Monte Carlo Simulation of Optical Properties of Phosphor-Screened Ultraviolet Light in a White Light-Emitting Device

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Monte Carlo Simulation of Optical Properties of Phosphor-Screened

Ultraviolet Light in a White Light-Emitting Device

Chien C. CHANG, Ruey-Lin CHERN, C. Chung CHANG, Chin-Chou CHU, Jim Y. CHI1, Jung-Chieh SU1, I-Min CHAN1 and Jih-Fu Trevor WANG1

Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China

1_{Opto-Electronics and Systems Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan, Republic of China} (Received May 11, 2005; accepted June 2, 2005; published August 5, 2005)

In this paper, we study the optical properties of phosphor-screened ultraviolet light emitted by a quantum well through a chamber. The chamber contains randomly distributed red, blue and green phosphors, and is top-covered with a layer of omnidirectional photonic bandgap material. A Monte Carlo ray tracing method is developed to model the absorption, reflection and transmission for the excited radiation of the ultraviolet light as well as the visible light by the individual phosphor particles. The efficiency of emitting white light by synthesizing the visible light through the top substrate is investigated with respect to the weight ratio, the size of phosphor particles, the dimension of the chamber and the reflectivity of the side wall and the bottom substrate. [DOI: 10.1143/JJAP.44.6056]

KEYWORDS: Monte Carlo method, white light-emitting diode, photonic bandgap materials, conversion efficiency, ultraviolet light source, reflecting wall

1. Introduction

It has gained an increasing interest to synthesize white light from a light-emitting diode (LED). In particular, white LEDs have practical applications such as full-color flat-panel displays when combined with color filters, and incandescent bulbs or even fluorescent lamps as alternative lighting sources or equipments. These applications are of great importance due to small size, long lifetime, and the lower energy consumption of LEDs. In the past years, there are intensive efforts devoted to the study of the transfer efficiency of white light from other light sources. Basically, white light can be obtained by mixing different colors with adequate intensities. The most commonly used method is to combine a phosphor wavelength converter with an excitation source, such as blue light with yellow phos-phors,1–4) and ultraviolet light with blue, green and red phosphors.5,6) Another method to achieve white LEDs is to use quantum-well structures which are able to emit red, green and blue light.7)

In the earlier study, we proposed a new design of white LEDs8) _{as shown in Fig. 1. The device consists of three}

parts: LED chip, LED chamber, and LED superstrate. On bottom of the chamber, ultraviolet light are emitted from an InGaN-based quantum well LED chip. The LED chamber contains a mixture of epoxy and red phosphor La2O2S:Eu,

green phosphor ZnS:Cu, Al, and blue phosphor (Sr,Ca, Ba,Mg)10(PO4)6Cl2:Eu used to re-emit visible light. The side

wall of the chamber and the bottom substrate excluding the LED chip are considered to be totally or partially reflecting. On the top of the chamber, a layer of omnidirectional photonic bandgap material9) is used to totally reflect the ultraviolet light and allow only passage of visible light. The efficiency of emitting white light by synthesizing the visible light through the top substrate depends on the number, size, shape, roughness, and optical properties of the phosphors such as index of refraction, absorption coefficient, quantum efficiency, as well as the optical properties of the matrix material and dimension of the chamber.

The phosphor particles considered here have a radius ranging from 2 to 6 mm. These are substantially larger than the wavelengths of the visible light, and therefore visible light as well as ultraviolet light can be modelled as rays of photons, each of which carries a photon energy with itself. By tracing a large number of rays of photons, the optical properties of phosphor-screened light can be obtained by counting the number and directions of the photons at the outlet of the chamber. Ray tracing includes free travelling of light, collisions of light photons with phosphors (reflection, transmission, absorption: emission and dissipation), reflec-tion of light by the side wall and bottom substrate, as well as total reflection of ultraviolet light by the top superstrate. All the optical processes are required to follow Snell’s law for plane waves incident on a flat surface, while the orientation of each collisional process, scattering direction of emitted light, and the chances of light to be reflected, refracted,

r

h

Fig. 1. Schematic of the white light-emitting diode with r ¼ 1:2 mm, h ¼ 1:5 mm. The ultraviolet light are emitted from the quantum well on the bottom substrate.

_{E-mail address: [email protected]} #2005 The Japan Society of Applied Physics

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transmitted or absorbed, emitted or dissipated are deter-mined by sampling a set of appropriate random numbers.

