a practical method to design reflector-based light...

Research ArticleA Practical Method to Design Reflector-Based Light-EmittingDiode Luminaire for General Lighting

Jinren Yan

Department of Optoelectronics and Communication EngineeringYunnan Open University (Yunnan Vocational amp Technical College of National Defense Industry) Xuefu Road 113Kunming 650223 China

Correspondence should be addressed to Jinren Yan renyj0720163com

Received 2 August 2018 Accepted 3 October 2018 Published 18 October 2018

Academic Editor Somenath N Sarkar

Copyright copy 2018 Jinren Yan This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A reflector-based light-emitting diode (LED) luminaire structure that can achieve a large cut-off angle for general lighting ispresented in this work The proposed lighting unit mainly consists of a spherical reflector and a primary packaging lens thatcontains an aspheric surface and a spherical surface The light rays emitted from the LED light source are well controlled by thespherical reflector and the aspheric surface of the lens for the purpose of obtaining a uniform illumination on the target surfaceBoth the ideal Lambertian LED and non-Lambertian LED light sources were employed to validate the proposed structure and theperformance of the designed lighting units was analyzed by optical simulationThe results show that the light utilization efficienciesand the estimated uniformities are 9296 and 9111 for ideal Lambertian LED-based lighting unit and 9331 and 9164 for non-Lambertian LED-based lighting unit respectively Further analysis shows that the tolerances of horizontal vertical and rotationaldeviation of the both lighting units were about 20 mm 10 mm and 10∘ respectively

1 Introduction

After years of development light-emitting diodes (LEDs)have been widely used in the fields of indoor lighting roadlighting automobile headlights backlight of liquid crystaldisplay and so forth [1ndash4] Different from the conventionallight sources such as the incandescent lamp and the flu-orescent lamp the LED light source is a Lambertian-likeemitter and the light intensity distribution curve (LIDC)follows a cosine power function [5] Thus a secondaryoptical element is generally needed to redistribute the lightenergy emitted from LED to meet the specific illuminationrequirements Over the past decade the freeform lenses andthe freeform reflectors have been widely used in the design ofLED secondary optics in illumination designs [5ndash10] In realapplications especially in outdoor lighting the reflector isconsidered as an important optical component which can notonly be used to modulate the light energy emitted from thelight source to achieve specific lighting but also be employedas the protector of the lighting equipment However it is

still a challenge to faultlessly control all of the light raysemitted from the LED chip by a single freeform reflectoralone The optimal cut-off angle of the reflector is limited bythe LIDC of the LED light source for the case of uniformlighting [11] For example for an ideal Lambertian LED thelargest cut-off angle of the freeform reflector is about 45∘ foruniform illumination [12] As the cut-off angle of the reflectorincreased larger than 45∘ the uniformity will deteriorateSuch restriction put up some obstacles in using LEDs in thefield of general lighting where the cut-off angle of the lightingunit typically defined between 55∘ and 65∘ [12 13]

In this work a reflector-based LED luminaire structurewhich can achieve a large cut-off angle for general lighting isdeveloped and the method for the design of the developedlighting unit structure is presented in detail The structureof the lighting unit only consists of a spherical reflector anda primary lens which comprises an aspheric surface and aspherical surface In the proposed lighting unit the reflectoris employed to modulate the side rays with an emitting anglelarger than the specified cut-off angle of the lighting unit and

HindawiAdvances in OptoElectronicsVolume 2018 Article ID 7138592 10 pageshttpsdoiorg10115520187138592

2 Advances in OptoElectronics

Aspheric surface

Spherical reflectorSpherical surface

x

O

R

z

cut i

ri

i

ri

O

Figure 1 Geometrical diagram of the developed LED lighting unit

the aspheric surface of the primary lens is used to modulatethe light rays with an emitting angle smaller than the cut-offangle Then the uniform illumination on the target surfaceis formed by superposition of the light energy reflected bythe spherical reflector and the light energy modulated by theaspheric surface of the lens

