Lighting technology (7S880)
Light and radiation
Energy of a photon: qph = h*v = h*(c0/λ)λ))
h = Planck’s constant = 6.626176*10-34ws
v = frequency
Monochromatic light: light emitted at only 1 wavelength, perceived as a chromatic colour.
Radiometric quantities: quantity that describes the radiation as a whole.
Photometric quantities: measure the radiation weighted by the human eye.
∂ Xλ ( λ )
Spectral distribution of the radiometric quantity: Xλλ=
∂λ
Radiometric quantity x: Xλ e=∫ Xλ eλ dλλ
λ
Index e indicates that it is a radiometric (energetic) quantity.
Y
Sensitivity of a receptor: S=
Xλ
Y λ dλλ
Spectral sensitivity: s ( λ ) =
Xλ λ dλλ
s (λ) Y λ ( λ) Xλ λ ( λ 0)
Relative spectral sensitivity: s(λ))rel = =
s ( λ 0) Xλ λ ( λ) Y λ (λ 0)
∞
Effect y related to a given spectral sensitivity: Y =s (λ0 )∫ Xλ eλ ∙ s rel ( λ ) dλ λ
0
y = effect corresponding to the given sensitivity
s( λ 0) scaling factor
Xλ eλ spectral distribution of the quantity x
s(λ) spectral sensitivity for the given effect
λ) wavelength
The maximum sensitivity occurs at a wavelength of 555nm. Can be determined by
comparing radiances of monochromatic light of various wavelengths to radiance at 555nm.
Le ( 555 nm )
Eye sensitivity function (v(λ))) for photopic vision = V ( λ ) =
Le ( λ )equal brightness
For scotopic vision the peak lies at 507nm.
Photopic curve is applicable for luminances
above 3.4cd/λ)m2. Scotopic curve v’ is valid for
luminance’s below 0.034cd/λ)m2. Between these
two values, a mix of the other two spectral
response curves occurs (= mesopic vision).
Spectral power distribution to photometric quantities: Xλ =K m ∙∫ Xλ eλ ∙ V ( λ ) dλλ.
,Km = 683lm/λ)w, it is the maximum luminous efficacy of radiation for photopic vision. For
scotopic vision use k’m = 1699lm/λ)w.
Luminous flux: φ with unit lumen (lm). It is the photometric quantity that relates to power
and the equivalent to the radiant power described by radiant flux. The luminous flux is used
to describe the light generated by lamps. Φ=K m ∙ Φ eλ ∙ V ( λ ) dλ λ.
∫
The spectral power density φeλ) is typically given as a mean value for small wavelength
bands. The luminous flux is then the summation over all intervals weighted by the v(λ))
N
function and km. Φ=K m ∙ ∑ Φ eλi ∙ V ( λ ) ∙ ∆ λ/λ)
i=1
N
Similarly, the calculation can be done for the radiant flux: Φ e =∑ Φeλi ∙ ∆ λ
i=1
Luminous efficacy of radiation: ratio between luminous and radiant flux of a light source:
k= φ/λ)φe
Luminous efficacy: η with unit lm/λ)w. It is a measure of how much luminous flux is
generated by an electrical light source in relation to the electrical power drawn. It is typically
used as an indicator for the energy efficiency of a lamp.
Illuminance: e with unit lux(lx). It is the areal density of the luminous flux and describes how
dλΦ
much luminous flux is received by a given receiving area. E=
dλ A2
dλΦ luminous flux on the surface element dλ A 2
dλ A 2 the illuminated area
- Horizontal illuminance: eh, measured on a horizontal surface.
- Vertical illuminance: ev, is measured on a vertical surface.
- Mean illuminance: É , is the average of series of illuminance measurements.
- Cylindrical illuminance: mean illuminance on the outer surface of a very small
cylinder that is put vertically at a point in the space, to obtain composite illuminance
produces by incident light from various directions.
2π
1 1
E z= ∫ Ev ( φ ) dλ φ E z ≈ ∙ ( E v ( 0 )+ Ev ( 90 ° )+ E v (180 ° ) + E v ( 270 ° ) )
2π 0 4
- Semi-cylindrical illuminance: mean illuminance on the outer surface of a very small
π
2
1
semi-cylinder that is put vertically at a point in space. E sc= ∫ E ( φ ) dλφ
π −π v
2
Planar angle: length of a circular arc around the vertex between the two rays relative to the
radius of the circular arc.
b
Angle α= the whole planar angle is 2π and is measured in radian.
r
Ak
Solid angle: (3 dimensional) Ω= 2 is measured in steradian (sr). The
r
dλ A k
whole solid angle is 4π. dλΩ= 2 en dω = sinε * dε * dα.
