Optimal design of retroreﬂection corner-cube sheets by...

Os

HBSSKSE

1

OstrrFcatpptTro

otbrmtsclapi

cgmemr

0

Optical Engineering 46�9�, 094002 �September 2007�

O

ptimal design of retroreflection corner-cubeheets by geometric optics analysis

wi Kimyoungho Lee, FELLOW SPIEeoul National Universitychool of Electrical Engineeringwanak-Gu Shinlim-Dongeoul 151-744, Korea-mail: [email protected]

Abstract. We describe the geometric optic design and analysis of theoptimal retroreflection corner-cube sheets with the effective retroreflec-tive area ratio of 100%. The maximum condition of the effective retrore-flective area of elementary retroreflection corner-cube is investigated,and as a result, the analytic design formulas of optimal retroreflectioncorner-cube structure are derived. We propose a novel packing methodfor the elementary retroreflection corner-cube to construct optimal ret-roreflection corner-cube sheets with the effective retroreflective area ra-tio of 100%. © 2007 Society of Photo-Optical Instrumentation Engineers.�DOI: 10.1117/1.2779030�

Subject terms: retroreflection corner-cube; geometric optics analysis; effectiveretroreflective area.

Paper 060830R received Oct. 24, 2006; revised manuscript received Mar. 10,2007; accepted for publication Apr. 10, 2007; published online Sep. 10, 2007.

Introduction

ptical reflection for engineering applications can be clas-ified into three types, as shown in Fig. 1: specular reflec-ion, diffusing reflection, and retroreflection. The speculareflection is a simple reflection that occurs when an opticalay bundle is incident on a mirror, as shown in Fig. 1�a�.igure 1�b� shows the diffusing reflection, which is theomplex backscattering of an incident optical ray bundle byreflection surface having coarse surface roughness or in-

ernal volume scatters. This diffusing reflection is the basicroperty of general projection display screens; high-erformance diffusing reflection is particularly important inhe backlight systems of liquid crystal displays �LCDs�.he retroreflection is a special reflection of ray bundle that

eturns back to the source origin along the inverse directionf the incident optical ray bundle, as indicated in Fig. 1�c�.

For realizing the retroreflection, a specifically designedptical device called a retroreflector is needed. Severalypes of retroreflectors and their optical properties haveeen intensively investigated.1–6 The most well-known ret-oreflector is the retroreflection corner-cube, which is sche-atically described in Fig. 2. They are used in many prac-

ical applications, such as positioning and guidanceystems,7–9 wavefront correction,10 opticalommunications,11 and traffic control signs.12–14 In particu-ar, retroreflectors are considered the key element of a newpplication, head-mounted projective displays, which incor-orate great technology advancements that have been maden recent years.15–21

Several design criteria of retroreflection corner-cubes in-lude retroreflectance, brightness, angularity, and diver-ence. But there are tradeoffs among these criteria, and theost important feature depends on each application. For

xample, in traffic applications, the intentional slight mis-atch or flaws of retroreflection direction to incidence di-

ection and a wide angularity of retroreflected light14 are

091-3286/2007/$25.00 © 2007 SPIE

ptical Engineering 094002-

desirable. However, the most fundamental design issue ofthe retroreflection corner-cube for most applications is themaximization of the effective retroreflective area. The ef-fective retroreflective area is the practical area on the inci-dence facet through which the incident ray bundle passesand is properly retroreflected.2 The effective retroreflectivearea of a retroreflection corner-cube is varied for the inci-dence direction of the optical ray bundle. An optimal struc-ture for the retroreflection corner-cube that has the maxi-mum effective retroreflective area exists for a specificincidence direction. In practical applications, a sheet formto join many identical retroreflection corner-cubes is oftenfabricated. An optimal sheet structure having maximum ef-fective retroreflective area also exists.

This paper describes a geometric optics analysis for theoptimal design of retroreflection corner-cube sheets. Amathematical theorem about the optimal structure of asingle retroreflection corner-cube with maximum effectiveretroreflective area is proposed, and from this theorem afew analytic design formulas of optimal retroreflectioncorner-cubes are derived. Based on this geometric opticsanalysis, a novel packing method of elementary retroreflec-tion corner-cubes for constructing the optimal retroreflec-tion corner-cube sheet having a 100% effective retroreflec-tive area is proposed.

This paper is organized as follows. In Sec. 2, a geomet-ric optics analysis on the effective retroreflective area of asingle retroreflection corner-cube is presented, and themathematical theorem about the optimal retroreflectioncorner-cube structure is proved. Section 3 describes theproposed packing method of elementary retroreflectioncorner-cubes with a related geometric optics analysis. Sec-tion 4 contains concluding remarks.

2 Geometric Optics Analysis on the EffectiveRetroreflective Area of a SingleRetroreflection Corner-Cube

In general, a retroreflection corner-cube has the form of a

trigonal pyramid composed of four facets, as shown in Fig.

September 2007/Vol. 46�9�1

2rda9cc

Imcrcfl

FR

Kim and Lee: Optimal design of retroreflection corner-cube sheets…

O

. The ceiling facet is the incidence facet through whichays enter into the structure. Three other facets reflect inci-ent rays. For the complete retroreflection, three dihedralngles between adjacent reflection facets must be exactly0 deg. There are basically two-types of retroreflectionorner-cubes: the total internal reflection �TIR� corner-ube, and the metal-coated mirror reflection corner-cube.