2 Design Method

In order to obtain a uniform far-field illumination on thetarget surface the LIDC of the lighting equipment shouldcomply with the following expression [14]

119868119897119904 (120593) = 1198681198970cos3 (120593) (1)

where 119868119897119904(120593) represents the luminous intensity distributionof the lighting equipment and 1198681198970 is the luminous intensityat 120593 = 0∘ Figure 1 shows the geometrical structure of thedeveloped lighting unit As depicted in Figure 1 the proposedstructure comprises a spherical reflector and a primary lensOne can note that the central part of the lens is an asphericsurface and the side part of the lens is a spherical surfacewith the center located at the origin O of the coordinatesThe LED light source is immersed into the primary lens andalso positioned at the origin O of the coordinates [15ndash17]1198741015840 is the center of the spherical reflector and R denotesthe radius of the reflector As can be noted from Figure 1the opening dimension of the spherical reflector equals itsdiameter 120593119888119906119905 represents the cut-off angle of the lighting unitand it is generally given by designers 120593119894 denotes the anglebetween the light ray that will be refracted by the asphericsurface of the lens and z-axis and 1205931015840119894 is the angle between thecorresponding refracted light ray and z-axis One can notethat since the side part of the lens is spherical surface the lightrays will directly pass through the spherical surface of thelens without deflection Thus 120593119903119894 denotes the angle betweenthe light ray that will be reflected by the spherical reflectorand z-axis and 1205931015840119903119894 is the angle between the correspondingreflected light ray and z-axis

x

O

z

ri

ri

ri+1

ri+1

i+1

i

i

cut

d

rmin

rmin

rmax

Figure 2 Diagrammatic sketch of the light energy distribution ofthe lighting unit

From the geometrical relationship illustrated in Figure 1the relationship between the light reflection angle 1205931015840119903119894 andthe light-emitting angle 120593119903119894 of the side rays can be writtenas

1205931015840119903119894 = 120587 minus 120593119903119894 minus 2acrsin [ sin (120593119903119894)tan (120593119888119906119905)] (2)

As illustrated in Figure 1 the light-emitting angle 120593119903119894ranges from 120593119888119906119905 to 1205872 Consider the reflection characteristicof the spherical reflector in this case the minimum value1205931015840119903119898119894119899 and the maximum value 1205931015840119903119898119886119909 of the light reflectionangle 1205931015840119903119894 can be obtained when the light-emitting angle 120593119903119894reaches 1205872 and 120593119888119906119905 respectively Since the reflection angleof the light ray reflected by the spherical reflector is invariantfor a specified emitting angle in this case the uniform lightingon the target surface should be achieved by redistributing thelight energy passing through the aspheric surface of the lensThe right side of Figure 2 shows the diagrammatic sketch ofthe light energy distribution of the developed lighting unitAccording to the geometrical relationship shown in Figure 2one can note that the light reflection angle 1205931015840119903119898119886119909 equals120593119888119906119905 As shown in Figure 2 the light energy emitted fromthe LED light source can be divided into three parts thefirst part contains the light energy in the angle range of 0∘to 120593119889 and the light energy in this part is refracted to theangle range of 0∘ to 1205931015840119903119898119894119899 by the aspheric surface of the lensthe second part contains the light energy in the angle rangeof 120593119889 to 120593119888119906119905 and the third part contains the light energy inthe angle range of 120593119888119906119905 to 1205872 The light energy in these twoparts is modulated to the angle range of 1205931015840119903119898119894119899 to 1205931015840119903119898119886119909 bythe aspheric surface of the lens and the spherical reflectorrespectively

Advances in OptoElectronics 3

Based on the contents aforementioned and the law ofenergy conservation the following two equations can beobtained respectively

2120587int1205931198890119868119904 (120593) sin (120593) 119889120593 = 2120587int120593

1015840

119903min

0119868119897119904 (120593) sin (120593) 119889120593 (3)

