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Experimental and Simulation Study of Optimal Illumination Systems 46

CHAPTER – 3

Modeling of Sources

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3.1 Introduction

Review on the requirements and specifications of typical illumination systems

have been presented in the previous chapter. Performance of an illumination

system depends on number of parameters such as source configuration and

optical properties of sources. Proper selection of these parameters leads to

an efficient design of an illumination system. Designing of these parameters is

the job of illumination system designer. He analyses the illumination system

to verify whether his design satisfies the need of the application and is giving

optimistic solution. The analysis can be carried out by three ways viz. using

analytical formulae or by discretized numerical solution or commercially

available simulation tool. Out of three, simulation tool gives faster results and

helps in finalizing optimized parameters of the system. In most of the

illumination applications simulation tool is used to assist the optical system

design.

Development of simulation tool needs precise and close-to-reality light source

models to perform realistic simulation of illumination systems. Modeling of

LED sources is different from other conventional sources [1-6]. Since each

type of LED has a specific radiation and spectral pattern it should be precisely

modeled else simulation results cannot be trusted [7].

Thus for development of OPTSIMLED tool, the need to thoroughly define and

describe light-source characteristics, especially the multiple LED source

geometry and the spatial distribution of luminous intensity is a must. These

two characteristics help in computing illuminance distribution over the target

surface. To generate the color illumination pattern spectral characteristic

needs to be considered. A wide range of illumination applications demands

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various color patterns. The homogenous color mixing of multiple LED source

depends upon their relative positions as well. This aspect is also studied.

Further colorimetry model is implemented to visualize the colored illumination

pattern.

The present chapter describes various optical models of light sources with

emphasis on LED as light source, reported in research papers. Suitable

models are selected for OPTSIMLED development and integrated together to

characterize the multielement illumination system. Based on these the

required analytical equations are derived which are used in computational tool

reported in the fourth chapter. The chapter also explains color rendering

required to generate an illumination pattern. Chapter begins with the

explanation of various illumination terms and optics laws used in further

development.

3.2 Illumination Terms and Basic Optics laws

Illumination applications use two measurement systems: Radiometric and

Photometric. Radiometry is concerned with the total energy content of the

optical radiation (visible, ultraviolet and infrared). On the other hand

photometry is concerned with humans’ visual response to the optical

radiation. Illumination system analysis is carried out in photometric terms.

The foundation of photometry was laid in 1729 by Pierre Bouguer who

discussed photometric principles in terms of the convenient light source of his

time: a wax candle [8]. Thus candela (cd) became the basic unit in

photometry, from which the other units are derived. It is the unit of luminous

intensity (I) of a light source in a specified direction. When a uniform isotopic

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point source of 1 cd is placed at the centre of a sphere of 1 m radius it emits

luminous flux (Ф) uniformly in all directions (figure 3.1). The unit of luminous

flux is the lumen (lm) and is the rate at which luminous energy is incident on 1

m2 surface at 1 m distance from uniform point source of 1 cd intensity.

Figure 3.1 Luminous Flux

As luminous flux travels outward from the source, it ultimately impinges on the

surface of the objects. The illuminance (E) on a surface is the density of

luminous flux incident on that surface. Thus

A

φE = ……. (3.1)

where A is the surface area. The unit of illuminance is the lux ( lx ) in the SI

system & the footcandle (fc) in the English system, a lux being a lumen per

square meter and a footcandle a lumen per square foot .

In practice, most light sources are not point sources nor do they have the

same intensity in all directions. One can observe the entire source at a

distance which is large compared with the maximum source dimension, so

that its luminous flux may be considered as radiation from a point. With either

case, concept of solid angle is required.

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Refer to figure 3.2. When A, the intercepted area on the sphere, equals π r 2,

where r is the radius of the sphere, the angle Ω is referred to as a solid angle

expressed in steradian (sr). We say that the area A subtends a solid angle of

1 sr at the centre of the sphere when A = 1m2 and r = 1m. Since the surface

area of the sphere is 4 π r2, there is 4π sr surrounding a point in space.

Figure 3.2 Illustrating Steradian

A light source of one candela that uniformly radiates in all directions has a

total luminous flux of 1 cd·4π sr = 4π ≈ 12.57 lumens (1 lm = 1 cd·sr ). The

intensity of a point source in a given direction is the ratio of the differential

luminous flux to the differential solid angle. Thus

I = dф / d Ω. ……. (3.2)

Lambert’s cosine law

The flux impinges on the receiver surface at an angle θ. The solid angle

subtended by the differential source element dA, at the receiver point is

d Ω = dA cos θ / r2 ……. (3.3)

where r is the distance between source and receiver. If source to receiver

distance is large as compared with source area, cos θ & r2 will remain

essentially constant as source area is traversed.

Thus,

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∫ ==Ωs

22 r

cos θAθ cos dA

r ….…. (3.4)

If dA was rotated so as to be perpendicular to the direction of luminance,

more flux would be intercepted & illuminance, E, would increase. In is normal

to the incident flux. In general, we can write

E = En cos θ ……. (3.5)

where En is the illuminance when the receiver surface is normal to the

incident flux. This result is known as Lambert’s cosine law. It states that the

irradiance or illuminance falling on any surface varies as the cosine of the

incident angle, θ.