The TIR corner-cube is based on the TIR on the facets.n this case, the facet is uncoated and the TIR conditionust be simultaneously satisfied on three facets so that a

omplete retroreflection occurs. In the metal-coated mirroreflection corner-cube, three reflection facets are metal-oated and the retroreflection occurs by three specular re-

ig. 1 �a� Specular reflection. �b� Diffusing reflection. �c�etroreflection.

ections on three metal-coated facets. In general, the TIR


corner-cube is considered to be more advantageous than themetal-coated corner cube because the TIR corner-cube ischeaper, more efficient, structurally simple, and easier tofabricate than the metal-coated corner-cube.

In this section, a geometric optics analysis on the effec-tive retroreflective area of a single retroreflection corner-cube is described. For convenience, the term “corner-cube”will be used to indicate a general retroreflection corner-cube. The analysis presented in this paper is so general thatit is equally pertinent to TIR corner-cubes and metal-coatedmirror reflection corner-cubes. Through this analysis, wecan prove a mathematical theorem and derive a few usefulformulas about the optimal corner-cube structure.

Assuming the dihedral angles between adjacent reflec-tion facets are exactly 90 deg, we can set the schematic ofa single corner-cube for the convenience of geometricanalysis as shown in Fig. 3�a�. Let the coordinate systemadopted here be considered as the local coordinate systemof the corner-cube. The corner-cube structure depicted onits local coordinate system is completely parameterized bythree positive real numbers, x0, y0, and z0, as shown in Fig.3�a�.

In the local coordinate system, the plane equation of theincidence facet of the corner-cube is given by

x

x0+

y

y0+

z

z0= 1, �1a�

where x, y, and z are in the regions of 0�x, 0�y, and 0�z, respectively. For convenience, x0, y0, and z0 are nor-malized as

� 1x0�2 + � 1

y0�2 + � 1

z0�2 = 1. �1b�

In Fig. 3�a�, two parallel rays, l1 and l2, are incident on theincidence facet of the corner-cube, where l1 is completelyretroreflected but l2 is not. For a clear comparison, the re-

flected rays of l1 and l2 are denoted by l̄1 and l̄2, respec-tively. An intuitive geometric analysis using mirror symme-try can be employed to analyze the respective internal raytraces of two rays. For easy understanding of this method, a2-D analogy of this ray-tracing analysis using mirror sym-metry is depicted in Fig. 3�b�, where a ray enters into thestructure as it passes point A, then the incident ray is re-fracted into the structure. The ray is reflected at point B and

¯

Fig. 2 Retroreflection corner-cube.

continues to strike the point C. This ray trace can be


ep

smc

rlardncb


O

quivalently described by the straight path connectingoints A, B, and C. Point C corresponds to the mirror-

ymmetric point C̄. This kind of geometric analysis usingirror symmetry may be useful for analyzing several spe-

ific optical geometries.22

We can apply this geometric analysis method using mir-or symmetry to the analysis of corner-cubes. At first, theozenge regions, P1, P2, and P3, are defined in the x-y, y-z,nd z-x planes, respectively, as shown in Fig. 3�c�. Theseegions together are called the “retroreflection region.” Theirection vector of the ray entering into the structure, de-oted by �kx ,ky ,kz�, is that of the ray in the medium of theorner-cube that already experienced the refraction on the

Fig. 3 �a� A single corner-cube depicted on itsof the geometric ray-tracing analysis using mirroP1, P2, and P3. �d� Three-dimensional ray-trasymmetry.

oundary of the corner-cube’s incidence facet and the sur-


round �i.e., air�. When a ray with the direction vector�kx ,ky ,kz� passes the retroreflection regions P1, P2, and P3,the direction vector components of the corresponding re-flected ray, kz, kx, ky, are changed to −kz, −kx, and −ky,respectively. Therefore, when the ray passes through P1,P2, and P3 successfully without an exception, the final re-flected ray has the direction vector of the retroreflection,�−kx ,−ky ,−kz�. Figure 3�d� shows the respective ray tracesof rays l1 and l2 with direction vectors �k1,x ,k1,y ,k1,z� and�k2,x ,k2,y ,k2,z�, respectively. Ray l1 passing by point Ameets point B in the region P2, the point C in the region P1,and finally, point D in the region P3, successively. Since all

ordinate system. �b� Two-dimensional analogymetry. �c� Definition of the retroreflection regionnalysis of a single corner-cube using mirror

local cor symcing a

cross points B, C, and D are inside the retroreflection re-


g

rorsi

�

ci�fzponbt�ive

n

If

Fgis


O

ion, the ray l1 is definitely retroreflected, so the reflected

ay l̄1 has the direction vector of �−k1,x ,−k1,y ,−k1,z�. On thether hand, ray l2 passing by point A� meets point B� in theegion P2, point C� in the region P3, and point D�, succes-ively. Point D� is outside the region P1. As a result, ray l2s not retroreflected. Since ray l2 passes through just regions

P2 and P3, the reflected ray l̄2 has the direction vector−k2,x ,−k2,y ,k2,z�.

