page_title: Fabry–Pérot interferometer
section_title: Scanning the Fabry-Pérot resonator: Airy linewidth and finesse
hierarchical_section_title: Fabry–Pérot interferometer / Theory / Scanning the Fabry-Pérot resonator: Airy linewidth and finesse
caption_attribution_description: English: Example of a frequency-dependent mirror reflectivity and resulting distorted mode profiles of the modes with indices q = 2000 , 2001 , 2002 {\displaystyle q=2000,2001,2002} and Airy distribution A &#x00B4; t r a n s {\displaystyle {\acute {A}}_{trans}} in a Fabry-Pérot resonator. [1] The vertical dashed lines denote the maximum of the reflectivity curve (black) and the resonance frequencies of the individual modes (colored).
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: When the Fabry-Pérot resonator is used as a scanning interferometer, i.e., at varying resonator length (or angle of incidence), one can spectroscopically distinguish spectral lines at different frequencies within one free spectral range. Several Airy distributions , each one created by an individual spectral line, must be resolved. Therefore, the Airy distribution becomes the underlying fundamental function and the measurement delivers a sum of Airy distributions. The parameters that properly quantify this situation are the Airy linewidth  and the Airy finesse . The FWHM linewidth  of the Airy distribution  is
  The Airy linewidth  is displayed as the green curve in the figure "Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry-Pérot resonator".
  The concept of defining the linewidth of the Airy peaks as FWHM breaks down at  (solid red line in the figure "Airy distribution "), because at this point the Airy linewidth instantaneously jumps to an infinite value for  function. For lower reflectivity values of , the FWHM linewidth of the Airy peaks is undefined. The limiting case occurs at
  For equal mirror reflectivities, this point is reached when  (solid red line in the figure "Airy distribution ").
  The finesse of the Airy distribution of a Fabry-Pérot resonator, which is displayed as the green curve in the figure "Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry-Pérot resonator" in direct comparison with the Lorentzian finesse , is defined as
  When scanning the length of the Fabry-Pérot resonator (or the angle of incident light), the Airy finesse quantifies the maximum number of Airy distributions created by light at individual frequencies  within the free spectral range of the Fabry-Pérot resonator, whose adjacent peaks can be unambiguously distinguished spectroscopically, i.e., they do not overlap at their FWHM (see figure "The physical meaning of the Airy finesse"). This definition of the Airy finesse is consistent with the Taylor criterion of the resolution of a spectrometer. Since the concept of the FWHM linewidth breaks down at , consequently the Airy finesse is defined only until , see the figure "Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry-Pérot resonator".
  Often the unnecessary approximation  is made when deriving from  the Airy linewidth . In contrast to the exact solution above, it leads to
  This approximation of the Airy linewidth, displayed as the red curve in the figure "Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry-Pérot resonator", deviates from the correct curve at low reflectivities and incorrectly does not break down when . This approximation is then typically also used to calculate the Airy finesse.
caption: Example of a Fabry-Pérot resonator with (top) frequency-dependent mirror reflectivity and (bottom) the resulting distorted mode profiles 
    
      
        
          
            γ
            
              q
              ,
              
                
                  t
                  r
                  a
                  n
                  s
                
              
            
            
              ′
            
          
        
      
      {\displaystyle \gamma _{q,{\rm {trans}}}^{\prime }}
    
   of the modes with indices 
    
      
        
          q
          =
          2000
          ,
          2001
          ,
          2002
        
      
      {\displaystyle q=2000,2001,2002}
    
  , the sum of 6 million mode profiles (pink dots, displayed for a few frequencies only), and the Airy distribution 
    
      
        
          
            A
            
              
                t
                r
                a
                n
                s
              
            
            
              ′
            
          
        
      
      {\displaystyle A_{\rm {trans}}^{\prime }}
    
  .[8] The vertical dashed lines denote the maximum of the reflectivity curve (black) and the resonance frequencies of the individual modes (colored).
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