Chapter 1 Introduction
1.3 Motivation and Objective
As the lighting technologies continuously progress, the classification between the light source and object in the triangle is gradually indeterminate. The emitted radiance and physical scenes should be both considered on some objects. For instant, as the object in Fig. 1-1 is the yttrium aluminum garnet (YAG) phosphor in a phosphor-converted light emitted diode (pcLED) (shown in Fig. 1-9) [4], which
scatters the illuminating flux and re-emits a fraction of the absorbed light at a different wavelength peak, measurement of the optical properties is prohibitively difficult, and produces massive quantities of data that are difficult to meaningfully utilize. The quantification indices characterizing the spectral and geometric relationship should be modified in an effective and a sensible way.
(b) (a)
Fig. 1-9 YAG phosphor layer in a pcLED: (a) the schematic graph of the interaction; (b) the radiant spectral power distribution
Basing on the energy balance equation, the thesis attempts to study the measurement instruments, mathematical relationships, and calculation methodologies for illumination systems, such as luminaire and liquid crystal display (LCD) backlighting modules. First, the experimental setup for the bidirectional scattering distribution function (BSDF) measurement was developed [5]. Here the measurement instrument is associated with a conoscopic imaging system [6,7]. Next, a new mathematical definition, dichromatic BSDF, is introduced for the characterization of photo-fluorescent materials [8,9]. Here the YAG phosphor layer in a pcLED is introduced for demonstration. Then, a simple but effective calculation methodology
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for the integration of the energy balance equation is described and demonstrated [10,11].
1.4 Organization
The rest of this thesis is organized as follows. In Chapter 2, for scene physics, the theory and operation principle of BSDF are introduced based on the photometry and radiometry. The BSDF measurement instrument was set up, and the measured bidirectional photometric data were analyzed. In Chapter 3, the proposed dichromatic BSDFs are introduced, and the YAG phosphor layer in a pcLED is considered as an example. After the BSDF, a calculation methodology of the integral equation, Eq. 1.6, is presented in Chapter 4. The concept was verified by the diffuser of a commercial available LCD backlighting module. In Chapter 5, by using the proposed methods, a novel planar lighting source by blue LEDs array with remote phosphor sheet was simulated as an application. Here the optical properties were predicted, and the geometric parameters were optimized for the demonstration of a 7-inch prototype.
Finally, the conclusions of this thesis and recommendations for the future works are given in Chapter 6.
1.5 References
[1] Mark. D. Fairchild. Color Appearance Models, (John Wiley & Sons, Chichester, England, 2005).
[2] William. R. McCluney, Introduction to Radiometry and Photometry, (Artech House, Boston, 1994)
[3] Francois X. Sillion, and Claude Puech, Radiosity and Global Illumination, (Morgan Kaufmann Publishers Inc., San Francisco, 1994)
[4] R. Mueller-Mach, G. O. Mueller, M. R. Krames, and T. Trottier, “High-power phosphor-converted light-emitting diodes based on III-nitrides,” IEEE J. Sel. Top.
Quantum Electron. 8, 339-345 (2002).
[5] John. C. Stover, Optical Scattering: Measurement and Analysis, (Mc Graw-Hill, New York, 1990).
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[6] M. E. Becker, “Evaluation and characterization of display reflectance,” Displays 19, 35-54 (1998).
[7] M. E. Becker, “Display Reflectance: Basics, Measurement, and Rating,” J. SID 14/11, 1003-1017 (2006).
[8] C.-H. Hung, C.-H. Tien, “Phosphor-converted LED modeling by bidirectional photometric data,” Opt. Express (to be published).
[9] C.-H. Hung, C.-H. Tien, “Phosphor Modeling for Phosphor-converted LEDs,”
SID Symposium Digest Tech., Papers 59.3, (2010).
[10] C.-H. Tien, and C.-H. Hung, “An iterative model of diffuse illumination from bidirectional photometric data,” Opt. Express 17, 723-732 (2009).
