Figure 1 Photopic Vision Function V(λ)
Photometry for Flashlights
Click here for a printable pdf version.
A.R. Gawthrop
Contents
2. The Visible Light Spectrum.. 2
4. Defining Luminous Intensity and Illuminance. 5
7. Calculation of the Luminous Flux and the Solid Angle. 11
8. Directivity and Beamwidth. 14
Appendix B – Units Conversion. 18
Appendix C – Method for Numerical Integration. 20
Heliotek Inc.
720 Hickory Road
P.O. Box 1000
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888-HELIOTEK (888-435-4683)
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Email: eng@heliotekinc.com
Version 1.0
2005
Heliotek Inc. strives to provide accurate performance data for its flashlight products. For customers not familiar with light measurements, this document introduces the common terminology used for photometry. Written descriptions and mathematical definitions of terms and the relationship between terms are presented to help give physical insight to their meaning. While the fundamental units of photometry are defined by differential and integral calculus, the reader competent in algebra should have little problem utilizing most of the material. An appendix dedicated to numerical integration describes a method for solving integral equations by using summations.
The portion of the electromagnetic spectrum responsible for visible light has wavelengths extending from 380 to 770 nanometers (nm). A nanometer is 1x10-9 meters or 1/1,000,000,000 of a meter (m). The Greek letter λ (lambda) is typically used to represent wavelength. Figure 1 shows the colors associated with wavelength.
Figure 1 Photopic Vision Function V(λ)
Shorter wavelengths just below the blue band of colors are called ultraviolet. Longer wavelengths just above the red band of colors are called infrared. While the ultraviolet and infrared bands can’t be seen, at sufficient levels they can be sensed as heat.
Sometimes light is given in terms of frequency in units of hertz (Hz, meaning cycles per second). In free space the velocity of propagation for light c is about 3 x 108 meters per second (m/s)[1]. The frequency f is the velocity of propagation of light divided by the wavelength. Thus, for a given wavelength, the frequency will be
f = c/λ. (Hz) (1a)
Or conversely, given the frequency, the wavelength will be
λ = c/f (m) (1b)
For example, given a wavelength of 555 nm [yellow-green], the frequency will be 540 x 1012 Hz. Notice that the frequency and wavelength are inversely proportional to each other. As the wavelength increases, the frequency goes down. As the frequency goes up, the wavelength decreases.
The human eye does not respond uniformly to every color across the visible light spectrum. Nor does the human eye respond the same in daylight as it does at night. The average response of the nighttime adjusted eye is called the Scotopic vision. The average response of the daytime adjusted eye is called the Photopic vision. Following international standards, all the definitions and measurements discussed herein will be given in terms of the photopic vision function. The CIE[2] has defined this as the V(λ) function or “the spectral luminous efficiency function for photopic vision”. Figure 1 shows this response curve[3]. Notice that Figure 1’s amplitude divisions are logarithmic (varying by a factor of ten) which is similar to the eye’s amplitude sensitivity. Also note that the function is normalized to one at 555 nm where the maximum occurs.
When the V(λ) function is multiplied by 683, the radiometric power (radiant flux in watts) can be directly related to the luminous flux in lumens. All the radiant flux across the visible spectrum is weighted by the V(λ) function to give the “human eye” perceivable luminous flux.
At a single wavelength, the V(λ) function can easily be used to relate radiant and luminous flux [1, pp. 30-31]. Thus, for a monochromatic light source
lumens = watts ∙ 683 ∙ V(λ) (2a)
Or conversely,
watts = lumens / (683 ∙ V(λ)) (2b)
For example, at 555 nm [yellow-green], the V(λ) function equals 1. So
683 lumens = 1 watt at 555 nm, and
0.001464 watt = 1 lumen at 555 nm
For many real world light sources having broad spectral distributions, the above procedure is not adequate. For these sources, the relationship between radiant flux and luminous flux requires the use of integration[4] across all the spectral components. Following the NIST [5][2, p. 4], the symbols for flux and luminous efficacy constant are given as:
Фe(λ) Radiant flux as a function of wavelength in watts per nanometer (W/nm),
Фv Luminous flux in lumens (lm), and
Km 683 is the maximum luminous efficacy constant[6] in lumens per watt (lm/W).