The Inverse Square Law

As shown in fig. 3.3 consider the general situation of a finite, non uniform

source, radiating luminous flux to a point, P on a receiver surface. The

illuminance at point P is

E = I cos θ / r 2 ……. (3.6)

This is the form of famous inverse square law for a point source.

Figure 3.3.Inverse Square Law

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The intensity of light observed from a source of constant intrinsic luminosity

falls off as the square of the distance from the object.

The inverse square law is applicable to point sources only. The law is

applicable to extended sources in far field case where the distance, r is large

as compared with the maximum dimension of the source. A general rule of

thumb to use point source approximation is the “five times rule”; the distance

to a light source should be greater than five times the largest dimension of the

source [8-11]. Thus in far-field photometry, a light source is regarded as a

point. In near field a source is modeled as an extended area, and it is usually

assumed that distance, r to the illuminated target is shorter than 5 times the

maximum source dimension [12-14].

3.3 Overview of Modeling of Light sources

All types of light sources are described as a combination of three parameters:

the geometry of the light source, its luminous intensity distribution and its

emitted spectral distribution [15]. There are other light-source properties, such

as reflectance and transmittance, but their effects are so overwhelmed by the

emissivity of a light source that they have minimal influence on the resultant

distribution of the light energy and can therefore be ignored. Therefore

modeling of light source includes three attributes viz. source geometry, spatial

and spectral modeling. Brief review of the contributions of various researchers

in this area is produced in following subsections.

3.3.1 Geometry

Primary light sources (lamps) have well defined geometries which greatly

affect the distribution of the light emitted from the source. Varieties of light

sources ranging from simple incandescent lamp to recent CFL are available

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in various sizes and shapes. According to Rykowski [16] LEDs, xenon arc

lamp, CFL, strobe lamps do not have any specific geometry.

The practical sources are formed by using one or more primary light sources

with reflector called luminaire. These luminaires too will have different

physical geometries. A single LED is normally not sufficient to fulfill the

requirement of illuminance level of application. Multiple LEDs are required to

be arranged in a cluster. The cluster can be of any shape. Standard LED

luminaire geometries are linear, triangular, circular and square [17]. It may

take any other form too.

According to Verbeck et. al. [15] these physical geometries can be easily

defined and depicted through any standard three-dimensional modeling

techniques. However, the emissive geometry of a light source is different than

the physical geometry. There are three types of emissive geometries which

are modeled as zero, one or two dimensional objects respectively for point,

linear or area light sources.

3.3.2 Radiation pattern or Luminous Intensity Distribution

A radiation pattern describes the relative intensity strength in any direction

from the light source. A point light source which radiates uniformly has a

luminance given by:

L = I cos θ / r 2 ……. (3.7)

where I is the intensity of the source, θ is the incident angle and r is the

distance between the source and the point where the luminance is computed.

Radiation pattern of uniform point source is circular in shape.

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Practical light sources have various forms of radiation patterns based on

different emission mechanism, materials used for fabrication of sources and

manufacturing techniques. Several approaches are used to model luminous

intensity distribution of these sources. The models currently employed are

based on either analytical approximations or on Monte Carlo ray tracing [1,18-

20].

Original radiation pattern of light source can be altered using secondary

optics. To study and design a source with secondary optics a ray tracing

model is useful. Designing of secondary optics using analytical equation is

described by P. Benitez using non-imaging optics methods [21]. An analytic

equation of the radiation pattern gives designers more flexibility in analyzing.

Light source can be modeled either in far field or in near field. In far-field

photometry, the angular distribution of a light source has been first

represented by an orthogonal curve, often called goniometric diagram (prefix

“gonio” means directional). It is simply a two-dimensional angular

representation of the directional information of luminous intensity. Figure 3.4

shows a typical goniometric diagram.

Figure 3.4 Goniometric Diagram

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On the other side, with near-field photometry a goniometric diagram is not

sufficient [22]. The volume of the light source and more generally its entire

geometry acts upon the resulted illuminance. Usually, near field distribution is

modeled analytically with spot light sources which are zero dimensional light

sources where the energy is reduced according to a cone [23]. For near-field

photometry, a first attempt was made by Houle and Fiume [24] for planar light

sources. After sampling the surface, a 2D goniometric diagram is linked to

each sample point. The resulting contribution is computed by interpolating

values between points. Therefore, it is not easy to establish a correlation

between the location and the variation of the luminous intensity distribution. In

[25], Shirley et. al. proposed a method with stochastic sampling. In 1999,

Zaninetti et. al. introduced a model based on an adaptive subdivision of a

planar rectangular surface [26]. This model works for planar diffuse light

sources, and is easily extended to non-uniform surfaces, as all sub-sources

are independent. In Brotman and Badler’s paper, light sources are modeled

with polygons to get polyhedral [27]. Then radiance is computed by a random

sampling of polygons.

