of radiometry and photometry radiometric analysis. . . . . asap technical guide 11 (eq 9) equation...

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Technical Guide

RADIOMETRIC ANALYSIS

Breaul t Research Organizat ion, Inc.

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This Technical Guide is for use with ASAP®.

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brotg0909_radiometry (April 18, 2007)

ASAP Technical Guide 5

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Radiometric Analysis 7

Fundamentals of Radiometry and Photometry 7Radiant energy and power 8Geometrical characteristics of radiation 8Spectral dependence of radiometric quantities 14Photometry 14Luminous flux 14Luminous intensity 17Luminance 17Illumination Concepts 17Colorimetry: CIE color coordinates 18

Radiometric Calculations in ASAP 21Coherent and incoherent calculations 21Rays and the ray-object relationship 22Radiometry 23Radiance 40Statistical error 43Photometry and colorimetry 46CIE numerical and graphical color analysis 47Examples 50

References 50Monte Carlo Simulations 51

Appendix: PHOTOMETRY.INR File i


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In this Technical Guide, you will learn how to perform various radiometric calculations in the Advanced Systems Analysis Program (ASAP®) from Breault Research Organization (BRO). Before you learn how to perform these calculations, we want to introduce certain radiometric definitions, nomenclature, and concepts to you.

The remainder of this ASAP technical guide is divided into two primary sections. The first section introduces radiometric and photometric definitions and concepts. In this section, you will be introduced to the radiometric definitions of energy, power, and geometric characterizations of power, such as irradiance, intensity, and radiance. Photometry and colorimetry are discussed at the end of the first section (see “Photometry” on page 15, and “Colorimetry: CIE color coordinates” on page 19).

The second section associates radiometric and photometric definitions and concepts with the equivalent ASAP procedures and commands for performing radiometric and photometric calculations. You will also learn about distribution data files, which store the results of radiometric calculations for later graphical viewing.

F U N D A M E N T A L S O F R A D I O M E T R Y A N D

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NOTE: If you have a background in radiometry, you may choose to skip this section.

Radiometry literally means the measurement of radiation. However, in the context of ASAP, radiometry means the calculation of radiation, or some specific geometric quantity thereof. We make a clear and immediate distinction between radiometry and photometry in this guide. Photometry is a normalized form of radiometry. Normalization is a process where a measurement or calculation is caused to conform to a standard, model, or established norm. In the area of photometry, the standard, or established norm, is the response of the human eye.


R A D I O M E T R I C A N A L Y S I S

Fundamentals of Radiometry and Photometry

Other forms of normalization exist, such as specific detector responses, but photometry is the most common and recognized form of normalization.

Radiant energy and powerWe describe radiation in terms of energy, power, and the geometric characterization of power. Energy is the ability to do work. Power is the time rate of change of energy, or the amount of energy expended per unit time. We use the International System (SI) of symbols, units, and dimensions to define energy and other radiometric quantities. Where appropriate, we mention their relationship to other units and dimensions of historical significance.

Q is the radiant energy. Power, denoted by , is the time rate of change of energy or,

(EQ 1)

Equation 2

The Greek symbol phi, , refers to the word “flux”, which means flow or flowing. Power is the flux or flow of energy in the stream of time. In SI units, energy is represented in joules (J) and power in watts (W).

Geometr ica l character ist ics of radiat ionGeometrical characteristics of radiation are used, along with power, to define radiometric terms. We are most interested in areas and solid angles. An area is the measure of a planar region or the surface region of a three-dimensional solid. We typically represent differential areas as dA. In SI units, areas are represented in square meters (m2), square millimeters (mm2), and so on.

An observer or a detector sees projected area (dAp) when the original area is tilted at some angle (), with respect to the line of sight of the observer. Projected area is reduced in size by the cosine of the angle between the area normal and the line of sight. Operationally, the projected area is,

(EQ 3)

Equation 4

8 ASAP Technical Guide

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. .R A D I O M E T R I C A N A L Y S I S


To understand the concept of projected area, tilt a piece of paper. The total projected area you see decreases as you tilt it. See .

Think of a solid angle as a two-dimensional (2D) angle. First, an angle is a subtense: one linear dimension divided by another linear dimension. Consider the projection of a line or curve in space onto a circle. Mathematically, the angle is the arc length of this projection divided by the radius of the circle. In SI units, angles are represented in radians. In a full circle, there are 2 radians; that is, the circumference, or total arc length of a circle, is 2 multiplied by the radius of the circle. The solid angle () is the projection of an area onto a sphere, or the surface area of the projection divided by the square of the radius of the sphere. See . Operationally, the differential solid angle is,

(EQ 5)

Equation 6

A useful relationship is obtained if the differential solid angle is integrated from 0 to 2 in and 0 to in .

Projected area




Angles and solid angles

(EQ 7)

Equation 8

Geometric characteristics of area and solid angle, when combined with power, lead to the radiometric quantities of flux density, radiant intensity, and radiance.

R A D I A N T F L U X D E N S I T Y

Flux density is the amount of radiant power emitted or received in a surface region. Later, we will see that this is the radiance integrated over all solid angles. But for now, we define the radiant exitance (M), a flux density, as


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(EQ 9)

Equation 10

This is the emitted radiant power per unit area. Its SI units are watts per square meter (W/m2).

The radiant incidence or irradiance (E)1, another flux density, is defined as,

(EQ 11)

Equation 12

This definition is the received radiant power per unit area. Its SI units also are watts per square meter (W/m2).

1. “E” comes from the French word, elcairage.




R A D I A N T I N T E N S I T Y

The radiant intensity (I) is an angular flux density.1 It is defined as,

(EQ 13)

Equation 14

This definition is the power per unit solid angle. Its SI units are watts per steradian (W/sr).

R A D I A N C E

The fundamental radiometric quantity is the radiance (L).2 It is an angular-area flux density. Radiance is the radiometric quantity that is conserved throughout an optical system. The radiance is defined as,

(EQ 15)

Equation 16

This is the power per unit area per unit solid angle. Its SI units are watts per square meter per steradian (W/m2/sr). Irradiance and intensity can be computed from the radiance, with appropriate integration and integration limits. Irradiance is the integral of the radiance over all solid angles. Intensity is the integral of the radiance over all areas.

Typically, you need to determine which radiometric quantity is necessary for your analysis. In most cases, you will be given a radiometric requirement to fulfill. In a situation where you are not given a radiometric requirement, the following considerations may help in determining the radiometric quantity to compute for your particular application.

1. “I” comes from the French word, intensité.

2. “L” comes from the French word luminosité.


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C O M P U T I N G I R R A D I A N C E

Irradiance is typically computed in those situations where a flux distribution over a surface region is the required calculation. Irradiance calculations integrate over all angles, and therefore remove all angular dependence from the computation. If you are interested in both the spatial and angular geometric characteristics of the radiation, a radiance computation may be more appropriate for your application.

