Lambert's Cosine Law

Lambert's cosine law states that the intensity of radiation along a direction which has angle with the normal to the surface is:

where In is the intensity of radiation in normal direction.


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Lambert's Cosine Law

Lambert’s Cosine Law holds that the radiation per unit solid angle (the radiant intensity) from a flat surface varies with the cosine of the angle to the surface normal (Fig. 4). Some Oriel^® Sources, such as arcs, are basically spherical. These appear like a uniform flat disk as a result of the cosine law. Another consequence of this law is that flat sources, such as some of our low power quartz tungsten halogen filaments, must be properly oriented for maximum irradiance of a target. Flat diffusing surfaces are said to be ideal diffusers or Lambertian if the geometrical distribution of radiation from the surfaces obeys Lambert’s Law. Lambert’s Law has important consequences in the measurement of light. Cosine receptors on detectors are needed to make meaningful measurements of radiation with large or uncertain angular distribution.

Fig. 4 Lamberts cosine law indicates how the intensity, I, depends on angle.



 



 

Basic  Principles

The Inverse Square Law

The inverse square law defines the relationship between the irradiance from a point source and distance.  It states that the intensity per unit area varies in inverse proportion to the square of the distance. E = I / d² In other words, if you measure 16 W/cm² at 1 meter, you will measure 4 W/cm² at 2 meters, and can calculate the irradiance at any other distance.  An alternate form is often more convenient:
E₁ d₁² = E₂ d₂² Distance is measured to the first luminating surface - the filament of a clear bulb, or the glass envelope of a frosted bulb.

Example:

You measure 10.0 lm/m² from a light bulb at 1.0 meter.  What will the flux density be at half the distance?

Solution:

E₁ = (d₂ / d₁)² * E₂ E_{0.5 m} = (1.0 / 0.5)² * 10.0 = 40 lm/m²  

Point Source Approximation

The inverse square law can only be used in cases where the light source approximates a point source.  A general rule of thumb to use for irradiance measurements is the “five times rule”:  the distance to a light source should be greater than five times the largest dimension of the source.  For a clear enveloped lamp, this may be the length of the filament.  For a frosted light bulb, the diameter is the largest dimension.  Figure 6.2 below shows the relationship between irradiance and the ratio of distance to source radius.  Note that for a distance 10 times the source radius (5 times the diameter), the error from using the inverse square is exactly 1 %, hence the “five times” approximation. Note also, that when the ratio of distance to source radius decreases to below 0.1 (1/20 the diameter of the source), changes in distance hardly affect the irradiance (< 1 % error).  This is due to the fact that as the distance from the source decreases, the detector sees less area, counteracting the inverse square law.  The graph above assumes a cosine response.  Radiance detectors restrict the field of view so that the d/r ratio is always low, providing measurements independent of distance.

Lambert’s Cosine Law

The irradiance or illuminance falling on any surface varies as the cosine of the incident angle, q.  The perceived measurement area orthagonal to the incident flux is reduced at oblique angles, causing light to spread out over a wider area than it would if perpendicular to the measurement plane. To measure the amount of light falling on human skin, you need to mimic the skin’s cosine response.  Since filter rings restrict off-angle light, a cosine diffuser must be used to correct the spatial responsivity.  In full immersion applications like the phototherapy booth shown above, off angle light is significant, requiring accurate cosine correction optics.

Lambertian Surface

A Lambertian surface provides uniform diffusion of the incident radiation such that its radiance or luminance is the same in all directions from which it can be measured.  Many diffuse surfaces are, in fact, Lambertian.  If you view this Light Measurement Handbook from an oblique angle, it should look as bright as it did when held perpendicular to your line of vision.  The human eye, with its restricted solid viewing angle, is an ideal luminance, or brightness, detector. Figure 6.4 shows a surface radiating equally at 0° and at 60°.  Since, by the cosine law, a radiance detector sees twice as much surface area in the same solid angle for the 60° case, the average incremental reflection must be half the magnitude of the reflection in the 0° case.
Figure 6.5 shows that a reflection from a diffuse Lambertian surface obeys the cosine law by distributing reflected energy in proportion to the cosine of the reflected angle. A Lambertian surface that has a radiance of 1.0 W/cm²/sr will radiate a total of p*A watts, where A is the area of the surface, into a hemisphere of 2p steradians.  Since the radiant exitance of the surface is equal to the total power divided by the total area, the radiant exitance is p W/cm2.  In other words, if you were to illuminate a surface with an irradiance of 3.1416 W/cm², then you will measure a radiance on that surface of 1.00 W/cm²/sr (if it is 100% reflective).
The next section goes into converting between measurement geometries in much greater depth.

