Analytical light-ray tracing in two-dimensional objects forlight-extraction problems in light-emitting diodes

Hisashi Masui,* Shuji Nakamura, and Steven P. DenBaarsMaterials Department, College of Engineering, University of California, Santa Barbara, California 93106-5050, USA

*Corresponding author: masui@engineering.ucsb.edu

Received 27 August 2007; accepted 22 October 2007;posted 24 October 2007 (Doc. ID 86896); published 20 December 2007

Light extraction from two-dimensional objects is discussed. Analytical calculations in terms of threedifferent parameters have been applied to equiangular polygons to trace light rays during multiplereflections in a polygon. Based on the result that there are a finite number of incident angles in a polygonfor a light ray, it was found that the triangle has the least chance to trap light rays among the polygons.The discussion has been extended to parallelograms, which have an advantage in light extraction torectangles. Placement of a possible light source in polygons is discussed. © 2008 Optical Society ofAmerica

OCIS codes: 250.0250, 160.6000.

1. Introduction

Light extraction was studied [1] as early as the emer-gence of light-emitting diodes (LEDs) and has becomea great concern in solid-state lighting technology [2]lately. This is due to the fact that semiconductor ma-terials inherently have higher refractive indices thanthat of free space, which results in trapped light inthe material. Techniques such as chip shaping [3,4]and surface roughening [5] utilize acute angles toassist incident light to escape and have improvedlight-extraction efficiency of LEDs. While the verticaldirection of the LED structure has attracted vastattention, the lateral direction has not been muchconsidered. A square is a conventional shape, andrectangles have been employed lately [6]. In laserdiode (LD) stripes, rectangles are known to confineoptical modes, so they may be inappropriate in termsof light extraction. Other shapes may be effective toextract such guided modes by avoiding parallel sides.The implementation of photonic crystals is an ap-proach to redirect the guided modes [7].

The aim of the present paper is to initiate geomet-rical studies of lateral light extraction for LED appli-cations. To understand loci of light rays duringmultiple reflections in two-dimensional (2D) objects,

we first define the problem by identifying three inde-pendent parameters and their roles. Understandingthese parameters enables simple analytic calcula-tions to be used on equiangular polygons. As a resultof the calculations, it is shown that shapes with acuteangles, triangles and parallelograms, for example,are advantageous to ones with obtuse angles.

In modern LED structures, active regions arecommonly very thin compared to their lateral di-mensions. Under this geometrical condition, three-dimensional (3D) light-propagation problems canbe separated into lateral and vertical problemsfor first-order approximation, since the light-propagation vector is nearly confined in the hori-zontal plane (i.e., cos � � 1, where � is the elevationangle of the propagation vector). When 3D problemsare reduced to 2D problems, it is often a temptingapproach to begin with a point source placed at thecenter of the 2D objects to obtain an elemental pros-pect. This approach however leads the solution ulti-mately to a circle, despite the fact that the real activeregion is hardly a point [8,9]. The present paper willconclude that acute angles are appreciated not onlyin the vertical direction but also in the horizontalplane, which excludes the conventional square designof active regions from the best options. In practicalconsiderations, especially in nitride-based LEDs, it isnot impractical to obtain horizontal angles of ��3 radrelying on the hexagonal structure of nitride crystals

0003-6935/08/010088-05$15.00/0© 2008 Optical Society of America

88 APPLIED OPTICS � Vol. 47, No. 1 � 1 January 2008

and widely used sapphire substrates. Furthermore,dry etching is an industrial technique commonly usedto define mesas of nitride-based LEDs and LDs, sothat 2D shape fabrication other than rectanglescomes to only a matter of photolithographic maskdesign.

2. Calculation and Discussion

A. Preparation

The geometrical objects we consider are equiangularpolygons (i.e., 2D objects) made out of an opticallytransparent material that may have a high refractiveindex compared to outside the polygons. When theyare considered in terms of the extraction process oflight that has been generated in the object, there arethree independent parameters: the internal angle ofequiangular polygons (geometrical parameter), theincident angle of light (light-ray parameter), and thecritical angle (material parameter). The followingdiscussion deals with a light ray impinging onto oneof the sides of a polygon with an initial incident angle�1, and traces the light ray during subsequent reflec-tions at different sides of the polygon. Any light raycan be then described by varying �1 from 0 to ��2. Theangle of reflection is always the same as the angle ofincidence at an event of reflection.

