There are different systems of units for optical radiation. The article follows the internationally agreed CIE system. The CIE system fits well with the SI system of units.
Radiometric, Photometric and Photon Quantities
Radiometric quantities are purely physical. How the human eye records optical radiation is more relevant when compared to the absolute physical values. This evaluation is described in photometric units and is limited to the small part of the spectrum called the visible. Photon quantities are essential for a number of physical processes. Table 1 lists radiometric, photometric and photon quantities.
Table 1. Commonly Used Radiometric, Photometric and Photon Quantities
Radiometric 
Photometric 
Photon 
Quantity 
Usual Symbol 
Units 
Quantity 
Usual Symbol 
Units 
Quantity 
Usual Symbol 
Units 
Radiant Energy 
Q_{e} 
J 
Luminous Energy 
Q_{v} 
Im s 
Photon Energy 
N_{p} 
* 

Radiant Power or Flux 
φ_{e} 
W 
Luminous Flux 
φ_{v} 
Im 
Photon Flux 
Φ_{p}=dNp/dt 
s^{1} 
Radiant Exitance or Emittance 
^{M}e 
W m^{2} 
Luminous Exitance or Emittance 
^{M}v 
Im m^{2} 
Photon Exitance 
^{M}p 
s^{1}s^{2} 
Irradiance 
E_{e} 
W m^{2} 
Illuminance 
E_{v} 
Ix 
Photon Irradiance 
E_{p} 
s^{1}s^{2} 
Radiant Intensity 
I_{e} 
W sr^{1} 
Luminous Intensity 
I_{v} 
cd 
Photon Intensity 
I_{p} 
s^{1}sr^{1} 
Radiance 
L_{e} 
W sr^{1} m^{2} 
Luminance 
Lv 
cd m^{2} 
Photon Radiance 
L_{p} 
s^{1}sr^{1}m^{2} 
* Photon quantities are expressed in number of photons followed by the units, eg. photon flux (number of photons) s^{1}. The unit for photon energy is number of photons.
The subscripts e,v, and p designate radiometric, photometric, and photon quantities respectively. While working with only one type of quantity, they are normally omitted.
Symbols Key:
J: joule
lm: lumen
W: watts
s: second
m: meter
cd: candela
sr : steradian
lx: lux, lumen m^{2}
Table 2. Some Units Still in Common Use
Units 
Equivalent 
Quantity 
Talbot 
Im s 
Luminous Energy 
Footcandle 
Im ft^{2} 
Illuminance 
Footlambert 
cd ft^{2} 
Luminance 
Lambert 
cd cm^{2} 
Luminance 
Sterance, aerance and pointance are used for supplementing or replacing the above terms.
 Sterance, means, related to the solid angle, so radiance may be described by radiant sterance.
 Areance, related to an area, gives radiant areance instead of radiant exitance.
 Pointance, related to a point, leads to radiant pointance instead of radiant intensit
Spectral Distribution
Using spectral before the tabulated radiometric quantities implies consideration of the wavelength dependence of the quantity. The measurement wavelength should be given when a spectral radiometric value is quoted. The variation of spectral radiant exitance (M_{eλ}), or irradiance (E_{eλ}) with wavelength is often shown in a spectral distribution curve.
Units for Spectral Irradiance
The preferred unit for spectral irradiance shown in this article is mW m^{2}nm^{1}. Conversion to other units, such as mW m^{2} μm^{1}, is straightforward.
Wavelength, Wavenumber, Frequency and Photon Energy
Wavelength is inversely proportional to the photon energy; shorter wavelength photons are more energetic photons. Wavenumber and frequency increase with photon energy.
The units of wavelength used are nanometers (nm) and micrometers(µm).
1 nm = 10^{9} m = 10^{3} μm
1 μm = 10^{9} m = 1000 nm
1 Angstom unit (A) = 10^{10} m = 10^{1} nm
Figure 1 shows the solar spectrum and 5800K blackbody spectral distributions against energy (and wavenumber), in contrast with the familiar representation.
Table 2 below helps in converting from one spectral parameter to another. The conversions use the approximation 3 x 10^{8} ms^{1} for the speed of light. For precise work, the actual speed of light in the medium must be used. The speed in air is based on the wavelength, humidity and pressure, but the variance is only important for interferometry and high resolution spectroscopy.
Table 3. Spectral Parameter Conversion Factors

Wavelength 
Wavenumber* 
Frequency 
Photon Energy** 
Symbol (Units) 
λ (nm) 
υ (cm^{1}) 
ν (Hz) 
E_{p} (eV) 
Conversion Factors 
λ 
10^{7}/ λ 
3 x 10^{17}/λ 
1,240/λ 