C O M P U T I N G I N T E N S I T Y

Intensity is typically computed when your source or output from your optical system may be considered a point source. In a point source situation, your computation need not resolve the spatial variations of the radiation from the source or optical system. Under this circumstance, analogous measurements are made with the detector at a very large distance from the source. This is the case in goniophotometer measurements of the luminous intensity from automotive lights. What, then, constitutes a large distance? Yamuti and Fock (IES Lighting Design and Analysis Handbook) outlined a general rule for Lambertian emitters to determine whether or not a source may be considered a point source. Their rule, originally called the “Rule of 5”, states that if the distance from the source to the next optical element or detector is greater than five times the maximum dimension of the source, the source may be considered a point source, with respect to that specific optical element or detector.

NOTE: Many optical and illumination engineers apply the “Rule of 10”, where a factor of 10 replaces the factor of 5. However, depending upon your source and optical systems, this rule may not apply.

C O M P U T I N G R A D I A N C E

The radiance is typically computed when you are concerned with the spatial and angular behavior of your source, or input or output of your optical system.




Graphical illustrations are shown in .

Radiant exitance (top left), radiant incidence (irradiance) (top right), radiant intensity (bottom left) and radiance (bottom right)

Radiant Exitance Radiant Incidence (irradiance)

Radiant Intensity Radiance


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Spectra l dependence of radiometr ic quant i t iesAll previous radiometric quantities can be distributed with respect to a spectral variable. Spectral variables are parameters such as wavelength, frequency, and wave number. A spectral radiometric quantity is a radiometric quantity per unit spectral variable. For example, spectral radiant power is,

(EQ 17)

Equation 18

This definition is the power per unit wavelength interval, also called a spectral power density. It is the power at a specific wavelength, in an interval d around the specified wavelength. The total power over the spectral emission bandwidth of the source is then,

(EQ 19)

Equation 20

The geometrical and spectral aspects of radiometric quantities are mathematically separable.

PhotometryAs mentioned previously, photometry is a normalized form of radiometry. All the concepts and basic definitions discussed in “Fundamentals of Radiometry and Photometry” on page 7 still apply, except that the spectral response of the detector, the standardized human eye, is now included in the radiometric quantities to form the photometric quantities. The eye actually has three different responses, based upon brightness conditions. The response of the eye under bright lighting conditions is called photopic response, the response under low lighting conditions is scotopic response, and the response in between these lighting conditions is mesopic response. For the purpose of our discussions here, we will concern ourselves with only the photopic response.




Luminous f luxThe response of any detector ()—the eye or another photonic device—is defined as the ratio of the detector output to the optical input. Typically, these units are voltage or current per watt. In the case of the eye, the units are lumens per watt. Detector responses are functions of the wavelength of light incident on them, (). They do not necessarily give the same response at all wavelengths. In other words, the output is weighted by the detector’s response. This leads to effective values, which are the values of a quantity that effects a response from a detector. For example, an effective watt is the detector response weighted output of spectral radiant power or spectral power density. Operationally, we can define the output of a detector as,

(EQ 21)

Equation 22

If the response is now the photopic response of the human eye, we write the luminous flux in terms of an eye watt or lumen (lm). The lumen may be thought of as the amount of spectral radiant power that elicits a response in the human eye. Operationally, it is defined as,

(EQ 23)

Equation 24

Here, Km is the luminous efficacy, and is equal to 683 lm/W at approximately 555 nm. V() is the luminous efficiency function, or visibility curve. The luminous efficiency function is shown in the figure, “Visibility curves for photopic and scotopic responses” on page 18. It is the normalized photopic response function of the human eye. The curve closely resembles a Gaussian curve, as many things in nature do, and is approximated by the function,

(EQ 25)

Equation 26


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when is in microns. Note that the luminous power is designated with a subscript “v”. In fact, all photometric quantities are designated with this special subscript, to distinguish them from their radiometric counterparts and other normalizations.

L U M I N O U S F L U X D E N S I T Y

The luminous flux density is the amount of photopically normalized (luminous) power emitted or received per unit area. The luminous exitance (Mv) is,

(EQ 27)

Equation 28

This is the emitted luminous power per unit area. Its SI units are lumens per square meter (lm/m2). A lumen per square meter is also called a lux (lx). A lumen per square foot (lm/ft2) is called a foot-candle (fc).

The luminous incidence or illuminance (Ev), the other flux density, is defined as,

(EQ 29)

Equation 30

This is the received luminous power per unit area. Its SI units also are lumens per square meter (lm/m2). See .

MddAv

v= Φ

EddAv

v= Φ




Visibility curves for photopic and scotopic responses

Luminous intensi tyThe luminous intensity (Iv) is a photopically normalized angular flux density. It is defined as,

(EQ 31)

Equation 32

This is the luminous power per unit solid angle. Its SI units are lumens per steradian (lm/sr). A lumen per steradian is also called a candela (cd).


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LuminanceLuminance is a photopically normalized angular-area flux density. Luminance is defined as,

Equation 33

This is the luminous power per unit area per unit solid angle. Its SI units are lumens per square meter per steradian (lm/(m2 sr). Its units also are referred to as candela per square meter (cd/m2).

I l luminat ion ConceptsNow that we have the formal radiometric definitions available, we want to relate these definitions to more familiar illumination concepts; in particular, those concepts that describe the quantity of illumination. We will consider only light output, light level, and brightness.

The total light output from a source is called luminous flux. This output is just the luminous flux density or luminous exitance integrated over the surface area of the source. Its SI unit is lumens.

The light level of a source is the total light measured on a plane, at a specific location, divided by the area of the plane. This quantity is the illuminance or luminous incidence, and is measured in SI units in lumens per square meter.

Brightness is another name for luminance, as defined earlier. Brightness measures light emitted from a unit surface area and going into a specific direction. It effectively accounts for the illuminance on the surface and the reflective properties of the surface. The brightness is measured in SI units of lumens per square meter per steradian.

Color imetry: CIE color coordinatesWe will briefly investigate concepts of colorimetry. Colorimetry literally means the measurement of color. However, this definition is a misnomer, because we perceive color, and do not directly measure it. Radiometry measurements are physical because we measure radiant power directly. Our observation, or measurement, of color is based upon perception, which classifies the measurement as psychophysical. Furthermore, individuals perceive color differently. Radiometry yields precise and accurate descriptions of the physical characteristics




of a source, such as its spectral power density, while two individuals observing the same source may say it looks different. A person might even say that two sources look the same though they are created with different spectral power densities. Different types of spectral power densities that lead to the same perception of color are called metamers. How, then, do we measure and compare color, taking into account human perception?

Empirically, colors may be compared with the color-matching experiment. This experiment explains the concept of tristimulus coordinates and CIE (International Commission on Illumination) chromaticity coordinates, and, eventually, how to compute them in ASAP. The experiment basically consists of matching colors in a bipartite field. One side of the bipartite field is a test field, which has a certain spectral power density. The other side is a matching field, with three adjustable, spectral power densities of three primary colors. You scale the spectral power densities of the three primary colors until the matching field color matches that of the test field. The scaling is done in such a way to change the overall power at each wavelength without altering the relationships between wavelengths. Those relationships remain invariant. It can be shown empirically and mathematically that only three primaries are needed to match colors. Doing so is outside the scope of this brief introduction to colorimetry. An excellent reference is in David Brainard's chapter on colorimetry, in the Handbook of Optics, Vol. I.