2120587int120593119888119906119905120593119889

119868119904 (120593) sin (120593) 119889120593 + 2120587120588int1205872120593119888119906119905

119868119904 (120593) sin (120593) 119889120593= 2120587int120593119888119906119905

1205931015840119903min

119868119897119904 (120593) sin (120593) 119889120593(4)

where120588 denotes the reflectivity of the reflector in thiswork itis supposed to be 95 119868119904(120593) represents the luminous intensitydistribution of the LED light source 119868119897119904(120593) represents theluminous intensity distribution of the lighting unit as it isexpressed by (1)

To construct the aspheric surface of the lens the relation-ship between the light-emitting angle 120593119894 and the correspond-ing refraction angle 1205931015840119894 should be obtained at first as shownin the left side of Figure 2 It is noted that 120593 shown in thefigure represents the preset angle increment of the incidentrays and it is set as 025∘ in this work Then 120593119894 = 119894 120593(119894 = 0 1 2 119873) can be obtained easily Using the lawof energy conservation and the edge-ray principle [18] thefollowing two equations are obtained respectively

2120587int1205931198940119868119904 (120593) sin (120593) 119889120593 = 2120587int120593

1015840

119894

0119868119897119904 (120593) sin (120593) 119889120593 (5)

2120587int120593119894120593119889

119868119904 (120593) sin (120593) 119889120593 + 2120587120588int1205872120593119903119894

119868119904 (120593) sin (120593) 119889120593

= 2120587int1205931015840 1198941205931015840119903min

119868119897119904 (120593) sin (120593) 119889120593(6)

It is worth noting that (6) implies the condition that1205931015840119894 equals 1205931015840119903119894 Thus (2) could be substituted into (6) toreplace 1205931015840119894 Substituting 120593119894 into (5) and (6) respectively thecorresponding 1205931015840119894 for the angle range of 0∘ to 120593119889 and theangle range of 120593119889 to 120593119888119906119905 could be obtained After obtainingthe relationships between 120593119894 and 1205931015840119894 the aspheric surface ofthe primary lens can be easily constructed using the methoddescribed in [15]

3 Design Examples

Using the method described in Section 2 an LED lightingunit with a cut-off angle of 120593119888119906119905 = 60∘ was designed as anexample Substituting 1205872 and 120593119888119906119905 into (2) the minimumangle 1205931015840119903min and maximum angle 1205931015840119903119898119886119909 of the reflectedlight rays are calculated as 195∘ and 60∘ respectively Inthe simulation it is assumed that the LED light source hasa dimension of 10 mmtimes10 mm The material of the lensis assumed to be PMMA (Polymethyl Methacrylate) with arefractive index of n = 149 It is assumed that the LED lightsource emits 100 lm light energy and 10 million rays wereused for simulation The lighting unit is 5 m away from thetarget surface

31 Design with Ideal Lambertian LED For an ideal Lamber-tian LED the LIDC of the light source can be expressed as

119868119904 (120593) = 1198680 cos (120593) (7)

where 1198680 denotes the luminous intensity at120593 =0∘ Substituting(1) and (7) into (3) and (4) the parameter 120593119889 and the ratio of1198680 to 1198681198970 can be attained

120593119889 = arcsin [ tan (1205931015840119903min)tan (120593119888119906119905) radic(1 minus 120588) sin2 (120593119888119906119905) + 120588] (8)

11986801198681198970 =tan2 (1205931015840119903min)sin2 (120593119889) (9)

Using (8) and (9) the value of parameter 120593119889 and the ratio of1198680 to 1198681198970 are about 116∘ and 308 in this caseThen substituting(1) (2) (7) (8) and (9) into (5) and (6) one can find thefollowing two expressions respectively

1205931015840119894 = arctan(radic 11986801198681198970 sin2 (120593119894)) 120593119894 isin [0 120593119889] (10)