So more formally, the total luminous flux is the integration of all the weighted radiant flux across all the wavelengths of interest
(lm) (3)
While the focus of this document will not be determining luminous flux (lumen) from the radiant flux (electromagnetic power in watts), it is important to understand the relationship. The lumen is a basic photometric quantity that will be used in the definition of a number of light intensity and illuminance quantities such as the candela, the lux, and the foot-candle. These quantities are essentially the measure of the luminous flux within a given area or solid angle.
The total luminous flux from a light source can be measured directly with an instrument called an Integrating Sphere. Basically, the inside of integrating sphere has reflective coatings which can collect all the light from a source. Ports for a detector and for injecting light are provided. A variety of integrating spheres are available for photometry, radiometry, and colorimetry measurements. These instruments are typically found in laboratory and industrial settings. Later, an alternative method for determining luminous flux will be described. But first, an understanding of luminous intensity, illuminance, the spherical polar coordinate system, and radiation patterns will be needed.
4. Defining Luminous Intensity and Illuminance
The luminous intensity quantity, the candela, is the base SI[7] unit for all photometry [2, pp. 3-4]. The formal definition for the candela is given as
“The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of (1/683) watt per steradian.”
By using equation (1b), the frequency of 540 x 1012 Hz can be expressed as a wavelength of 555 nm. This is the wavelength where the V(λ) function equals unity. Referring to the relationship of (2a), at this wavelength, 1/683 of a watt produces one lumen of luminous flux. The formal definition also specifies this luminous intensity over a steradian, the unit solid angle of a sphere.
Thus, the candela (cd) is a quantity of emitted luminous intensity in units of lumens per steradian (lm/sr). The steradian is the measure of a solid angle of a sphere [3, p. 24]. One steradian is defined as a solid angle whose vertex is at the center of a sphere that is subtended by a surface area equal to the radius squared r2. See Figure 2 for an aid in visualizing the steradian. Be aware that on the surface of a sphere of radius r, any shape surface area of r2 will subtend a solid angle of one steradian.
Figure 2 Defining the steradian
For an arbitrary amount of surface area A, the solid angle Ω subtended can be found from
(sr). (4)
From geometry, it is given that a sphere of radius r has a total surface area of 4πr2. Since one steradian is subtended by an area of r2, there are 4π steradians of total solid angle in a sphere (4πr2/r2 = 4π). Thus, 4π is the maximum value for the solid angle Ω in steradians.
Since the candela is defined in terms of a sphere, further mathematical analysis requires the use of the spherical polar coordinate system. See Figure 3 for an illustration of this coordinate system. The three orthogonal coordinates are
r radius, in meters (m),
θ theta, the polar angle measured from the z-axis, in radians (rad), and
f phi, the azimuth angle measured from the x-axis, in radians (rad).
A radiation pattern for light intensity will be measured in terms of the spherical polar coordinate system with a constant radius r, which represents the measurement distance. Frequently, for better detail, a representation of the radiation pattern will be shown on a rectangular graph. For this sort of graph, the axis values will still be in terms of the spherical polar coordinate system.
Figure 3 Spherical polar coordinates
Further analysis also requires the mathematical description of a point on a sphere. Figure 3 shows the (infinitesimally small) differential surface area dA. The sides of this small area consist of the infinitesimally short segments rּdθ and rּsinθּdφ. Thus, for a sphere with a constant radius r, an element (or point) of surface area is
(m2). (5)
By definition of steradians (4) and the differential surface area dA (5), an element of the solid angle dΩ is
(sr). (6a)
It is interesting to note that integrating over the entire spherical surface reveals
(sr) (6b)
Therefore, as noted before, there are 4π steradians of solid angle in a full sphere. Thus, 4π (≈12.57) is the maximum value for the solid angle Ω in steradians. In characterizing a light source, this value would only occur for an isotropic source, one in which the light was emitted evenly in every direction.