Adjusting three primary colors is what your computer screen does to produce color. Each pixel has three primary phosphors—red, green, and blue. Each phosphor color has its own spectral power density. By adjusting the number of electrons that bombard each phosphor, the spectral power density of each phosphor can be scaled so that their additions yield a perceived color. In both cases, the scale adjustments may be considered a type of color coordinate. If the spectral power densities are standardized, a color may be specified by its associated color coordinate or scaling factor.

The color-matching experiment determines the scale factors for each primary that are needed to match color. This is a metameric match, and not necessarily a physical match. The scale factors are called color or tristimulus coordinates. Once the tristimulus coordinates are known for a set of equally spaced monochromatic lights at different wavelengths, we can compute the coordinates for any light with a broad spectrum. The tristimulus coordinates for a given set of primaries, as a function of wavelength, are denoted by,


. . .



(EQ 34)

Equation 35

The computed tristimulus values for the spectral power density of a test sample or source are denoted by X, Y, and Z. X, Y, and Z are the scaling factors of the primaries needed to obtain a metameric match between the primaries and the spectral power density of a test sample or source. Furthermore, the ratios of X, Y, and Z, with their sum, determine the CIE color or chromaticity coordinates.

The tristimulus coordinates for a given primary, as a function of wavelength, collectively form a color-matching function. Therefore, three color-matching functions exist, one for each primary. The functions are not spectral power densities. In fact, they are more like the visibility curves used for photopic and scotopic normalization. In this sense, they reflect the scaling factors required of the cones in the human eye to metamerically match a single wavelength of light with a set of primary colors. The visible spectrum of light extends approximately from 380nm to 780nm.

The standard you use to compute tristimulus values determines the type of the primaries. For example, a set of color-matching functions is shown in the . This set of color-matching functions was standardized for large field sizes. The 10-degree color-matching functions are used when the observed field of view is greater than four degrees.

NOTE: CIE first standardized a set of color-matching functions in 1931, to describe tristimulus values for people with normal color vision. This set of primaries is slightly different from the 1964

standard. The color-matching function of this standard is

the photopic response of the eye.




CIE 1964 standard observer color-matching functions for viewing angles greater than four degrees

Operationally, the tristimulus coordinates for a test sample or illuminant and source are given by,

(EQ 36)

Equation 37

where is the spectral power density of the source or standard illuminant, and is the spectral reflectivity of the sample. The standard illuminate is used to illuminate materials that are not self-luminous. The spectral reflectivity is set to 1, to calculate the tristimulus values for a source only. The computed tristimulus values are the scaling factors of the primary that are needed to produce a metameric color match of the actual spectral power density to the primaries.

X K x

Y K y

Z

m

m

=

=

=

=

=

∑

∑

Φ Δ

Φ Δ

λλ

λλ

ρ λ λ λ

ρ λ λ λ

( ) ( )

( ) ( )

380

780

380

780

KK zm Φ Δλλ

ρ λ λ λ ( ) ( )=∑

380

780


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Radiometric Calculations in ASAP

Computing the tristimulus coordinates for a source or reflective sample bears a mathematical resemblance to photopic normalization.

Tristimulus values are often plotted on the chromaticity diagram. The chromaticity diagram is a plot of the ratio of the X and Y tristimulus coordinates to the sum of the tristimulus coordinates. In other words, the ratio of X to X+Y+Z and Y to X+Y+Z yield chromaticity coordinates. Normalizing the tristimulus quantities reduces the dimensionality of the data set. Chromaticity coordinates are defined this way so that any two lights with the same tristimulus coordinates have the same chromaticity coordinates, but may now be viewed two-dimensionally, instead of three-dimensionally. A similar ratio of the tristimulus values of the CIE 1931 standard primaries produce the spectrum locus on the chromaticity diagram. The spectrum locus is the boundary of monochromatic light on the chromaticity diagram.

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Coherent and incoherent calculat ionsASAP has the ability to simulate incoherent and coherent sources of radiation. When we speak of the coherence of a source of radiation, we are usually referring to its amount of spatial and temporal coherence. Spatial coherence has to do with the extent or size of the source. (How well do the point emitters of the extended source, emitting radiation at the same wavelength, emit in spatial phase with respect to each other?) A point source is, by definition, spatially coherent. Temporal coherence has to do with the monochromaticity or bandwidth of the source. (How well matched in phase is the radiation from a point source or points of an extended source, emitting at different wavelengths?) A monochromatic source is, by definition, temporally coherent.

Incoherent sources typically are extended sources, such as filaments, that emit radiation over a broad spectral band. Coherent sources typically are point sources, such as lasers, that emit radiation over a very narrow spectral band. The initial degree of spatial and temporal coherence of the ASAP source, and the radiometric calculation method used to compute the geometrical characterization of that radiative transfer, determine the resultant degree of spatial and temporal coherence.

In this section, we concern ourselves with only spatially and temporally incoherent sources. We will superpose sources linearly in terms of their energy densities. This




is equivalent, for example, to adding flux densities, radiant intensities, and radiances directly. Coherent superposition of point sources where the amplitude and not the energy density is added linearly is discussed in the Technical Guide, Wave Optics in ASAP. Such superpositions give rise to diffraction and interference phenomena.

Rays and the ray-object re lat ionshipAll sources in ASAP—incoherent or coherent—are modeled with rays. A ray is a purely geometrical concept. It is basically a vector that simulates radiative transfer. The spatial point of the ray vector is its location in space or the optical system. The direction of the ray vector is the propagation direction of the radiation. Each ray has an individual power or flux that contributes to the total flux of the source. In fact, each ray in ASAP can have over 40 pieces of information associated with it. The concept of the individual flux of a ray is important, because we will count the number of rays and their associated fluxes in area bins and angular bins, to compute radiometric quantities.

Rays also may be considered normals to wavefronts. Wavefronts originate from points of an extended source or from a point source. A wavefront is a mathematical surface of constant spatial phase. Wavefronts are surfaces over which the optical path lengths (OPL) of rays from a point source have the same value. The sum of the lengths of the vector path of the ray, multiplied by the refractive index of the materials in which the ray has propagated, is its OPL. The OPL and optical path difference (OPD) maps are used in analyzing the aberrations of imaging optical systems.

A monochromatic source of radiation in ASAP has a single wavelength assigned to it with the WAVELENGTH command. A polychromatic source is made up of individual monochromatic sources with wavelengths interspersed in the bandwidth or spectrum of the source. The SPECTRUM command is used to relatively spectrally weight the flux of the individual monochromatic sources comprising the polychromatic source. The FLUX command with the TOTAL option is used to set the total flux of the polychromatic source.

