Chapter 1 Introduction and Objective
1.5 Organization of this thesis
This thesis is organized as follows: The principles of illumination systems are presented in Chapter 2. In Chapter 3, the design concepts and mathematical model are described. Practical design case and simulation results are illustrated in Chapter 4. In Chapter 5, the fabrication methods and results of the prototype are shown, and experimental results are discussed. Finally, conclusions and recommendations for the future work are presented in Chapter 6.
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Chapter 2
Principles of Illumination Systems
For the purpose of designing and analyzing illumination systems, several optical principles are described in this chapter. Rays emitted from light source pass through an optical system following the principles of geometrical optics. Ray-tracing results are mainly interpreted by Radiometry and Photometry quantities, and illumination uniformity on the target plane is defined to evaluate the performance of the illumination system.
2.1 Laws of refraction and reflection
Light is an electromagnetic wave phenomenon. As the wavelength of light is much smaller than surrounding objects it propagates through and around, the behavior of light can be approximated and described by ray optics (geometrical optics). In geometrical optics, light travels in the form of rays and obeys a set of geometrical rules including laws of refraction and reflection.
Law of refraction (Snell’s law)
When a light ray is incident on a boundary surface of two media where the refractive indices of the two media are ni and nt, the ray is split into a reflected ray and a refracted (transmitted) ray as shown in Fig. 2-1. The incident ray, the normal direction of the surface, and the refracted ray all lie in the same plane called incident plane. The propagating direction of the refracted ray follows the relationship,
11 where i is the angle between the incident ray and the normal direction of the surface
(the incidence angle) and t is the angle between the refracted ray and the surface normal (the refraction angle). This relationship is called Snell’s law and can be proved by Fermat’s Principle. [7]
Fig. 2-1 Reflection and Refraction on a boundary surface
Law of reflection
The reflected light lies in the incident plane and possesses an angle r with respect to the normal direction of the surface. The angle of reflection is equal to the angle of incidence,
i r,
(2.2) which is called the law of reflection.
ni
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Therefore, the propagating direction of a light ray in an optical system can be traced and calculated.
2.2 Radiometry and Photometry quantities
2.2.1 Radiometry
Radiometry is the science of measurement of electromagnetic radiation. Some fundamental quantities which characterize the energy content of radiation are summarized in Table 2-1. [8]
Table 2-1 Energy-based units
Quantity Symbol Definition SI Units
Radiant energy Qe -- Joule
From the above table, Qe is the energy of a collection of photons while the energy of a single photon is hν. Radiant exitance Me pertains to radiation leaving a surface;
Irradiance Ee pertains to radiation incident on a surface. Radiant intensity Ie is the radiant flux Φe emitted per unit of solid angle Ω by a point source in a given direction.
The Radiance Le indicates the radiant intensity per unit of projected area perpendicular to the light of sight as shown in Fig. 2-2.
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Fig. 2-2 Definition of projected area
2.2.2 Photometry
Compared to Radiometry that measures all radiant energy, Photometry applies only to the visible portion of the optical spectrum for the human eye. Since the human eye does not respond with equal sensitivity at all wavelengths of visible light, the radiant power at each wavelength is weighted by CIE luminous efficiency curve which models human brightness sensitivity. The standard model that represents response or sensation of brightness for the eye versus wavelength is reproduced in Fig. 2-3.