Again, following the NIST [2, p. 4], the symbols for defining luminous intensity and illuminance are given as:
Iv Luminous intensity in candela (cd) or (lm/sr)
Ev Illuminance in lux (lx) or (lm/m2)
Luminous intensity Iv, which is the luminous flux emitted from a point source in a given direction per unit solid angle, is defined as
(cd) (7)
Illuminance Ev, which is the luminous flux density incident at a given point on a surface or plane, is defined as
(lx) (8)
Notice that the definitions state that the luminous intensity Iv is the luminous flux emitted from a point solid angle while the illuminance Ev is the density of luminous flux incident on a point area. The luminous intensity Iv (luminous flux per solid angle) is the same regardless of the distance from the point source[8]. The illuminance Ev (luminous flux density) incident on a given size measurement area A will vary inversely with r2 the square of the distance from the point source.
If the measurement radius (in meters) is known, the relationship given in (6a) can be used to relate the illuminance Ev in lux to the luminous intensity Iv in candela. Thus,
(cd) (9a)
Conversely, if the luminous intensity Iv in candela is known, then at a given measurement radius (in meters), the illuminance Ev in lux will be
(lx) (9b)
If the measurement radius r = 1 meter, then the value for Ev in lux will be the same as the Iv value in candelas.
A typical industrial light meter will measure the illuminance in lux (lx) or in foot-candles (fc). The lux is defined as lumen per square meter (lm/m2). The foot-candle is defined as lumen per square foot (lm/ft2). To convert the English measure of foot-candles to the SI unit of lux, use the relationship
Lux = 10.764 ∙ Foot-candles (10)
To convert feet to meters use
Meters = 0.3048 ∙ Feet (11)
Appendix B presents units conversion for a number of other light and distance measurement quantities.
Illuminance measurements, such as lux or foot-candles obey the inverse square law. For a given light source, the illuminance value varies inversely in proportion to the distance squared[9]. For two sets of measurements, E1 at distance r1 and E2 at distance r2, this relationship is expressed as
(12)
Given the measurement of E1 at distance r1, to project an illuminance measurement E2 for a different distance r2, (12) is rearranged to give
(12b)
Alternately, to project a distance r2 for a different illuminance value E2, (12) is rearranged to give
(12c)
Applications of the inverse square law would be analyzing performance of a flashlight and the verifying the accuracy of a measurement system.
A quantity related to illuminance and luminous intensity is luminance (Lv). Luminance is measure of brightness that is independent of the measurement distance. Luminance expressed in SI terms is lumens per steradian per square meter (lm/sr/m2) or equivalently, candela per square meter (cd/m2). Typical luminance measurements are for diffuse (Lambertian) surfaces [1, pp. 27-28] such as light diffusers and television screens. The derivation and use of luminance Lv is beyond the scope of this paper. Appendix B presents units conversion for a number of luminance quantities. Furthermore, with knowledge of measurement distance and the beam solid angle, a luminance quantity can be converted to an illuminance Ev quantity. Alternately, with knowledge of the source (and/or detector) area, a luminance quantity can be converted to luminous intensity Iv.
An important description of a light source is its radiation pattern. If the direction of the main beam of a light source is considered oriented on the z-axis (see Figure 3), this will be at the zero polar angle θ. If illuminance measurements are taken at sufficient intervals over the polar angle θ from 0 to π (0 to 180º) and over the azimuth angle φ from 0 to 2π (0 to 360º), then these measurements can be used to describe the radiation pattern of a light source. When the luminous intensity of a light source is symmetric around the main beam (symmetric in the azimuthal angle φ), a single set of measurements along the polar angle θ is sufficient to describe the radiation pattern.
(a) Linear Display
(b) Logarithmic Display
Figure 4 Radiation Pattern for the HTE-1 flashlight (Undiffused)
Figure 4 shows two such representations of the radiation pattern plotted on rectangular graphs. Figure 4(a) has a linear scale while Figure 4(b) has a logarithmic illuminance scale. The logarithmic scale shows more detail at the lower levels and more closely represents the eye’s sensitivity. The maximum value of illuminance on the beam is 2420 lux measured at 1 meter. The illuminance values at angles greater than 30 degrees are taken to be zero. For a flashlight, this is a very narrow light beam.