All sources and, therefore, all rays initially begin at OBJECT 0. By default, all rays begin in MEDIA 0 with a default refractive index of 1. When the rays are created, they are stored in a file with a default file name of virtual.pgs. As rays propagate through the optical system, they intersect the objects and are optically transformed by the optical elements in the system model. The ray information is then updated in the virtual.pgs file.


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Rays normally terminate when they encounter a totally absorbing object and stop on that object. Ray cessation warnings occur when rays do not reach a totally absorbing object. For example, after a ray intersects an object, it may not be able to physically intersect another object in its path, because no other objects are in the ray path. ASAP displays a message that the ray remains on the last intersected object, and has ceased tracing because it missed intersecting any further objects.

NOTE: Rays are always associated with objects in ASAP, whether they terminate normally or cease for another reason.

You may use the CONSIDER command to isolate rays on objects of interest. Recall that the CONSIDER command is used to turn on and off objects in your optical system database model. The SELECT command isolates individual sources or rays. The TRACE command, with the object range option, traces rays from an initial object to a final object. You may even trace rays through n-object intersections. The point here is that you may increment the ray trace on an object-by-object basis, temporarily stopping rays and isolating them at an object of interest, while next performing any of the ASAP radiometric calculations. You may continue the ray trace by issuing another TRACE command. In other words, you do not have to individually and separately change an object’s interface to make it totally absorbing, and then repeatedly restart the ray trace to look at the radiometric quantities on the object of interest.

Radiometry

R A D I A N T P O W E R A N D F L U X

When ASAP computes radiant power, it computes the amount of incoherent flux, using all the rays currently CONSIDERed and SELECTed. The STATS command and the LIST command are the actual commands that print radiant flux. The LIST command prints flux on a ray-by-ray basis, including such things as the ray number; ray position (POSITION option) or direction vector (DIRECTION option) in the global coordinate system; the current object the ray is on; and, in certain cases, the ray’s optical path length (OPL). OPLs are computed only when the ray storage, set with the XMEMORY command, is not set to MIN. If you use the LIST command for very large ray sets, the screen display time will be quite long.

As shown in , the STATS command prints statistical ray information, including the total flux of currently considered objects. The STATS command is different from the LIST command in the sense that STATS prints summary ray information, but not information on a ray-by-ray basis. STATS by itself produces a one-line-per-




object summary of the flux and number of rays on the objects that have rays. STATS ALL lists the same as STATS for all CONSIDERed objects, whether or not there are rays on the object. The total number of rays and total flux from all currently considered objects are summarized on the last line. The POSITION and DIRECTION options produce, on a considered object basis, object number, total flux on the object, and total number of rays on the object. The percentage of the total number of rays is shown for each object. It also produces a statistical analysis of the positional or directional (ray vector) data along all coordinate axes, which includes centroids, root-mean-square (RMS) deviations, and maximum spreads of the ray’s positional or directional coordinate.

For more information on detailed ray data, see the related ASAP commands FOCUS, GET, PUT, and EXTREMES.


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Example of STATS, STATS ALL, STATS POSITION, STATS DIRECTION,

and LIST POSITION output




R A D I A N T F L U X D E N S I T Y

The flux density was defined to be the amount of flux emitted or received in a surface area. In ASAP, we use the same procedure to compute both exitance and irradiance. We already know that the ray flux determines the flux or flux density. However, how do we combine the geometric characteristic of area with the flux, to compute the flux density? The answer is found in the relationship of total projected area and projected differential area to the size of the object, type, and orientation. This relationship is determined by the WINDOW command and the PIXEL command.

NOTE: ASAP computes all its incoherent radiometric quantities by a process called ray binning. As an analogy, imagine shooting arrows or darts at a target. Some arrows do not fall in the bull’s eye. Some arrows fall in rings outside the small center ring, and they are binned in these rings. Imagine a target with square or rectangular bins, instead of annular bins, and arrows that are now rays. The rays intersect the target, and are binned at various locations. An individual bin is an element of differential area. The total number of rays and their total amount of flux in a bin are an element of differential flux. The ratio of the two is the differential irradiance at that bin location.

The WINDOW command determines primarily the total area of the computation. This is the same WINDOW command that you use, along with your plot and profile commands, to create system geometry plots. ASAP windows can be only rectangular in shape. The window also is aligned with the X-Y-Z coordinate axes of the global coordinate system; that is, ASAP windows cannot be tilted in the global coordinate system.

The PIXEL command determines the total number of bins along the vertical portion of your plot window. This is the axis first specified on the WINDOW command. If you use an odd number of pixels, you always get a pixel in the center of the plot. If your window is square, you get the same number of pixels or bins along the second or horizontal axis specified on the WINDOW command. If your window is rectangular, ASAP uses the window aspect ratio to determine the number of pixels in the other direction. In the event that the aspect ratio times the number of pixels is not an integer, ASAP slightly sizes the pixels in the horizontal direction. It does this to accommodate the smallest integer number of pixels that fit the horizontal window direction.


. . .



If the normal of your detector plane is parallel to a global coordinate axis, you can always find a pair of window axes to match the axes of the detector. However, what should you do if your detector is tilted, or not planar? For example, what if it is spherical or cylindrical? In these cases, ASAP parallel-projects the rays onto the object and into the window and pixels.

To understand this concept, think of the window and pixels as a lengthy set of adjacent mail boxes at a post office, with your object sitting behind the mail boxes. The mail boxes (see ) cut through your object and the rays, regardless of whether or not they are on a planar, tilted planar, or curved surface. The rays are binned in a particular pixel—or mail box. In the case of a tilted detector plane, the projected area corresponds exactly to the projection term of the flux density equations. In general, the total number of rays in a particular pixel increases as the detector plane is tilted, thereby increasing the differential flux density in that pixel location. You can always rotate your ray data with linear transformation commands to remove the projection, if your detector is tilted with respect to the window and global coordinate axes.

The ASAP command to compute the flux density is SPOTS POSITION, which generates a geometric spot diagram. However, at the same time that the spot diagram is generated on your computer screen, a distribution file, with a default name of BRO009.DAT, is generated with the spot diagram.

A spot diagram is a 2D plot of spot position as a function of spatial coordinates, which appears on your computer screen. The distribution data file contains 3D data. The 3D data is the flux density (the third dimension) as a function of two spatial coordinates.

The WINDOW command determines the physical area of the flux density calculation and, along with the PIXEL command, sets the resolution (bin size), and hence the number of data values in the distribution data file. This file may be read into a special graphical viewing area in ASAP called DISPLAY, which has its own set of commands. In DISPLAY, the graphic commands allow you to plot the results of your flux density calculation, such as an isometric plot, contour plot, grayscale, or false-color plot. DISPLAY also allows you to average (smooth) and modify, and numerically output the results of your flux density calculation. However, if you want to see the flux density calculation at a different resolution, you must change the PIXEL command and re-run the SPOTS POSITION command.

, , , and illustrate SPOTS, PIXEL and the distribution data file in DISPLAY.




