### Photometry of Strobes

BY Glenn Birket, 2004

The purpose of this paper is to review photometry concepts which are applicable to strobes, to better understand how strobes are specified. The issue is considered from the perspectives of energy in and light out, then also from the photographer’s approach to quantifying light.

The light output from a strobe is usually quantified in one of three ways.

**Joules.**The electrical energy input to the strobe may be stated in joules. These figures can only be compared if one may assume that the strobes compared convert energy to light with the same efficiency.**Candela.**The light output may be specified in candela, a measure of luminous intensity. Candela must be specified for a specified direction. It is therefore usually stated in one of two ways: (a) Mean Spherical Candela is the average of the light output in all directions. (b) Beam Candela is the light output in a specific direction, usually the maximum. Ideally a luminous intensity diagram (usually a polar plot) is provided for each of two mutually perpendicular planes, thus indicating the candela output in all directions.**Guide Number.**For photographic purposes, light output from a strobe is specified as a Guide Number. Given a particular film speed, the guide number may be divided by the flash-to-subject distance to obtain the required camera aperture for a proper exposure.

This paper explains the relationship of these three methods of quantifying light from a strobe. It is intended to serve as an aid to interpreting strobe specifications. To begin, a introduction to photometry is provided. Topics related to strobes are introduced later in the paper.

### Photometry

Photometry is the measurement of the intensity of light and its illuminating power. *Luminance* is light leaving a light source. *Illuminance* is light *incident* upon a surface. When considering the light from a luminaire, including a strobe, we are interested in luminance. When lighting a subject we are more interested in illuminance. Either way, we must understand luminance first, because it is the luminance that creates the illuminance.

### Luminous Intensity

In both the SI and English measurement systems the basic measure of *luminous intensity* is the *candela* (cd). (The predecessor of the candela was the candle. One candela = 0.98 candle.)

To understand the candela, assume that we have the ideal candle. It would be an isotropic point source of light, meaning that all light comes from a central point and radiates uniformly outward in all directions. For measurement purposes, imagine the light passing through an imaginary sphere centered on the light source.

Area on the surface of the sphere may be measured in steradians (sr). A steradian is an area equal to the square of the radius of the sphere. For example, if the radius of the sphere is one meter, one steradian is the solid angle projected on one square meter of the surface of a sphere with a one meter radius. Similarly a steradian yields one square foot at one foot from the point source. A steradian is independent of distance from the center of the sphere; it is the measure of solid angle. Since the surface area of a sphere is 4π times the square of the radius of the sphere, there are always exactly 4π steradians in a sphere.

A *candela* is a point source of light yielding 1/683 watt (W) of light energy per steradian.

1 cd = 1/683 W/sr

Just as the steradian is independent of the length of the radius of the sphere, the *candela is independent of the distance* from the light source.

Think of candela as an emission from a source which loses interest in what happens to the photons it emits. The photons may diverge or may have been focused in a single direction.

An isotropic one candela source will radiate a total of 4π candelas (12.57 candelas) which is 4π/683 watts (1.84 x10^{-2}watts).

Light from a real source is of course not uniform in all directions. Therefore luminous intensity in *candela is always specified in a specific direction* from the source. A luminous intensity diagram, usually a polar plot, may be provided showing the intensity in all directions. Often two diagrams are provided, in each of two perpendicular planes.

### Luminous Flux

Once light has left its source it is luminance and is quantified by the unit of luminous flux which is the *lumen* (lm). Think of a lumen as light that has left the source but has not yet arrived at a destination.

By definition, an isotropic one candela light source emits one *lumen* per steradian. Thus one lumen has 1/683 watt of illuminating power.

1 cd = 1 lm/sr

1 lm = 1/683 W = 1.464 x 10^{-3} W

Imagine the one candela isotropic point source above. Since the luminous flux in one steradian is one lumen, then the luminous flux over one square meter at a distance of one meter from the source is one lumen. The same can be said of the luminous flux over one square foot at one foot from the source. The sum of the luminous flux in all directions then is 4π lumens, or 12.57 lumens, or 1.84 x10^{-2} watts of luminous power for an isotropic one candela source.

Thus, the candela and the lumen are interchangeable in a sense, because one results directly from the other. The candela describes light from its source while the lumen describes light flux in transit. The key difference is in how the measurements relate to distance from the source. Since light usually spreads out as it travels, as you move farther from the source it takes more surface area to capture the one lumen that resulted from the one candela at the source. A candela is independent of distance; a lumen is not.

Even though practical light sources do not emit light uniformly, the output from a lamp is usually quoted in lumens. This is accomplished by summing the lumens in all directions with consideration for the varying intensity in different directions. Mathematically this is

F = ∫I dR

over the entire sphere where F is luminous flux in lumens, I is intensity in candela, and dR is an element of solid angle. In practice, this value is measured by placing the light source in the center of an integrating sphere which captures and measures all of the light leaving the source, in all directions.