Measurement  Geometries

Solid  Angles

One of the key concepts to understanding the relationships between measurement geometries is that of the solid angle, or steradian.  A sphere contains 4p steradians.  A steradian is defined as the solid angle which, having its vertex at the center of the sphere, cuts off a spherical surface area equal to the square of the radius of the sphere.  For example, a one steradian section of a one meter radius sphere subtends a spherical surface area of one square meter. The sphere shown in cross section in figure 7.1 illustrates the concept.  A cone with a solid angle of one steradian has been removed from the sphere.  This removed cone is shown in figure 7.2.  The solid angle, W, in steradians, is equal to the spherical surface area, A, divided by the square of the radius, r. Most radiometric measurements do not require an accurate calculation of the spherical surface area to convert between units.  Flat area estimates can be substituted for spherical area when the solid angle is less than 0.03 steradians, resulting in an error of less than one percent.  This roughly translates to a distance at least 5 times greater than the largest dimension of the detector.  In general, if you follow the “five times rule” for approximating a point source, you can safely estimate using planar surface area.

Radiant and Luminous Flux

Radiant flux is a measure of radiometric power.  Flux, expressed in watts, is a measure of the rate of energy flow, in joules per second.  Since photon energy is inversely proportional to wavelength, ultraviolet photons are more powerful than visible or infrared. Luminous flux is a measure of the power of visible light.  Photopic flux, expressed in lumens, is weighted to match the responsivity of the human eye, which is most sensitive to yellow-green.  Scotopic flux is weighted to the sensitivity of the human eye in the dark adapted state.

 

Units Conversion: Power

RADIANT FLUX:
1 W (watt)
= 683.0 lm at 555 nm
= 1700.0 scotopic lm at 507 nm
1 J (joule)
= 1 W*s (watt * second)
= 10⁷ erg
= 0.2388 gram * calories
LUMINOUS FLUX:
1 lm (lumen)
= 1.464 x 10^-3 W at 555 nm
= 1/(4p) candela  (only if isotropic)
1 lm*s (lumen * seconds)
= 1 talbot (T)
= 1.464 x 10^-3 joules at 555 nm

 

l  nm	Photopic  Luminous  Efficiency	Photopic  lm / W  Conversion	Scotopic  Luminous  Efficiency	Scotopic  lm / W  Conversion
380  390  400  410  420  430  440  450  460  470  480  490  500  507  510  520  530  540  550  555  560  570  580  590  600  610  620  630  640  650  660  670  680  690  700  710  720  730  740  750  760  770	0.000039  .000120  .000396  .001210  .004000  .011600  .023000  .038000  .060000  .090980  .139020  .208020  .323000  .444310  .503000  .710000  .862000  .954000  .994950  1.000000  .995000  .952000  .870000  .757000  .631000  .503000  .381000  .265000  .175000  .107000  .061000  .032000  .017000  .008210  .004102  .002091  .001047  .000520  .000249  .000120  .000060  .000030	0.027  0.082  0.270  0.826  2.732  7.923  15.709  25.954  40.980  62.139  94.951  142.078  220.609  303.464  343.549  484.930  588.746  651.582  679.551  683.000  679.585  650.216  594.210  517.031  430.973  343.549  260.223  180.995  119.525  73.081  41.663  21.856  11.611  5.607  2.802  1.428  0.715  0.355  0.170  0.082  0.041  0.020	0.000589  .002209  .009290  .034840  .096600  .199800  .328100  .455000  .567000  .676000  .793000  .904000  .982000  1.000000  .997000  .935000  .811000  .650000  .481000  .402000  .328800  .207600  .121200  .065500  .033150  .015930  .007370  .003335  .001497  .000677  .000313  .000148  .000072  .000035  .000018  .000009  .000005  .000003  .000001  .000001	1.001  3.755  15.793  59.228  164.220  339.660  557.770  773.500  963.900  1149.200  1348.100  1536.800  1669.400  1700.000  1694.900  1589.500  1378.700  1105.000  817.700  683.000  558.960  352.920  206.040  111.350  56.355  27.081  12.529  5.670  2.545  1.151  0.532  0.252  0.122  .060  .030  .016  .008  .004  .002  .001

  Spectroradiometry  is the calibrated analysis of light from radiant sources, e.g. the sun, lamps and other light sources.
Photometry  involves measurement of radiation visible to the human eye.

