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Experimental and Simulation Study of Optimal Illumination Systems 46
CHAPTER – 3
Modeling of Sources
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Experimental and Simulation Study of Optimal Illumination Systems 47
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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Experimental and Simulation Study of Optimal Illumination Systems 48
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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Experimental and Simulation Study of Optimal Illumination Systems 49
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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Experimental and Simulation Study of Optimal Illumination Systems 50
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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Experimental and Simulation Study of Optimal Illumination Systems 51
∫ ==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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Experimental and Simulation Study of Optimal Illumination Systems 52
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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Experimental and Simulation Study of Optimal Illumination Systems 53
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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Experimental and Simulation Study of Optimal Illumination Systems 54
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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Experimental and Simulation Study of Optimal Illumination Systems 55
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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Experimental and Simulation Study of Optimal Illumination Systems 56
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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Experimental and Simulation Study of Optimal Illumination Systems 57
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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Experimental and Simulation Study of Optimal Illumination Systems 58
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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Experimental and Simulation Study of Optimal Illumination Systems 59
( 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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Experimental and Simulation Study of Optimal Illumination Systems 60
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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Experimental and Simulation Study of Optimal Illumination Systems 61
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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Experimental and Simulation Study of Optimal Illumination Systems 62
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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Experimental and Simulation Study of Optimal Illumination Systems 63
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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Experimental and Simulation Study of Optimal Illumination Systems 64
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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Experimental and Simulation Study of Optimal Illumination Systems 65
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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Experimental and Simulation Study of Optimal Illumination Systems 66
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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Experimental and Simulation Study of Optimal Illumination Systems 67
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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Experimental and Simulation Study of Optimal Illumination Systems 68
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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Experimental and Simulation Study of Optimal Illumination Systems 69
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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Experimental and Simulation Study of Optimal Illumination Systems 70
∫=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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Experimental and Simulation Study of Optimal Illumination Systems 71
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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Experimental and Simulation Study of Optimal Illumination Systems 72
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
_____________________________________________________________________
Experimental and Simulation Study of Optimal Illumination Systems 73
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
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Experimental and Simulation Study of Optimal Illumination Systems 74
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.
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Experimental and Simulation Study of Optimal Illumination Systems 75
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).
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Experimental and Simulation Study of Optimal Illumination Systems 76
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.
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Experimental and Simulation Study of Optimal Illumination Systems 77
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.
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Experimental and Simulation Study of Optimal Illumination Systems 78
( ) 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)
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Experimental and Simulation Study of Optimal Illumination Systems 79
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)
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Experimental and Simulation Study of Optimal Illumination Systems 80
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.
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Experimental and Simulation Study of Optimal Illumination Systems 81
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.
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Experimental and Simulation Study of Optimal Illumination Systems 82
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
_____________________________________________________________________
Experimental and Simulation Study of Optimal Illumination Systems 83
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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Experimental and Simulation Study of Optimal Illumination Systems 84
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