Fig. 2-3 Scotopic and Photopic spectral sensitivities
Photometric quantities are related to Radiometric quantities through the dA
θ cos dA dA
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luminous efficiency curve. One watt of radiant energy at the wavelength of maximum visual sensitivity (555 nm) is defined to be 683 lumens. Thus, the luminous flux emitted from a source with a radiant flux( ) is given by
( ) 683 lm/w ( ) ( )
v V d
(2.3)where
V( ) is the normalized Luminous efficiency depicted in Fig. 2-4. Some Photometric units parallel to stated Radiometric units are illustrated in Table 2-2.Fig. 2-4 1988 CIE Photopic Luminous Efficiency Function [9]
Table 2-2 Photon-based Units
Quantity Symbol Definition SI Units
Luminous energy Qv -- lumen‧s
Luminous flux Φv dQv/dt lumen (lm)
Luminous exitance Mv dΦv/dA lumen /m2
Illuminance Ev dΦv/dA lumen /m2 or lux
Luminous intensity Iv dΦv/dΩ lumen /sr or candela
Luminance Lv dIv/dA⊥ lumen/sr‧m2 or nits
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That is, the illuminance on the screen is proportional to cos3 where cos z
r . The
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2.3 Etendue theorem
Etendue or étendue is an optical invariant of an optical system. It specifies geometric capability of an optical system that transmits radiation, or its optical throughput. In a perfect optical system without any losses from reflections, absorptions, or scattering, the etendue is a constant. The radiance of an optical system is equal to the derivative of the radiant flux with respect to the étendue.
Generalized Etendue theory which was developed by Welford (1974) is introduced in the following. [10] It is assumed that any ray is traced through an optical system as illustrated in Fig. 2-6. Any two points P and P’ are located in the entry and exit spaces at Cartesian coordinates (x, y) and (x’, y’), respectively. Eikonal V is defined as the optical path length from P to P’ along the physically possible paths. V is multi-valued if there are more than one ray passing through P and P’ and then V is a function of x, y, x’, and y’. The direction cosines of the ray in the two spaces are described as (L, M, N) and (L’, M’, N’).
Fig. 2-6 Proof of Etendue theorem
According to the fundamental property of the eikonal, we have
17 where n and n’ are refractive indices of entry and exit spaces, respectively.
Extend to two-dimensional and differentiate Eq. (2.7) again, we obtain
' ' After a sequence of mathematical manipulation, these terms can be put into a matrix form, Eq. (2.10) can be denoted as matrices A, B and the column vectors M,
' .
BM AM (2.11) Multiplying through by the inverse of B,
' 1 . which transforms the differential four-volumedxdydpdq. Thus,
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( ', ', ', ')
' ' ' '
( , , , ) x y p q
dx dy dp dq dxdydpdq dxdydpdq x y p q
(2.15)
because of the unity value of the Jacobian.
Therefore, the optical invariant of any system is proved,
2 2
' ' ' ' ' .
n dx dy dL dM n dxdydLdM (2.16) The invariant in Eq. (2.16) shall be interpreted in another way. (x, y, p, q) is treated as coordinates in a four-dimensional space and U is any enclosed volume in the space. Then U is given by
, dU dxdydpdq
(2.17) which indicates the invariant property.2.4 Illumination uniformity evaluation index
The illumination uniformity can be evaluated by means of two methods. The first method is familiar to the evaluation rule for LCD backlight systems.[11] As shown in Fig.
2-7, thirteen positions on the target plane are chosen as measuring points to calculate the uniformity where
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Fig. 2-7 Positions of measuring points
The other evaluation method characterizes illumination uniformity by calculating the uniformity deviation δ. [12]
1 where E is the average illuminance of number n samplings on the target plane. The illumination uniformity is superior when the uniformity deviation δ is smaller.
The first evaluation method is applicable on the basis of that there are no apparent dark or bright regions on the target plane, so the uniformity evaluated from these chosen measuring points are meaningful. In this research, the second method is preferable since more measuring points can ensure the uniformity evaluation to be more accurate.
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Chapter 3
Front Lighting Modeling
The Luminaire is put in front and above the target display to provide downward illumination. (Fig. 3-1) Vertical light distribution on the target plane is not uniform with the upper plane receiving more light. To simplify optical analysis and reflector design, a two-dimensional analysis is investigated in the following.
Fig. 3-1 Sketch of Luminaire and illuminated target plane
3.1 Design model in terms of Etendue
Modeling of front light reflector with a free-form curvature for uniform illumination is described. The luminous flux from the light source is adjusted by variable curvatures of the reflector curve and then the flux is redistributed as an
Luminaire
θ
Target plane z
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expected light distribution on the target plane.
