BOX I: Metrics of optical radiation and (bio-)effectiveness
Here follows a concise representation of the physical and bio-effective metrics of optical
radiation as discussed in 3.4.2.1 and 3.4.2.2 in mathematical formulae with physical
dimensions given in square brackets,”[ ]”.
Given the spectral irradiance E(λ) in [W/m2 /nm] at the wavelength λ in [nm], we find
the total irradiance, E, in [W/m2] by integration over the wavelengths:
E = ∫ E(λ) dλ,
If the spectral irradiance varies with time, we find the spectral exposure, H(λ) in [J/ m2
/nm] at wavelength λ by integration over time, t in [s]:
H(λ) = ∫ E(λ,t) dt
and the exposure, H, by integration over λ:
H = ∫ ∫ E(λ,t) dλdt = ∫ H(λ) dλ.
which simplifies to:
H = ε ∫ E(λ) dλ = ε.E
if E(λ) is constant over the exposure time ε in [s].
To ascertain the bio-effectiveness of the radiation we define a dimensionless action
spectrum S(λ) to weight the spectral exposure. S(λ) is inversely proportional to the
exposure Hr(λ) required at a certain wavelength λ for a certain level of biological
response (level), and normalized to equal “1” at λmax, the most effective wavelength to
induce this response with smallest Hr(λ), i.e.
S(λ) = Hr(λmax)/Hr(λ).
The effectiveness spectrum is then defined as:
Ee(λ) = S(λ).E(λ),
The (bio-) effective irradiance as:
Ee = ∫ S(λ) E(λ) dλ,
and the (bio-) effective exposure or photobiologic “dose” as:
He = ∫ S(λ) H(λ) dλ,
where Ee(λ), Ee and He are then given in equivalents of [W/m2 nm], [W/m2] and [J/m2],
respectively, at wavelength λmax.
Note that this procedure of spectral weighting requires “additivity” to hold, i.e. that
separate effective doses from different wavelengths can be added up to a total effective
dose.
The irradiance at the retina of the eye from a radiant source is strongly dependent on the
imaging, i.e. focusing, of the source onto the retina which can cause up to a 200,000-fold
increase in irradiance from the surface of the eye to the retina3. When we consider a
source (a lamp or a reflecting surface), its spectral radiant power in [W/nm] emitted per
radiant surface area in [m2] into a solid angle in steradian [sr], i.e. the spectral radiance,
L(λ), in [W/sr m2 nm], towards the pupil of the eye is crucial to the retinal irradiance Eret in [W/m2]:
Eret = (π/4) (dp/f)2 ∫ L(λ) τ(λ) dλ,
where τ(λ) is the transmittance from the source to the retina at wavelength λ, dp denotes
the pupil diameter in [m] and f is the focal length of the eye in [m]. Note that the
distance of the source from the eye drops from the equation (as the source is moved
away from the eye the image spot on the retina becomes smaller but the irradiance
remains constant if the radiance is constant over the surface of the source). L is also
referred to as the source’s “intensity”. The eye tremor (around 80 Hz with amplitudes of
0.2 to 2.5 μm) spreads a point source over a larger image area on the retina (about 25
μm in diameter in 1 sec and 190 μm in 100 s, subtending about 11 mrad). It should also
be noted that in a realistic assessment, for example of a human who is reading, the
source is a page at a certain distance from a lamp, i.e. not the lamp itself. The eyes
rarely focus on a primary light source, a lamp, and strongly evade looking into bright
ones (eyes closing and causing the face to turn away from the source).
To ascertain the bio-effectiveness on the retina Eret (λ) and L(λ) can be weighted by an
action spectrum S(λ). For example, taking S(λ) as the photopic* luminous efficiency
function, V(λ), (normalized to 1 at λmax = 555 nm) one converts a physical radiant power
spectrum, or spectral flux Φ(λ) in [W/nm], into a luminous flux, Φv, in lumens [lm]
representing the visual effectiveness:
Φv = 683.002 [lm/W] ∫ V(λ) Φ(λ) dλ.
Illuminance (the photometric equivalent of irradiance) equals the luminous flux per
surface area in lux, [lx] = [lm/m2].
The photopically weighted radiance, luminance Lph, of a source is given in candela per
surface area [cd/m2] = [lm/sr m2]. Luminance is also loosely referred to as the
“brightness” of a radiant source, i.e. the visual intensity.
* Photopic refers to daylight vision, i.e. with a light-adapted eye; scotopic, on the other hand,
refers to night vision, i.e. with a dark-adapted eye.
Source: SCENIHR, Health effects of artificial light, 19 March 2012,
3.4.1 Optical radiation and 3.4.2 Radiant energy absorption, pp. 22-31.