Patterns can also be presented in polar (circular) form. For narrow beam lights, rectangular (x-y) plots, such as those in Figure 4, can give more detail. Frequently, relative (or normalized) radiation patterns will be presented. Since this gives only the shape of the pattern, other information, such as the maximum beam luminous intensity Iv or total luminous flux Фv must be specified.
If care is taken, a light meter can be used to measure the pattern (as was the case for Figure 4). This can be an economical method, especially when the illuminance is symmetric in φ around the main beam. Thus, a single set of measurements along the polar angle θ is sufficient to describe the radiation pattern. However, if the pattern varies in the azimuthal angle φ as well, then numerous series of measurements will be needed. Alternately, the process can be automated with the use of a goniophotometer. A simple goniophotometer has two rotating stages turned by stepping motors. One stage rotates the lamp under test while the other stage rotates the detector. To collect the radiation pattern data, one stage is rotated over the azimuth angle φ while the other stage is rotated over the polar angle θ. Some goniophotometers use moving mirrors instead of rotating stages. A variety of goniophotometers are commercially available (or one can be shop built).
Flashlights are typically classified by the size of their battery power sources and the luminous intensity of the main beam. Flashlights range in size and weight from miniature single cell penlights to large spotlights powered by lantern batteries. If the light beam is narrow (spotlight), it may have a high intensity value, but may not be suitable for situations where a broader beam (floodlight) is needed. Narrow beam flashlights (as most flashlights are) are generally compared by the luminous intensity (or illuminance measured) of the main beam. For broad beam lights, comparisons of the total luminous flux (in lumens) would be more informative.
The luminous efficacy, expressed in lumens per watt, is an effective way to compare light sources. Thus, comparing flashlights can also be done on the basis of efficacy -- the luminous flux (lumens) output from the power (watts) consumed.
Additionally, since flashlights are usually directive light sources, a practical way to compare flashlights is by its “throw”. The throw is the distance in meters the main beam can be measured at the illuminance level of 1 lux. As an example, the undiffused Heliotek HTE-1 flashlight[10] has a throw of 50 meters.
Spill is another common characteristic of a flashlight. The spill is the useful light extending at an angle beyond that of the main beam of a flashlight.
Figure 5 shows a target illuminated by an undiffused Heliotek HTE-1, which is a highly directional flashlight. While the main beam’s half-power beamwidth is ±1.6 degrees, the spill angle of ±20 degrees has been found to make walking at night safe and comfortable.
Figure 5 Target with Main Beam and Spill Angles
Later, procedures for calculating the directivity and determining beamwidth will be presented. These engineering measures help quantify the effectiveness of the reflector/lens system.
7. Calculation of the Luminous Flux and the Solid Angle
As previously discussed, to directly measure the total luminous flux Фv from a light source, an Integrating Sphere measurement system can be used. However, if the radiation pattern is known, the luminous flux Фv can be calculated. First, when the illuminance pattern (in lux) is known, equation (5) is substituted in equation (8) and equation (8) is rearranged and integrated. Then the following expression for luminous flux is obtained:
(lm) (13a)
where Ev(θ,φ) is the illuminance pattern data (in lux) expressed as a function of the polar angle θ and the azimuthal angle φ. The angles are in radians and r is the measurement distance in meters. In words, the total luminous flux Фv can be found by summing up all the illuminance values over the surface of a sphere of constant radius centered on the light source.
Alternately, when the intensity radiation pattern (in candela) is known, equation (6a) is substituted in equation (7) and equation (7) is rearranged and integrated. Then the following expression for luminous flux is obtained:
(lm) (13b)
where Iv(θ,φ) is the intensity pattern data (in candela) expressed as a function of the polar angle θ and the azimuthal angle φ. The angles are in radians. Equation (13b) could also be derived by substituting equation (9b) into equation (13a).
Seldom is Ev(θ,φ) or Iv(θ,φ) available in a form[11] that can be directly integrated. Typically, Ev or Iv values are given as a set of discrete measurements. Later, a practical numerical method for solving these equations will be presented. But first, an examination of how to compute the beam solid angle will be helpful.