We pointed out in the discussion of the WINDOW command in the section, “Radiant Flux Density” on page 28 that ASAP windows cannot be tilted in the global coordinate system. However, it is possible to show the SPOTS information projected toward the normal of a specific surface. The following script shows how the projection direction is specified:

AXIS LOCAL [the object’s name or entity number]

SPOTS POSITION XY [or XZ or YZ]

SPOTS POSITION spot diagram


. . .



SPOTS POSITION overlaid pixels




SPOTS POSITION projection into pixels


. . .



SPOTS POSITION distribution data file

By default, every new SPOTS command overwrites any existing BRO009.DAT distribution data file. You may write data to a different file unit number, but do not use an existing unit. You may change this unit so that the results of your current SPOTS calculation are added or subtracted, on a pixel-by-pixel basis, to the existing BRO009.DAT file. You may also plot EVERY nth ray in the screen spot diagram, instead of all the rays. The rays are still used in the flux density calculation and stored in the distribution data file (not all of them appear on the screen to prevent screen saturation). As with all plot files, spot diagram information is plotted to the 3D vector file and the plot file. If you want to suppress the spot diagram entirely but still generate a distribution data file, use the ATTRIBUTE 0 option with the SPOTS POSITION command. This approach is most useful for illumination system flux density calculations that require more than hundreds of thousands, if not millions, of rays.




R A D I A N T I N T E N S I T Y

Radiant intensity is the radiant flux per solid angle. Again, the ray flux determines the flux or the angular flux density. How do we combine the geometric characteristic of solid angle with the flux to compute the angular flux density or intensity? ASAP offers two ways to perform this calculation, each with its own method for computing the solid angle. One method uses the SPOTS command, but this time with the DIRECTION option to compute the intensity in direction cosine space. The other method uses the RADIANT command to compute the intensity in angle space.

C O M P U T I N G I N T E N S I T Y I N D I R E C T I O N C O S I N E S P A C E

The direction cosine space is the space of vectors. Each ray is essentially a line in the global coordinate system whose direction is the propagation direction of the ray. The line—that is, the ray’s propagation direction—is described by a normalized vector whose coordinates are direction cosines. A direction cosine is defined as the cosine of the angle of the ray with respect to a coordinate axis. The cosine of an angle can vary from 1 to 1. Our direction cosines are therefore bound between 1 and 1.

Imagine our vector plotted in such a bound coordinate system, one that runs from 1 to 1 along the X, Y, and Z axes. Designate the axes as A or , B or , and C or , the direction cosine axes. One end of this vector is always located at the center of this new coordinate system. The other end of the vector lies somewhere on a sphere with a radius of 1, centered at the origin of the new coordinate system. Therefore, the sum of the squares of the vector components equals 1. This is the direction cosine sphere. The vector components locate the end point of the vector on this sphere. As you look along a particular direction cosine axis, you see a point parallel projected in a plane formed by the remaining direction cosine axes. In the parallel projection, you are looking at the sine components of the vector with respect to the projected axes. See the example in .


. . .



Concepts of direction cosine space

We can define direction cosine axes with the WINDOW command, just as we did for the X-Y-Z axes, except the coordinates do not extend beyond 1 to 1. This is the equivalent of a 180-degree angle. Whenever you define direction cosine coordinate extents, do so with the sine function, as it is the complement of the cosine function and automatically accounts for the parallel projection. For example, for radiation propagating primarily down the Z-axis, you want to set the Y- and X-axes in the WINDOW command.




To define the angular window extending from +/-10 degrees along the Y-axis and +/-20 degrees along the X-axis, issue the following WINDOW command or parameters:

WINDOW Y -SIN[10] SIN[10] X -SIN[20] SIN[20]

or,

WINDOW Y -2@SIN[10] X -2@SIN[20]

The SPOTS command accepts angle values as an option in its syntax. For example, to set the SPOTS command window in the previous case, type:

SPOTS DIRECTION YX -2@10D -2@20D

The D syntax instructs ASAP to automatically compute the sine of the angles. Therefore, the result is represented in direction cosine coordinates.

The PIXEL command works in exactly the same way as with flux densities, but now it sets up the number of bins in a direction cosine space. By issuing the SPOTS command with the DIRECTION option, ASAP computes the radiant intensity in direction cosine space. As with our flux density (exitance or irradiance) calculations, we see a geometric spot diagram, which is plotted in direction cosine space. Notice that the axes are labeled with A, B, and C, rather than X, Y, and Z, to indicate direction cosine space. A distribution data file, defaulted to BRO009.DAT, is created, and contains the radiant intensity as a function of direction cosine coordinates. Similar to the flux density calculations, this distribution data file may be read into DISPLAY and viewed with a variety of display commands.

DISPLAY has some special, reserved commands for viewing and changing the direction cosine space data. If you do not use these commands, all the other plot commands for viewing the intensity show the intensity in direction cosine space. First, the DIRECTION command creates either a polar or Cartesian (unwrapped polar plot) diagram of the intensity pattern in angle space. Polar plot diagrams are commonly used to display intensity information. The ANGLES command automatically converts your intensity data from direction cosine space to angle space and a spherical-polar coordinate system. Now, any intensity plots that you view with other DISPLAY graphics commands are displayed in an angular coordinate system. The angular plot axes on subsequent plots are renamed “vertical” and “horizontal”, in deference to the nomenclature of the WINDOW command. The polar axis of the spherical-polar coordinate system is in the horizontal direction.1


. . .



Direction cosines provide a convenient and powerful, mathematical coordinate system for intensity calculations. However, conversion of this coordinate system to an angular coordinate system is not linear. The ANGLES command basically applies a cosine factor to convert from direction cosine coordinates to angular coordinates. Therefore, square pixels in direction cosine space become somewhat rectangular in angle space. This effect becomes more noticeable with pixels towards the edges of plots with very wide angular coordinates. Angle space pixel sizes are increased, by a cosine factor, the further away they are from the origin. , , and illustrate SPOTS DIRECTION, direction cosine space, and the conversion to angle space.

SPOTS DIRECTION (intensity) spot diagram (Plot Viewer)

1. Corresponds to type B photometry of the Illumination Engineering Society (IES).




SPOTS DIRECTION distribution data file, in direction cosine space (Plot Viewer)

SPOTS DIRECTION distribution data file, converted to angle space (Chart Viewer)


. . .



C O M P U T I N G I N T E N S I T Y I N A N G U L A R S P A C E

The second way to compute the radiant intensity uses a form of the RADIANT command. This command performs its radiant intensity calculation in angle space, referenced to a spherical-polar coordinate system. In this calculation, the ray vector may be thought of as located at the origin of a sphere pointing into space. To compute the radiant intensity, we divide the sphere into solid angles, and essentially count the number of rays in a bin pointing into a particular direction. The WINDOW and PIXEL commands are not used to specify the angular extent and resolution. All this information is entered on the RADIANT command itself.