3.1.1 Target light intensity distribution
Referred to cos3 Law illustrated in section 2.2.3, the illuminance on the target plane is a function of the angle θ,
3
If the illuminance on the target plane is defined to be a constant E0 everywhere, that is, uniform illumination, then the luminous intensity distribution is derived as a function of the inverse of the cosine θ cube, where
2 illustrated in Fig. 3-2. It is assumed that the linear light source is placed at the origin and perpendicular to the drawing plane. Light rays emit uniformly from the source and then reflect on the surface of the reflector.
The radiuses of curvature for different points on the reflector curve are not the same, resulting in variant ray diverging control. Luminous flux (ΔΦ) which is contained in an angle dθ falls onto the reflector curve, then it reflects and projects on the target plane with a diverging angle dθ’. Using Etendue concept, because of flux conservation, luminance is inversely proportional to the projected area and diverging angle. Thus, the luminance can be adjusted by the diverging angle corresponding to the arbitrary curvature of the reflector.
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Fig. 3-2 Geometric configurations of the light source and the reflector with free-form curvature
The original luminous flux from the source is a conservation of the flux after being reflected from the reflector curve, so
' ', I ds I d
(3.3)where I’ is the target light intensity distribution derived from Eq. (3.2). I’ was approximated as emitting from a point source.
3.2 Linear model
Compared to the Etendue model, the linear model is easier for calculation but is subjected to some conditions. The light source is assumed to emit equal light intensity in all directions. An example of that is the CCFL (cold cathode fluorescent lamp), which can be regarded as a point source for two-dimensional analysis.
Geometric configurations of the light source and the shape of the reflector are illustrated in Fig. 3-3. The linear light source is placed at point S (0,-d) and
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perpendicular to the drawing plane. Light rays emit uniformly from the linear source and then reflect from the surface of the reflector. The incident ray angle θ is defined beginning on the y-axis. The illuminated target region Q is put at distance L from the origin.
Fig. 3-3 Geometric analysis for linear design model
Using vector form and the assumed point-source for the two-dimensional analysis, incident and reflected ray vectors can be described as SP( ,x yd) and
PQ( ( )X x, L y) , respectively. The tangent vector at a moving point P on the reflector curve can be derived from
T= SP PQ ,
SP PQ (3.4) so the differential equation for the reflector curve is written as
2 2 2 2
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After substitution and simplification, Eq. (3.5) yields
2
For uniform illumination on the target region, X(θ) is linear to θ, that is, ( ) 0
X c X (3.8) where c is a constant and X(0) X0.
As long as initial conditions, d, L, and the uniformly illuminated region Q is determined, the resulting reflector curve can be numerically calculated by the Runge-Kutta Method, which is a numerical analysis and will be introduced in the next section.
3.3 The Runge-Kutta method
The common fourth-order Runge–Kutta method [13][14] is a family of implicit and explicit iterative methods for approximation of solutions for ordinary differential equations.
The initial value problem is specified as
0 0
25 referred to as “RK4”. It is a fourth-order method, meaning that the error per step is on the order of h5, while the total accumulated error has order h4.
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Chapter 4
Practical Design Case
Based on the mathematical model explained in section 3.2, the shape of the reflector could be derived using determined conditions. First, the flow chart for design is illustrated. Then selection of parameters and solutions for a free-form reflector curve for the illuminating target is presented. Simulation results together with discussions are shown as follows.
4.1 Flowchart
Illuminated target: Q
Parameters: d, L
Solutions for reflector
Simulation
Uniformity evaluation
Completed δ>10%
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4.2 Illuminating target
A practical design case which is aimed at the uniform illumination region whose area is 50 cm x 40 cm2 is shown in Fig. 4-1.