[11] C.-H. Hung, C.-H. Tien, “Modeling Diffuse Components by Bidirectional Scatter Distribution Function for LCD Applications,” SID Symposium Digest Tech., Papers 36.4, (2009).
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Chapter 2
Scene Physics
Light radiated from light sources is reflected, refracted, scattered, or diffracted by objects. As a result of all these light-matter interactions, light is redistributed spatially and temporally to create physical scenes that we can see. Thus, the study of illumination system should begin with the light-field formation process of physical scenes. This is what we mean by scene physics. BSDF is the most general expression of the light energy behavior passing through a surface or thin film. Although various materials in nature induce light in very different ways, which explains why their appearance can be dramatically different, the BSDFs can completely describe such optical phenomena. In this chapter, we first introduce the optical scattering, which is the basic of global illumination. Then, the BSDFs are defined basing on the radiometry. Finally, the BSDF measurement is presented and demonstrated.
2.1 Optical Scattering
To begin a discussion of scene physics, the light reflected by an object is considered. The optical properties of reflector materials can be specular, spread, or diffuse as shown in Fig. 2-1 [1]. Specular materials permit precise redirection of light rays and sharp cutoffs, while painted reflectors produce diffuse, scattered, or widespread light distribution. The high-reflectance materials with about 98%
reflectivity can enhance lighting efficiency.
The scattered radiance distribution for an ideal diffuse surface is Lambertian.
The distribution of scattered intensity is given by the following equation,
( ) 0 cos( )
I θ = ⋅I θ (2.1) as illustrated in Fig. 2-2(a). The Gaussian distribution for a rough surface is shown in Fig. 2-2(b), and the distribution of scattered intensity varies according to the equation
2
( ) 0 exp[ ( )
I I 21
θ ],
θ = ⋅ − σ (2.2) where I is the intensity in the specular direction, 0 σ is the standard deviation of
e Gaussian distribution.
Fig. 2-1 Reflector materials and optical properties th
Polished surface, specular Rough surface, spread Matte surface, diffused
Diffuse and specular Diffuse and spread Specular and spread
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(a)
(b) I0
0cos( )
I θ
I0
2 0
exp[ 1( ) ]
I 2 θ
− σ
Fig. 2-2 (a) Lambertian distribution for a diffuse surface (b) Gaussian distribution for a rough surface
Scatter from transmission and reflection optics is mainly caused by several mechanisms including surface topography, surface contamination, bulk index fluctuations, and bulk particulates [2]. These scatter characteristics are introduced as the following.
Surface topography
By diffraction theory, surface topography induces phase deviations to the incident wavefront. In general, surface topography contributes multiple refractions and reflections on a ray. Thus, it’s difficult to analyze the wavefront behavior through primary optics by the random surface vectors.
Surface contamination
The intensity and polarization of the scatter patterns by surface contamination
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depend on particulate size, shape, and material constants. Particulates play a significant role in producing scatter.
Bulk index fluctuations
Index fluctuations induce a phase change to the transmitted beam through the film or the reflected beam by subsurface. As a diffraction effect, scatter from these flaws has a dependence on spot size similar to that of surface fluctuations.
Bulk particulates
Bulk particulate flaws are due to small bubbles, inclusions, and contamination.
Bulk scatter caused by isolated particulates which are small compared to the wavelength is called Rayleigh scatter.
