---
title: "Cherenkov radiation"
chunk: 2/4
source: "https://en.wikipedia.org/wiki/Cherenkov_radiation"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T10:54:47.273838+00:00"
instance: "kb-cron"
---

In their original work on the theoretical foundations of Cherenkov radiation, Tamm and Frank wrote, "This peculiar radiation can evidently not be explained by any common mechanism such as the interaction of the fast electron with individual atom or as radiative scattering of electrons on atomic nuclei.  On the other hand, the phenomenon can be explained both qualitatively and quantitatively if one takes into account the fact that an electron moving in a medium does radiate light even if it is moving uniformly provided that its velocity is greater than the velocity of light in the medium."

=== Emission angle ===

In the figure on the geometry, the particle (red arrow) travels in a medium with speed 
  
    
      
        
          v
          
            p
          
        
      
    
    {\displaystyle v_{\text{p}}}
  
 such that 

  
    
      
        
          
            c
            n
          
        
        <
        
          v
          
            p
          
        
        <
        c
        ,
      
    
    {\displaystyle {\frac {c}{n}}<v_{\text{p}}<c,}
  

where 
  
    
      
        c
      
    
    {\displaystyle c}
  
 is speed of light in vacuum, and 
  
    
      
        n
      
    
    {\displaystyle n}
  
 is the refractive index of the medium. If the medium is water, the condition is 
  
    
      
        0.75
        c
        <
        
          v
          
            p
          
        
        <
        c
      
    
    {\displaystyle 0.75c<v_{\text{p}}<c}
  
, since 
  
    
      
        n
        ≈
        1.33
      
    
    {\displaystyle n\approx 1.33}
  
 for water at 20 °C.
We define the ratio between the speed of the particle and the speed of light as 

  
    
      
        β
        =
        
          
            
              v
              
                p
              
            
            c
          
        
        .
      
    
    {\displaystyle \beta ={\frac {v_{\text{p}}}{c}}.}
  

The emitted light waves (denoted by blue arrows) travel at speed 

  
    
      
        
          v
          
            em
          
        
        =
        
          
            c
            n
          
        
        .
      
    
    {\displaystyle v_{\text{em}}={\frac {c}{n}}.}
  

The left corner of the triangle represents the location of the superluminal particle at some initial moment (t = 0). The right corner of the triangle is the location of the particle at some later time t. In the given time t, the particle travels the distance

  
    
      
        
          x
          
            p
          
        
        =
        
          v
          
            p
          
        
        t
        =
        β
        
        c
        t
      
    
    {\displaystyle x_{\text{p}}=v_{\text{p}}t=\beta \,ct}
  

whereas the emitted electromagnetic waves are constricted to travel the distance

  
    
      
        
          x
          
            em
          
        
        =
        
          v
          
            em
          
        
        t
        =
        
          
            c
            n
          
        
        t
        .
      
    
    {\displaystyle x_{\text{em}}=v_{\text{em}}t={\frac {c}{n}}t.}
  

So the emission angle results in

  
    
      
        cos
        ⁡
        θ
        =
        
          
            1
            
              n
              β
            
          
        
      
    
    {\displaystyle \cos \theta ={\frac {1}{n\beta }}}
  

=== Arbitrary emission angle ===
Cherenkov radiation can also radiate in an arbitrary direction using properly engineered one dimensional metamaterials. The latter is designed to introduce a gradient of phase retardation along the trajectory of the fast travelling particle (
  
    
      
        d
        ϕ
        
          /
        
        d
        x
      
    
    {\displaystyle d\phi /dx}
  
), reversing or steering Cherenkov emission at arbitrary angles given by the generalized relation:

  
    
      
        cos
        ⁡
        θ
        =
        
          
            1
            
              n
              β
            
          
        
        +
        
          
            n
            
              k
              
                0
              
            
          
        
        ⋅
        
          
            
              d
              ϕ
            
            
              d
              x
            
          
        
      
    
    {\displaystyle \cos \theta ={\frac {1}{n\beta }}+{\frac {n}{k_{0}}}\cdot {\frac {d\phi }{dx}}}
  

Note that since this ratio is independent of time, one can take arbitrary times and achieve similar triangles. The angle stays the same, meaning that subsequent waves generated between the initial time t = 0 and final time t will form similar triangles with coinciding right endpoints to the one shown.

=== Reverse Cherenkov effect ===
A reverse Cherenkov effect can be experienced using materials called negative-index metamaterials (materials with a subwavelength microstructure that gives them an effective "average" property very different from their constituent materials, in this case having negative permittivity and negative permeability). This means that, when a charged particle (usually electrons) passes through a medium at a speed greater than the phase velocity of light in that medium, that particle emits trailing radiation from its progress through the medium rather than in front of it (as is the case in normal materials with both permittivity and permeability positive). One can also obtain such reverse-cone Cherenkov radiation in non-metamaterial periodic media where the periodic structure is on the same scale as the wavelength, so it cannot be treated as an effectively homogeneous metamaterial.

=== In vacuum ===
The Cherenkov effect can occur in vacuum. In a slow-wave structure, like in a traveling-wave tube (TWT), the phase velocity decreases and the velocity of charged particles can exceed the phase velocity while remaining lower than 
  
    
      
        c
      
    
    {\displaystyle c}
  
. In such a system, this effect can be derived from conservation of the energy and momentum where the momentum of a photon should be 
  
    
      
        p
        =
        ℏ
        β
      
    
    {\displaystyle p=\hbar \beta }
  
 (
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 is phase constant) rather than the de Broglie relation 
  
    
      
        p
        =
        ℏ
        k
      
    
    {\displaystyle p=\hbar k}
  
. This type of radiation (VCR) is used to generate high-power microwaves.

=== Collective Cherenkov ===
Radiation with the same properties of typical Cherenkov radiation can be created by structures of electric current that travel faster than light. By manipulating density profiles in plasma acceleration setups, structures up to nanocoulombs of charge are created and may travel faster than the speed of light and emit optical shocks at the Cherenkov angle. Electrons are still subluminal, hence the electrons that compose the structure at a time t = t0 are different from the electrons in the structure at a time t > t0.

== Characteristics ==
The frequency spectrum of Cherenkov radiation by a particle is given by the Frank–Tamm formula:

  
    
      
        
          
            
              
                
                  d
                
                
                  2
                
              
              E
            
            
              
                d
              
              x
              
              
                d
              
              ω
            
          
        
        =
        
          
            
              q
              
                2
              
            
            
              4
              π
            
          
        
        μ
        (
        ω
        )
        ω
        
          
            (
            
              1
              −
              
                
                  
                    c
                    
                      2
                    
                  
                  
                    
                      v
                      
                        2
                      
                    
                    
                      n
                      
                        2
                      
                    
                    (
                    ω
                    )
                  
                
              
            
            )
          
        
      
    
    {\displaystyle {\frac {\mathrm {d} ^{2}E}{\mathrm {d} x\,\mathrm {d} \omega }}={\frac {q^{2}}{4\pi }}\mu (\omega )\omega {\left(1-{\frac {c^{2}}{v^{2}n^{2}(\omega )}}\right)}}