We can analyze the effective retroreflective area of theorner-cube within the framework of Fig. 3�d�. Before do-ng this task, let us define the global coordinate systemx� ,y� ,z��. In the global coordinate system, the incidenceacet of the corner-cube is placed on the x�-y� plane and the� axis is set to direct the opposite direction of the apexoint of the corner-cube. Figure 4�a� shows the global co-rdinate system within the framework of the local coordi-ate system, where point C of the incidence facet is set toe on the positive part of the y� axis. Then we can establishhe relationship between this local coordinate systemx ,y ,z� and the global coordinate system �x� ,y� ,z�� shownn Fig. 4�b�. In the local coordinate system, the normalector n to the incidence facet of the corner-cube is param-terized by �s and �s as

= �1/x0,1/y0,1/z0� = �cos �s sin �s,sin �s sin �s,cos �s� .�2�

n Eq. �2�, cos �s, sin �s, cos �s, and sin �s can be solved

ig. 4 �a� Relationship between the local coordinate system and thelobal coordinate system. �b� An incident optical ray bundle with the

ncidence angle �� and the azimuth angle �� in the global coordinateystem.

or x0, y0, and z0. In the configuration shown in Fig. 4�a�, a


spatial point �x ,y ,z� in the local coordinate system is rep-resented by a point �x� ,y� ,z�� in the global coordinate sys-tem as

�x − xmy − ymz − zm

� = �cos �s cos �s − sin �s cos �s sin �ssin �s cos �s cos �s sin �s sin �s− sin �s 0 cos �s

��0 − 1 01 0 0

0 0 1��x�y�

z�� , �3�

where �xm ,ym ,zm� is the crossing point of the normal vectorn of Eq. �2� passing through the origin of the coordinatesystem and the incidence facet of the corner-cube of Eq.�1a�, which is given by

�xm,ym,zm� = �1/x0,1/y0,1/z0� �4�

and corresponds to the origin of the global coordinate sys-tem. In addition, a vector representation �kx ,ky ,kz� in thelocal coordinate system is connected to the vector represen-tation �kx� ,ky� ,kz�� in the global coordinate system throughthe relation

�kxkykz� = �cos �s cos �s − sin �s cos �s sin �ssin �s cos �s cos �s sin �s sin �s

− sin �s 0 cos �s�

��0 − 1 01 0 00 0 1

��kx�ky�kz�� . �5�

Figure 4�b� shows that an optical ray bundle with the inci-dence angle of �� and the azimuth angle of �� is incidenton the incidence facet of a corner-cube. The direction vec-tor �ki,x� ,ki,y� ,ki,z� � of the incident optical ray bundle is repre-sented by

�ki,x� ,ki,y� ,ki,z� � = �cos �� sin ��,sin �� sin ��,− cos �� . �6�

This incident ray bundle goes to and is refracted into thecorner-cube. The direction vector of the refracted raybundle �kx� ,ky� ,kz�� in a material with refractive index n isobtained by Snell’s law:

�kx�,ky�,kz�� =1

n�cos �� sin ��,sin �� sin ��,− �n2 − �sin ��2

= �cos � sin �,sin � sin �,− cos �� , �7�

where � and � are the refraction incidence angle and therefraction azimuth angle, respectively. This direction vectorof the refracted ray is represented as �kx ,ky ,kz� in the localcoordinate system through the relation of Eq. �5�.

As previously mentioned, corner-cubes can be classifiedinto TIR corner-cubes and metal-coated mirror reflectioncorner-cubes. We can see that in the case of the TIR corner-cube, to achieve the proper TIRs at the retroreflection re-gions P1, P2, and P3, the angles between the incident ray

and the normal vectors of the x-y�P1�, y-z�P2�, and z-x�P3�


pc

wrctHimc

ocvott

�

�

�

Ttrr

−

−

−

−


O

lanes must be larger than the TIR critical angle of theorner-cube material. These conditions are represented as

kx � cos �c, �8a�

ky � cos �c, and �8b�

kz � cos �c, �8c�

here �c is the TIR critical angle of the corner-cube mate-ial. In the case of the metal-coated mirror reflectionorner-cube, any ray with an arbitrary direction incident onhe metal-coated facet is specular-reflected unconditionally.ence, in this case, it can be stated that the critical angle �c

s zero. Therefore, Eqs. �8a�–�8c� are valid for both theetal-coated mirror reflection corner-cube and the TIR

orner-cube.Within this framework, we can proceed to the analysis

n the effective retroreflective areas of both types oforner-cubes. An incident refracted ray with the directionector �kx ,ky ,kz� originated from an incidence point �x̄ , ȳ , z̄�n the incident facet meets three cross points at three or-hogonal x-y, y-z, and z-x planes, which are given, respec-ively, as:

xxy,yxy,zxy� = �− kxz̄kz + x̄,− kyz̄kz + ȳ,0�, in the x − y plane,�9a�

xyz,yyz,zyz� = �0,− kyx̄kx + ȳ,− kzx̄kx + z̄�, in the y − z plane,and �9b�

xzx,yzx,zzx� = �− kxȳky + x̄,0,− kzȳky + z̄�, in the z − x plane.�9c�

he necessary condition for the proper retroreflection ishat three points of Eqs. �9a�–�9c� must be inside the ret-oreflection regions P1, P2, and P3, respectively, which isepresented by the following inequalities:

1 � −xxyx0

+yxyy0

� 1, �10a�

1 �xxyx0

+yxyy0

� 1, �10b�

1 � −zyzz0

+yyzy0

� 1, �11a�

1 �zyz +

yyz� 1, �11b�

z0 y0


− 1 � −zzxz0

+xzxx0

� 1, and �12a�

− 1 �zzxz0

+xzxx0

� 1. �12b�

By substituting Eqs. �9a�–�9c� into Eqs. �10a�, �10b�, �11a�,�11b�, �12a�, and �12b�, we can obtain a set of six inequali-ties with respect to �x̄ , ȳ� as

�kx/x0��kz/z0�

+�ky/y0��kz/z0�

− 1 � �� kx/x0��kz/z0� + 1 + �ky/y0��kz/z0� ��x̄/x0�+ �� ky/y0��kz/z0� + 1 + �kx/x0��kz/z0� ��ȳ/y0�

� 1 +�kx/x0��kz/z0�


, �13a�

−�kx/x0��kz/z0�


− 1 � �− � �kx/x0��kz/z0� + 1 + �ky/y0��kz/z0� ��x̄/x0� + �� ky/y0��kz/z0� + 1−

�kx/x0��kz/z0�

��ȳ/y0� � 1 − �kx/x0��kz/z0�+

�ky/y0��kz/z0�

, �13b�

0 � � �kz/z0��kx/x0� + �ky/y0��kx/x0� + 1�x̄/x0� � 2, �13c�0 � � �kz/z0��ky/y0� + �kx/x0��ky/y0� + 1�ȳ/y0� � 2, �13d�0 � � �kz/z0��kx/x0� + 1 − �ky/y0��kx/x0��x̄/x0� + 2�ȳ/y0� � 2, and

�13e�

0 � � �kz/z0��ky/y0� + 1 − �kx/x0��ky/y0��ȳ/y0� + 2�x̄/x0� � 2. �13f�Here the main task is to find the condition that maximizesthe area of �x̄ , ȳ� and satisfies Eqs. �13a�–�13f�.

The first case to be investigated is that of the symmetricTIR corner-cube. The refractive indices of the corner-cubematerial and the surround are set to 1.55 and 1, respec-tively. Figure 5 displays the effective retroreflective areasof the symmetric corner-cube �x0=�3, y0=�3, z0=�3� forseveral incidence directions. The calculated effective ret-roreflective areas are the images projected onto the inci-dence facet of the corner-cube in the areas of �x̄ , ȳ� thatsatisfy Eqs. �13a�–�13f�. Let the effective retroreflectivearea ratio � be defined by the ratio �%� of the effective

retroreflective area to the total area of the incidence facet of


ttefabsra


O

he corner-cube. Figure 5�a� shows the effective retroreflec-ive area ratio distribution for optical ray bundles with sev-ral incidence angles and azimuth angles. We can see thator this corner-cube, the maximum effective retroreflectiverea ratio of 67% is obtained for the incident optical rayundle with the normal incidence angle of �=0 deg. Ashown in Fig. 5�b�, the shape of the maximum effectiveetroreflective area is a hexagon. We can vary the incidence

Fig. 5 �a� Effective retroreflective area distribuThe maximum retroreflective area is obtained foareas for �b� ��=0 deg, ��=0 deg; �c� ��=30=30 deg, ��=270 deg; and �f� ��=50 deg, ��=

ngles and the azimuth angles of the incident ray bundle to


inspect the changes in the shape and the area ratio of theeffective retroreflective area. Some of the results are pre-sented in Figs. 5�c�–5�f�. From several simulations, we candetermine that the shapes of the effective retroreflectiveareas are classified into just lozenge and hexagon.

Based on the simulation result of the symmetric corner-

cube �x0=�3, y0=�3, z0=�3�, we can derive the more gen-eral optimal condition of corner-cube structure. The follow-

several incidence angles and azimuth angles.deg and ��=0 deg. The effective retroreflective�=90 deg; �d� ��=50 deg, ��=90 deg; �e� ��

g.

tion forr ��=0deg, �

270 de


i

dce�

comtatsTWdTr

D

ra

S

a

S

FSfatf


O

ng theorem settles up this task.Theorem: For an incident refracted ray bundle with the

irection vector of k= �kx ,ky ,kz� represented in the localoordinate system, the optimal corner-cube with maximumffective retroreflective area is that ofx0 ,y0 ,z0� /�x02+y02+z02= �−kx ,−ky ,−kz�.

Proof: It has already been shown with the aid of numeri-al simulation that the optimal corner-cube structure for theptical ray bundle of k= �−1,−1,−1� /�3 is that of the sym-etric corner-cube �x0 ,y0 ,z0�= ��3,�3,�3�. Let us remind

hat the ratio of the areas of two arbitrary triangles �ABCnd �DEF is invariant even with a scaling coordinateransformation such as �x ,y ,z�→ �ax ,by ,cz�. Let’s con-ider the two triangles �ABC and �DEF shown in Fig. 6.he triangle �ABC circumscribes the triangle �DEF.ithout loss of generality, we can let the three apexes be

enoted by A= �x1 ,y1 ,z1�, B= �x2 ,y2 ,z2�, and C= �0,0 ,0�.he dividing points, denoted by D, E, and F, are given,

espectively, by

= ��x1 + �1 − ��x2,�y1 + �1 − ��y2,�z1 + �1 − ��z2, E

= �x2,y2,z2�, and F = �x1,y1,z1� ,

espectively. The areas of the triangles �ABC and �DEFre given, respectively, by

ABC =1

2��x2y1 − y2x1�2 + �y2z1 − z2y1�2 + �x2z1 − z2x1�2,

�14a�

nd

DEF =�� − � + �1 − ��

2

��x2y1 − y2x1�2 + �y2z1 − z2y1�2 + �x2z1 − z2x1�2.�14b�

rom Eqs. �14a� and �14b�, we can confirm that the ratio ofDEF and SABC is invariant for the scaling coordinate trans-ormation �x ,y ,z�→ �ax ,by ,cz�. Since general polygonsre composed of elementary triangles, we can conclude thathe ratio of the areas of two arbitrary polygons is invariant

Fig. 6 Two triangles, �ABC and �DEF.

or the scaling coordinate transformation.