By default, the polar axis of the RADIANT command is the Z-axis, but you may choose another polar axis. See . The zenith or latitude angles are from the polar axis. The zero angle begins at a point on the positive direction of the polar axis. The 180 degrees correspond to the negative direction of the polar axis. The number of subdivisions is the number of bins along this axis.

TIP The zero angle location of any azimuth angle can be determined by the right-hand rule. For example, if the polar axis is the Z-axis, the zero angle of the azimuth axis starts along the X-axis and is positive in the direction of the Y-axis.




Zenith and azimuth angles (Z-axis as polar axis)

The azimuth or longitude angles are around the polar axis. The number of subdivisions is the number of bins around the polar axis. The angular extents in the zenith and azimuth axes, along with the number of subdivisions along each respective axis, determine the number and resolution of the differential solid angle bins. Any ray vector, and its associated flux, falling in one of these bins is used to compute the radiant intensity in that solid angle and in that direction. The polar axis specification, the zenith and azimuth extents, and the number of subdivisions along each axis are the equivalents of the WINDOW and PIXEL commands used

+180 Degrees

X

Y

Z

0Degrees

Azimuth Angle andSubdivisions

Polar Axis

Zenith Angle andSubdivisions

+


. . .



with SPOTS DIRECTION. However, now the computation is done in angle space.

NOTE: When using the RADIANT command, the differential angles bins can become more compressed around the poles.

Consider a geographic analogy in the following discussion. The lines of longitude and latitude combine to form a solid angle referenced to the center of the sphere. When you observe lines of longitude and latitude, and the solid angles they form at the equator, they are larger than at the poles. The error in a flux density calculation is related to the number of rays that are in a bin. The larger the number of rays, the smaller the error. Smaller bins usually have a smaller number of rays in them, so their errors can be much larger. We refer to this effect as noise, and we address sampling issues at the end of this section.

The RADIANT command also produces a distribution data file that may be viewed with the DISPLAY subcommands. The RADIANT distribution data file coordinates are spherical-polar angles, unlike the distribution file created by the SPOTS DIRECTION command, whose coordinates are direction cosines. The coordinate axes of the RADIANT distribution data file are labeled “Angle from axis” and “Angle about axis”. These labels refer to the zenith and azimuth directions, respectively, specified on the RADIANT command.

A commonly asked question is, “When do I use SPOTS DIRECTION, and when do I use RADIANT to compute the radiant intensity?” The answer depends on your specific application. Generally, SPOTS DIRECTION is used for radiant intensity computations over smaller window dimensions—for example, less then a hemisphere, such as those corresponding to the fields of view of imaging optical systems and intensity requirements of automotive lighting systems. Computations with the current form of the RADIANT command are also used to compute radiance. Most commonly, these computations compute the radiant intensity patterns of extended sources emitting into 4 steradians—that is, into the entire sphere. If RADIANT creates a distribution data file over the entire sphere, after loading the distribution file into DISPLAY, you can use the MESH command to create a three-dimensional pattern in the 3D vector file that you can view with the 3D Viewer. A vector from the center of a meshed intensity pattern to the meshed surface represents the intensity from the source in that specific direction.

In both intensity computations, ASAP is operating only on the ray vectors, which are the propagation directions of the rays. In doing so, reference to spatial coordinates is removed from the angular flux density calculation. Both




computations remove or, in a sense, integrate the radiance over space, thereby leaving only flux and angular terms. Previously, we mentioned that intensity measurements were made at very large distances from the source, effectively making the source appear as a point. This is what the SPOTS DIRECTION and RADIANT calculations do, by using only the ray vector information. This concept is exemplified by the fact that the ray propagation vectors may be plotted in their own direction cosine or angle coordinate systems. In both cases, the starting end of the ray vector is at the origin of the coordinate system. The other end is on the direction cosine sphere, or pointing outward from the origin of the spherical-polar coordinate system. Physically, it is as if we moved our detector so far from the source that we no longer can resolve the size of the source. That is, it now looks like a point in space, at the origin of the coordinate system.

RadianceRadiance was previously defined to be the flux per unit area per unit solid angle. The flux portion of the computation is determined by the ray flux. The WINDOW and PIXEL commands define the differential projected area, and the angular syntax of the RADIANT command defines the differential solid angle.

There are two radiance forms of the RADIANT command, both of which are available in the ASAP Builder and in the script language. The first form of the RADIANT command, the MAP option, computes the radiant intensity as a function of spatial coordinates. The second form, RADIANT...AREA, computes the actual radiance. Both forms are illustrated in and . The source used for these calculations is an EMITTING RECTANGLE in the X-Y plane with the rays constrained to a cone half-angle of 60 degrees in the X-direction and 30 in the Y-direction.


. . .



First form of the RADIANCE command with RADIANT...MAP

Second form of RADIANCE command with RADIANT...AREA (Chart Viewer)

RADIANT...MAP produces a 3D, polar plot diagram at the center of each spatial PIXEL location of the currently defined spatial WINDOW. It is not a true radiance in this sense and because we will remove any projections from the calculation.




Therefore, it is the WINDOW and PIXEL commands that define the differential area, not differential projected area. The RADIANT command itself defines the differential solid angle in exactly the same way as when RADIANT was used to compute the radiant intensity. The polar axis is chosen to be normal to the detector plane and window.

You may have to rotate your ray data so that it appropriately aligns spatially with the WINDOW axes and, subsequently, the global coordinate axes. This, of course, removes any projections. The zenith angle of the polar axis typically runs from 0 to 90 degrees, while the azimuth angle runs from 0 to 360 degrees. You set the number of subdivisions along the axes. The intensity is computed over a hemisphere, under these conditions. ASAP uses only the rays bounded by the pixels in that particular area to compute each intensity plot. Since a polar plot diagram is plotted at each PIXEL location, use a small number of pixels, usually less than 3x3 or 9 total, to minimize the sampling error. Although a distribution data file is created during this calculation, you do not have to load it into DISPLAY to see the results. In fact, ASAP automatically places the results of the calculation in the 3D vector file, which can be plotted with system geometry.

RADIANT...AREA calculates the true radiance. This is the flux per unit area per unit solid angle. The WINDOW and PIXEL commands again define the differential projected area. The RADIANT command itself defines the differential solid angle, in exactly the same way as was done when RADIANT was used to compute the radiant intensity. With this form of the RADIANT command, you can significantly increase the number of pixels over what is recommended with RADIANT...MAP. However, now you want to set the zenith angular range to be coordinated with your optical system or performance requirements. For example, set the zenith angle to correspond to the field angle of a radiometer used to measure the radiance of an actual system, so you can compare your simulation with actual measurements. You must set the number of subdivisions in the zenith and azimuth directions to be only 1. This setting essentially tells ASAP to compute the radiance at a given pixel location, by using only those rays that fall into a single cone defined by the zenith and azimuth angles of the RADIANT command.

RADIANT...AREA produces a distribution data file, again defaulted to BRO009.DAT, which you may load into DISPLAY. All the graphics and processing commands of this command are available for you to use, with the exception of the ANGLES and DIRECTION commands, which are exclusively reserved for intensity calculations.