r
, Luminous intensity: i, measured in candela (cd). 1 candela is 1 lm/λ)sr. It relates to the
luminous flux emitted into a solid angle and describes how much luminous flux is emitted in
dλΦ
a certain direction. It is used to characterize the light distribution of a luminaire. I = .
dλ Ω1
Luminance: l, measured in cd/λ)m2. It is the luminous intensity in a given direction related to
the emitting area. It is the photometric quantity that impacts the brightness perception under
normal circumstances. It is used to evaluate the glare rating in a given scene.
dλ 2 Φ
L=
dλ Ω 1 ∙ dλ A 1 ∙ cos ( ε 1 )
luminous exitance: m, measured in lm/λ)m2. It is a measure for the luminous flux
dλΦ
emitted per surface area. M =
dλ A 1
Luminous energy: q, measured in lmh. It is the time integral of luminous flux:
Q=∫ Φ ( t ) dλt it is often used to describe the total amount of light that a lamp can
∆t
produce over its lifetime by weighing its luminous flux with the lamp life.
Luminous exposure: h, measured in lxh. It is the time integral of illuminance (e): :
H=∫ E ( t ) dλt
∆t
Photon characteristics:
- Energy of a photon: eph = h*v
Considering the medium with the refractive index n: λ)*v = n*c0.
dλ N p
- Photon flux: Φ́ p=
dλt
- Photon irradiance: quotient of the photon flux incident on an element of the surface
dλ Φ́ p
containing the point, by the area da, of that element. É p =
dλ A 2
- Photon intensity: quotient of the photon flux, leaving the source and propagated in
the element of solid angle containing the given direction, by the element of solid
dλ Φ́ p
angle: Í p=
dλ Ω1
- Photon radiance: photon flux transmitted by an elementary beam passing through
the given point and propagating in the solid angle containing the given direction.
dλ Φ́ 2 p
Ĺ p=
dλ Ω1 ∙ dλ A 2 ∙cos ε 1
- Photon exitance: quotient of the photon flux leaving an element of the surface
dλ Φ́ p
containing the point, by the area of that element. Ḿ p=
dλ A 1
- Number of photons: integral of the photon flux over a given duration: N p=∫ Φ́ p dλt
∆t
Light and radiation
Energy of a photon: qph = h*v = h*(c0/λ)λ))
h = Planck’s constant = 6.626176*10-34ws
v = frequency
Monochromatic light: light emitted at only 1 wavelength, perceived as a chromatic colour.
Radiometric quantities: quantity that describes the radiation as a whole.
Photometric quantities: measure the radiation weighted by the human eye.
∂ Xλ ( λ )
Spectral distribution of the radiometric quantity: Xλλ=
∂λ
Radiometric quantity x: Xλ e=∫ Xλ eλ dλλ
λ
Index e indicates that it is a radiometric (energetic) quantity.
Y
Sensitivity of a receptor: S=
Xλ
Y λ dλλ
Spectral sensitivity: s ( λ ) =
Xλ λ dλλ
s (λ) Y λ ( λ) Xλ λ ( λ 0)
Relative spectral sensitivity: s(λ))rel = =
s ( λ 0) Xλ λ ( λ) Y λ (λ 0)
∞
Effect y related to a given spectral sensitivity: Y =s (λ0 )∫ Xλ eλ ∙ s rel ( λ ) dλ λ
0
y = effect corresponding to the given sensitivity
s( λ 0) scaling factor
Xλ eλ spectral distribution of the quantity x
s(λ) spectral sensitivity for the given effect
λ) wavelength
The maximum sensitivity occurs at a wavelength of 555nm. Can be determined by
comparing radiances of monochromatic light of various wavelengths to radiance at 555nm.
Le ( 555 nm )
Eye sensitivity function (v(λ))) for photopic vision = V ( λ ) =
Le ( λ )equal brightness
For scotopic vision the peak lies at 507nm.
Photopic curve is applicable for luminances
above 3.4cd/λ)m2. Scotopic curve v’ is valid for
luminance’s below 0.034cd/λ)m2. Between these
two values, a mix of the other two spectral
response curves occurs (= mesopic vision).
Spectral power distribution to photometric quantities: Xλ =K m ∙∫ Xλ eλ ∙ V ( λ ) dλλ.
,Km = 683lm/λ)w, it is the maximum luminous efficacy of radiation for photopic vision. For
scotopic vision use k’m = 1699lm/λ)w.