In general, rays scattered by a material behave as the combination of the four mechanisms. As Fig. 2-3 shows, we take rays scattered from a surface as example. As two light rays (r1 and r2) are incident on the surface between air and an object of some inhomogeneous material, part of their energy is reflected (r3 and r7) and part of it penetrates into the surface (r4 and r6). The reflected part may undergo many-times reflection (r7). The penetrating light may undergo multiple scattering by particles in the material, and then is absorbed (r6) or reflected back into the air (r5). This example shows the incident light may undergo the combination of the refraction, reflection, scattering, and absorption. The four material properties we mentioned may affect the incident light, simultaneously. Only in the ideal situation of a perfectly smooth surface of homogeneous material with infinite spatial extent, the reflected and refreacted electromagnetic fields can be neatly treated by the Fresnel equations [3]. For nonideal surfaces, the real physical description is much more complicated. If the surface roughness is small compared with the wavelength of the incident light, it can be treated by the perturbation method for the ideal solution for smooth surfaces. If the scale of the surface toughness is of about the same order as the light wavelength, the
problem is very difficult to deal theoretically. Numerical calculations can only handle it satisfactorily. When the surface irregularity is of a scale much larger than the
Fig. 2-3 Different components of reflected and refracted light rays.
2.2 Bidirectional Scatter Distribution Function
Basing on radiometry and photometry, the bidirectional transmittance and reflectance distribution functions (BTDF and BRDF) are developed to characterize the optical energy behavior passing through a surface [4-6]. This bidirectional function is a general and useful concept to describe the optical efficiency through geometric relationship between the incoming and outgoing flux.
The geometric relationship of BTDF and BRDF is shown in Fig. 2-4, where the subscripts i, t, and r denote the incident, reflected and transmitted quality, respectively.
Referring to Fig. 2.5, the BTDF and BRDF are defined as the ratio of the outgoing radiance in the viewing direction to the irradiance in the direction of the incident light,
( , ) ( , )
angles with respect to the surface normal.
Fig. 2-4 Schematic diagram of BTDF and BRDF.
Although the bidirectional function is quite general and useful, its limitation should be pointed out as well:
1. It is based on the approximation where light propagation is treated as rays.
2. The two quantities, the outgoing radiance L and the incident irradiance E, should be proportional.
3. The definitions of the BTDF and BRDF assume that the outgoing radiance depends only on the irradiance on the infinitely small area.
In general, fluorescent material cannot be treated by the bidirectional function concept.
However, under some conditions, the bidirectional function was presented to be applied on fluorescent material. We proposed this concept to describe the optical characteristics of phosphor in white light emitting diodes (LEDs) in Chapter 3.
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2.3 Non-paraxial Scalar Diffraction Theory
Before proceeding to measure and analyze the BSDF, we briefly introduce the non-paraxial scalar diffraction theory to illustrate that the diffracted radiance is shift invariant in spherical coordinate. The scattering phenomena are caused by the combination of the diffracted wave, so the scalar diffraction theory can briefly characterize the geometrical properties. It’s well known that the irradiance distribution on a plane in the far-field of a diffracting aperture is obtained by the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the aperture :
where is the complex amplitude distribution emerging from the diffracting aperture with complex amplitude transmittance t1(x1, y1). Implicit in the Fraunhofer approximation is a paraxial limitation which restricts the diffraction angles and incidence angles in a small range. However, the diffracted radiance of non-paraxial diffraction is the fundamental quantity which exhibits shift-invariant behavior.
0( ,1 1) 0( ,1 1) ( ,1 1 1) U x y+ =U x y t x y−
Fig. 2-5 Geometrical configuration used to demonstrate the fundamental theory of radiometry from which the quantity radiance is obtained..
Referring to Eq. 1.2, which defines the radiance, and noting that cos / 2
2 cos
where r is the distance between the source and the collector, as illustrated in Fig. 2-5.
This equation can be considered to be the fundamental theorem of radiometry, as it describes the radiant-power transfer between an elemental source and an elemental collector. If we consider a diffracting aperture as the source and the observation hemisphere as the collector, we can integrate with respect to to the projected source area to obtain the diffracted intensity:
( , ) cos . If we integrate the double differential of Eq. 2.2 with respect to the source’s solid angle ∂ , the irradiance on the observation hemisphere (ωs θc=0) is just the intensity divided by the square of the radius of the hemisphere:
2 From Eq. 1.1, the total radiant power diffracted into the complete hemisphere can be expressed as conventional spherical coordinates. The Parseval theorem form Fourier transform theory states that the integral over all space of the squared modulus of any function is equal to the integral over all space of the squared modulus of the Fourier transform of that function. We can write
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{ }
so the Eq. 2.9 is equal to the right-hand side of this equation:{ }
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( , ) 0( , ) .