Subsequent paragraphs describe radiation pattern modeling of LEDs. LEDs

are small extended directional sources with extra optics added to the chip,

resulting in a complex intensity distribution difficult to model. Each LED has its

own specific intensity pattern owing to the differences in chip structure,

package and other factors. Therefore lot of variety is observed in radiation

patterns of LEDs and frequently new patterns get added to the list.

LEDs can be modeled in far-field or in near field zone depending upon the

distance between the target surface and the source. C.C.Sun et. al. have

proposed the concept of mid-field zone [4]. In case of high power LED with a

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1 mm2 chip, near field zone occurs till 5 mm and mid-field zone is found upto

20 mm. Then far field zone begins where the normalized radiation pattern

does not change with the LED-target distance. The radiation pattern in far

field is characterized by the angular intensity distribution [28].

Datasheets of LED manufacturers provide radiation patterns in terms of

relative intensity versus angle. The patterns may be symmetrical or

nonsymmetrical. In case of symmetric radiation patterns, manufacturers

mention view angles. Symmetrical patterns are classified as Lambertian,

Batwing and Side Emitter according to radiation characteristics. The patterns

are shown in figure 3.5.

Figure 3.5 Standard Radiation Patterns

(a) Lambertian pattern

(b) Batwing pattern (c) Side emitting pattern

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Lambertian pattern shows bell shaped curves with maximum emitted power in

the direction perpendicular to the emitting surface. Lambertian patterns are

characterized by angle θ1/2, view angle at which luminous intensity is half of its

maximum value. Datasheets review shows that LEDs available in market

have wide range of view angles varying from 10⁰ to 160⁰.

Batwing LED radiation pattern has lobes on either side of centre with

increasing light intensity with angle, up to a limit, where after it falls off

sharply. Side emitting pattern shows two distinct peaks of intensities.

Some LEDs have radiation patterns which do not resemble with any of the

above standard shapes. Due to varieties of radiation patterns useful modeling

algorithm, which is suitable for most LEDs is necessary. Most widely used

models for radiation are analytical and ray tracing. Realistic radiometric model

for the emitted radiation distribution of LED was not available upto 2006. The

optical modeling of LEDs was carried out by means of Monte Carlo ray

tracing methods [29-32]. In this method during simulation of radiation pattern

LED is considered as a light source that has multiple emitting faces instead of

Lambertian active layer inside the chip. Modeling is done by recording the

distance of the ray vectors of the six exit faces of the LED die and then using

commercial software. These techniques randomly simulate 1 to 10 million

light rays and the output ray-density distribution serves as an indirect value

for the radiated pattern. In addition to the time consumed by these

techniques, the lack of an analytic expression for the output intensity and

irradiance reduces the optimization process to a trial and error procedure.

First realistic model providing an analytical relationship between the radiated

pattern and the main parameters of LEDs, such as: chip shape, chip

radiance, encapsulant geometry, encapsulant refractive index, chip location,

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and cup reflector was put forward by Moreno using radiometric approach [33].

LEDs have integrated optics and therefore analytical models are preferred.

Ivan Moreno has introduced analytical expressions for irradiance & intensity

considering radiation patterns of LEDs for far-field [28]. According to him,

mathematically the pattern is sum of Gaussian distributions or cosine-power

functions. Equation for sum of Gaussian distribution is

( ) ∑

−−=

i

2

i

i

iG3

G2θ2lnexpG1θ I ……... (3.8)

The equation for sum of cosine power functions is

( ) ( ) i C3

i

i 2Cθcos1CθI i−=∑ …..…. (3.9)

where ‘i’ represents number of terms dependent on LED type

G1i, G2i, G3i and C1i, C2i, C3i are constants.

Function requires nonlinear regression to fit the model to realistic data.

Luminous Intensity Distribution Model used in OPTSIMLED:

To begin with, equation (3.8) was used to model the radiation pattern of

LEDs. Figure 3.6 (a) shows the results of warm white LED of Edison

Company with view angle of 135⁰. Luminous intensities were computed for

polar angle varying from -180⁰ to +180⁰. The normalized intensity curves are

plotted against polar angle. As suggested by researchers, numerical

adjustments have been made in coefficients of Gaussian equation to obtain

radiation pattern as given in the datasheet [34-39]. Similarly, figure 3.6 (b)

shows modeled and datasheet batwing patterns for Luxeon 1-W star LED.

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( a) Lambertian pattern using 5 terms in equation 3.8 show good match

for LED of Edison company with θ1/2 = 135°

(b) Batwing pattern computed using two Gaussian terms in equation 3.8 show

a good match with the data sheet for an LED of Luxeon company

Figure 3.6 Spatial Distribution Patterns

It may be noted that modeled result matches with the data sheet results. The

spatial distribution of various types of commercially available LEDs of Philips,

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Edison and Cree Companies were also tried and found the good matching of

modeled curves with the realistic data curves. Thus the sum of Gaussians

equation is seen to be suited for describing the radiation pattern. Therefore

equation 3.8 is chosen for the modeling of spatial distribution of LED sources

in OPTSIMLED.

3.3.3 Spectral Power Distribution

The spectral power distribution (SPD) serves as the starting point for

quantitative analysis of color [40]. In color science SPD describes the power

per unit area per unit wavelength of an illumination or more generally, the per-

wavelength contribution to any radiometric quantity. The distribution curves

are different for various light sources and explain how various lamps differ in

the color composition of their light output.