Now we are prepared to find the optimal corner-cubestructure for an arbitrary incident optical ray bundle. Let kx�,ky�, kx�, x̄�, ȳ�, and z̄� be defined by kx�=kx /x0, ky�=ky /y0,kz�=kz /z0, x̄�= x̄ /x0, ȳ�= ȳ /y0, and z̄�= z̄ /z0, respectively. Bysubstituting these variables into Eqs. �13a�–�13f�, the fol-lowing inequalities are obtained:

kx�

kz�+

ky�

kz�− 1 � � kx�

kz�+

ky�

kz�+ 1x̄� + � kx�

kz�+

ky�

kz�+ 1ȳ�

�kx�

kz�+

ky�

kz�+ 1, �15a�

−kx�

kz�+

ky�

kz�− 1 � �− kx�

kz�+

ky�

kz�− 1x̄� + �− kx�

kz�+

ky�

kz�+ 1ȳ�

� −kx�

kz�+

ky�

kz�+ 1, �15b�

0 � �1 + ky�kx�

+kz�

kx��x̄� � 2, �15c�

0 � � kx�ky�

+ 1 +kz�

ky��ȳ� � 2, �15d�

0 � �1 − ky�kx�

+kz�

kx��x̄� + 2ȳ� � 2, and �15e�

0 � 2x̄� + �− kx�ky�

+ 1 +kz�

ky��ȳ� � 2. �15f�

Comparing Eqs. �15a�–�15f� and Eqs. �13a�–�13f�, we candetermine that Eqs. �15a�–�15f� have the same form as Eqs.�13a�–�13f� when �x0 ,y0 ,z0� in Eqs. �13a�–�13f� is set to��3,�3,�3�. Therefore, the optimal value of �kx� ,ky� ,kz�� tomaximize the effective retroreflective area ratio is given by�kx� ,ky� ,kz��= t�−1,−1,−1�. This condition is equivalent tothe relation

�− kx,− ky,− kz� = t�x0,y0,z0� ↔ �x0,y0,z0�/�x02 + y02 + z02= �− kx,− ky,− kz� . �15g�

This concludes the proof of the theorem.With the condition �x0 ,y0 ,z0� /�x02+y02+z02= �−kx ,−ky ,

−kz� satisfied, the inequalities of Eqs. �15a�–�15f� read as

1

3� x̄� + ȳ� � 1 ↔

1

3�

x̄

x0+

ȳ

y0� 1, �16a�

− 1 � − x̄� + ȳ� � 1 ↔ − 1 � −x̄

x0+

ȳ

y0� 1, �16b�

0 � x̄� �2

↔ 0 �x̄

�2

, �16c�

3 x0 3


0

0

0

TF6

c

tiasar

x

t

t

w

t

�

Tf�E


O

� ȳ� �2

3↔ 0 �

ȳ

y0�

2

3, �16d�

� x̄� + 2ȳ� � 2 ↔ 0 �x̄

x0+ 2

ȳ

y0� 2, and �16e�

� 2x̄� + ȳ� � 2 ↔ 0 � 2x̄

x0+

ȳ

y0� 2. �16f�

he area constrained by the above inequalities is plotted inig. 7. In this case, the effective retroreflective area ratio is7%.

By extending the theorem, we can obtain the followingorollary about the corner-cube design formulas.

Corollary: When an optical ray bundle with the refrac-ion incidence angle � and the refraction azimuth angle � isncident on the corner-cube in the global coordinate system,s shown in Fig. 4, we can build up the functional relation-hip between the optimal corner-cube structure �x0 ,y0 ,z0�nd the incidence direction parameter �� ,�� of an opticalay bundle as follows:

0 =1

, �17a�

cos��s��,��sin��s��,��

q. �5� and using the relation of Eq. �2�, we can obtain


y0 =1

sin��s��,��sin��s��,��, and �17b�

z0 =1

cos��s��,��, �17c�

where �s and �s are obtained with respect to � and � from

Fig. 7 Optimal corner-cube structure and its effective retroreflectivearea.

the following relations:

an �s =

3 cos��2

+ ��sin � +�9 cos2��2

+ ��sin2 � + 8 cos2 �2 cos �

, �18a�

an �s=

sin��2

+ ��sin �sin �st

+��sin��2


2 + 42

, �18b�

here t is given by t = cos �s sin �s cos��2 + ��sin �+ cos �s cos �s cos � . �18c�

Proof: From the theorem, we know that the condition ofhe optimal corner-cube is given by

kx,ky,kz� = − t�x0,y0,z0�, where t is give by t

= 1/�x02 + y02 + z02. �19�

he direction vector of the optical ray bundle with the re-raction incidence angle � and the refraction azimuth angle

is given by Eq. �7�. By substituting Eqs. �19� and �7� into

t�−

1

cos �s sin �s

−1

sin �s sin �s

−1

cos �s

� = �cos �s − sin �s 0sin �s cos �s 00 0 1 �� cos �s 0 sin �s0 1 0

− sin �s 0 cos �s��0 − 1 01 0 0

0 0 1�

��cos � sin �sin � sin �− cos �

� . �20�


B

�A

t

t

a

t

Fca

x

a

z

T

�


O

y further arrangement, Eq. �20� reads as

− 2t

sin �s

�tan �s − 1tan �s� tsin �s−

t

cos �s

� = �cos �s cos��2 + ��sin � − sin �s cos �

sin��2

+ ��sin �− sin �s cos��2 + ��sin � − cos �s cos � � . �21�

t this stage, we can find the solution to the above equations, �s and �s, as follows:

an �s =

3 cos��2

+ ��sin � +�9 cos2��2

+ ��sin2 � + 8 cos2 �2 cos �

, �tan �s � 0� �22a�

= cos �s sin �s cos��2 + ��sin � + cos �s cos �s cos � , �22b�nd

an �s =

sin��2


+��sin��2


�2 + 42

, �tan �s � 0� . �22c�

rom Eq. �2�, the optimal corner-cube structure for the in-ident optical ray bundle with the refraction incidencengle � and the refraction azimuth angle � is derived as

0 =1

cos��s��,��sin��s��,��, �23a�

y0 =1

sin��s��,��sin��s��,��, �23b�

nd

0 =1

cos��s��,��. �23c�

his concludes the proof of the corollary.

cos �s

sin �s

sin �s

sin �s


Using the corollary, we can obtain the design formulasabout the lengths of edges l1, l2, and l3 of the corner-cube,and the height h of the apex point, respectively, by

l1 = �x02 + y02, �24a�l2 = �y02 + z02, �24b�l3 = �z02 + x02, and �24c�h =

1

�� 1x0�2 + � 1

y0�2 + � 1

z0�2 . �24d�

In addition, we can extract the refraction incidence angle �and the refraction azimuth angle � of the optimal incidentoptical ray bundle to a given corner-cube structure by usingthe inversion of the corollary given by

cos � sin �

sin � sin �

− cos �� =

� 0 1 0− 1 0 00 0 1

��cos �s cos �s cos �s sin �s − sin �s− sin �s cos �s 0sin �

scos �

ssin �

ssin �

scos �

s

��−

1

cos �s

sin �s

−1

sin �s

sin �s

−1

cos �s

�� 1 �2 + � 1 �2 + � 1 �2

, �24e�

cos �s


wr

to

r=st


O

here �s and �s are the functions of the corner-cube pa-ameters �x0 ,y0 ,z0� defined by Eq. �8�.

By substituting Eq. �24e� into Eq. �7�, we can also findhe incidence angle �� and the azimuth angle �� of theptimal incident optical ray bundle.

Next we will present an example to be compared to theesults shown in Fig. 5. For the incidence angle ��50 deg and the azimuth angle ��=270 deg, the optimal

tructure parameters x0, y0, and z0 of the corner-cube struc-

Fig. 8 �a� Effective retroreflective area distribuThe maximum retroreflective area is obtained fflective areas for �b� ��=0 deg, ��=0 deg; �c� ��=30 deg, ��=270 deg; and �f� ��=50 deg, ��

ure with the refractive index of 1.55 is obtained by

ptical Engineering 094002-1

x0=1.5228, y0=1.5228, and z0=2.6965, respectively, whichare calculated from Eq. �7� and Eqs. �17a�–�17c� and �18a�–�18c� of the corollary. Figure 8�a� shows the effective ret-roreflective area ratio distribution for optical ray bundleswith several incidence angles and azimuth angles. As ex-pected from the described theory, we can see that thiscorner-cube has the maximum effective retroreflective arearatio of 67% when the incidence angle � and the azimuthangle � of the incident optical ray bundle are �=50 deg

several incidence angles and azimuth angles.50 deg and ��=270 deg. The effective retrore-eg, ��=90 deg; �d� ��=50 deg, ��=90 deg; �e�eg.

tion foror ��=�=30 d=270 d

and �=270 deg. The retroreflective areas obtained for sev-


es5t

¯

Fp


O

ral cases of the incidence angle and the azimuth angle arehown in Figs. 8�b�–8�f� for comparison with Figs.�b�–5�d� and 5�f�. In the optimal case shown in Fig. 8�f�,

ig. 9 Regions actually contributing the retroreflection in �a� the x-ylane, �b� the y-z plane, and �c� the z-x plane.

he shape of the retroreflective area is also a hexagon.


3 Optimal Packing Method of RetroreflectionCorner-Cube Sheets

In practice, retroreflection corner-cubes are used in a sheetform with the periodic array of elementary corner-cubes.2

In this section, based on the corollary derived in Sec. 2, anovel packing method of optimally designed elementarycorner-cubes for constructing optimal corner-cube sheets isproposed. As analyzed in Sec. 2, the maximum effectiveretroreflective area ratio � of a single corner-cube is 67%.However, it can be shown that by constructing the sheetform of tightly and periodically arranged identical corner-cubes, we can obtain the maximum effective retroreflectivearea ratio of 100%. This optimization problem can besettled with an additional simple geometric analysis contin-ued from Sec. 2.