. . .



We typically perform a radiance calculation with the WINDOW and polar axis normal to the detector plane. However, the radiance of a source or distribution in reality is a six-dimensional function. ASAP directly computes four of those dimensions, which form the projected area and solid angle of the radiance function. The other two dimensions are line-of-sight or viewing angles, typically referenced to a spherical-polar coordinate system. The line-of-sight angles are needed to quantify the non-isotropic radiance behavior of most sources. Non-isotropic sources do not have the same radiance when you observe them from different viewing angles. The only source that has an isotropic radiance is a Lambertian emitter, which is a non-physical, idealized source. Its radiance is isotropic, but its intensity and irradiance exhibit a cosine fall off with line-of-sight angle, or viewing angle. To fully quantify the radiance of a ray distribution, you must perform the RADIANT...AREA calculation at different viewing angles. Rotate your ray distributions and perform the RADIANT...AREA calculation at the different viewing angles, while using the same WINDOW, PIXEL, and RADIANT...AREA options, especially the polar axis specification, that you used for the normal incidence case.

Stat ist ical errorIncoherent, geometric simulations of extended ray sources, and the interaction with their associated optical systems, are Monte Carlo simulations (see the sidebar, “Monte Carlo Simulations” on page 54).

We will first estimate the error of our calculations, based entirely on ray statistics, to determine how many rays we will need for a simulation. We will then present a succinct equation that accurately predicts the error of the simulation when we include other cascading effects, such as Fresnel transmission and reflection losses.

As an estimate of error, based entirely on ray statistics, let us first consider the Bernoulli trials, a probability law that describes whether an event occurs or not. In the present situation, the event is whether or not a ray gets to the detector (similar to a big energy counting bin), and into what smaller energy counting bin it is distributed across the detector, which may represent either a differential area or solid angle. The signal-to-noise ratio for a probability density law is the ratio of the mean to the square root of the variance of the distribution.

If we invert this relationship, we have a noise-to-signal ratio, which we use as our error metric. Although the signal-to-noise ratio is the more common metric for noise analysis, it is convenient to use the reciprocal for the following discussion




relating the number of required rays to a given measurement uncertainty. For a Bernoulli trial, the noise to signal ratio is,

(EQ 38)

Equation 39

where

(EQ 40)

When the probability of a ray getting to a bin is small—for example, p<<1, the statistical error reduces to the mean and variances of a Poisson distribution. Low efficiencies in illumination systems are not desirable, but are usually the norm. The Poisson distribution is just the limiting case of the Bernoulli distribution in this situation. The Poisson noise-to-signal ratio is,

(EQ 41)

Equation 42

What does this relationship imply? Let’s consider the example of radiant flux. Assume that the number of initial rays was 10,000. If p=0.001, then only 10 rays get to the detector or energy counting bin. According to our error equation, this is approximately 32% error. If these 10 rays contained 1 watt of power, our simulation predicts that the power at the detector is 1 W 0.32 W, which is a significant error. Now, if the total number of initial rays is 1,000,000, and the same 0.1% (1000) of the rays get to the detector, our estimated error is 3.2%, which is a much less significant simulation error.


. . .



Note that we are talking about energy counting bins. Let us estimate the error for a flux density calculation. In the previous example, our bin was the entire detector. But now we have many differential bins across the detector. We can use our error equation to estimate the error per bin, which translates into the error on the differential flux density in each bin. Again, start with 1,000,000 total initial rays. This time, 30% of the rays get to the detector, or 300,000 rays. Let us assume that the detector is divided into 51x51 bins, as set by the PIXEL command, to compute the flux density. Let us further assume that the rays are distributed uniformly across the detector plane. Under these circumstances, we can expect approximately 115 rays per bin. The error of the flux density calculation on a bin-by-bin basis is then 9%.

When other cascading effects, such as source apodization, Fresnel reflection, and transmission losses are accounted for, the noise-to-signal ratio in a bin is,

(EQ 43)

Equation 44

where

max = maximum flux of a ray in the energy-counting bin

total = total flux of the rays in the energy-counting bin

We can obtain most of the information for these calculations on the detector as a whole, or as a single energy-counting bin from the STATS command (number of rays and ray flux), and also the EXTREMES command with the FLUX option. To do this on a pixel-by-pixel basis, you must isolate equivalent areas or solid angles with the SELECT command, and apply the STATS and EXTREMES commands on the selected ray set.

You must trace a large number of rays for illumination systems to achieve suitable radiometric accuracy. In fact, to perform a true Monte Carlo analysis, you should perform multiple ray traces and analyses with a different SEED number, and then average the results. This alleviates any artifacts of the random ray generation that might manifest themselves in the simulation results. Using the same SEED number and QUASI sequence yields the same sequence of random numbers and, hence,

Noise

Signal

max

total=

Φ

Φ




random rays. This approach allows for precision or repeatability in your simulations, but large numbers of rays are still required for accuracy.

CAUTION To keep your ray trace times down and minimize the disk space used for the ray trace and analyses, use XMEMORY MIN, SEED...QUASI, and ACCURACY LOW to set up the sources and ray trace them. It is not advisable to do a TRACE PLOT with this many rays. Use the SPOTS...ATTRIBUTE 0 option to suppress the geometric spot diagram for flux density and intensity calculations.

Photometry and color imetryTo convert from radiant to luminous quantities, we must take into account the response of the human eye. The conversion is normalization, which requires multiplying together the spectral power density of the source by the visibility curve, luminous efficacy, and a uniform spectral bandwidth, and integrating the result. However, when ASAP sets up the spectral apodization, it assigns powers, and not spectrally-distributed powers, to individual sources at discrete wavelengths. In other words, ASAP does not divide each source by the uniform spectral bandwidth to achieve the spectral power density. You could, after setting the entire source flux with the FLUX TOTAL command, divide each source’s power by the uniform spectral bandwidth to create the spectral power density. In photometry and colorimetry, uniform steps between 1nm and 10nm are the most common. However, this is not really necessary, if we think about the multiplication and integration of the normalization. In a discrete normalization, the spectral power density (a spectrally-distributed variable) is multiplied by the visibility curve, the luminous efficacy, uniform bandwidths at each wavelength, and integrated or summed.

We can adopt a similar convention in ASAP that is commonly used in photometry and colorimetry, where the uniform bandwidth is already combined with the spectral power density. This quantity is the total power in the bandwidth around a particular wavelength, and forms the spectral power distribution, not density. This, of course, is the way ASAP distributes power with the SPECTRUM command, assuming you use uniform bandwidths. If you do this, you do not need to express the bandwidth explicitly in the photopic normalization over wavelength.


. . .



Operationally,

(EQ 45)

Equation 46

In other words, we have converted the spectral power density into a spectral power distribution. You can perform photopic normalization by using the SELECT command to isolate sources by number or wavelength, and multiplying each source by the ASAP intrinsic visibility curve function—the EYE function.