Fig. 4-1 Sketch of the practical design case
4.3 Selection of parameters
As shown in Fig. 4-2, according to the linear model described in section 3.2, the shape of the reflector can be derived from the following three initial conditions. d is the distance between the source and the origin. L is the distance between the target plane and the origin. Q is the illuminated region.
z=L-d
50 cm 20 cm
40 cm
50 cm
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Fig. 4-2 Geometric analysis for linear design model
d is the distance between the source and the starting point (or the origin) of the reflector, designated as 5 cm. For the practical case, the illuminated region Q is 50 cm.
L is the distance from the origin to the target plane and can be determined by the following. According to numerical analysis for solving the reflector curve, the relationship between distance L and the width of the reflector (or extension of the reflector in the positive x direction) is depicted in Fig. 4-3. Since the illuminated region is 50 cm (in the positive x direction), the reflector width at distance L=55 cm is considered to be the preferable choice for appropriate reflector width.
S (0,-d)
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Fig. 4-3 Reflector widths at different distances L
4.4 Numerical solutions for a free-form reflector
When L=55 cm and d=5 cm, the free-form reflector curve could be solved by using numerical analysis, the Runge–Kutta method, which was introduced in section 3.3. The numerical solutions for Eq. (3.6), Eq. (3.7), and Eq. (3.8) were fitted through spline interpolation. Thus, the curve shown in Fig. 4-4 is the result of the free-form reflector shape, where the source is located at point S (0,-5).
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Fig. 4-4 Solutions for the free-form reflector shape
4.5 Simulation results and analysis
Optical simulation software LightTools® [15] was used to verify the mathematical model and evaluate optical performance of illumination systems. Cold Cathode Fluorescent Lamp (CCFL) was applied in the illumination system. With the assumption of the two-dimensional design and analysis model explained in section 3.2, light emitted from the smaller diameter (2 mm) of the CCFL tube was regarded as a point source relative to the reflector and the target plane. CCFL and the free-form mirror surface reflector were constructed by LightTools. Based on Monte Carlo ray tracing, reflected light was detected by the receiver on the target plane, and illuminance was calculated. Configurations of the CCFL source (located at the origin) and the free-form reflector as well as ray-tracing results are shown in Fig. 4-5.
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Fig. 4-5 (a) Configurations of the CCFL source (located at the origin) and the free-form reflector (b) ray-tracing results
The illuminance distribution chart of the free-form reflector illumination system is shown in Fig. 4-6. Compared to the illuminance distribution chart in Fig. 4-7, where CCFL was directly illuminating the plane, illumination uniformity was improved by the free-form reflector redistributing light from the CCFL source. The uniformity deviation was 8 %, where the uniformity deviation was defined in section 2.4. In addition, the average illuminance E of the free-form illumination system was enhanced to 111 Lux while the average illuminance of CCFL directly illuminating case was 63 Lux. Thus, the mathematical model was verified.
Source
Free-form reflector
(a) (b)
Target plane
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Fig. 4-6 Illuminance chart of the illumination system composed of the CCFL and the free-form reflector
Fig. 4-7 Illuminance chart of the case where CCFL is directly illuminating (Lux)
x
(Lux) x
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Moreover, the illuminance on the target plane was improved through increasing the use of light emitted from the source. A half circular lamp reflector shown in Fig. 4-8 (a) was included to reflect and reuse the light, and the circular reflector did not affect the output light distribution from the source. Configurations of the lamp refletor and the free-form reflector are shown in Fig. 4-8 (b).
Fig. 4-8 (a) Half circular lamp reflector (b) Configurations of the circular lamp reflector and the free-form reflector
Two-dimensional and cross-sectional illuminance distribution charts are shown
(a)
(b)
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in Fig. 4-9. The uniformity deviation was within 5% and the average illuminance was 167 (Lux), which was brighter and more uniform than the previous case without the lamp reflector.