A light source may be fully specified by its output in lumens and two mutually perpendicular plots of the luminous intensity (candela) distribution.

### Illuminance

When luminous flux strikes a surface, illumination is provided. The unit of illuminance is the *lux* (lx). It is defined as one lumen per square meter.

1 lx = 1 lm/m^{2}

In the English system the similar unit is the *foot-candle*, which is defined as one lumen per square foot. One foot-candle equals 10.76 lux.

A spherical surface will receive one lux of illuminance from a point light source that emits one candela of *luminous intensity* in its direction from a distance of one meter.

### Inverse Square Law

The density of flux (illumination) radiating out from a point source diminishes by the reciprocal of the square of the distance. Beginning with an isotropic one candela source, we know that at one meter we will have one lumen per square meter of spherical surface, i.e. one lux. According to the inverse square law we will have one quarter lux at two meters, one ninth lux at three meters, and so on.

To apply this relationship, we must know the effective point of the origin of the light. For a simple lamp, it will be the lamp. If the light is somehow focused though, the virtual source maybe elsewhere, such as behind the apparent source. If the light is perfectly collimated, the source will be at infinity and the level of illumination will not change with distance.

Practical sources are never truly a point. The inverse-square law will however give good results if the distance from the source is at least five times the diameter of the source. Similarly, when the distance to the source decreases to less than 1/20 of the diameter of the source, changes in distance no longer affect the flux density.

For different distances from the same source,

E_{1} r_{1}^{2} = E_{2} r_{2}^{2} *or* E_{1} = E_{2} (r_{2 }/ r_{1})^{2}

Using this relationship, we may measure the flux at one distance from the source and know the flux at another distance from the source.

If E is flux density in lux from a source of intensity I, and r is the distance from the source in meters, then:

E = I/r^{2} *or* I = E r^{2}.

Using this relationship we may determine the intensity of the source in candela simply by multiplying the flux density by the square of the distance.

### Luminance from a Surface

Once light from a lamp has struck a surface, some part of the light is reflected. Only a polished reflector will reflect nearly all of the light. Dark matte surfaces reflect almost no light. Other surfaces are somewhere in between.

Thus an illuminated surface becomes a new source of light. It is an area source though, not a point source. The surface is called a Lambertian Surface if it provides uniform diffusion; luminance which is the same in all directions. If the surface is an extended surface, meaning that it extends a great distance in all directions compared to the viewing distance from the surface, the luminance from the surface is again independent of distance from the surface.

Light from such a source is measured in *candela per square meter* (cd/m^{2}), or *foot-lamberts* (fL)in the English system. One foot-lambert equals 10.76/π candela/square meter.

Just as the candela and the lumen are related, so are the candela/square meter and the lux. The first is a source specified in a way which is independent of distance, and the other is illuminance which may decrease with distance from the source.

### Summary of Units

Summarizing the units of measurement to this point,

- Intensity of the source is measured in
*candelas*(cd) - Flux transmitted through space is measured in
*lumens*(lm) - Light illuminating a surface is measured in lux (lx = lm/m
^{2}) or foot-candles - Light reflected from an area (luminance) is measured in cd/m
^{2}or foot-lamberts (fL).

### Color and Luminous Efficiency

Throughout this discussion color has not been mentioned. A monochromatic light source was assumed. White light (a combination of a variety of light colors) is now introduced to complete the topic of light measurement.

Consider that the eye is more efficient for some colors, i.e. wavelengths or frequencies, than others. Because of this, it takes more energy to create a blue or red light that seems as bright to the eye as a green or yellow light. Likewise, the eye will perceive a blue light as less intense than a green light even when an instrument reports that the two are emitting the same amount of energy.

The V-lambda curve describes the CIE standard photometric observer, i.e. a human. It provides luminous efficiency vs. color. It is a bell-shaped curve with a maximum efficiency at 555 nm, which is a yellow-green light color. Thus, all of the units of measure for light are stated at 540 x 10-12 Hz, which is about 555 nm, a yellow-green light to which the eye is most sensitive. When dealing with other colors of light which contain many parts of the spectrum, such as “white” light, the instrument must weigh each distinct color of light according to its efficiency on the V-lambda curve. This is usually done with a filter in front of the sensor. The filter makes the sensor respond similar to the human eye. Throughout this discussion it has been assumed that we are either measuring light at 555 nm, or that we are measuring white light using an instrument provided with a V-lambda filter.

### Photographic Light Meters, Exposure Value and Guide Number

As might be predicted from the discussion above, units for luminance and illuminance are regularly confused and interchanged. If it is desired to use a photographic light meter to measure light, this confusion will likely be encountered. Light meters report the luminance of a subject in lux or foot-candles (units of illuminance) when they should report in candela/square meter or foot-lamberts. The light meter is reporting the illumination which if reflected from a perfectly reflective surface would produce the same value in luminance.