An SPD diagram of sunlight at midday shows that it is an exceptionally

balanced light source - all wavelengths of visible light are present in nearly

equal quantities. (figure 3.7). Logically, it indicates outstanding color

rendering ability. When compared to artificial light sources, sunlight exhibits

large amounts of energy in the blue and green portions of the spectrum,

making it a cool light source with a high color temperature (5500K).

Incandescent and halogen lamps exhibit smooth SPD curves. Incandescent

lamps have very high Color Rendering Index rating. This does not mean,

however, that they render all colors in an identical manner. Standard

incandescent lamps produce very little radiant energy in the short wavelength

end of the spectrum and therefore do not render blues very well. Discharge

lamps produce narrow bands of energy at specific wavelengths. Fluorescent

lamps produce a combined spectrum – a continuous or broad spectra from

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their phosphor, plus the line spectra of the mercury discharge. SPD of some

of light sources are shown in figure 3.7.

Figure 3.7 SPD curves of daylight and conventional sources

Spectral power distribution curves for LEDs are different from conventional

sources. With advancements in fabrication technologies of LED sources,

various types of color and white LEDs are available in the market. Color LEDs

have specialized domain of application as discussed in previous chapters.

People are even trying to generate white light from several narrow-spectral-

band sources [41,42].

Spectrum of white LEDs are nonsymmetrical and show uneven spread of

intensity distribution over a wavelength range 380 to 780 nm. The curve

shape depends upon fabrication technique and has different CCTs. Colored

LEDs show single peak in intensity distribution. This peak value is provided

by manufacturer either in radiometry or in photometry units. If datasheets

provide dominant wavelength derived from CIE chromaticity diagram then it is

in photometry unit. It represents single wavelength of monochromatic light

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that defines color of the device as perceived by the eye. Some of the SPDs of

the LEDs are shown in figure 3.8.

Figure 3.8 Spectral Power Distribution of LEDs

Frank Reifegerste et al. [43] have given various approximation functions to

model SPD of LEDs as given in the table 3.1. Out of these, sum of Gaussian

is found to describe the color as well as white LEDs most suitably. Color

sources require single Gaussian function while white sources require addition

of two or more terms.

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Table 3.1 Mathematical functions to model SPD of LED sources

Gaussian

Split Gaussian

With W=W1 for λ<C, W=W2 otherwise

Sum of Gaussians

Second order

Lorentzian

Logistic Power Peak

Asymmetric logistic

peak A

Pearson VII

Split Pearson VII

With w=w1 s=s1 for λ< C and W=W2 s=s2 otherwise

Piecewise 3rd

order

Polynomial (spline) x

Piecewise definition for n ranges xk-1≤x<xk k=1…n

In OPTSIMLED Gaussian distribution functions are used after verifying

modeled results. The results are given later on.

Sometimes datasheets provide peak wavelength which is the single

wavelength where the radiometric emission spectrum of light source reaches

its maximum. Simply stated, it does not represent the perceived emission of

light source by the human eye. Here SPD is represented in radiometric units

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and needs to be converted into photometric form. One has to consider eye

response for this conversion.

The human visual system is complex and has highly nonlinear response for

wavelengths ranging from 380 to 780 nanometers (nm). Its sensitivity to light

varies with wavelength. Therefore a light source with a radiance of one watt /

m2

- steradian of green light, for example, appears much brighter than the

same source with a radiance of one watt / m2

- steradian of red or blue light. In

photometry, we do not measure watts of radiant energy. Rather, we attempt

to measure the subjective impression produced by stimulating the human

eye-brain visual system with radiant energy. In 1924, the Commission

Internationale d’Eclairage (International Commission on Illumination, or CIE)

published CIE photometric curve shown in figure 3.9. It shows the photopic

luminous efficiency of the human visual system as a function of wavelength.

Figure 3.9 CIE Photometric Curve

Curve provides a weighting function that can be used to convert radiometric

into photometric measurements [44]. The relation between radiometric to

photometric for monochromatic light of 555 nm is 1 watt = 683 lumen. For

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light at other wavelengths, the conversion between watt and lumen is given

by

( )λV*k *unit cradiometri unit cphotometri m= ……. (3.10)

where Km = 683 lm / W ; maximum spectral efficacy for photopic vision

and V (λ) = luminous efficacy

For polychromatic sources, conversion requires multiplying the spectral

distribution curve by the photopic response curve, integrating the product

curve and multiplying the result by a conversion factor of 683. This is

illustrated in the following formula:

( ) ( ) dλλφλVK λ

780

380

mvφ ∫=

……. (3.11)

where Фv = photopic luminous flux (lm)

Фλ(λ) = spectral radiant flux (W)

Photopic luminous efficiency values V (λ) for visible wavelength range of 380

nm to 770 nm are provided in appendix 2.