Light source	Accessory	Radiometric unit	Photometric unit
Tungsten halogen lamp	Integrating sphere	Radiant power [W/nm]	Luminous flux [lm]
LED	LED adapter	Radiant intensity [W/sr nm]	Luminous intensity [cd]
Sun	External optical probe	Irradiance [W/m2 nm]	Illuminance [lux]
Display	Telescope head	Radiance [W/cm2 sr nm]	Luminance [cd/m2]

 

  Irradiance   and   Illuminance:

Irradiance  is a measure of radiometric flux per unit area, or flux density.
Irradiance  is typically expressed in W/cm² (watts per square centimeter) or W/m² (watts per square meter). Illuminance  is a measure of photometric flux per unit area, or visible flux density.
Illuminance  is typically expressed in lux (lumens per square meter) or foot-candles (lumens per square foot).
In figure 7.4, above, the lightbulb is producing 1 candela.  The candela is the base unit in light measurement, and is defined as follows:  a 1 candela light source emits 1 lumen per steradian in all directions (isotropically).  A steradian is defined as the solid angle which, having its vertex at the center of the sphere, cuts off an area equal to the square of its radius.  The number of steradians in a beam is equal to the projected area divided by the square of the distance. So, 1 steradian has a projected area of 1 square meter at a distance of 1 meter.  Therefore, a 1 candela (1 lm/sr) light source will similarly produce 1 lumen per square foot at a distance of 1 foot, and 1 lumen per square meter at 1 meter. Note that as the beam of light projects farther from the source, it expands, becoming less dense. In fig. 7.4, for example, the light expanded from 1 lm/ft² at 1 foot to 0.0929 lm/ft² (1 lux) at 3.28 feet (1 m).

Cosine Law

Irradiance measurements should be made facing the source, if possible.  The irradiance will vary with respect to the cosine of the angle between the optical axis and the normal to the detector.

Calculating Source Distance

Lenses will distort the position of a point source.  You can solve for the virtual origin of a source by measuring irradiance at two points and solving for the offset distance, X, using the Inverse Square Law: E₁(d₁ + X)² = E₂(d₂ + X)² Figure 7.5 illustrates a typical setup to determine the location of an LED’s virtual point source (which is behind the LED due to the built-in lens).  Two irradiance measurements at known distances from a reference point are all that is needed to calculate the offset to the virtual point source.

Units Conversion: Flux Density

IRRADIANCE:

1 W/cm² (watts per square centimeter)
= 104 W/m² (watts per square meter)
= 6.83 x 10⁶ lux at 555 nm
= 14.33 gram*calories/cm²/minute
 
ILLUMINANCE:
 
1 lm/m² (lumens per square meter)
= 1 lux (lx)
= 10^-4 lm/cm²
= 10^-4 phot (ph)
= 9.290 x 10^-2 lm/ft²
= 9.290 x 10^-2 foot-candles (fc)

Radiance and Luminance:

Radiance is a measure of the flux density per unit solid viewing angle, expressed in W/cm²/sr.  Radiance is independent of distance for an extended area source, because the sampled area increases with distance, cancelling inverse square losses. The radiance, L, of a diffuse (Lambertian) surface is related to the radiant exitance (flux density), M, of a surface by the relationship:
L = M / p Some luminance units (apostilbs, lamberts, and foot-lamberts) already contain p in the denominator, allowing simpler conversion to illuminance units.
 

Example:

Suppose a diffuse surface with a reflectivity, r, of 85% is exposed to an illuminance, E, of 100.0 lux (lm/m²) at the plane of the surface.  What would be the luminance, L, of that surface, in cd/m²?
 

Solution:

1.) Calculate the luminous exitance of the surface:
M = E * rM = 100.0 * 0.85 =   85.0 lm/m²
2.) Calculate the luminance of the surface:
L = M / p
L = 85.0 / p = 27.1 lm/m²/sr =  27.1 cd/m²

Irradiance From An Extended Source:

The irradiance, E, at any distance from a uniform extended area source, is related to the radiance, L, of the source by the following relationship, which depends only on the subtended central viewing angle, q, of the radiance detector: E = p L sin²(q/2) So, for an extended source with a radiance of 1 W/cm²/sr, and a detector with a viewing angle of 3°, the irradiance at any distance would be 2.15 x 10^-3 W/cm².  This assumes, of course, that the source extends beyond the viewing angle of the detector input optics.

