleading, because it assumes that interference takes place between the outcoupled beams after mirror 2, outside the resonator, rather than the launched and circulating beams after mirror 1, inside the resonator. Since it is interference that modifies the spectral contents, the spectral intensity distribution inside the resonator would be the same as the incident spectral intensity distribution, and no resonance enhancement would occur inside the resonator.
caption: Airy distribution 
    
      
        
          
            A
            
              trans
            
            
              ′
            
          
        
      
      {\displaystyle A_{\text{trans}}^{\prime }}
    
   (solid lines), corresponding to light transmitted through a Fabry-Pérot resonator, calculated for different values of the reflectivities 
    
      
        
          
            R
            
              1
            
          
          =
          
            R
            
              2
            
          
        
      
      {\displaystyle R_{1}=R_{2}}
    
  , and comparison with a single Lorentzian line (dashed lines) calculated for the same 
    
      
        
          
            R
            
              1
            
          
          =
          
            R
            
              2
            
          
        
      
      {\displaystyle R_{1}=R_{2}}
    
  .[8] At half maximum (black line), with decreasing reflectivities the FWHM linewidth 
    
      
        
          Δ
          
            ν
            
              Airy
            
          
        
      
      {\displaystyle \Delta \nu _{\text{Airy}}}
    
   of the Airy distribution broadens compared to the FWHM linewidth 
    
      
        
          Δ
          
            ν
            
              c
            
          
        
      
      {\displaystyle \Delta \nu _{c}}
    
   of its corresponding Lorentzian line: 
    
      
        
          
            R
            
              1
            
          
          =
          
            R
            
              2
            
          
          =
          0.9
          ,
          0.6
          ,
          0.32
          ,
          0.172
        
      
      {\displaystyle R_{1}=R_{2}=0.9,0.6,0.32,0.172}
    
   results in 
    
      
        
          Δ
          
            ν
            
              Airy
            
          
          
            /
          
          Δ
          
            ν
            
              c
            
          
          =
          1.001
          ,
          1.022
          ,
          1.132
          ,
          1.717
        
      
      {\displaystyle \Delta \nu _{\text{Airy}}/\Delta \nu _{c}=1.001,1.022,1.132,1.717}
    
  , respectively.
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