When calculating luminous flux Фv from a light source, another important measure is the light-beam’s solid angle Ω. Equation (6b) calculated the maximum solid angle for a sphere as 4π (≈12.57) steradians. This maximum value would only occur for an isotropic light source, one in which the light was emitted evenly in every direction. To determine the beam solid angle for a directional pattern, the normalized pattern is multiplied by the differential solid angle dΩ and the product is then integrated over the entire spherical surface [3, p. 32]. In general, the solid angle for any radiation intensity pattern can be expressed as
(sr) (14a)
where
(14b)
By dividing the function F(θ,φ) by its maximum value, a normalized function with a maximum value of 1 is the result. The normalized function represents the shape of the radiation pattern. It is the shape of the radiation pattern that determines the beam solid angle. The beam solid angle has the physical significance of being the solid angle that all the luminous intensity would flow through if the intensity were uniformly constant at the maximum value.
Equations (13a) and (13b) for directly determining luminous flux are closely related to the expression for determining the light-beam’s solid angle. If the illuminance or intensity pattern is normalized and integrated over the entire spherical surface, the result is the beam solid angle. The following equations are used to find the beam solid angle from an illuminance pattern (in lux) or from an intensity radiation pattern (in candela) respectively:
(sr) (15a)
(sr) (15b)
By comparing equations (13a) and (13b) to (15a) and (15b), a number of useful relationships can be derived. If equation (13a) is used to find the luminous flux Фv, then the beam solid angle Ω can be found from
(sr) (16a)
where the luminous flux Фv (lm) is divided the maximum illuminance value of Ev (lx) and the square of the measurement distance r (m). Alternately, if equation (13b) is used to find the luminous flux Фv, then the beam solid angle Ω can be found from
(sr) (16b)
where the luminous flux Фv (lm) is divided the maximum luminous intensity value of Iv (cd).
If the datasheet for a given flashlight provides the luminous flux Фv in lumens and the maximum luminous intensity Iv|max value in candela (or Ev|max in lx and r in m), then equation (16b) or (16a) can be used to compute the beam solid angle Ω. In this case, no numerical integration will be required. However, if the datasheet only provides the radiation pattern and the maximum luminous intensity Iv|max value in candela (or Ev|max in lx and r in m), then determining the luminous flux requires integration. If an Integrating Sphere measurement system is not available, then numerical integration can be used. Appendix C describes a method for performing the numerical integration of the pattern data.
Continuing with the comparisons of equations (13a) and (13b) to (15a) and (15b), if the beam solid angle Ω and maximum illuminance Ev or luminous intensity Iv values are known, then the luminous flux Фv can be found from either
(lm) (17a)
or
(lm) (17b)
Furthermore, if the luminous flux Фv and beam solid angle Ω are known, then the maximum illuminance Ev or luminous intensity Iv value can be found from either
(lx) (18a)
or
(cd) (18b)
A flashlight and nearly every other light source exhibit directionality. To quantify the directionality of a given light, its beam solid angle Ω is compared to the solid angle of an isotropic light source.
An isotropic light source is one in which the light is emitted evenly in every direction. Recall that equation (6b) calculated the solid angle for a sphere as 4π (≈ 12.57) steradians. This is the maximum solid angle that would only occur for an isotropic light source.
Directivity is defined as the maximum possible solid angle (4π) divided by the beam solid angle of a given light source. This is expressed as
(19a)
The directivity can also be expressed in terms of decibels by
(19b)
As an example, the undiffused Heliotek HTE-1 flashlight has a beam solid angle of about 0.00828 steradians. Thus, its directivity is
D0 = 1518
D0(db) = 31.8 db
The beamwidth is the angle the main beam fans out to some specified amplitude level. Common amplitude levels for specifying beamwidth are the half-power (-3 db) level and the 1/10-power (-10 db) level [3, p. 46]. Thus, in a plane containing the direction of the beam maximum, the angle between the two directions of the specified luminous intensity defines the beamwidth. Frequently, the pattern data will have to be interpolated to find the angles of interest.