In previous studies, Monte Carlo ray tracing methods have been developed for calculating the destinations and inten-sities of luminescent emission in a phosphor screen,10) modelling the optical properties of fluorescent powders,11) and investigating the optical emission properties of a fluorescent display.12) Unlike these studies which deal with phosphors with fixed positions, the presently proposed Monte Carlo ray tracing method is mainly developed to simulate a randomly distributed phosphor particles, which means that the size, location, density, and even the shape of the phosphors can be arbitrarily distributed.

2. Monte Carlo Method

As mentioned in the previous section, the phosphor size ranges from 2 to 6 mm, which are much larger than the wavelengths of visible light. Therefore, the Monte Carlo ray tracing method is qualiﬁed for the simulation. Each ray carrying a photon energy undergoes a sequence of funda-mental processes, which are repeated until the photon comes out of the chamber or is absorbed by a phosphor. The Monte Carlo ray tracing method is implemented in the following steps:

(1) Emission of the ultraviolet photons. First, we consider an ultraviolet photon emitted from bottom of the chamber according to the directivity pattern sin i,

where i is the polar angle measured from the z-axis.

By choosing a random number Z1 between 0 and 1 as

the probability of directivity Pd, the polar angle of the

emitted ultraviolet photon is determined by i¼

sin1Pd. The azimuthal angle i is determined by a

random number Z2 between 0 and 2.

(2) Collision on the phosphors. Initially, the photon is randomly directed, and after travelling a distance of a mean free path, the photon hits the surface of a phosphor particle. The mean free path lmean is

determined as

lmean¼

V Np

; ð1Þ

where V is the chamber volume, Np is the number of

the particles, and ¼ r2

p is the cross section of the

particle with radius rp.

(3) Orientation of the phosphors. Figure 2 shows the schematic diagram for the orientation of the phosphor

particle upon collision by the photon. Let i, j, k refer to the standard base vectors of a ﬁxed coordinate system. In order to determine the location of the phosphor center, we deﬁne a local coordinate system with unit vectors k0¼ rnrn1 jrnrn1j ; j0¼ k 0_i jk0ij; i 0_¼_j0_k0_{; ð2Þ}

where rn is the position vector of the photon at step n,

and also the point of collision on phosphor. By choosing a random number Z3 between =2 to =2

as the incident angle 1 between the incident photon

direction and the inward normal of the phosphor particle at the point of incidence, and a random number Z4between 0 to 2 as the azimuthal angle 1about the

normal direction, the center of the sphere rcis given as

rc¼rnþrpði0sin 1cos 1

þj0sin 1sin 1þk0cos 1Þ:

ð3Þ

(4) Construction of the incident plane. Figure 3 shows the schematic diagram for the reflection, refraction and transmission of the photon from the phosphor particle. In order to construct the incident plane on which the reflection, refraction or transmission occur, we define a local coordinate system with unit vectors

e1¼ rcrn jrcrnj ; e3¼ e1k0 je1k0j ; e2¼e3e1: ð4Þ

Then, the direction of reﬂection is given as

u1¼ e1cos 1þe2sin 1; ð5Þ

and the direction of refraction as

u2 ¼e1cos 2þe2sin 2: ð6Þ

(5) Reﬂection of the photons. Let ne and np denote the

refractive indices of the epoxy and phosphors, respec-tively. The average intensity of reﬂection for unpolar-ized light is given as Rs¼ ðREs þR

H sÞ=2, where RE_s ¼ npcos 2necos 1 npcos 2þnecos 1 2 ; ð7Þ RH_s ¼ npcos 1necos 2 npcos 1þnecos 2 2 ð8Þ

are reﬂection coeﬃcients for E and H polarizations, respectively, and the angle of refraction 2 is then

r

i

j

k

r

n

r

c 1

θ

1

φ

p

r

i'

j'

k'

i

j

k

n−1

Fig. 2. Schematic of the orientation of the phosphor particle upon collision by the photon.

1

θ

1

θ

2 θ 2 θ n+1

r

n₊₁

r

n+1

r

2

e

1

e

r

c n

r

t

r

m

S

u

3 3

e

1

θ

2

'

e

1

'

e

3

'

e

1

u

₂

r

_p p

r

Fig. 3. Schematic of the reﬂection, refraction and transmission of the photon from the phosphor particle.

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determined from Snell’s law:

nesin 1¼npsin 2: ð9Þ

By choosing a random number Z5 between 0 and 1 as

the probability of reﬂection, a photon is reﬂected if Z5 Rs, or refracted if Z5> Rs. If the photon is

reﬂected, it travels with another mean free path, and hits another phosphor particle according to the same process described above.