SPD Model used in OPTSIMLED:

OPTSIMLED accepts monochromatic as well as polychromatic LEDs. They

are modeled using Gaussain distribution functions. The equations are

validated before adaptation in tool. Using Gaussian equation the SPDs for

LEDs with different peak wavelengths and FWHM are plotted and are

compared with the datasheet SPDs. Sample result is produced in figure 3.10.

In (a,b) SPD of yellow LED having dominent wavelength as 595 nm and

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FWHM of 15.95 nm is shown. Figure 3.10 (a) shows characteristics obtained

using Gaussian equation and graph (b) is SPD available in datasheet of

superbright LED. SPDs obtained for warm white Edixeon LED is shown in

figure (c,d). Figure (c) shows plot using Gaussian equations while (d) is SPD

curve measured using Ocean Optics spectrometer. (Measurement details in

chapter 5).

Figure 3.10 Spectral Distribution Characteristics

The two characteristics match with each other and hence Gaussian

distribution functions are used for SPD modeling in OPTSIMLED. If

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radiometric data is available the equations 3.10 and 3.11 are used to convert

radiometry data into photometry before SPD computation.

3.3.4 Colour Rendering

Once the spectrum of light impinging upon a point on target plane is known,

the eye’s perception of that spectrum may be determined. The present

subsection describes how to determine the CIE X, Y and Z values/ color

cordinates which characterise a standard human observer’s perception of

color from spectral intensity distribution. Further calculations of R,G and B

values generate colored illumination pattern.

The pattern output of system must try to achieve as normal vision as possible.

This requires accurate color perception. To obtain this effect a detailed study

of the colour perception mechanism was done. To visualize color of

illumination pattern, it becomes necessary to consider a complicated

psychophysical response of human vision [45].

The human eye has photoreceptors (called cone cells) for medium- and high-

brightness color vision, with sensitivity peaks in short (S, 420–

440 nm), middle (M, 530–540 nm), and long (L, 560–580 nm) wavelengths.

There are also the low-brightness monochromatic "night-vision" receptors,

called rod cells, with peak sensitivity at 490-495 nm. Thus, in principle, three

parameters describe a color sensation. The tristimulus values of a color are

the amounts of three primary colors in a three-component additive color

model needed to match that test color. Using reference stimuli at specified

wavelengths, CIE has defined a standard set of tristimulus values to match

different wavelength of the spectrum. This necessity is fulfilled by ‘trichromatic

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theory’. Any specific method for associating tristimulus values with each color

is called a color space. CIE XYZ (also known as CIE 1931 color space), one

of many such spaces, is the first mathematically defined color space

commonly used as standard and serves as the basis from which many other

color spaces are defined. These data were measured for human observers

for a 2-degree field of view. In 1964, supplemental data for a 10-degree

field of view were published. The details are given below:

CIE 1964 Method

In 1964, CIE (Commission International deL’Eclairage) drew up the (X, Y, Z)

system of colour specifications that has been adopted internationally. The tri-

stimulus values X, Y and Z describe the amount of three matching stimuli in a

given trichromatic system required to match the stimulus concerned.

In 1964, experiments were performed to determine how a standard human

observer perceives color. The experiments were done by projecting lights

onto a screen and having an observer match the light using a combination of

red, green and blue lights. These experiments were done with the observer

having field of view ten degrees.

The curves generated from this data were mathematically manipulated so that

all the curves were positive and the λy was equal to the luminosity function

(the way humans perceive brightness). The resulting curves, λx , λy and λz

are referred to as the CIE 10 degree Standard Observer Functions. The

curves are shown in figure 3.11.

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Figure 3.11: Tristimulus Color Matching Functions CIE 1964

The CIE Tristimulus values (XYZ) are calculated from these CIE Standard

Observer Functions, taking in to account the type of illumination and

reflectance of the sample. At each wavelength λx , λy and λz are multiplied

by the spectral energy emitted by the light source. Then that value is

multiplied by reflectance of the sample at each wavelength. The values for the

entire wavelength of 380 nm to 780 nm are then summed. The XYZ values

are calculated based on the luminosity of a perfect reflecting diffuse which

has a reflectance of one at each wavelength. The sums are divided by the

sum of the spectral energy time’s λy at each wavelength because Y for the

perfect white must equal to one by definition. CIE publication 15.2 (1986)

contains information on the XYZ color scale and CIE Standard Observer

Functions.

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∫=780

380

_

X λλ dRIx , ∫=780

380

_

Y λλ dRIy , ∫=780

380

_

Z λλ dRIz

.. (3.12)

where 380nm and 780nm are accepted limiting wavelengths of human vision

of normal conditions;

λx , λy and λz are the tristimulus values (Table in appendix 3);

‘I’ is the spectral energy distribution curve for illuminant;

‘R’ is the spectral reflectance or transmittance factor for the coloured sample

under observation.

Performing these triple integration gives tristimulus to be the amounts of each

of the three primary responses that, when combined in specified amounts,

produce a total colour sensation. Spectral Tristimulus functions defining CIE

1964 supplementary standard colorimetric observer are given in the appendix

3. For most applications sampling wavelength bands 5 or 10 nm apart is

adequate; (CIE 1971) and (CIE 1986) provide colour matching tables with 1

nm resolution.