As an example, Figure 6 shows the illuminance pattern for the undiffused Heliotek HTE-1 flashlight. Figure 6 is essentially Figure 4a with the angular displacement axis expanded. The maximum value of the beam is 2420 lux, so the half-power (-3 db) level is 1210 lux and the 1/10-power (-10 db) level is 242 lux. The markers in Figure 6 show where these levels intercept the illuminance pattern. At these intercept points, the beamwidth angles can be found. Thus, the half-power beamwidth is ±1.6 or 3.2 degrees. The 1/10-power beamwidth is ±3.8 or 7.6 degrees.
Figure 6 Graphical Method for Determining Beamwidth
In the case presented above, the pattern is independent in φ, that is, symmetric about the main beam along the polar axis z. For patterns that are not symmetric, beamwidth readings for specific angles of φ can be given. Typically, readings for the widest and narrowest beamwidth angles verses φ would be recorded.
The directivity and beamwidth are useful engineering measures for comparisons and quality control of the reflector/lens system.
[1] Ryer, A., Light Measurement Handbook, International Light, Newburyport, MA, (1997). http://www.intl-light.com/handbook.html
[2] Ohno, Y., NIST Measurement Services: Photometric Calibrations, NIST Special Publication 250-37, National Institute of Standards and Technology, Gaithersburg, MD (July 1997). http://physics.nist.gov/Divisions/Div844/products/products.html
[3] Balanis, C.A., Antenna Theory: Analysis and Design, Harper and Row, New York (1982).
The candela (cd) is the SI base unitA1 for luminous intensity. The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of (1/683) watt per steradian. 1 candela = 1 lumen per steradian.
The lumen (lm) is the SI unit for the luminous flux emitted into a unit solid angle (one steradian) by a uniform point source having an intensity of one candela. 1 lumen = 1 candela per steradian.
The luminous efficacy is the effectiveness of a light source expressed in lumens per watt (lm/W). Specifically, it is the luminous flux (lm) produced by the light source for a given amount of electrical input power (W).
The luminous flux is the luminous equivalent of radiometric power in watts weighted to match the standard eye response. One lumen of luminous flux is equivalent to 1/683 watt of light at a wavelength of 555 ∙10-9 meters. (See lumen).
The lux (lx) is the SI unit of illuminance equal to one lumen per square meter (lm/m2).
The meter (m) is the SI base unit of length. The meter is the length equal to 1,650,763.73 wavelengths in a vacuum of the orange-red radiation of the krypton-86 atom. Alternately, the meter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second.
The radian (rad) is the angle subtended (at the center of a circle) by an arc with a length equal to the radius (r) of the circle. There are 2π radians of arc in a full circle.
The spill is the useful light extending at an angle beyond that of the main beam of a flashlight.
The steradian (sr) is defined as the solid angle, whose vertex is at the center of a sphere, that is subtended by a surface area equal to the radius squared (r2). The surface of a complete closed sphere subtends a solid angle of 4π steradians.
The throw is the distance in meters the main beam of a flashlight can be measured at the illuminance level of 1 lux.
The following are conversions for Illuminance (Ev) quantities. The SI unit for illuminance is the lux (lx) which is lumen per square meter (lm/m2).
| To find: | From: | Multiply by: | |||
| lux | footcandle | 10.764 | lx - lumen per square meter (lm/m2) | ||
| lux | phot | 10,000 | |||
| footcandle | lux | 0.0929 | fc - lumen per square foot (lm/ft2) | ||
| phot | lux | 1 · 10-4 | ph - lumen per square centimeter (lm/cm2) |
The following are English - Metric linear distance conversions. The SI unit for distance is the meter.
| To find: | From: | Multiply by: | |||
| meters | feet | 0.3048 | |||
| feet | meters | 3.281 | |||
| centimeters | inches | 2.54 | |||
| millimeter | inches | 25.4 | |||
| inches | centimeters | 0.3937 |
The following are surface area conversions.
| To find: | From: | Multiply by: | |||
| m2 | ft2 | 0.0929 | m2 - square meter | ||
| m2 | cm2 | 1 · 10-4 | |||
| cm2 | m2 | 10,000 | cm2 - square centimeter | ||
| cm2 | ft2 | 929 | |||
| ft2 | m2 | 10.764 | ft2 - square foot | ||
| ft2 | cm2 | 1.076 · 10-3 |
The following are conversions for Luminance (Lv) quantities. The SI derived unit for luminance is candela per square meter (cd/m2), which is lumen per steradian per square meter (lm/sr/m2).