Units Conversion: Radiance & Luminance

RADIANCE:
 
1 W/cm²/sr (watts per sq. cm per steradian)
= 6.83 x 10⁶ lm/m²/sr at 555 nm
= 683 cd/cm² at 555 nm

 
LUMINANCE:
 
1 lm/m²/sr (lumens per sq. cm per steradian)
= 1 candela/m² (cd/m²)
= 1 nit
= 10^-4 lm/cm²/sr
= 10^-4 cd/cm²
= 10^-4 stilb (sb)
= 9.290 x 10^-2 cd/ft²
= 9.290 x 10^-2 lm/ft²/sr
= p apostilbs (asb)
= p cd/p/m²
= p x 10^-4 lamberts (L)
= p x 10^-4 cd /p/cm²
= 2.919 x 10^-1 foot-lamberts (fL)
= 2.919 x 10^-1 lm/p/ft²/sr

Radiant and Luminous Intensity:

Radiant Intensity is a measure of radiometric power per unit solid angle, expressed in watts per steradian.  Similarly, luminous intensity is a measure of visible power per solid angle, expressed in candela (lumens per steradian).  Intensity is related to irradiance by the inverse square law, shown below in an alternate form: I = E * d² If you are wondering how the units cancel to get flux/sr from flux/area times distance squared, remember that steradians are a dimensionless quantity.  The solid angle equals the area divided by the square of the radius, so d²=A/W, and substitution yields:
I = E * A / W The biggest source of confusion regarding intensity measurements involves the difference between Mean Spherical Candela and Beam Candela, both of which use the candela unit (lumens per steradian).  Mean spherical measurements are made in an integrating sphere, and represent the total output in lumens divided by 4p sr in a sphere.  Thus, a one candela isotropic lamp produces one lumen per steradian.
Beam candela, on the other hand, samples a very narrow angle and is only representative of the lumens per steradian at the peak intensity of the beam.  This measurement is frequently misleading, since the sampling angle need not be defined. Suppose that two LED’s each emit 0.1 lm total in a narrow beam: One has a  10° solid angle and the other a 5° angle.  The 10° LED has an intensity of 4.2 cd, and the 5° LED an intensity of 16.7 cd.  They both output the same total amount of light, however -- 0.1 lm.
A flashlight with a million candela beam sounds very bright, but if its beam is only as wide as a laser beam, then it won’t be of much use.  Be wary of specifications given in beam candela, because they often misrepresent the total output power of a lamp.
 

Units Conversion: Intensity

RADIANT INTENSITY:
 
1 W/sr (watts per steradian)
= 12.566 watts (isotropic)
= 4*p W
= 683 candela at 555 nm

 
LUMINOUS INTENSITY:
 
1 lm/sr (lumens per steradian)
= 1 candela (cd)
= 4*p lumens (isotropic)
= 1.464 x 10^-3 watts/sr at 555 nm

Converting Between Geometries

Converting between geometry-based measurement units is difficult, and should only be attempted when it is impossible to measure in the actual desired units.  You must be aware of what each of the measurement geometries implicitly assumes before you can convert.  The example below shows the conversion between lux (lumens per square meter) and lumens.

Example:

You measure 22.0 lux from a light bulb at a distance of 3.162 meters.  How much light, in lumens, is the bulb producing?  Assume that the clear enveloped lamp is an isotropic point source, with the exception that the base blocks a 30° solid angle.

Solution:

1.) Calculate the irradiance at 1.0 meter:
E₁ = (d₂ / d₁)² * E₂
E_{1.0 m} = (3.162 / 1.0)² * 22.0 = 220 lm/m²
2.) Convert from lm/m² to lm/sr at 1.0 m:
220 lm/m² * 1 m²/sr =  220 lm/sr
3.) Calculate the solid angle of the lamp:
W = A / r² = 2ph / r = 2p[1 - cos(a / 2)]
W = 2p[1 - cos(330 / 2)] = 12.35 sr
4.) Calculate the total lumen output:
220 lm/sr * 12.35 sr = 2717 lm