We start discussions by considering ray tracing inequiangular polygons with a condition that light doesnot exit a polygon (Subsection 2.B) where we inves-tigate the incident angles upon multiple reflections.The notion of critical angle is introduced in Subsec-tion 2.C, and we identify conditions under which lightcan escape from the polygon. The discussion is ex-tended to an isosceles triangle in Subsection 2.D,where the advantage of parallelograms over rectan-gles is found.

B. Equiangular Polygons with Reflective Sides

With sides that are perfectly reflective, a set of twosuccessive reflections in a polygon (the number ofsides is n) is categorized by the bounce mode m. Whentwo successive reflections occur at sides next to eachother, this is defined as m � 1. When two successivereflections occur at two sides that are one side apart,this is m � 2, and so on (Fig. 1). Under this circum-stance, we can derive the master equation:

�1 � �2 � 2�m�n � �, (1)

for incident angles of two successive reflections, �1and �2, as shown in Fig. 2. Incident angles are taken

between 0 and ��2. Because of the nature of poly-gons, m � n�2.

We show first that there are only two incident an-gles possible for a light ray when the same bouncemode is maintained. The master equation for thesecond bounce is

�2 � �3 � 2�m�n � �, (2)

and by combining Eqs. (1) and (2),

�3 � �1. (3)

For a circle, �1 � �2, and there is always only oneincident angle for a light ray.

Then the interest goes to cases where m changesupon every incident. Combine Eqs. (1) and (2) withdifferent modes, m1 and m2 (the suffix indicates theorder of the bounce),

�3 � �1 � 2��m1 � m2��n. (4)

Similarly,

�4 � �2 � 2��m2 � m3��n, (5)

and so on. Due to the restriction in m, the number ofincident angles is limited to a finite number for onetrace of light. In rectangles �n � 4�, there are twoincident angles, or one if the two happen to be thesame ���4�. That there are two incident angles is thebasis of standing-wave modes in LD stripes [10], andthe standing modes are not appreciated in terms oflight extraction.

C. Role of the Critical Angle

A simple assumption is taken here that whenever anincident angle becomes smaller than or equal to acritical angle, the light ray exits the polygon. Weconsider first a case where light can be trapped in apolygon due to the total internal reflection (incidentangles larger than the critical angle). We seek a con-dition under which the smallest incident angle dur-

Fig. 1. Definition of bounce mode m in the octagon as an exam-ple.

Fig. 2. Derivation of the master equation. The sum of the threeangles is �.

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ing multiple reflections should be maintained largerthan the critical angle. This occurs when possibleincident angles are all equal to each other in thelowest bounce mode (i.e., �1 � �2 and m � 1) [11]. Thisangle �max is from Eq. (1),

�max � ��12 �

1n�. (6)

If �max is larger than a critical angle �c, the light rayis trapped in the polygon. �max becomes larger when nbecomes larger, which means there is more likelihoodfor a light ray to be trapped under a fixed �c. Thewhispering gallery modes are resonant modes of thistype of trapped light interfering with itself upon re-flection. If the critical angle happens to be larger than�max, all the light rays will exit within two reflectionevents. This is because there will be only two possibleincident angles for the lowest bounce mode and one ofthem is smaller than �c, and the higher bounce modeshave smaller angles of incident. �max becomes mini-mum at n � 3, so the triangle requires the smallestcritical angle ���6� to let all the light (i.e., light rayswith any incident angles) exit. A practical example ofmaterial combinations to give a critical angle largerthan ��6 may be GaN surrounded by a common en-capsulant resin, whose refractive index is 2.3 (GaN)and 1.4 (resin). The result from Eq. (6) is plotted inFig. 3, where some examples of material combina-tions are shown with critical angles indicated. FromFig. 3, it is understood that GaN squares surroundedby resin is disadvantageous to triangle ones in lightextraction.

The above calculation can be utilized in the follow-ing way. We place a 2D light source in a polygon witha condition that light emitted from the light source inall directions exits the polygon upon the first inci-dence. We take a square as an example (Fig. 4). Con-sider two ends of a side, from where an extractioncone for �c � ��4 has been drawn in Fig. 4. Theshaded part in Fig. 4 is the area from where light is

emitted and a light ray will exit upon the first inci-dence at the side. Whenever �c � ��4, the two extrac-tion cones intersect at a point at the center of thesquare or closer to the side. By repeating the proce-dure for the rest of the three sides with the criticalangle, we obtain an area in the middle where a lightsource can be placed. All the light rays from this areawill exit upon the first incidence. This area does notexist in cases where �c ��4. When the same processis applied to the equiangular triangle, such an areaexists when �c � ��3. This technique of finding thelight-source area in 2D objects can be applied, forexample, to determine resin package shapes and di-mensions as the light source being an LED chip [9].