| To find: | From: | Multiply by: | |||
| cd/m2 | nit | 1 | cd/m2 – candela per square meter | ||
| cd/m2 | stilb | 10,000 | |||
| cd/m2 | apostilb | 0.3183 | |||
| cd/m2 | Lambert | 3183 | |||
| cd/m2 | foot-Lambert | 3.4263 | |||
| nit | cd/m2 | 1 | nt – nit (10-4 lm/sr/cm2) or (cd/m2) | ||
| stilb | cd/m2 | 1 · 10-4 | sb – stilb (10,000ּcd/m2) | ||
| apostilb | cd/m2 | 3.1416 | asb – apostilb (π-1 cd/m2) | ||
| Lambert | cd/m2 | 3.1416 · 10-4 | L – Lambert (π-1 cd/cm2) | ||
| foot-Lambert | cd/m2 | 0.29186 | fL – foot-Lambert (π-1 cd/ft2) |
Note, with knowledge of measurement distance and beam solid angle, a luminance Lv quantity can be converted to an illuminance Ev quantity. Alternately, with knowledge of the source (and/or detector) area, a luminance quantity can be converted to luminous intensity Iv.
Appendix C – Method for Numerical Integration
If a radiation pattern is known in terms of illuminance, the luminous flux Фv can be calculated (as derived in the main text) by
(lm) (C-1a)
where Ev(θ,φ) in lux is expressed as a function of the polar angle θ and the azimuthal angle φ. The angles are in radians and r is the measurement distance in meters. Similarly, if a radiation pattern is known in terms of luminous intensity, the luminous flux Фv can be calculated (as derived in the main text) by
(lm) (C-1b)
where Iv(θ,φ) in candela is expressed as a function of the polar angle θ and the azimuthal angle φ. The angles are in radians.
When the radiation pattern of a light source is symmetric around the main beam (symmetric in the azimuthal angle φ), a single set of measurements along the polar angle θ is sufficient to describe the radiation pattern. For radiation patterns symmetric in φ, equations (C-1a) and (C-1b) can be simplified to
(lm) (C-2a)
(lm) (C-2b)
Seldom is Ev(θ) or Iv(θ) available in a form that can be directly integrated. Typically, Ev or Iv values are given as a set of discrete measurements. If care is taken, a light meter can be used to measure the radiation pattern. This can be a practical method when the radiation pattern is symmetric in φ.
Consider the following two arrays which represent the simplified radiation pattern for an undiffused Heliotek HTE-1 flashlight. The Θ array contains a discrete set of θ polar angles in degrees. The E array contains the corresponding set of illuminance readings (in lux at 1 meter) taken at the polar angles given in the Θ array. Both arrays are of length N = 10. The last entry at 180 degrees was shown to emphasize the complete θ interval ranges from 0 to 180 degrees.
r = 1 m (C-3)
Figure C-1 shows a plot of the main portion (0 to 10 degrees) of the simplified radiation pattern. It is helpful to plot the pattern to check for errors. Note that lines are drawn between the data points to help visualize the complete pattern.
Figure C-1 Simplified HTE-1 Radiation Pattern
The following method for numerical integration of the pattern data has been developed to utilize discrete data such as that presented in the Θ and E arrays. This summation method approximates a definite integral by using the trapezoidal rule with variable length sub-intervals. Given that the data is in illuminance values (lx), the appropriate equation (C-2a) for the luminous flux Фv is approximated by
(lm) (C-4a)
where the intermediate functions
(C-4b)
and
(C-4c)
Also, N is the length of the arrays Θ and E, which elements are indexed by i in the range from 0 to N-1. The π/180 term multiplying the angle values converts the angle to radians from degrees.