(6) Refraction of the photons. If the photon is refracted, then there are two possibilities: transmitted or absorb-ed. According to the Beer–Bouguer–Lambert’s expo-nential law of attenuation,13) _{the transmitted distance}

S associated with a transmission probability Ptis given

by

Pt¼eSkp; ð10Þ

where kpis the absorption coeﬃcient of the particle. By

sampling a random number Z6 between 0 and 1 as the

probability of transmission, the transmitted distance S is given by S ¼ 1 kp ln 1 Pt : ð11Þ

The photon is transmitted through the phosphor if S is larger than the maximum possible distance Sm¼

2rpcos 2 of the refracted light inside the particle,

and is absorbed otherwise.

(7) Transmission of the photons. In order to obtain the transmission direction, we construct a local coordinate system with unit vectors

e0₁¼ rtrc jrtrcj ; e0₃¼ e 0 1u2 je0 1u2j ; e0₂¼e0₃e0₁; ð12Þ

where rt¼rnþSmu2 is the position vector of the

transmitted point. According to Snell’s law, the direction of the transmission photon is given as

u3¼e01cos 1þe02sin 1: ð13Þ

If the photon is transmitted, the travelling and colli-sional processes are repeated.

(8) Absorption of the photons. If the photon is absorbed, it is either re-emitted as a visible light and transmitted out of the phosphor particle, or dissipated into heat. By choosing a random number Z7 between 0 and 1 as the

probability of emission, the ultraviolet photon is emitted if Z7 is smaller than the overall quantum

eﬃciency Q, or dissipated otherwise. The overall

quantum eﬃciency is equal to the sum of internal and external quantum eﬃciencies, and here we use the value 0.674, which is obtained from experiments. If the photon is already a visible one, then it is dissipated into heat.

(9) Emission of the photons. In the emission process, the ultraviolet photon is converted to a visible one, and re-emitted in a random direction out of the particle. The emission direction can be determined by the polar angle eby sampling a random number Z8 between 0

and , and the azimuthal angle e by sampling a

random number Z9 between 0 and 2. The emission

point is then given as

re¼rcþrpðsin ecos ei þsin esin ej þcos ekÞ:

ð14Þ Figure 4 shows the schematic diagram for the absorption and emission of the photon by the phosphor particle. The direction of emission is determined by

u4¼

rera

jreraj

; ð15Þ

where ra¼rnþSu2 is the position vector of the absorption

point. Construct a local coordinate system with unit vectors

e00₁¼ rerc jrercj ; e00₃¼ e 00 1u4 je00 1u4j ; e00₂ ¼e00₃e00₁: ð16Þ

The angle between the emission photon direction and the outward normal 0

2 is then given by

0₂¼cos1ðu4e001Þ: ð17Þ

The direction of transmission of the re-emitted photon is given as u5¼e001cos 0 1þe 00 2sin 0 1; ð18Þ

where, according to Snell’s law,

₁0 ¼sin1ðnpsin 20=neÞ: ð19Þ

The color of the visible photon is determined by choosing a random number Z10 between 0 and 3. Red, blue or yellow

color is assigned to the re-emitted photon if the integer part of Z10 is equal to 0, 1, or 2, respectively.

In the method described above, we have totally ten random numbers in the Monte Carlo simulation, which are summarized in the following:

Z1: sampled to determine the initial polar angle i of the

photon;

Z2: sampled to determine the initial azimuthal angle iof

the photon;

Z3: sampled to determine the incident angle 1 of the

photon upon collision with the phosphor;

Z4: sampled to determine the azimuthal angle 1 of the

photon upon collision with the phosphor;

Z5: sampled to determine the reﬂection of the photon from

the phosphor;

Z6: sampled to determine the transmission distance

through the phosphor;

c

r

e

r

e θ p

r

i

j

k

e φ 2 c

r

1 θ1 θ 2' θ 2 θ n+1

r

n+1

r

n+1

r

e

2 1

e

n

r

e

r

S

1

u

5

u

2

u

3

e

1' θ 2

''

e

1

''

e

3

''

e

a 4

u

p

r

p

r

Fig. 4. Schematic of the absorption and emission of the photon by the phosphor particle.

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Z7: sampled to determine the emission of the photon from

the phosphor;

Z8: sampled to determine the polar angle e for the

re-emitted photon;

Z9: sampled to determine the azimuthal angle efor the

re-emitted photon;

Z10: sampled to determine the color for the re-emitted

photon.

3. Results and Discussion

A large number (104_{) of ultraviolet photons are emitted in}

random directions from the bottom center of the chamber. The number of escaped visible photons from top of the chamber is counted, and the ratio of the number of visible photons to that of ultraviolet photons is the conversion eﬃciency Ec of the light-emitting device. First of all, we

consider the side wall and the bottom substrate to be perfectly reﬂecting.