In photography one “stop” or one Exposure Value (EV) is related to luminance measured in foot-lamberts. When using a photographic light meter that reports lux, divide by 10.76 to get foot-lamberts. When using a light meter that reports in foot-candles, use the value as if it were in foot-lamberts.

An understanding of EV is useful because it relates to the Guide Number which is a useful means of quantifying light from a strobe. EV is used to express the amount of exposure to light required by film. For this use, EV is:

EV = SV + BV

where SV (the film speed value) = log_{2}(0.32S) where S is the ISO (ASA) film speed, and BV (brightness value or luminance) = log_{2}(B) where B is in foot-lamberts. BV = 0 at 1 foot-lambert.

EV is also used to express the amount of light that will be admitted to the film. For this use, EV is:

EV = AV + TV

where AV (aperture value) = log_{2}(N^{2}) = 2log_{2}(N) where N is the aperture *f*-number, and TV (time value) = log_{2}(1/t) where t is the shutter speed in seconds.

A proper exposure is obtained when the EV admitted equals the EV required to properly expose the film, i.e. SV + BV = AV + TV.

The light meter measures luminance which is the BV component of the EV. Therefore, each stop of light change reported by the light meter (one *f*-number) corresponds to one unit of log_{2}(B) where B is in foot-lamberts. Since we are using base two logarithms, this means that *each one stop increase represents a doubling of the luminance* in cd/m^{2} or foot-lamberts.

The maximum power of a photographer’s strobe is reported as a Guide Number (GN) for the strobe. The GN is always specified for a particular film speed, and in either feet or meters. The GN is the product of the flash-to-subject distance and the aperture *f*-number.

GN = (aperture f-number) * (strobe-to-subject distance)

For example, a better than average photographic strobe has a GN of 66 in feet for ASA 100 film. A good camera may have an aperture as large as *f*-2.8. The greatest distance at which a subject can be adequately illuminated by this strobe is then 66/2.8 = 23.5 feet. Typical disposable cameras have weaker strobes and apertures with larger apertures (smaller aperture openings) yielding even shorter distances. This explains the great number of disappointing photographs taken from the bleachers and the back of the auditorium.

As can be seen, if the GN of a strobe is known the *f*-number can be calculated for a particular distance, and from that a luminance value can be calculated. The math is left as an exercise.

### Strobes: Energy in vs. Light Out

Light from a strobe is produced by dumping charge from a capacitor through a xenon flashtube. The intensity in candelas of the light from the flashtube is proportional to the energy supplied to the flashtube. The energy supplied is the initial energy in the capacitor less the energy that remains

E = ½CV_{1}^{2} – ½CV_{2}^{2} = ½C(V_{1}^{2} – V_{2}^{2})

Only a portion of the energy in the capacitor will be converted to radiant energy, perhaps half. Of that radiant energy, only about 30% is visible. Of that which is visible, much of it must be discounted due to its low luminous efficiency on the V-lambda curve. (The flash meter takes this into consideration.)

One watt of power, if converted into light with 100% efficiency would yield 683 lumens. Efficient mercury-vapor lamps may achieve 100 lumens per watt because their radiant energy output is concentrated near the 555 nm wavelength which is most efficient for the human eye. The practical limit for incandescent lamps is 40 lumens per watt. The luminous efficiency of a xenon-filled flashtube is in the range of 10 to 50 lumens per watt, less than 10% efficient.

A strobe is such a brief light source that it is most useful to express its output by summing all of the light coming in a single flash, essentially a report of the total number of photons emitted by the flash. This is accomplished by integrating (summing) the illuminance over the duration of the flash. The result is *lux-seconds*. Another useful value is *candela-seconds*, essentially a measure of the light energy released by a single flash. One watt equals one joule/second. A candela is 1/683 watts. Thus a candela-second represents 1/683 joule.

Consider an example. Suppose our flashtube emits an efficient 50 lumens total luminous flux per watt. This is 50/683 (about 7%) of the ideal. If this power is radiated uniformly in all directions it is from a 50/4π = 3.98 candela/watt source. Thus, we can say that this efficient strobe produces 50 lumen-seconds per joule from a source of about 4 candela-seconds per joule. This value can be multiplied by the actual duration of the flash (or better, integrate the area under the lumen v. time curve) to obtain joules output. Then, given the anticipated efficiency of the flashtube, this result may be compared to the joules input as a check.

Peak (initial) intensity and duration are the two aspects of a strobe flash that may be controlled. Typically the maximum intensity is achieved within a few microseconds of triggering. As the capacitor voltage drops, so does the light output. Watching the light output with a scope across a photo-detector, the strobe’s decay is seen to be exponential, but appears almost linear for much of the decay. The maximum light intensity produced and the shape of the light pulse will be affected by several factors including the inductance and resistance of the wire between the capacitor and the flashtube.