Programming these equations and performing the numerical integration on the data given in the Θ and E arrays, the result for luminous flux is
Фv = 20.0 lm
As derived in the main text, the beam solid angle Ω can be found from
(sr) (C-5a)
where the luminous flux Фv (lm) is divided the maximum illuminance value of Ev (lx) and the square of the measurement distance r (m). Then, for the above example
Фv = 20.0 lm
Ev|max = 2420 lx, and
r = 1 m
thus
Ω = 0.00826 sr
In the cases where the pattern data is given as luminous intensity (candela), the equations of (C-4) can be used by changing the illuminance E array to a luminous intensity I array. Also, just set r in (C-4a) to 1. Then to find the beam solid angle Ω, use the previously derived relationship
(sr) (C-5b)
where the luminous flux Фv (lm) is divided the maximum luminous intensity value of Iv (cd).
Another common situation arises in cases where the relative or normalized pattern is given. The normalized function represents the shape of the radiation pattern. It is the shape of the radiation pattern that determines the beam solid angle. In general, the solid angle for any radiation intensity pattern can be expressed as
(sr) (C-6a)
For radiation patterns symmetric in φ, equations (C-6a) can be simplified to
(sr) (C-6b)
To numerically approximate this equation, consider the following two arrays which represent the simplified relative radiation pattern for a Nichia NSPWR70SS LED. The Θ array contains the angles (in degrees) while the F array contains the corresponding normalized pattern data. Both arrays are of length N = 9.
(C-7)
Figure C-2 shows a plot of the simplified radiation pattern. Note that lines are drawn between the data points to help visualize a complete pattern. This LED without an external lens or reflector has a much broader pattern than the previous flashlight example. With a half-power beamwidth of ±52 (or 104) degrees, this LED was designed for use in display backlighting.
Figure C-2 Simplified Relative NSPWR70SS Pattern
The following method for numerical integration can be used on the discrete data such as the Θ and the corresponding F array with relative pattern data. As before, this summation method approximates a definite integral by using the trapezoidal rule with variable length sub-intervals. So, the equation for the beam solid angle (C-6b) is approximated by
(sr) (C-8a)
where the intermediate functions
(C-8b)
and
(C-8c)
Also, N is the length of the arrays Θ and F, which elements are indexed by i in the range from 0 to N-1. The π/180 term multiplying the angle values converts the angle to radians from degrees.
Note that the equations of (C-8) are nearly the same as those of (C-4). However, (C-8) computes solid angle from the pattern shape while (C-4) computes the luminous flux from the actual pattern.
Programming these equations (C-8) and performing the numerical integration on the data given in the Θ and F arrays, the result for beam solid angle is
Ω = 2.49 sr
Now that the beam solid angle Ω can be computed from a relative pattern, the luminous flux Фv can be found from either
(lm) (C-9a)
or
(lm) (C-9b)
Alternately, if the luminous flux Фv is already known, then the maximum illuminance Ev or luminous intensity Iv value can be found from either
(lx) (C-10a)
or
(cd) (C-10b)
For the case of the NSPWR70SS LED, illuminance measurements at r = 1 m on the beam axis revealed Ev|max to be 1.85 lux. Knowing the beam solid angle and the maximum illuminance, equation (C-9a) can be used to find the luminous flux. So, with
Ω = 2.49 sr
Ev|max = 1.85 lx, and
r = 1 m
thus
Фv = 4.61 lm
[1] More precisely, c, the velocity of propagation in free space, is 2.9979 x 108 m/s
[2] CIE (or the ICI in English) is the International Commission on Illumination.
[3] This function is referred to by many names including the 1931 CIE Luminosity function, the CIE Photopic sensitivity curve, and the CIE Spectrum curve.
[4] Integration is a mathematical procedure from calculus that sums infinitesimally small differentials across a given interval. Integration can find the area under a curve, the area of a surface, and the volume within a closed surface.
[5] The NIST is the National Institute of Standards and Technology.
[6] More precisely, Km is 683.002 lm/W.
[7] SI is the International System of measurement.
[8] Assuming an ideal lossless medium.
[9] For the inverse-square law to hold, measurements need to be made at sufficient distances from the source. The light should effectively appear as a point source to the meter probe.
[10] The HTE-1 is a 2-AA cell, 20 lm, 1W flashlight with a typical 2400 cd beam intensity.
[11] Needed is a closed form mathematical expression which is a continuous function on the closed intervals of θ[0, π] and φ[0, 2π]. Even then, the form may be too intricate for conventional integration.
A1 SI is the International System of measurement.