D. Other Two-Dimensional Objects

Two more 2D objects are considered in this Subsec-tion. Discussions start with an isosceles triangle witha vertical angle � ( ��3, as always at least one ofthree angles becomes so) and extends the discussionsto a parallelogram. In the case of an isosceles triangleshown in Fig. 5, the discussion in Subsection 2.Bapplies. One possible situation that is different from

Fig. 3. Relationship between the maximum angle of incidenceamong two successive reflections in the lowest mode, �max, and thenumber of sides of the polygon, n (10 n has been omittedbecause of the obviousness). On the vertical axis on the right,critical angles of some of the material combinations are shown.

Fig. 4. Extraction cones in a square. The shaded area is in the twocones, therefore a light ray emitted from this area will exit thesquare at the first incidence upon the side.

Fig. 5. Isosceles triangle with a vertical angle �. The two succes-sive reflections fall in the condition described in Eq. (11). Notethe sequent reflection (not indicated by the light-ray line) woulddirect the ray toward the bottom, resulting in a bounce mode withm � 1.

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the discussion in Subsection 2.B is that two succes-sive reflections may go toward the vertical angle (thatis m � 1, nevertheless). In this case, the master equa-tion is slightly altered and appears as

��

2 � �1�� �2 ��

n � �, (7)

which can be reduced to

�1 � �2 � �. (8)

In the case that the light ray is reflected toward thebottom, the altered master equation gives

�1 � �2 � �. (9)

Transition between these two conditions described byEqs. (8) and (9) occurs at �1 � � (i.e., �2 � 0). A lightray will reach the condition in Eq. (9) within a finitenumber of successive reflections (see Appendix A).One of the incident angles of the last two successivereflections is smaller than or equal to ��2 accordingto Eq. (9). If ��2 has been made smaller than or equalto the critical angle, the light ray can exit before beingreflected back toward the bottom.

The same condition can be applied to parallelo-grams, which have bounce modes up to m � 3 (not 2,because of their lower symmetry than the square).There exists a parallelogram that has all the angleslarger than ��3, nevertheless, the same analysis ap-plies. When the acute angle of a parallelogram ismade smaller than twice the critical angle, a light rayalways has a chance to exit. Rectangles do not havethis advantage in light extraction. It is equivalent tostate that standing-wave modes are less likely to ap-pear in parallelograms than in rectangles, which isthe technique often used in superluminescent diodes[12].

3. Summary

We studied light-ray tracing during multiple reflec-tions in 2D objects in relation to light extraction forLED application purposes. This subject of laterallight extraction has importance to achieve ultimatelight-extraction efficiency, while vertical light extrac-tion has already attracted vast interest. Analytic cal-culation became feasible and was carried out byidentifying three independent parameters. It wasfound that equiangular polygons raise only a finitenumber of incident angles for a light ray during itsmultiple reflections at the sides. Under this circum-stance, it was shown that n � 3 traps the leastamount of light rays, and is therefore the most effec-tive for light extraction. We calculated the geometri-cal area and a placement of the light source in the 2Dobjects from where emitted light escapes upon thefirst incidence. Light-ray tracing was extended to an-alyze two other 2D objects, where it was shown thatparallelograms would trap light rays less likely thanrectangles.

Appendix A

We seek the number of reflections required for anincident angle to become an angle such that the lightray takes the condition in Eq. (9) at the sequent re-flection. An incident angle �f is assumed to be anangle after whose reflection the light ray falls in thecondition in Eq. (9), that is �f �. Until then Eq. (8)applies, therefore �f � �1 � ��f � 1���. Hence,

�1 � ��f � 1��� �, (A1)

which reduces to �1�� f, where f is the number ofreflections. The maximum of �1 is ��2 by definition.As a consequence, when f becomes larger than���2��, Eq. (9) becomes applicable for a light rayimpinging at any angle initially (that means the lightray goes back away from the vertical angle). �2 in Fig.5 corresponds to �f in this particular case, since thenext reflection (not indicated by the light-ray line inFig. 5) would be reflecting the light back down, whereEq. (9) would apply.

The authors acknowledge The Solid-State Lightingand Display Center at University of California, SantaBarbara and the Exploratory Research for AdvancedTechnology (ERATO) Program of the NICP�JapanScience and Technology Agency for the financialsupport.

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