In a three-meter fiber optic with 2 percent (0.02) attenuation per meter, the first meter transmits 98 percent of the Entrance Flux. This transmitted flux, represented by 0.98 x Entrance Flux, is then passed on to the second meter of fiber, which in turn undergoes the same attenuation, 0.98 x (0.98 x Entrance Flux). Similarly, the flux emerging from the third meter of fiber optic is given by 0.98 x 0.98 x 0.98 x Entrance Flux. This repeated multiplication by the transmittance per unit length can be mathematically simplified by using an exponent function (e.g. 0.98 x 0.98 x 0.98 = 0.983). To calculate the total transmission, for any length of fiber optic, the following formula can be used:
Total flux transmitted = Entrance Flux x tN
t = transmittance per unit length in decimal form (this can be determined by one attenuation per unit length), and
N = number of unit lengths, which is simply the length of the fiber optic given in the same units (e.g. feet or meters) as the units used for transmittance and attenuation.
Example: 1,500 Lumens of flux enter a fiber optic. The fiber optic is 2.5 meters long and has an attenuation of 3.4 percent per meter. How many lumens exit from the terminal end?
Solution: First the transmittance is determined. Transmittance = 1 - Attenuation. Thus the transmittance is 0.966 per meter (1–0.034). The total flux transmitted is then given by 1,500 Lumens x (0.966)2.5, which is 1,376 Lumens.
The above factors and considerations tell a designer how much flux will exit an end-emitting fiber optic system. To calculate or estimate illuminance on a surface from a particular source, information is needed on the direction of the light output from the fiber. Conventional lighting system design relies on measuring candlepower distributions under laboratory conditions, which indicate both the intensity and direction of light emission. These measurements can be used to generate coefficients of utilization, which are published factors that tell a designer how much light actually arrives at a work plane given particular conditions (room dimensions and surface reflectance). Designers can also use distribution information for a conventional light source, typically expressed as a candlepower distribution. These same photometric approaches can be applied effectively to fiber optic lighting systems.
As fiber optic systems become more common, "coefficients of utilization" will become available based on measured candlepower distributions. At this time, designers need to rely on published or measured candlepower distributions and to use point-to-point illuminance calculation techniques.
To determine actual illuminance at a given point using any of these fiber optic lighting systems, a designer needs to know the light distribution characteristics for each light-emitting component of the system. All three systems can be easily characterized, resulting in standard candela plots of the light-emitting portion of the fiber optic. For end emitters, a generalized candela plot can be measured for each distinct type of end emitter connected to the central illuminator or light source. These characterizations can be generated by a photometric test laboratory or by the manufacturer of the optical distribution head, and can be published so that they are available to designers.
Standard photometry techniques generate a spherical characterization of the light-emitting section of a fiber. This characterization, expressed in candelas, represents the candlepower of the emitter as measured in a particular direction. This technique usually involves measuring the illuminance under laboratory conditions over a large extended spherical area surrounding the source. From these measurements, a candlepower plot is generated and expressed in luminous intensity over an entire 360-degree vertical plane. Multiple planes can be measured, depending on the degree of the area's asymmetry. For rotationally symmetric emitters like fiber optic end emitters, a single plane of measurement will be appropriate for most calculations. This calculation generates a distribution of intensity as a function of direction, expressed in candelas. This characterization allows a designer to calculate illuminance at any point within the space, based on the geometric relationship between task and source.
Using published candlepower distributions, a designer can easily determine illuminance at a surface using a point-to-point illuminance calculation.
Illuminance is calculated using the basic equation:
E= I/ D2
E= illuminance (in foot-candles),
I= intensity directly under the source (in candelas), and
D= is the distance from source (in feet).
This relationship can only be used when surface is directly under the source and normal (perpendicular) to the light ray.
For all other positions a more generalized formula is:
E=I x cosØ/D2
E = illuminance (in foot-candles),
I = intensity of the source (in candelas) in the direction toward the point on the illuminated surface,
Ø = the angle between the line joining the source to the point on the illuminated surface and a line normal (perpendicular) to the illuminated surface.
This can also be expressed as the angle between the light ray and a vertical through the center of the source known as the nadir.
cos = the trigonometric cosine function (found on most all hand calculators).
D = distance between source and task point.
If multiple sources are used, the illuminance contributed by each emitter must be determined and then summed. For continuous emitters or series emitters, the designer must use the candlepower for each section and then conduct a summation for total illuminance. In continuous, light-emitting fiber optic systems, light is emitted along the entire length of the fiber optic. In order to characterize this type of system photometrically, one practical approach is to break the entire length into smaller unit lengths and conduct photometric measurements to generate a series of unique candlepower distributions for each length. This approach is particularly useful because the luminous intensity down the length of the fiber optic may vary. The unit characterization and resulting candlepower distributions would be done in a photometric testing laboratory. The resulting candlepower distribution could then be used by a designer for any type of application. These calculations that use simple point-to-point estimates do not include reflectance effects from secondary sources such as walls or adjacent surfaces. For more detailed estimates, a designer can take the candela power distribution and use computer-based calculations to determine illuminance at a point from single or multiple fiber optic point sources.