Fig. 4-9 Two-dimensional and cross-sectional illuminance distribution charts of the illumination system composed of circular lamp reflector and the free-form reflector
Table 4-1 Summary of average illuminance and uniformity deviation Average
illuminance (Lux)
Uniformity deviation (%)
CCFL directly illuminating 63 34
Free-form reflector 111 8
Free-form reflector and lamp reflector 167 5
(Lux) x
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4.6 Faceted analysis
To implement the free-form surface reflector in manufacturing, the feasibility of combining faceted surface reflectors instead of continuous surface reflector was analyzed. The performance of faceted reflectors was modeled and analyzed through LightTools, which significantly reduced reliance on prototypes and design costs.
The shape of the free-form reflector shown in Fig. 4-4 was analyzed. This free-form continuous surface was divided into several planar faceted surfaces corresponding to the partition angle ∆α as defined in Fig. 4-10. From the two- dimensional analysis viewpoint, each point is connected by a straight line.
Fig. 4-10 Faceted surface reflectors
Illuminance distribution charts as well as illumination uniformity deviation of the faceted reflectors for different partition angles ∆α, ∆α is 10°, 5°, 2°, 1°, are shown in Fig. 4-11 and Table 4-2.
αααα
source
faceted reflector
∆α=5°
36 (Lux)
(a) α=10°
x
(Lux)
(b) α=5°
x
(Lux)
(c) α=2°
x
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Fig. 4-11 Illuminance distribution charts of faceted reflectors for different partition angles ∆α, ∆α is 10°, 5°, 2°, 1°, respectively
Table 4-2 Average illuminance and uniformity deviation of faceted surface reflectors for different partition angles ∆α
∆α (degree) Average illuminance (Lux) Uniformity deviation (%)
1 168 5
2 167 6
5 167 7
10 168 13
According to the simulation results, the uniformity deviation is more close to the continuous surface when the resolution of partition angle ∆α is higher. The uniformity deviation is around 5% for the 1 degree partition, which is almost the same as the uniformity deviation of the continuous surface.
Cross-sectional illuminance distribution between faceted surface reflectors (the 1 degree partition) and the continuous surface reflector are compared in Fig. 4-12.
According to Fig. 4-11 and Fig. 4-12, some dark and bright stripes appear across the illuminance distribution, due to the reasons that rays striking the edge of the faceted
(d) α=1°
(Lux) x
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reflector reflects to stray directions, and the faceted segments seriously deviate from continuous path when the angle θ is larger. Smaller partition angle ∆α might relieve dark and bright stripes problems.
Fig. 4-12 Comparison of cross-sectional illuminance distribution between faceted surface reflectors (∆α=1°) and the continuous surface reflector
4.7 Discussion
In terms of simulation results in section 4.5, in order to improve illuminance on the target plane, some free-form reflector solutions for the same illuminated target plane were analyzed.
According to the mathematical model in Fig. 4-2, the reflector curve starts from the origin and extends in the positive x direction. The reflector only utilizes the source-emitted light which ranges from 0 degree to around 80 degrees. To increase the
0
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use range of emitted light, another reflector extends in the negative x direction, which uses the light ranging from 0 degree to around -80 degree. Thus, the reflectors extend in the positive and negative x directions. Both negative and positive x direction reflectors redistribute light to the entire illuminated region, and the reflector curves are plotted in Fig. 4-13. The superposition of the illuminated light from the two reflectors is shown in Fig. 4-14. However, this illumination system suffers the serious issue of shadow of CCFL.
Fig. 4-13 Reflector curves extend in both the positive and negative x directions
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Fig. 4-14 Illuminance distribution charts of superposition of the illuminated light from the two reflectors which extend in both the positive and negative x directions
Furthermore, to solve the lamp shadow issue, by interchanging X(0°) and X(-80°), the negative x direction reflector reflects light and crosses light rays to redistribute on the illuminated region as shown in Fig. 4-15. The negative x direction reflector curve changes slightly (Fig. 4-16) and the simulation results are shown in Fig.
Furthermore, to solve the lamp shadow issue, by interchanging X(0°) and X(-80°), the negative x direction reflector reflects light and crosses light rays to redistribute on the illuminated region as shown in Fig. 4-15. The negative x direction reflector curve changes slightly (Fig. 4-16) and the simulation results are shown in Fig.