Figure 5 shows the conversion eﬃciency Ecof ultraviolet

photons to visible photons for diﬀerent weight ratios of phosphor particles with radius from 2 to 6 mm with and without a top layer. In the case with a top layer, the conversion eﬃciency Ecis seen to reach a maximum 65% at

the weight ratio w ¼ 6%, but changes little over the range from 4 to 7%. Below 4% the conversion eﬃciency Ec

decreases sharply with decreasing the weight ratio, while above w ¼ 7% the conversion eﬃciency Ec also decreases

quite rapidly with further increasing the weight ratio. At lower weight ratios, ultraviolet light photons have relatively little chance to be screened by the phosphors, while at large weight ratios, phosphor-screened photons have little chance to escape the chamber to reach the top layer. In the case without the top layer, the conversion eﬃciency Ec changes

only slightly in a wide range of weight ratios, maintaining at about 20%. By increasing the weight ratio above w ¼ 30%, the conversion eﬃciency Ecdecreases as more light photons

will bounce between phosphors and eventually dissipate into heats. In either case, the conversion eﬃciency Ec depends

mildly on the phosphor size, but interestingly we still observe a conspicuous eﬀect of size that smaller phosphors

are more eﬃcient in conversion at lower weight ratios (relative to that at maximum conversion), while larger phosphors are more eﬃcient at larger weight ratios.

Figure 6 shows the maximum percentages of red, green and blue photons converted from ultraviolet photons with and without top-covered omnidirectional photonic crystal for phosphors of size 3 mm. For a comparison, the line at 382 nm denotes the percentage of ultraviolet photons that escape the chamber in the case without the top layer. These light photons can be reused if a top layer of omnidirectional photonic bandgap material is inserted. The maximum conversion eﬃciency ðEcÞmax with the top layer improves

by more than 200% compared the that without the top layer. The remarkable improvement is attributable to the fact that the ultraviolet light, when reaching the top layer, are reflected back into the chamber and can be reused to emit useful visible light. The conversion efficiency can be readily translated in terms of the unit of luminous flux (lumen)14)to denote the strength of luminescence.

Furthermore, we investigate the cases in which the side wall and the bottom substrate are non-perfectly reflecting. This effect can be taken into account by sampling a uniformly distributed random number over ½0; 1 to deter-mine whether a light photon, when hitting a wall, is reflected by the wall. Figure 7 shows the conversion efficiency Ec

versus the reflectivity R of the side wall and the bottom substrate for a fixed chamber radius r ¼ 5 mm but with varying the chamber height from h ¼ 0:5 to 5 mm. In all the instances, the conversion efficiency Ec increases with

increasing the reflectivity. The increasing rate for the case with a top layer of photonic bandgap materials is more rapid than for the case without a top layer. This is certainly due to the single effect that the photons are reused more efficiently in the case with a top layer, considering that all the other conditions are identically the same.

At lower wall reﬂectivity R, the best conversion eﬃciency Ec is observed to reach at the chamber height h ¼ 1 mm.

There are three eﬀects that strike a good balance at this chamber height. The chamber height cannot be too small as the ultraviolet photons would not have been eﬃciently converted to visible light by the phosphors. On the other Weight Ratio w (%) Con v ersion Efficiency E c (%) 0 10 20 30 40 50 0 20 40 60 80 100 r_p=2µm rp=3µm rp=4µm r_p=5µm rp=6µm With top layer

Without top layer

Fig. 5. The eﬃciency Ec of ultraviolet photons converted to visible photons for diﬀerent weight ratios of phosphor particles with radius r varying from 2 to 6 mm with and without a top layer of omnidirectional photonic bandgap material.

Wavelength (nm) Con v esion Efficiency E c (%) 300 400 500 600 700 0 20 40 60 0

Fig. 6. Maximum percentages of red, green and blue photons converted from ultraviolet photons with (light, scaled from the bottom) and without (heavy) a top layer of omnidirectional photonic bandgap material.

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hand, the chamber height h cannot be too large as a large portion of photons would have been absorbed during their travelling in the chamber. Moreover, the high leakage of the side wall and bottom substrate at low reflectivity also tends to require a smaller chamber height. Consider the other extreme of perfectly reflecting side wall and bottom substrate. In the case with a top layer, the smallest chamber height h ¼ 0:5 mm gives a very high conversion efficiency Ec¼60% (compared to only 10% at the low wall

reﬂectivity). However, the conversion eﬃciency Ec

decreas-es rapidly with increasing the chamber height from h ¼ 0:5 to 5 mm as a result of the higher probability of absorption of a photon at a larger chamber height. At the larger chamber height, the conversion eﬃciency Ec increases only slightly

from its value at low wall reflectivity, for most useful photons are lost in their travelling in the chamber. At about wall reflectivity R ¼ 90%, we observe a level crossing of the curves of conversion efficiency Ecfor the chamber heights

h ¼ 0:5 and 1.0 mm. This indicates as the wall reﬂectivity is no longer an issue, the chamber with a top layer takes the small chamber height h ¼ 0:5 mm for better conversion of ultraviolet photons.

