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Radiant flux

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A flow chart describing the relationship of various physical quantities, including radiant flux and exitance

In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), one joule per second (J/s), while that of spectral flux in frequency is the watt per hertz (W/Hz) and that of spectral flux in wavelength is the watt per metre (W/m)—commonly the watt per nanometre (W/nm). Radiant flux is sometimes called luminosity, especially in astronomy contexts.

Radiant flux, denoted Φe ('e' for "energetic", to avoid confusion with photometric quantities), is defined as

              Φ
              
                
                  e
                
              
            
          
          
            
            =
            
              
                
                  d
                  
                    Q
                    
                      
                        e
                      
                    
                  
                
                
                  d
                  t
                
              
            
          
        
        
          
            
              Q
              
                
                  e
                
              
            
          
          
            
            =
            
              ∫
              
                T
              
            
            
              ∫
              
                Σ
              
            
            
              S
            
            ⋅
            
              
                
                  
                    n
                  
                  ^
                
              
            
            
            d
            A
            d
            t
          
        
      
    
  

{\displaystyle {\begin{aligned}\Phi _{\mathrm {e} }&={\frac {dQ_{\mathrm {e} }}{dt}}\\[2pt]Q_{\mathrm {e} }&=\int _{T}\int _{\Sigma }\mathbf {S} \cdot {\hat {\mathbf {n} }}\,dAdt\end{aligned}}}

where

Qe is the radiant energy passing out of a closed surface Σ in time interval T;
t is time;
A is the area of the surface Σ;
S is the Poynting vector, representing the directional flow of energy per unit time, per unit area;
n is the unit normal vector to the differential area element dA.

The rate of energy flow through the surface fluctuates at the frequency of the radiation, but radiation detectors only respond to the average rate of flow. This is represented by replacing the Poynting vector with the time average of its norm, giving

      Φ
      
        
          e
        
      
    
    ≈
    
      ∫
      
        Σ
      
    
    ⟨
    
      |
    
    
      S
    
    
      |
    
    ⟩
    cos
    ⁡
    α
     
    d
    A
    ,
  

{\displaystyle \Phi _{\mathrm {e} }\approx \int _{\Sigma }\langle |\mathbf {S} |\rangle \cos \alpha \ dA,}

where ⟨-⟩ is the time average, and α is the angle between n and S.

Spectral flux in frequency, denoted Φe,ν, is defined as

      Φ
      
        
          e
        
        ,
        ν
      
    
    =
    
      
        
          ∂
          
            Φ
            
              
                e
              
            
          
        
        
          ∂
          ν
        
      
    
    ,
  

{\displaystyle \Phi _{\mathrm {e} ,\nu }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \nu }},}

where ν is the frequency.

Spectral flux in wavelength, denoted Φe,λ, is defined as

      Φ
      
        
          e
        
        ,
        λ
      
    
    =
    
      
        
          ∂
          
            Φ
            
              
                e
              
            
          
        
        
          ∂
          λ
        
      
    
    ,
  

{\displaystyle \Phi _{\mathrm {e} ,\lambda }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \lambda }},}

where λ is the wavelength.

Comparison of photometric and radiometric quantities

Quantity	Unit	Dimension	Notes
Qe	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
we	joule per cubic metre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Φe	watt	W = J/s	M⋅L2⋅T−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in astronomy.
Φe,ν	watt per hertz	W/Hz	M⋅L2⋅T −2	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
Φe,λ	watt per metre	W/m	M⋅L⋅T−3
Ie,Ω	watt per steradian	W/sr	M⋅L2⋅T−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Ie,Ω,ν	watt per steradian per hertz	W⋅sr−1⋅Hz−1	M⋅L2⋅T−2	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
Ie,Ω,λ	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Le,Ω	watt per steradian per square metre	W⋅sr−1⋅m−2	M⋅T−3	Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also called "intensity".
Le,Ω,ν	watt per steradian per square metre per hertz	W⋅sr−1⋅m−2⋅Hz−1	M⋅T−2	Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also called "spectral intensity".
Le,Ω,λ	watt per steradian per square metre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
Ee	watt per square metre	W/m2	M⋅T−3	Radiant flux received by a surface per unit area. This is sometimes also called "intensity".
Ee,ν	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Irradiance of a surface per unit frequency or wavelength. This is sometimes also called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
Ee,λ	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Je	watt per square metre	W/m2	M⋅T−3	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also called "intensity".
Je,ν	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also called "spectral intensity".
Je,λ	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Me	watt per square metre	W/m2	M⋅T−3	Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also called "intensity".
Me,ν	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also called "spectral intensity".
Me,λ	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
He	joule per square metre	J/m2	M⋅T−2	Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
He,ν	joule per square metre per hertz	J⋅m−2⋅Hz−1	M⋅T−1	Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
He,λ	joule per square metre, per metre	J/m3	M⋅L−1⋅T−2
See also: SIRadiometryPhotometry

Luminous flux
Heat flux
Power (physics)
Radiosity (heat transfer)
Boyd, Robert (1983). Radiometry and the Detection of Optical Radiation (Pure & Applied Optics Series). Wiley-Interscience. ISBN 978-0-471-86188-1.

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