Color modeling is important in case of SSL as colored LED are available in

market. A source made up of multicolor LEDs allows user to generate

illumination of his choice. In addition to this color adjustability for some

applications LED arrays can be designed to produce different color patterns.

For most of the applications uniform color distribution is desired. Ivan has

reported color mixing methods of multicolor LED sources [46]. The output

color distribution from multicolor LED arrays shows distinctive color patterns

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which is a function of array configuration and panel-target distance. According

to Ash color distribution is an important issue in SSL design [47].

The concepts of spatial, spectral and color modeling are used for

development of various modules of OPTSIMLED tool. The integration of

modules so as to analyze the multielement LED illumination system is

described in next section.

3.4 Integration of Models into OPTSIMLED

Considering extensive use of LEDs in illumination system OPTSIMLED tool is

developed for analysis of LED based illumination system. The tool should be

capable of analyzing LED source system consisting of multiple sources

arranged in any shape and size. It should take into account various radiation

and spectral patterns of LEDs. Keeping this in mind various models described

above are selected and are integrated with proper modifications into the

OPTSIMLED. The output is available in numerical and pictorial form.

The details of the tool OPTSIMLED are described in the following paragraphs.

For multielement LED source system, the system performance depends

upon following factors:

• Geometry of source panel : Number, locations and orientation of LEDs

• Optical characteristics : Spatial and Spectral power distributions of

individual LED

• Target distance from source panel

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Analytical equations for LED system analysis need to be developed

considering above factors. It is expected that tool should be able to

characterize the system in terms of parameters such as illuminance

distribution, uniformity of illuminance and illumination pattern over target

surface. The equations to carry out these analyses are obtained using far field

photometric approach. Tool first computes illuminance at a point on the target

surface due to single LED in the array. The luminous intensity due to selected

source in the direction of target point is computed. Use of inverse square law

and Lambersian law enables us to find illuminance at point under

consideration. Using illuminance value and SPD of source colorimetry

coordinates are found out. The equations derived for this procedure are

repeated for all sources and effect is integrated. The whole target surface is

analyzed in a similar way. Ultimately average illuminance, uniformity, diversity

are calculated as per their definitions stated in chapter 2.

3.4.1 Mathematical Modeling of individual LED Source

LED source panel can have a cluster of LEDs; source type, number and their

relative placement is decided by specifications of illumination application.

These can be assembled in a variety of shapes depending on available

space, aesthetic design and cost. LEDs are available in different dimensions

but they are treated as point sources since target plane to source distance is

greater than 5 times source dimension in most of the applications.

The illumination system under consideration is shown in figure 3.10. It

consists of a flat LED source plane having ‘n’ number of LEDs arranged in a

two dimensional array. It is illuminating a flat target plane parallel to source

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plane at height ‘h’. The task is to find illuminance distribution on target plane

due to ‘n’ number of LEDs.

Let us consider a point source ‘ i S ’. Let it be located on the source plane at

location ( i x , i

y , h) with inclination angle ‘Φ’ with z - axis and angle ‘δ’ with x

– axis. ‘h’ is perpendicular distance of source plane from target plane, i.e. XY

– plane. Consider a point D(x, y, 0) on target plane. It sustains an angle γ

with the source, i S . (Refer figure 3.12). Illuminance at a point D due to source

i S depends on emitted intensity in γ direction.

Figure 3.12 Geometry of Illumination System

_____________________________________________________________________


Mathematically angle ‘γ’ can be computed as

( ) ( )

( ) ( ) ( ) ( )

+−+−×+−+−

−+−+−

=

−++

−

222222

222

1coshynxmhyyxx

iyynixxmi

yyi

xxhyx

iiii

ii

γ

…….(3.13)

where ( ) ( )δφ costanhm = and ( ) ( )δφ sintanhn =

3.4.2 Computation of Luminous Intensity

Illuminance distribution on target surface is mainly dependent on spatial

distribution of LEDs. In the present work, LED is modeled as point source in

far field condition. For this analytical approach is used. Luminous intensity at

a point on the target plane due to single source is computed by two different

methods as explained below:

A] Use of analytical equation

Lambertian LEDs having rotationally symmetric radiation patterns with single

intensity peak along the optical axis are simulated using Gaussian

distribution. One dimensional equation is selected as most of the LEDs

radiation patterns are rotationally symmetric in far field. According to Ivan

[28], when manufacturer provide view angle in their datasheet only one

Gaussian term is sufficient. Single intensity peak parameter of datasheet is

‘G3i’ in equation 8. This value is zero in most of the cases.

_____________________________________________________________________


Therefore the luminous intensity at point D, Ii (x, y, 0), (i.e. far field flux

distribution in lumen / steradian) due to ith LED is the divergence of optical

power at angular direction ‘γ ’. Simplified version of equation 8 gives

( )

( )

∑×

−−×

=m m

2

m

2

mpeak

mmax

iσ2π4π

2σ

θγExpI

y,0x,I ……. (3.14)

where 355.2

θ2

1m

m =σ , mmaxI = source intensity in lumen (lm) at mpeakθ . Here

‘m’ represents number of Gaussian terms.

