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 λ_max, the most effective wavelength to induce this response with smallest Hr(λ), i.e.

S(λ) = Hr(λ_max)/Hr(λ).

The effectiveness spectrum is then defined as:

E_e(λ) = S(λ).E(λ),

The (bio-) effective irradiance as:

E_e = ∫ S(λ) E(λ) dλ,

and the (bio-) effective exposure or photobiologic “dose” as:

H_e = ∫ S(λ) H(λ) dλ,

where E_e(λ), E_e and H_e are then given in equivalents of [W/m² nm], [W/m²] and [J/m²], respectively, at wavelength λ_max. 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 E_ret in [W/m²]:

E_ret = (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 E_ret (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 λ_max = 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.