### Strobes: Energy in vs. Guide Number

The relationship between the energy delivered to the flash the photographic guide number G is given by:

GN = K(E)^{½}

The guide number is proportional to the square root of the flash energy. K is a constant that varies widely depending on the design of the reflector and any losses which may occur due to light absorption.

### Strobes Underwater

Air is so thin that for any reasonable distance light loss due to attenuation in air can be ignored. When light passes through water there are noticeable losses. Attenuation over a given distance can be calculated using the Beer-Lambert law,

I = I_{0} e^{–}^{eL}

where I is the final intensity, I_{0} is the initial intensity, e is the absorbtivity or extinction coefficient, and L is the path length. It says that light intensity decays exponentially with distance in an absorbing medium. This decay is in addition to the angular dilution effect (inverse square law) which applies in all mediums. The extinction coefficient e varies with wavelength. In water, the attenuation is much greater at the red end of the spectrum than at the blue end, so flash illumination only works over a very short range underwater.

In photographic terms, water’s effect is to reduce the effective guide number of a flash by a factor of 3. The water also acts as a filter with a density of about 0.12 red per meter of light path; which means that you lose a whole stop of red for every 2.5m, and since the light must travel from flash to subject, and then from subject to camera, you lose 1 stop of red when you are only 1.25m away from the subject.

### Mean Spherical Candela vs. Beam Candela

Candela is a measure of light which is independent of the distance from the source of the light. It is therefore perhaps the best value to use in the specification of the light from a source. The light output from a real-world source varies according to the direction from the source. Candela measurements are therefore only meaningful if the direction from the source is given for the measurement. This fact is typically addressed in two ways. Light output is specified as either Mean Spherical Candela or Beam Candela.

Mean Spherical measurements are made in an integrating sphere. The light in all directions is collected and then divided by the 4π steradians in a sphere yielding lumens/steradian, which is candela. This then is an average value of candela for that fixture, over all directions.

Beam Candela samples only a very narrow angle of light from the source. Typically the manufacture will measure light in the brightest direction from the fixture. The figure is often misleading because the direction is not specified, nor is the spread over which the value applies.

To fully and fairly characterize light from a fixture, polar plots of luminous intensity are required, one for each of two mutually perpendicular planes. In this way the candela output can be expressed in all directions.

### Strobe Intensity Measurement Example

The intensity of a representative Birket strobe is measured with a Quantum Instruments Calcu-Flash II. The strobe and flash meter are positioned in a large dark area such that the center of the strobe’s flash-tube is 305mm or about 1 foot from the incident light dome of the instrument. An increment of one in the instrument’s digital display (d) corresponds to 1/3 EV and 0.78125*2^{d/3} lux-sec. A 5 corresponds to *f*-1.0 @ ASA 100 or 2.5 lux-sec.

First, an oscilloscope is also used to read the duration of the flash. Maximum intensity occurs within 100 μsec of trigger and the intensity decays to ten percent within 2 msec. From an energy perspective (area under the curve) the strobe can be considered to be at maximum intensity for 1 msec.

With the strobe vertical, readings of 15 are observed from all radial positions except for one area of about 20 degrees which reads 14. This is the area in which the anode wire partially blocks the view of the flash-tube.

With the end of the strobe pointing toward the instrument from the same distance, a reading of 8 is recorded. It is observed that the low of 8 is only present when the tube is pointed directly at the instrument. Tilting the tube to partially expose a side to the instrument gives an abrupt rise to a reading of 14 or 15.

A 15 corresponds to 25 lux-seconds. An 8 corresponds to 5 lux-seconds.

From most directions, the strobe emits 25 lux-seconds of radiant energy for 1 msec. Thus the strobe can be characterized as emitting 25000 lux in most directions, at a distance of 305 mm. By the inverse-square law, this represents about 25000/(.305)^{2} = 2326 candela. Considering the dark ends of the strobe, we will estimate that the average value over all directions is about 1200 candela. Totaling this over a sphere we have 1200 * 4π = ~15,000 cd. Multiplying this by 1/683 watts/candela, we get about 22 watts. As the flash lasts for only one millisecond, this would be about 22 millijoules.

Energy into the flash equals ½CV^{2} = ½ (68μF) (300)^{ 2} = 3.1 joules. Efficiency of the flash is then about .022/3.1 = 0.7%, roughly what might be expected, considering the several assumptions made throughout this example.

### Suggested Reading

http://www.intl-lighttech.com/services/light-measurement-handbook/

http://www.schorsch.com/kbase/glossary/

http://www.electro-optical.com

http://www.camerasunderwater.info/engineering/flash/flash_tp.html