If manufacturer’s data sheet provides intensity in candela, lumen conversion

is needed which is carried out as;

Lumen = candela × solid angle in steradian ....… (3.15)

Solid angle in steradian = 2 × Π × [1- cos (θ1/2 / 2)] …… (3.16)

B] Use of database:

The luminous intensity at point D, Ii (x, y, 0) is computed using intensity

values stored in database. Database has luminous intensity values against

polar angle stored in tabular form as shown in table 3.2. The intensity values

I18’ to I18 are measured experimentally for polar angle varying from -180 ⁰ to

180 ⁰and values are entered manually in the table. The values can also be

found from radiation pattern given in datasheet. The table is stored in

Microsoft Data Access. Intensity value Ii (x, y, 0) in ‘γ ’ direction is computed

using interpolation instead of equation (3.14).

_____________________________________________________________________


Table 3.2 Database table for luminous intensity for step angle of 10⁰

Polar Angle (⁰) Luminous Intensity in lumen

-180 I18’

-170 I17’

|

|

|

|

-10 I1’

0 I0

10 I1

|

|

|

|

170 I17

180 I18

This method is useful when radiation pattern of LED does not resemble with

Gaussian curve like in case of batwing, side emitter etc. or in case of

asymmetric radiation pattern. The method also helps in increasing accuracy

of modeling as experimentally measured spatial data is available for

calculation. One need not rely on accuracy of manufacturers data. Benavides

et al [47] measured radiation patterns of LEDs at 10⁰ intervals and concluded

that the manufacturer’s specifications are not adequate if detailed

characterization of an LED is required.

_____________________________________________________________________


3.4.3 Computation of Illuminance

Luminious intensity, Ii (x, y, 0) in ‘γ ’ direction helps in calculating Illuminance

(photometric flux spatial distribution) at a target point at distance ‘r’ from

source. It is given by inverse-square law of the distance between point source

and detector, Ei (γ ) = Ii (γ ) / r 2 and is measured in lm / m 2, also called as

lux. Thus

( )) ( )

2r

cos0,,(

0,, i

E

γ×Ι=

yxi

yx

……………… (3.17)

where ‘γ ’ is calculated from equation 3.13.

3.4.4 Computation of Illuminance due to array of LEDs

Equation (3.17) gives illuminance at a point D (x,y,0) on target plane due to

single LED source. If illumination system consists of ‘n’ number of sources

then the illuminance at a target point is summation of illuminance due to all

LEDs. Therefore one can write,

( ) ( )0,, i

E

n

1i0,,

totalE yxyx

=Σ=

………….. . (3.18)

The illuminance distribution of finite target plane of dimension (a X b) square

unit is computed by considering all points on target plane.

Total lumen on target surface is computed using 2-d trapezoidal rule.

_____________________________________________________________________


( ) dydxyx 0,, total

E

00

surface on targetlumen Total

ba ∫∫=

…….. (3.19)

Equations 3.13 to 3.19 compute illuminances at all target points and total

lumen on the surface. Based on these values analysis of multielement LED

source system can be carried out.

3.4.5 Analysis of Illumination System

One of the important parameters of illumination system which play major role

in deciding performance is uniformity of illumination. It can be specified by

uniformity and diversity ratio. These are calculated using the definitions given

in chapter 2.

a) Average illuminance on target plane is given by

plane target of Area

plane on targetlumen Total lux Average =

.... (3.20)

b) Uniformity of illumination is calculated as

eilluminanc Maximum

eilluminanc Average Ratio Uniformity =

..... (3.21)

c) Diversity of illumination is computed using equation

eilluminanc Maximum

eilluminanc Minimum RatioDiversity =

..... (3.22)

_____________________________________________________________________


3.4.6 Color Estimation of Multielement LED Source

Spectral power distribution of sources is responsible for generation of various

illuminated patterns on target plane. For color rendering proper mathematical

function is required to model LED spectra [49]. Colored monochromatic LEDs

show symmetric distribution around single dominant wavelength, λ peak with

specific full power spectral width (FWHM). These two quantities are specified

in manufacturers’ data sheets. Such types are simulated using single

Gaussian function as explained in section 3.3.3. Spectral illuminance

distribution Eλ (x, y, 0) in visible range, 380 nm to 780nm at target point D is

given by

( )

( )( )

∑×

−−×

=m

yx

m

2

m

2

mpeak

total

σ2π

2σExp0,,E

y,0x,E

λλ

λ …… (3.23)

where 355.2

FWHM2

1m

m =σ and ‘m’ is number of Gaussian terms.

( )0,,E total yx is computed using equation 3.18.

For non-symmetrical , polychromatic, especially white LEDs the SPD

database of LED is prepared using experimentally measured values or from

available graphical spectral distribution from manufacturers. Here absolute

intensity values in lumen against visible wavelength of 380 nm to 780 nm are

used for color rendering. Equation 3.24 is used instead of equation 23.

( ) ( ) ( )λtotalλ intensityDatabase0,,E0,,E ×= yxyx ………. (3.24)

_____________________________________________________________________


For colour rendering relative spectral contribution is converted to CIE 1964

tristimulus values X, Y and Z using equation 3.25 – 3.27 [50].