Luminous flux: φ with unit lumen (lm). It is the photometric quantity that relates to power
and the equivalent to the radiant power described by radiant flux. The luminous flux is used
to describe the light generated by lamps. Φ=K m ∙ Φ eλ ∙ V ( λ ) dλ λ.
∫
The spectral power density φeλ) is typically given as a mean value for small wavelength
bands. The luminous flux is then the summation over all intervals weighted by the v(λ))
N
function and km. Φ=K m ∙ ∑ Φ eλi ∙ V ( λ ) ∙ ∆ λ/λ)
i=1
N
Similarly, the calculation can be done for the radiant flux: Φ e =∑ Φeλi ∙ ∆ λ
i=1
Luminous efficacy of radiation: ratio between luminous and radiant flux of a light source:
k= φ/λ)φe
Luminous efficacy: η with unit lm/λ)w. It is a measure of how much luminous flux is
generated by an electrical light source in relation to the electrical power drawn. It is typically
used as an indicator for the energy efficiency of a lamp.
Illuminance: e with unit lux(lx). It is the areal density of the luminous flux and describes how
dλΦ
much luminous flux is received by a given receiving area. E=
dλ A2
dλΦ luminous flux on the surface element dλ A 2
dλ A 2 the illuminated area
- Horizontal illuminance: eh, measured on a horizontal surface.
- Vertical illuminance: ev, is measured on a vertical surface.
- Mean illuminance: É , is the average of series of illuminance measurements.
- Cylindrical illuminance: mean illuminance on the outer surface of a very small
cylinder that is put vertically at a point in the space, to obtain composite illuminance
produces by incident light from various directions.
2π
1 1
E z= ∫ Ev ( φ ) dλ φ E z ≈ ∙ ( E v ( 0 )+ Ev ( 90 ° )+ E v (180 ° ) + E v ( 270 ° ) )
2π 0 4
- Semi-cylindrical illuminance: mean illuminance on the outer surface of a very small
π
2
1
semi-cylinder that is put vertically at a point in space. E sc= ∫ E ( φ ) dλφ
π −π v
2
Planar angle: length of a circular arc around the vertex between the two rays relative to the
radius of the circular arc.
b
Angle α= the whole planar angle is 2π and is measured in radian.
r
Ak
Solid angle: (3 dimensional) Ω= 2 is measured in steradian (sr). The
r
dλ A k
whole solid angle is 4π. dλΩ= 2 en dω = sinε * dε * dα.
r
, Luminous intensity: i, measured in candela (cd). 1 candela is 1 lm/λ)sr. It relates to the
luminous flux emitted into a solid angle and describes how much luminous flux is emitted in
dλΦ
a certain direction. It is used to characterize the light distribution of a luminaire. I = .
dλ Ω1
Luminance: l, measured in cd/λ)m2. It is the luminous intensity in a given direction related to
the emitting area. It is the photometric quantity that impacts the brightness perception under
normal circumstances. It is used to evaluate the glare rating in a given scene.
dλ 2 Φ
L=
dλ Ω 1 ∙ dλ A 1 ∙ cos ( ε 1 )
luminous exitance: m, measured in lm/λ)m2. It is a measure for the luminous flux
dλΦ
emitted per surface area. M =
dλ A 1
Luminous energy: q, measured in lmh. It is the time integral of luminous flux:
Q=∫ Φ ( t ) dλt it is often used to describe the total amount of light that a lamp can
∆t
produce over its lifetime by weighing its luminous flux with the lamp life.
Luminous exposure: h, measured in lxh. It is the time integral of illuminance (e): :
H=∫ E ( t ) dλt
∆t
Photon characteristics:
- Energy of a photon: eph = h*v
Considering the medium with the refractive index n: λ)*v = n*c0.
dλ N p
- Photon flux: Φ́ p=
dλt
- Photon irradiance: quotient of the photon flux incident on an element of the surface
dλ Φ́ p
containing the point, by the area da, of that element. É p =
dλ A 2
- Photon intensity: quotient of the photon flux, leaving the source and propagated in
the element of solid angle containing the given direction, by the element of solid
dλ Φ́ p
angle: Í p=
dλ Ω1
- Photon radiance: photon flux transmitted by an elementary beam passing through
the given point and propagating in the solid angle containing the given direction.
dλ Φ́ 2 p
Ĺ p=
dλ Ω1 ∙ dλ A 2 ∙cos ε 1
- Photon exitance: quotient of the photon flux leaving an element of the surface
dλ Φ́ p
containing the point, by the area of that element. Ḿ p=
dλ A 1
- Number of photons: integral of the photon flux over a given duration: N p=∫ Φ́ p dλt
∆t