11986801198681198970 [sin2 (120593119894) minus sin2 (120593119889) + 120588 (1 minus sin2 (120593119903119894))]

minus tan2 (120593119903119894 + 2 arcsin( sin (120593119903119894)tan (120593119888119906119905))

+ tan2 (1205931015840119903min)) = 0 120593119894 isin (120593119889 120593119888119906119905](11)

It can be seen that for the angle range of 0∘ to 120593119889 1205931015840119894can be obtained easily by iteratively substituting 120593119894 into (10)However the relationship between 120593119894 and 1205931015840119894 in the anglerange of 120593119889 to 120593119888119906119905 is expressed by (11) which is a nonlinearequation and is solved by using numerical method [19] Itshould be emphasized that solving (11) can only get therelationship between 120593119894 and 120593119903119894 the corresponding refractionangle 1205931015840119894 of 120593119894 should be obtained by substituting 120593119903119894 into (2)

Figure 3 shows the calculated geometrical profile ofthe proposed lighting unit for the ideal Lambertian LEDlight source and the corresponding geometrical model Thesimulation results of the developed lighting unit are shownin Figure 4 Figure 4(a) shows the irradiance map of thedeveloped lighting unit It can be seen that a uniformillumination is obtained on the target surface The dot line inthe figure represents the slicing position of the slicing chartFigure 4(b) shows the light intensity distribution of the idealLambertian LED and the designed lighting unit it can beseen that the intensity peak of the designed lighting unit islocated at about plusmn 58∘ with the light energy sheared almostat plusmn 60∘ the simulation results are in good agreement withexpectations

The light utilization efficiency which is defined as theratio of the light energy within the circle region with theradius of 8660 mm to the light energy emitted from the LEDlight source is about 9296 The illuminance uniformitywhich is defined as the ratio of the minimum illuminance tothe average illuminance (119880 = 119864119898119894119899119864119886V119890) on the target regionis estimated as 9111


300

25

50mm

10mm

minus30 0

(a) (b)

Figure 3 (a) Geometrical profile of the designed lighting unit for the ideal Lambertian LED light source and (b) the correspondinggeometrical model

0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0

Lambertian LEDDeveloped lighting unit

(b)

Figure 4 (a) Irradiance map and (b) the angular intensity distribution of the developed lighting unit for the ideal Lambertian LED

32 Design with Non-Lambertian LED Figure (5) shows theLIDC of a non-Lambertian LED light source [13] As shownin the figure the LIDCof the LEDcannot be simply expressedby a simple cosine power function In general the LIDCsof the non-Lambertian LED light sources can be fitted bya polynomial function [20] Gaussian function or cosinefunction [21] In this study for the purpose of simple thepolynomial function was employed to represent the LIDC ofthe non-Lambertian LED light source

119868119904 (120593) = 1198680 (1198860 + 11988611205932 + 11988621205934 + 11988631205936 + 11988641205938) (12)

where 119886119894 (119894 = 0 1 2 3 4) represent the coefficients of thepolynominal function 1198680 is the luminous intensity at 120593 = 0∘After curve fitting the coefficients of the polynomial function

used to represent the LIDC of the non-Lambertian LEDshowed in Figure 5 are shown in Table 1 and the attainedLIDC model is shown in Figure 5

Substituting (1) and (12) into (3) and (4) the value ofparameter 120593119889 and the ratio of 1198680 to 1198681198970 can be obtained theobtained results are shown in Table 1Then these two attainedparameter values and (1) and (12) are substituted into (5)and (6) respectively the relationship between the lightingemitting angle 120593119894 and the corresponding refraction angle 1205931015840119894can be obtained It is worth noting that substituting (12) into(4) (5) and (6) will lead to nonlinear equations thus thenumerical method should be employed to solve these [19]

Figure 6 shows the calculated profile of the lightingunit for the non-Lambertian LED light source and the


30

210

60

240

90

270

120

300

150

330

180 0

LIDC of non-Lambertian LEDFitted LIDC model

Figure 5The LIDC of a non-Lambertian LED light source and the corresponding model fitted by a polynomial function