Let us inspect the meaningful regions that actually con-tribute the retroreflection at the retroreflection regions, P1,P2, and P3. From Eq. �1a�, we know that

z/z0 = 1 − �x̄/x0 + ȳ/y0� . �25�

By substituting Eq. �25� into Eqs. �9a�–�9c�, the followingrelations hold:

�xxy/x0,yxy/y0,zxy/z0� = �− 1 + 2x̄/x0 + ȳ/y0,− 1 + x̄/x0+ 2ȳ/y0,0� in the x-y plane, �26a�

�xyz/x0,yyz/y0,zyz/z0� = �0,− x̄/x0 + ȳ/y0,1 − 2x̄/x0− ȳ/y0� in the y-z plane, �26b�

and

�xzx/x0,yzx/y0,zzx/z0� = �− ȳ/y0 + x̄/x0,0,1 − x̄/x0− 2ȳ/y0� in the z-x plane. �26c�

By manipulating Eq. �26a� as

� x̄/x0ȳ/y0

= 13� 2 − 1

− 1 2��xxy/x0 + 1

yxy/y0 + 1

=1

3� 2xxy/x0 − yxy/y0 + 1

− xxy/x0 + 2yxy/y0 + 1

�27�

and substituting Eq. �27� into Eqs. �16a�–�16d� and �16f�,we can obtain the inequalities that indicate the image of theeffective retroreflective area projected onto the x-y plan:

− 1 � xxy/x0 + yxy/y0 � 1, �28a�

− 1 � 2xxy/x0 − yxy/y0 � 1, and �28b�

− 1 � 2yxy/y0 − xxy/x0 � 1. �28c�

From the symmetry, we can also obtain the images of theeffective retroreflective area projected onto the y-z planeand the z-x plane, respectively, as:

− 1 � yyz/y0 + zyz/z0 � 1, �29a�

− 1 � 2yyz/y0 − zyz/z0 � 1, and �29b�


−

a

−

−

−

TpscTwhdtrsFcr

dc


O

1 � 2zyz/z0 − yyz/y0 � 1; �29c�

nd in the z-x plane, we obtain

1 � zzx/z0 + xzx/x0 � 1, �30a�

1 � 2zzx/z0 − xzx/x0 � 1, and �30b�

1 � 2xzx/x0 − zzx/z0 � 1. �30c�

he regions in the x-y plane, the y-z plane, and the z-xlane are plotted in Figs. 9�a�–9�c�. From the above analy-is, we know that the area near the apex points of theorner-cube do not actually contribute the retroreflection.he effective volumetric region of the corner-cube throughhich rays pass can be sufficiently constrained by the poly-edron indicated in Fig. 10�a�. Theoretically, this polyhe-ron structure has a 100% effective retroreflective area ra-io while conventional sheets2 have an effectiveetroreflective area ratio less than 67%. Fortunately, thishape is appropriate for the tight and periodic arrangement.igure 10�b� illustrates the periodic arrangement of theorner-cubes with a 100% effective retroreflective areaatio.

In Fig. 11, some design examples of the corner-cubes forifferent incident optical ray bundles are presented for a

Fig. 10 �a� Polyhedron structure having the efarrangement of the corner-cubes with the effec

omparison. They include the optimal corner-cube struc-


tures for the incident optical ray bundles with the incidenceangle ��=0 deg and the azimuth angle ��=0 deg; the inci-dence angle ��=30 deg and the azimuth angle ��=90 deg;the incidence angle ��=50 deg and the azimuth angle ��=90 deg; and the incidence angle ��=30 deg; and the azi-muth angle of ��=270 deg.

4 Conclusion

In conclusion, an optimal retroreflection corner-cube sheetstructure with an effective retroreflective area ratio of 100%has been designed based on a geometric optics analysis. Amathematical theorem and a corollary about the optimalcorner-cube structure were also derived, and the analyticdesign formulas of optimal corner-cube were obtained. Anovel packing method was proposed of optimally designedelementary corner-cubes for constructing optimal retrore-flection corner-cube sheets with the effective retroreflectivearea ratio of 100%. Since the retroreflection corner-cubesheet structure considered in this paper is the perfect ret-roreflector, the proposed structure is more promising fordisplay applications such as head-mounted projective dis-plays that require exact retroreflection than for conven-tional traffic safety applications, wherein intentional flaws

retroreflective area ratio of 100%. �b� Periodicroreflective area ratio of 100%.

fectivetive ret

in retroreflection are a key design factor.


R

1

1

1

1

1

1

0 deg,


O

eferences

1. Commission International de L’Eclairage, “Retroreflection: definitionand measurement,” Publication CIE 54.202001 �2001�.

2. R. B. Nilsen and X. J. Lu, “Retroreflection technology,” Proc. SPIE5616, 47–60 �2004�.

3. D. C. O’Brien, G. E. Faulkner, and D. J. Edwards, “Optical propertiesof a retroreflecting sheet,” Appl. Opt. 38, 4137–4144 �1999�.

4. G. H. Seward and P. S. Cort, “Measurement and characterization ofangular reflectance for cube-corners and microspheres,” Opt. Eng. 38,164–169 �1999�.