TIP A similar technique may be used to compute tristimulus values and CIE chromaticity coordinates. However, do not photopically normalize the results first. Instead, compute the CIE tristimulus and chromaticity coordinates directly from radiometric data with proper attention to the units used. These calculations, by definition, photopically normalize the results. Data that are photopically normalized have lost the original spectral power distribution that is needed to compute the CIE tristimulus and chromaticity coordinates!

The techniques for computing luminous quantities, such as luminous flux, illuminance, luminous intensity, and luminance, are exactly the same as for computing similar radiant quantities. The key is to first convert the spectrally-distributed radiant power into luminous power or luminous flux, and then use the same procedures as for radiometric quantities.

Finally, when dealing with some illumination systems, we can “cheat” by not doing the normalization. This usually is the case when the optical system exhibits virtually no dispersion, and, therefore, does not appreciably change the spectral power density of the source. We are not even concerned with the spectral power density of the source. In this case, you can define a source of radiation as a single wavelength, and assume that the power units are luminous power units—that is, lumens and not watts.




CIE numerical and graphical color analysisASAP can track multiple wavelengths for light rays, and has a visualization mechanism for determining their positions in the CIE (International Commission on Illumination) color coordinate space. CIE color analysis tests for color content or uniformity of projected light on a surface. The output is displayed in the ASAP

Display Viewer, a pane that includes a CIE chromaticity plots and a text pane with tabular data. This tool is ideally suited for projection and display designers.

The procedure starts with the usual definitions of geometry and polychromatic sources in either the ASAP Builder on in an ASAP INR file. Then TRACE the rays to some selected target. Next we choose whether to do the analysis from a Command Input window or an ASAP dialog box.

If we use the command input approach, the object with the rays to be analyzed is chosen with the CONSIDER command. Set the PIXELS and WINDOW values next. The analysis is started with the $GUI CALCULATECIE command. This command provides several options for setting up the analysis. The first option is for calling, or not calling, a dialog box to set the remaining options. The remaining options include setting a Standard Observer Model to either 2 or 10 degrees; assigning a Color Appearance Model with either the 1931, 1964, or 1976 CIE models; assigning a display color system; and, finally, setting the Observer White

Point. This last option selects spectral data sets that describe the spectral components of the reference white light source. It is used in conjunction with tristimulus values such as XYZ (see “Colorimetry: CIE color coordinates” on page 19) to define a color. The following example shows the $GUI CALCULATECIE command with all of the options selected:

$GUI CALCULATECIE -ND -SOM 2 -CAM 1976 -CS 0 -WP E

NOTE: See the online Help topic, $GUI as it applies to CALCULATECIE.

If you began the analysis by selecting Calculate CIE from the Analysis menu on the ASAP main menu bar, or if the CIE Color Analysis dialog box opens from the Command Input window, the above options and the CONSIDER, PIXELS, and WINDOW commands can all be specified in the dialog box as shown in .


. . .



CIE Color Analysis dialog box (on the Analysis menu)

The results of the CIE colorimetry analysis are shown in a CIE Results window, as illustrated in . This window shows chromaticity coordinates of each pixel in the Chromaticity Diagram pane.

The next pane, a Display Viewer pane, allows a number of colorimetric measures to be shown in physical space; that is, on the currently CONSIDERed object; the specific quantity to view may be selected via a drop-down menu. Finally, a text pane contains the table of results for all pixels including physical Cartesian coordinates, tri-stimulus coordinates (X, Y, Z), and CIELAB coordinates (L*, a*, b*). The latter forms the basis for DeltaE*ab, a color difference measure that is among the results available in the Display Viewer pane. An example of this is shown, with grayscale mapping, in .




NOTE: Use appropriate colorimetric values to examine the behavior of systems. For example, if uniformity is important, the Delta E*ab may be a more appropriate metric than a direct comparison of chromaticity. The chromaticity RGB Display view is not typically the best tool to judge the “whiteness” of illumination, and further analysis of the data provided in the text table may be required for some critical applications. These data may be exported to another file for additional analysis by right-clicking in the table for the option to copy it to the Clipboard. You cane pasted into a text editor or other application to save it.

CIE Results Window: Shows the data plotted in chromaticity space (left), displayed in physical space (right), and recorded in a text table (bottom)


. . .


References

Spatial distribution of Delta E values: displayed in physical space with grayscale image mapping, as measured on the current object

ExamplesSeveral example files are included with your ASAP installation, which demonstrate radiometric and photometric calculations. These files are in the directory, <asap install folder>\Projects\Examples. All example files in this directory are also accessible from the Quick Start toolbar in ASAP on the Examples page.

An ASAP technique for converting from radiant flux to luminous flux is available in the file PHOTOMETRY.INR. See “Appendix: PHOTOMETRY.INR File” for the complete text.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R E F E R E N C E S

• IES Lighting Design and Analysis Handbook.



References

• Wolfe, W. L., 1998. Introduction to Radiometry. SPIE Optical Engineering Press. Washington.

• Bass, M., Editor, 1995. Handbook of Optics. Vol. II. McGraw-Hill. New York.

• Frieden, B. R., 1991. Probability, Statistical Optics, and Data Testing. Springer-Verlag. Berlin.

• Lighting Fundamentals: Lighting Upgrade Manual; US EPA Office of Air and Radiation 6202J; EPA 430-B-95-003; January 1995.

M O N T E C A R L O S I M U L A T I O N S

These simulations are used in cases where, due to the complexity of the system, we cannot obtain an analytic solution. The Monte Carlo approach is based upon the law of large numbers. This law says that the probability of an event occurring may be determined by observing the number of times, n, the event occurs, out of a large number of trials, N. The probability of an event occurring is really the probability of whether or not the ray gets to the detector plane. The number of times the event occurs is the number of rays that get to the detector plane. The large number of

trials is the total number of rays you start with, in the simulation. Probabilities, or probability density, or distribution functions, in our situation, are really flux density, radiant intensity, and radiance distributions. For example, the flux density is a probability density function on ray position over area. After this brief discussion of a Monte Carlo simulation, you should get the idea that you will have to start with a large number of rays, or initial trials, in order to accurately predict the performance of your optical system. The big question is just how many?


. . .

. .A P P E N D I X : P H O T O M E T RY. I N R F I L E

. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .APPENDIX: PHOTOMETRY.INR FILE A

This appendix includes the complete file, PHOTOMETRY.INR. The spectral power distribution is used, and not the spectral power density. A blackbody emitting at 6500K is used as the source.

The file also contains a method for computing CIE color coordinates, with the CIE 1964 Standard Observer observing a field of view greater than four degrees.

A blackbody emitter at a temperature of 6500K was used as the source. In this case, the computed CIE chromaticity coordinates should closely match those of a D65 standard illuminant. The D65 standard illuminant has a correlated color temperature of 6500K.

The correlated color temperature of a source is the absolute temperature of the blackbody, the color of which most nearly matches the color of the source.

The color temperature of a source is the absolute temperature of a blackbody, which produces a color equal to the color of the source.

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