( ) ( ) λλ

λ

λλ xyx ereflectanc spectral materialtarget 0,,E X

nm 780

nm 380

××= ∑=

=

... (3.25)

( ) ( ) λλ

λ

λλ yyx ereflectanc spectral materialtarget 0,,E Y

nm 780

nm 380

××= ∑=

=

... (3.26)

( ) ( ) λλ

λ

λλ zyx ereflectanc spectral materialtarget 0,,E Z

nm 780

nm 380

××= ∑=

=

... (3.27)

Here λx , λ

y , and λz are CIE 1964 tristimulus values given in appendix 2.

The CIE Y value is a measure of the perceived luminosity of the light source :

how bright it appears to an observer. The X and Z components give the color

or chromaticity of the spectrum. Since the perceived color depends only upon

the relative magnitudes of X, Y, and Z we define its chromaticity coordinates

x, y and z as

ZYX

X

++=x ,

ZYX

Y

++=y and

ZYX

Z

++=z …… (3.28)

These coordinates when combined in specified amounts produce a total

colour sensation.

Above calculations are performed for individual LED source and repeated for

all others at targeted point.

_____________________________________________________________________


Some output devices accept CIE color specifications directly and transform

them into color gamut, however, displays and printers require device-specific

color specifications in systems such as RGB or CMYK, requiring us to convert

the CIE perceptual color into device parameters. The color pattern is

generated from chromaticity coordinates by converting them into three

primary color components R, G, and B using following matrix [,51,52]:

−

−

−−

=

1Ζ1Υ

1Χ

0570.12040.00557.0

0.0415 1.8758 9689.0

4986.05372.12406.3

B

G

R

……. (3. 29)

where 95047.0X1 ×= x , Y1 = y, 08883.1Z1 ×= z .

Normalised R, G, B values are plotted for coloured illumination pattern as

bitmap image.

The equations from 23 to 29 are used in implementation of OPTSIMLED

algorithm for color estimation.

3.5 Conclusion

The chapter introduces basics of photometry and radiometry system. It takes

review of the models required for illumination system analysis. For analysis

source characterization is needed. Any source is defined by three attributes:

geometry, radiation pattern and spectral distribution. The model

developments for these attributes are discussed for conventional light

sources. It was found that attributes differ for multielement LED illumination

system and needs special considerations.

_____________________________________________________________________


Geometry of multiple-LED-source system has various sizes and shapes. The

sources are arranged in various forms of arrays as per the need of

illumination application. The array shape may be linear, circular, square etc.

Array size is decided by number of LEDs constituting a source and dimension

of individual source.

Radiation pattern of LEDs decide illuminance level at target point. Its

modeling depends upon source – target distance and source dimension. For

illumination applications generally far-field models are required in which LED

can be treated as a point source. Various types of radiation patterns are

available in LED. They are modeled either as sum of Gaussian distribution or

as sum of cosine power functions. For OPTSIMLED Gaussian distribution

function is considered. The radiation patterns were plotted for various LEDs

having different radiation patterns using Gaussian equation. It is found that

the modeled patterns match with the patterns provided in the manufacturers

datasheet and hence Gaussian equation is adopted in OPTSIMLED. To

model nonsymmetrical radiation patterns database facility is provided

Database stores luminous intensity values against polar angle varying from

-180⁰ to +180⁰ in steps. The intensity values at any polar angle are computed

using interpolation.

Spectral power distribution curves of LEDs are different from conventional

sources. Color LEDs have narrow band symmetric spectral distributions. For

their modeling Gaussian distribution equation is useful with peak wavelength

depending upon color of LED. The equation is verified by comparing

simulated spectral power distribution using Gaussian equation with datasheet

curves. SPDs of white LEDs depend upon materials and method used for

manufacturing of LED. White LEDs are characterized with different CCT

having unique spectral curve. The SPD curves for these LEDs are given in

_____________________________________________________________________


datasheets. Their modeling is done by preparing database of intensity versus

wavelength ranging from 380 nm to 780 nm. In case of unavailability of

datasheet measurement of intensity distribution as a function of wavelength

becomes necessity. One should remember that for illumination system

analysis photometric data is needed. If datasheet provides radiometric data it

has to be converted into photometric units before further computations.

After discussion of geometrical and optical models, one has to consider color

models. Spectral domain is converted in color domain using various

techniques, out of which CIE 1964 is most common. One gets CIE data

corresponding spectral intensity contribution of target surface.

The chapter presents integration of discussed models in the development of

OPTSIMLED. The tool considers source consisting of many LED sources with

different spatial and spectral characteristics. The illuminance values are

computed first for an individual LED and results are extended further for

illuminance calculation due to all LEDs on the source panel. For computation

of illuminance 3-D geometrical considerations of source panel and radiation

pattern of individual LEDs are required. Equations for parameters of analysis

of illumination system are put forward. Spectral distribution conversion to CIE

color domain is explained. Required conversion from chromaticity coordinates

to RGB color space for visualization of color pattern on display is discussed.

The integration of various models discussed in section 3.4 have been

adopted in OPTSIMLED tool, software structure of which is described in detail

in the next chapter.

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