5. S. E. Segre and V. Zanza, “Mueller calculus polarization change inthe cube-corner retroreflector,” J. Opt. Soc. Am. A 20, 1804–1811�2003�.

6. W. H. Venable and N. L. Johnson, “Unified coordinate system forretroreflectance measurements,” Appl. Opt. 19, 1236–1241 �1980�.

7. K. Hyyppa, “Signal response of a laser beam scanner,” Opt. Eng. 33,2770–2776 �1994�.

8. O. Nakamara, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, and T.Nakamata, “A laser tracking robot-performance calibration systemusing ball-seated bearing mechanisms and a spherically shaped cat’seye retro-reflector,” Rev. Sci. Instrum. 65, 1006–1011 �1994�.

9. B. E. Hines, “Optical truss and retroreflector modeling for picometerlaser metrology,” Proc. SPIE 1947, 198–208 �1993�.

0. R. A. Chipman, J. Shamir, H. J. Caulfield, and Q. Zhou, “Wave-frontcorrecting properties of corner-cube arrays,” Appl. Opt. 27, 3203–3209 �1996�.

1. K. S. J. Pister, D. S. Gunawan, and L. Lin, “Micromachined cornercube reflectors as a communication link,” Sens. Actuators, A 46–47,580–583 �1995�.

2. ASTM International, “Standard specification for retroreflective sheet-ing for traffic control,” Publication ASTM D 4956-01 �2001�.

3. 3M Corp., Product Catalog for Traffic Control Materials, 3M TrafficControl Materials Division, St. Paul, MN, http://www.3M.com/tcm�2003�.

4. T. L. Hoopman, “Cube-corner retroreflective articles having wide an-gularity in multiple viewing planes,” U.S. Patent 4,588,258 �1986�.

Fig. 11 Optimal corner-cube sheet structures f��=0 deg; �b� ��=30 deg, ��=90 deg; �c� ��=5

5. R. Fisher, “Head-mounted projection display system featuring beam


splitter and method of making same,” U.S. Patent 5,572,229 �1996�.16. H. Hua, A. Girardot, C. Gao, and J. P. Rolland, “Engineering of

head-mounted projective displays,” Appl. Opt. 39, 3814–3824�2000�.

17. J. Fergason, “Optical system for head mounted display using retro-reflector and method of displaying an image,” U.S. Patent 5,621,572�1997�.

18. R. Kijima and T. Ojika, “Transition between virtual environment andworkstation environment with projective head-mounted display,”Proc. IEEE VR 1997, 130–137 �1997�.

19. M. Inami, N. Kawakami, D. Sekiguchi, Y. Yanagida, T. Maeda, andS. Tachi, “Visuo-haptic display using head-mounted projector,” Proc.IEEE VR 2000, 233–240 �2000�.

20. H. Hua, C. Gao, F. Biocca, and J. P. Rolland, “An ultra-light andcompact design and implementation of head-mounted projective dis-plays,” Proc. of IEEE VR 2001, 175–182 �2001�.

21. H. Hua, L. Brown, and C. Gao, “SCAPE: supporting stereoscopiccollaboration in augmented and projective environments,” IEEEComput. Graphics Appl. 24, 66–75 �2004�.

22. H. Kim and B. Lee, “Geometric optics analysis of light transmissionand reflection characteristics of metallic prism sheets,” Opt. Eng. 45,084004 �2006�.

Hwi Kim received the B.S., M.S., and Ph.D.degrees in the School of Electrical Engi-neering from Seoul National University, Ko-rea, in 2001 and 2003, and 2007, respec-tively. He is currently a post-doc researcherin the National Creative Research Centerfor Active Plasmonics Applications Systems,Seoul National University. His primary re-search interests are in the areas of plas-monics, nanophotonics, and transformationoptics.

incident optical ray bundles with �a� ��=0 deg,��=90 deg; and �d� ��=30 deg, ��=270 deg.

or the


F�o


O

Byoungho Lee received the B.S. and M.S.degrees in 1987 and 1989, respectively,from Seoul National University, Korea, inelectronics engineering. He received aPh.D. degree in 1993 from the University ofCalifornia at Berkeley in electrical engineer-ing and computer science. In 1994, hejoined the faculty of the School of ElectricalEngineering, Seoul National University,where he is now an associate professor. Hebecame a Fellow of the SPIE in 2002 and a

ellow of the OSA in 2005. He is now serving as Director-at-largeBoard of Directors� of OSA and a member of the Award Committee

f Board of Directors of OSA. He is also serving as a member of the


Engineering, Science and Technology Policy Committee of SPIE.He has served as a committee member for various internationalconferences. He has authored or coauthored more than 190 papersin international journals and more than 300 international conferencepapers. In 1999, his laboratory was honored as a National ResearchLaboratory by the Ministry of Science and Technology of Korea. In2002 he received the Presidential Young Scientist Award of Korea.In 2007, his laboratory is honored as a National Creative ResearchCenter for Active Plasmonics Applications Systems by the Ministryof Science and Technology of Korea. His research fields are holo-gram applications, three-dimensional displays, optical fiber gratings,plasmonics, and nanophotonics. Currently he is on the editorialboards of Applied Optics, Optical Fiber Technology, and Journal of

the Society for Information Display.


Optimal design of retroreﬂection corner-cube sheets by...

Documents

Transcript of Optimal design of retroreﬂection corner-cube sheets by...