300

25

10mm

50mm

minus30 0

(a) (b)

Figure 6 (a) Geometrical profile of the non-Lambertian LED-based luminaire and (b) the corresponding three dimensional model

Table 1 The coefficients of the polynomial function the ratio of 1198680to 1198681198970 and the value of parameter 120593119889Parameter 119886

11989411986801198681198970

120593119889

Valuea0 = 09778 a1 = 005183

489 13∘a2 = -02492 a3 = -005671a4 = 00346

corresponding geometrical model The simulation results ofthe non-Lambertian LED-based lighting unit are shown inFigure 7 Figure 7(a) shows the irradiance map of the devel-oped lighting unit One can note that a uniform illuminationis obtained on the target surface Figure 7(b) shows theangular intensity distribution of the non-Lambertian LED

and the designed lighting unit one can note that the intensitypeak of the designed lighting unit is also located at aboutplusmn 58∘ with the light energy sheared almost at plusmn 60∘ Thelight utilization efficiency and the estimated uniformity of thedesigned unit are about 9331 and 9164 respectively

33 Tolerance Analysis The installation error is one of themost important issues for optical system because the opticalperformance of the optical system will be deteriorated byposition deviation of the components In this work thetolerance analysis will be focused on discussing the positionmigration and rotation of the spherical reflector Figures8(a) 9(a) and 10(a) show three possible installation errorsof the developed two lighting units in real applicationswhich represent the position migrations of the reflector in


0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0

Non-Lambertian LEDDeveloped lighting unit

(b)

Figure 7 (a) Irradiance map and (b) the angular intensity distribution of the lighting unit for non-Lambertian LED

Table 2 The optical performance of the SFRB luminaires and the newly developed luminaires for both LED light sources

Structure LED Type Uniformity Efficiency

Single Reflector Lambertian LEd 4752 9770Non-Lambertian LEd 5663 9735

Developed Structure Lambertian LEd 9111 9296Non-Lambertian LEd 9164 9331

horizontal direction (x-axis direction) vertical direction (z-axis direction) and rotational direction (rotate about y-axis) respectively Figures 8(b) 9(b) and 10(b) show thecorresponding effects on the light utilization efficiencies ofthe two lighting units and the uniformities on the targetsurfacesOne cannote that these three installation errors havelittle influence on the light utilization efficiencies of the twolighting units However the uniformities on the target sur-faces are deteriorated as long as any of these three installationerrors increases One can note that the installation deviationΔ120579 performs the highest sensitivity to the uniformities onthe target surface and the deviation of Δx performs thelowest sensitivity According to the lighting standards suchas GB 50034ndash2013 ldquoStandard for lighting design of buildingsrdquoand EN 12464ndash12002 ldquoLight and lighting-Lighting of workplacesrdquo to meet most of general lighting applications theilluminance uniformity on the target surface should begreater than 70 Thus after checking the simulation resultsof the two lighting units the uniformity on the target surfaceis better than 70 when the installation errors Δx Δzand Δ120579 less than or equal to 20 mm 10 mm and 10∘respectively

34 Discussion As a comparison the luminaires with asingle freeform reflector and a bare LED light source weredesigned using themethod described in [11] Figures 11(a) and11(b) show the geometrical profiles of the freeform reflectorsfor the ideal Lambertian LED and the non-LambertianLED respectively Both of the freeform reflectors have asimilar cross section thus only the geometrical model of thefreeform reflector for the ideal Lambertian LED is showed inFigure 11(c)

Figures 12 and 13 show the simulation results of the singlefreeform reflector-based (SFRB) luminaires for the idealLambertian LED and the non-Lambertian LED As shownin these two figures the illuminance uniformities on thetarget region are deteriorated a lot compared with the newlydeveloped LED luminaire structure Table 2 shows the opticalperformances of the SFRB luminaires and the developedluminaires for both the ideal Lambertian LED and the non-Lambertian LED light sources One can note from the tablethat the light energy utilization efficiencies of the SFRBlighting units are better than the newly developed lightingunits for both LED light sources however the uniformitiesare unacceptable for the SFRB lighting units Due to Fresnel


x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)