I α β =γλ ℑ U x y (2.11) From Eq. 2.7, we obtain the general equation:
{ }
2If the source is a uniformly illuminated diffracting aperture, there is no dependence on position in the aperture:
( , ) ( , , , ) cos ( , ) cos ,
s s s s
I α β =
∫
A Lα β x y θ ∂ ≈A L α β A θs (2.13)Substituting expression 2.13 into Eq. 2.12, dividing by the area of the diffracting aperture As, and recalling that γ cos θs, we obtain the following expression for the diffracted radiance:
{ }
We have thus shown that the squared modulus of the Fourier transform of the complex amplitude distribution emerging from the diffracting aperture yields the diffracted radiance and not the irradiance or the intensity. Furthermore, Eq. 2.14, unlike the more familiar Eq. 2.5, is not restricted to small diffraction angles. From the shift theorem of Fourier transform theory1 it is clear that changes in the angle of incidence of the radiation illuminating the diffracting aperture will merely result in a shift of the radiance function in direction cosine space and an attenuation by the factor γ0=cos θ0: Because the functional form does not change, the diffracted radiance is shift invariant in direction cosine space, and the simple Fourier techniques that have proven to be so
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useful in paraxial applications can be used for non-paraxial applications as well.
2.4 BSDF Measurement
To measure the BSDF of a scattering material, there have been two methods to measure the scattering angular field [7]. Analysis of the intensity of scattered light versus direction of light propagation in the hemispherical solid angle above the measurement spot on the sample requires either a mechanical system for motorized scanning of the of viewing directions (or source directions, gonioscopic) or a conoscopic system that projects a collimated beam of light through its front lens on the sample while simultaneously catching all rays reflected from the spot of measurement. In our study, we used the conoscopic approach to implement the BSDF measurement.
2.4.1 Conoscopic approach
A conoscopic system is utilized to measure the angular light distributions of an emitting surface [8]. The optical radiations within wavelength ranges between 380 nm to 780 nm, known as visible light, can be measured. The measuring mode of the conoscopic approach can be classified into transmissive and reflective modes.
Transmissive mode
In transmissive mode (as shown in Fig. 2-6), a Fourier transform lens is adopted to transform the received light into a two-dimensional pattern. Each light ray emitted from the test area in the emitting direction angle, (θ, φ) can be focused on the focal plane. Therefore, the directional information of the radiated energy is transferred onto the corresponding position in the focal plane. Then, a relay system is utilized to project this transformed pattern onto a CCD-array detector. In this transformed pattern, each area corresponds to one light radiating direction. Thus, as the pattern is recorded by the CCD-array detector, the optical energy flux and chromaticity in radiating
direction angle (θ, φ) are obtained efficiently without mechanical movement.
Fig. 2-6 The schematic graph of the conoscopic system in transmissive mode.
Reflective mode
Except the transmissive mode, a reflective mode in the conoscopic system is available to measure the angular distributions of light reflected from samples. In the reflective mode (as shown in Fig. 2-7), a light source emits back from the Fourier transform plane and is inversely focused on the sample under the stage. Here the focused light can be a collimated beam at an incident angle (θ, φ) or a diffuse field in all direction by controlling the pupil position and size of the back emitting source on the Fourier transform plane. The light illuminating the sample surface within a tiny illuminated area is reflected back into the Fourier lens. Therefore, the radiance angular distribution of the reflected light is obtained by the CCD-array detector as the transmissive mode.
Fig. 2-7 The schematic graph of the conoscopic system in reflective mode.