In the case without a top layer of photonic bandgap materials, the best conversion eﬃciency Ecis also observed

to reach at the chamber height h ¼ 1:0 mm. Even with a perfectly reflecting side wall, a smaller chamber height for less absorption does not overcome the leakage of useful photons out of the top substrate. The key observation is that none of these conversion efficiencies can be improved drastically by increasing the wall reflectivity. For example, the chamber with height 1.0 mm has the conversion efficiency Ec¼15% for the case of perfectly reflecting

walls compared to 5% at low wall reﬂectivity.

Figure 8 shows a different perspective of the effects of non-perfectly reflecting walls (side wall and bottom sub-strate) by fixing the chamber height. Again, we see the advantage by inserting a top layer of photonic bandgap materials which increases the conversion efficiency Ec less

substantially at low wall reflectivity, but quite significantly at large wall reflectivity. In particular, for small chamber

radius r (say, 1 mm), inserting the top layer of photonic bandgap materials does not improve at all the conversion efficiency in a wide range of lower wall reflectivity because the majority of photons leak out of the side wall and bottom substrate. As expected, the conversion efficiency at any wall reflectivity increases with increasing the chamber radius. However, we observe saturation of conversion at a certain chamber radius rs (¼ 3 mm in this case), above which the

light-emitting device can no longer improve the conversion efficiency. The radius of saturation can be considered a index for the maximum activity region of the light photons as we have considered a point source of emission at the center of the bottom substrate, and thus it is of no use increasing further the chamber radius above it. It is further observed that as long as the side wall and the bottom substrate are perfectly reflecting, the conversion efficiencies Ec for different chamber radii converge to the same value

(40% for the cases with the top layer, and 18% for the cases without the top layer). Notice that there are two cases with radii (1 and 2 mm) smaller than the radius of saturation (rs¼3 mm). This indicates that if the side wall and the

bottom substrate are made perfectly reflecting, the lateral dimension of the chamber can thus be made smaller without loss of the conversion efficiency. However, this is not the case if the wall reflectivity R deviates from 1 by a few percents, and at lower wall reflectivity it would be best to design the LED chamber at the radius of saturation rs for a

given chamber height h.

4. Concluding Remarks

In summary, we presented a very efficient Monte-Carlo method for simulating optical properties of a light-emitting chamber filled with randomly distributed phosphor particles. The present study indicates that it is crucial to use a top layer of omnidirectional photonic bandgap materials, with which the LED chamber can improve the conversion efficiency Ec

drastically by increasing the wall reﬂectivity above 90%, and at lower sidewall reﬂectivity, the chamber with the radius of saturation would be the most economic design.

Wall Reflectivity R (%) Con v ersion Efficiency E c (%) 0 20 40 60 80 100 0 20 40 60 0.5mm 1mm 2mm 3mm 5mm 0.5mm 1mm 3mm 5mm LED height h

Without top layer With top layer

}

Fig. 7. The conversion efficiency Ecvs wall reflectivity for weight ratio 25% of phosphor particles with fixed chamber radius 5 mm and chamber heights from 0.5 to 5 mm with and without a top layer of omnidirectional photonic bandgap material.

Wall Reflectivity R (%) Con v ersion Efficiency E c (%) 0 20 40 60 80 100 0 20 40 60 1mm 2mm 5mm 10mm 1mm 2mm 5mm LED radius r

Without top layer With top layer

Fig. 8. The conversion efficiency Ecvs wall reflectivity for weight ratio 25% of phosphor particles with fixed chamber height 1.5 mm and chamber radii from 1 to 10 mm with and without a top layer of omnidirectional photonic bandgap material.

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Acknowledgments

This work was supported in part by the Industrial Technology Research Institute under Contract No. 92S-23-N0, and the National Science Council of the Republic of China under Contract No. NSC 91-2212-E-002-072, and the Ministry of Economic Aﬀairs of the Republic of China under Contract No. MOEA 92-EC-17-A-08-S1-0006. The support from the National Center for Theoretical Sciences at Taipei is also acknowledged.

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