Uniformity of Lambertian LEDEfficiency of Lambertian LEDUniformity of non-Lambertian LEDEfficiency of non-Lambertian LED

1

08

06

04

02

00 05 10 15 20 25 30

(b)

Figure 8 (a) Schematic of horizontal deviation Δx and (b) influence of horizontal deviation on the illumination uniformity and the lightutilization efficiency

x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)

Figure 9 (a) Schematic of vertical deviationΔz and (b) influence of vertical deviation on the illumination uniformity and the light utilizationefficiency

reflection on the primary packaging lens surface both of thedeveloped lighting units suffers about 4 more light energyloss compared with the SFRB lighting luminaires even so theuniformities of the developed lighting units are much betterthan the SFRB lighting units

4 Conclusions

A reflector-based LED lighting unit structure that can achievea large cut-off angle for general lighting has been discussed inthis workThe unit only consists of a spherical reflector and aprimary packaging aspheric lens All of the light rays emittedfrom LED light source can be well modulated by the reflectorand the aspheric surface of the primary lens The developedlighting unit structure overcomes the shortcomings that exist

in the SFRB lighting unit in which the light energy close tothe optical axis of the lighting system cannot be modulatedby the freeform reflector and the illuminance uniformity willdeteriorate when the cut-off angle of the reflector is largerthan 45∘ In addition the freeform surface is only designedin the primary packaging lens which is widely used in LEDencapsulation and the reflector is a spherical reflector themanufacturing cost of the developed luminaire structure doesnot obviously increased compared with the SFRB luminairestructure

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request


x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)

Figure 10 (a) Schematic of rotational deviation Δ120579 and (b) influence of rotational deviation on the illumination uniformity and the lightutilization efficiency

150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)

Figure 11 Geometrical profiles of the single freeform reflectors for (a) the ideal Lambertian LED and (b) for non-Lambertian LED and (c)the geometrical model of the freeform reflector for the ideal Lambertian LED


10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00

Figure 12 Irradiance map of the SFRB lighting unit for the idealLambertian LED light source

(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00

Figure 13 Irradiance map of the SFRB lighting unit for the non-Lambertian LED light source

Conflicts of Interest

The author declares that there are no conflicts of interestregarding this paper

Acknowledgments

The author would like to thank the support of the ScientificResearch Fund of YunnanOpenUniversity (no 18YNOU01)

References

[1] MG Craford ldquoLEDs for solid state lighting and other emergingapplications status trends and challengesrdquo Proc SPIE vol5941 Article ID 594101 2005

[2] M H Crawford ldquoLEDs for solid-state lighting performancechallenges and recent advancesrdquo IEEE Journal of Selected Topicsin Quantum Electronics vol 15 no 4 pp 1028ndash1040 2009

[3] X Long J He J Zhou et al ldquoA review on light-emitting diodebased automotive headlampsrdquo Renewable amp Sustainable EnergyReviews vol 41 pp 29ndash41 2015

[4] T-X Lee and B-S Chen ldquoHigh Uniformity and Toler-ance Design for Direct-Lit LED Backlight Illumination UsingLagrange Interpolationrdquo Journal of Display Technology vol 12no 11 pp 1403ndash1410 2016

[5] Z Zhenrong H Xiang and L Xu ldquoFreeform surface lens forLED uniform illuminationrdquo Applied Optics vol 48 no 35 pp6627ndash6634 2009

[6] A Gitin ldquoDesigning illuminating devices with faceted reflec-torsrdquoOptik - International Journal for Light and Electron Opticsvol 124 no 22 pp 5801ndash5806 2013

[7] Z Zhuang P Surman and F Yu ldquoA freeform optics design withlimited data for extended LED light sourcesrdquo Journal of ModernOptics vol 63 no 21 pp 2151ndash2158 2016