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2.4.2 Experimental setup of BSDF measurement
As Section 2.2 mentioned, BSDF is classified into BRDF and BTDF. From their definitions, a collimated beam induces the sample as an impulse signal, and the measured angular spreading function would relate to the individual incident angle.
For BRDF measurement, the illuminating angle θ of the built-in source is easily varied by controlling the position of the light source with limited pupil size on Fourier transform plane, as Fig. 2-6 shows. In our study, the BRDFs of a scattering surface were obtained by measuring the angular spreading function of different incident angle under the collimated illumination mode of conoscopic system.
To measure the BTDFs, an external light source should be set up to illuminate the back surface of a scattering film. The external light source is schematically shown in Fig. 2-8. Here the collimated beam is produced by a LED combined with a total internal reflection (TIR) lens, and two apertures with 2-mm diameter are used to limit the beam divergent angle within +/- 1 degree. The LED module can slide on an arc track to provide variable incident angle θ. Thus, the geometric relationship of the light source and the angular spreading function is illustrated in Fig. 2-9 (a) and (b), respectively. For a specimen, the collimated beams at different incidence are incident from the illumination hemisphere and the scattering patterns projected on the observing hemisphere are collected by the objective lens of conoscopic system. Fig.
2-10 shows the experimental setup, where the designed light source module is integrated with the conoscopic system in transmissive mode. Through the definition, the BTDF of a scattering film can be recorded and calculated.
Slippery track
LED stage Aperture
(a) Front view (b) Side view Fig. 2-8 The scheme of the external light source module for BTDF measurement
Fig. 2-9 (a) Schematic measurement setup of BTDFs, (b) the measured angular spread functions of an available specimen.
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Conoscopic system
Measured sample (e.g. phosphor film) Our designed light
source device
Fig. 2-10 The photograph of BTDF measurement setup.
2.5 BSDF Analysis
BSDFs represent the angular spread function of the scattering specimen, which have a variety of optical features due to various manufacturers’ recipes about the refractive index, the density, and the size of scattering particles. In the following, we use the measured BTDFs to analyze two commercial diffusers with different bead sizes. As the tops of Fig. 2-11 (a) and (b) show, the diffusive layer on the top of Polyethylene Terephthalate (PET) substrate has some beads buried in an acrylic binding layer and other diffusing beads protrude partially out of the binder layer. Here the average diameters of the beads are about 10 and 25 μm, respectively. The bottoms of Fig 2-11 illustrate the one-dimensional BTDFs. From the envelope of every transmitted peak, the diffusing power and particle size has a reciprocal relationship.
For the case of a laser beam illuminating the diffusing structure with respectively different incident angles, the transmitted angular spectrum can be roughly found by the Fourier transform of the amplitude transmittance function of the bead aperture [9-11]. Thus, the smaller the bead size, the broader the scattering power after passing through the diffuser will be caused. This effect is entirely analogous to the broadening of the angular spectrum with the corresponding special frequency.
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(a) (b)
Fig. 2-11. The measured BTDFs and their corresponding OM pictures with (a) low spatial frequency and (b) high spatial frequency.
In addition to the dependence of scattering power on the particle size, there is another issue needed to be mentioned. As the definition, BTDF is simultaneously a function of the incident direction and the transmitted direction. J. E. Harvey [12] has mentioned that a general scattering surface has the shift-invariant behavior, which just requires one set of numerical data to completely characterize the scattering properties of a surface. The most conventional diffusers are circularly symmetric, so the variations of the incident impulse along azimuthal angle φ would be ignored.
However, as shown in Fig 2-11, the BTDFs exhibit a discrepant scattering shape and peak shift with different inclining illumination. Especially, the phenomena are more observable in larger angles of incidence. Thus, in the incident side, BTDF is a
However, as shown in Fig 2-11, the BTDFs exhibit a discrepant scattering shape and peak shift with different inclining illumination. Especially, the phenomena are more observable in larger angles of incidence. Thus, in the incident side, BTDF is a