10^{7}/υ 
υ 
3 x 10^{10} υ 
1.24 x 10^{4}υ 

3 x 10^{17}/ν 
3.33 x 10^{11}ν 
ν 
4.1 x 10^{15}ν 

1,240/E_{p} 
8,056 x E_{p} 
2.42 x 10^{14}E_{p} 
E_{p} 
Conversion Examples 
200
500
1000 
5 x 10^{4}
2 x 10^{4}
10^{4} 
1.5 x 10^{15}
6 x 10^{14}
3 x 10^{14} 
6.20
2.48
1.24 
* The S.I. unit is the m1. Most users, primarily individuals working in infrared analysis, adhere to the cm^{1}.
** Photon energy is usually expressed in electron volts to relate to chemical bond strengths.The units are also more convenient than photon energy expressed in joules as the energy of a 500 nm photon is 3.98 x 10^{19} J = 2.48 eV
Similarly by dividing by the beam impact area, photon irradiance can be easily calculated.
Irradiance:Spatial Dependence
Most radiometric quantities as well as irradiance have values that are defined at a point even though the units, mW m^{2} nm^{1}, imply a large area. The spatial map of the irradiance is required in the full description. Often average values over a defined area are most useful. Peak levels can greatly exceed average values.
Figure 1. Unconventional display of solar irradiance on the outer atmosphere and the spectral distribution of a 5800K blackbody with the same total radiant flux.
Converting from Radiometric to Photon Quantities
It is important to express radiation in photon quantities when irradiation results are described in terms of cross section, number of molecules excited or for many detector and energy conversion systems, quantum efficiency.
Monochromatic Radiation
It is easy to calculate the number of photons in a joule of monochromatic light of wavelength λ as the energy in each photon is given by E = hc/λ joules
Where:
h = Planck's constant (6.626 x 10^{34} Js), c = Speed of light (2.998 x 10^{8} m s^{1}) λ = Wavelength in m
Hence the number of photons per joule is
N_{pλ} = λ x 5.03 x 10^{15} where λ is in nm^{+} Since a watt is a joule per second, one Watt of monochromatic radiation at λ corresponds to N_{pλ} photons per second. The general expression is:
dN_{pλ}/dt = P_{λ} x λ x 5.03 x 10^{15}
where P_{λ }is in watts, λ is in nm
Table 4. Photopic Response
Wavelength (nm) 
Photopic Luminous Efficiency V(λ) 
Wavelength (nm) 
Photopic Luminous Efficiency V(λ) 
380 
0.00004 
580 
0.870 
390 
0.00012 
590 
0.757 
400 
0.0004 
600 
0.631 
410 
0.0012 
610 
0.503 
420 
0.0040 
620 
0.381 
430 
0.0116 
630 
0.265 
440 
0.023 
640 
0.175 
450 
0.038 
650 
0.107 
460 
0.060 
660 
0.061 
470 
0.091 
670 
0.032 
480 
0.139 
680 
0.017 
490 
0.208 
690 
0.0082 
500 
0.323 
700 
0.0041 
510 
0.503 
710 
0.0021 
520 
0.710 
720 
0.00105 
530 
0.862 
730 
0.00052 
540 
0.954 
740 
0.00025 
550 
0.995 
750 
0.00012 
555 
1.000 
760 
0.00006 
560 
0.995 
770 
0.00003 
570 
0.952 


The spectral distribution of the radiations must be known for conversions. Conversion from a radiometric quantity (in watts) to the corresponding photometric quantity (in lumens) simply requires multiplying the spectral distribution curve by the photopic response curve, integrating the product curve and multiplying the result by a conversion factor of 683.
Mathematically for a photometric quantity (PQ) and its matching radiometric quantity (SPQ).
PQ = 683 ∫ (SPQ_{λ}) • V(λ)dλ
Since V(λ) is zero except between 380 and 770 nm, simply sum the product values over small spectral intervals, Δλ:
PQ ≈ (Σ_{n} (SPQ_{λn}) • V(λ_{n})) • Δλ
Where:
(SPQ_{λn}) = Average value of the spectral radiometric quantity in wavelength interval number "n"
Broadband Radiation
For conversion from radiometric to photon quantities, you need to know the spectral distribution of the radiation. For irradiance you need to know the dependence of E_{eλ} on X. The curves will have different shapes as shown in Figure 2.
Figure 2. The wavelength dependence of the irradiance produced by the 6283 200 W mercury lamp at 0.5 m. (1) shown conventionally in mW m^{2} nm^{1} and (2) as photon flux.
Figure 3. The normalized response of the "standard" light adapted eye.
Conversion from Radiometric to Photometric Values
It is possible to convert from radiometric to matching photometric quantity.The photometric measure depends on how the source appears to the human eye.
Mathematically for a photometric quantity (PQ) and its matching radiometric quantity (SPQ).
PQ = 683 ∫ (SPQ_{λ}) • V(λ)dλ
Since Vλ is zero except between 380 and 770 nm, you only need to integrate over this range. Most computations simply sum the product values over small spectral intervals, Δλ
PQ ≈ (Σ_{n} (SPQ_{λn}) • V(λ_{n})) • Δλ
Where:
(SPQ_{λn}) = Average value of the spectral radiometric quantity in wavelength interval number “n”
The smaller the wavelength interval and the slower the variation in SPQ_{λ}, the higher is the accuracy.
Figure 4. Lamp Irradiance, V (λ), and product curve.
Figure 5 shows the irradiance curve multiplied by the V(λ) curve. The unit of the product curve that describes the radiation is the IW, or light watt, a hybrid unit bridging the transition between radiometry and photometry. The integral of the product curve is 396 mIWm^{2}, where a IW is the unit of the product curve.
Table 5. Light Watt Values
Wavelength Range (nm) 
Estimated Average Irradiance (mW m^{2} nm^{1}) 
V(λ) 
Product of cols 1 & 2 x 50 nm (mIW m^{2} ) 
380  430 
3.6 
0.0029 
0.5 
430  480 
4.1 
0.06 
12 
480  530 
3.6 
0.46 
83 
530  580 
3.7 
0.94 
174 
580  630 
3.6 
0.57 
103 
630  680 
3.4 
0.11 
19 
680  730 
3.6 
0.0055 
1.0 
730  780 
3.8 
0.0002 
0.038 
To get from IW to lumens requires multiplying by 683, so the illuminance is:
396x683mlumens m^{2}= 270lumensm^{2} (or 270 lux)
Since there are 10.764 ft^{2} in a m^{2}, the illuminance in foot candles (lumens ft^{2}) is 270/10.8 = 25.1 foot candles.
Conclusion
The article offers an overview of the different units of optical radiation and conversion from one system to another.
This information has been sourced, reviewed and adapted from materials provided by Oriel Instruments.
For more information on this source, please visit Oriel Instruments.