[8] M A Moiseev and L L Doskolovich ldquoDesign of TIR opticsgenerating the prescribed irradiance distribution in the circleregionrdquo Journal of the Optical Society of America A Optics andImage Science and Vision vol 29 no 9 pp 1758ndash1763 2012

[9] J S Yang J-H Park O Beom-Hoan S-G Park and S G LeeldquoDesign method for a total internal reflection LED lens withdouble freeform surfaces for narrow and uniform illuminationrdquoJournal of the Optical Society of Korea vol 20 no 5 pp 614ndash6222016

[10] X Zhang and D Liu ldquoFreeform lenses for illumination ofspecific shapes in (uv) coordinate systemrdquo Optical Review vol23 no 1 pp 77ndash83 2016

[11] J Yan ldquoDesign and analysis of reflectors for non-lambertianlight-emitting diodesrdquo Optical and Quantum Electronics vol48 no 12 2016

[12] A Pawlak and K Zaremba ldquoReflector luminaire with highpower light-emitting diodes for general lightingrdquo AppliedOptics vol 47 no 3 pp 467ndash473 2008

[13] J-W Pan S-H Tu W-S Sun C-M Wang and J-Y ChangldquoIntegration of nonndashlambertian LED and reflective opticalelement as efficient street lamprdquo Optics Express vol 18 no 102pp A221ndashA230 2010

[14] J Chaves Introduction to nonimaging optics vol 2nd CRCPress Boca Raton Fla USA 2015

[15] SWangKWang FChen andS Liu ldquoDesignof primaryopticsfor LED chip array in road lighting applicationrdquoOptics Expressvol 19 no 104 pp A716ndashA724 2011

[16] S Yu Y Tang Z Li Y Chen B Yu and G Liang ldquoFreeformillumination lens design combining energy and intensity map-pingrdquo Optical Engineering vol 56 no 4 p 045101 2017

[17] A Wei J R Sze and J L Chern ldquoLight-emitting diodes oncurved ceramic substrate with primary optics for modificationof luminous intensityrdquo Journal of Photonics for Energy vol 2 no1 Article ID 026501 2012

[18] H Ries and A Rabl ldquoEdge-ray principle of nonimaging opticsrdquoJournal of the Optical Society of America A Optics and ImageScience and Vision vol 11 no 10 pp 2627ndash2632 1994


[19] J HMathews andK D FinkNumericalMethods UsingMatlabvol 4th Pearson Education Inc 2004

[20] K Wang S Liu F Chen Z Qin Z Liu and X Luo ldquoFreeformLED lens for rectangularly prescribed illuminationrdquo Journal ofOptics A Pure and Applied Optics vol 11 no 10 p 105501 2009

[21] I Moreno and C-C Sun ldquoModeling the radiation pattern ofLEDsrdquo Optics Express vol 16 no 3 pp 1808ndash1819 2008

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Aspheric surface

Spherical reflectorSpherical surface

x

O

R

z

cut i

ri

i

ri

O

Figure 1 Geometrical diagram of the developed LED lighting unit

the aspheric surface of the primary lens is used to modulatethe light rays with an emitting angle smaller than the cut-offangle Then the uniform illumination on the target surfaceis formed by superposition of the light energy reflected bythe spherical reflector and the light energy modulated by theaspheric surface of the lens

2 Design Method

In order to obtain a uniform far-field illumination on thetarget surface the LIDC of the lighting equipment shouldcomply with the following expression [14]

119868119897119904 (120593) = 1198681198970cos3 (120593) (1)

where 119868119897119904(120593) represents the luminous intensity distributionof the lighting equipment and 1198681198970 is the luminous intensityat 120593 = 0∘ Figure 1 shows the geometrical structure of thedeveloped lighting unit As depicted in Figure 1 the proposedstructure comprises a spherical reflector and a primary lensOne can note that the central part of the lens is an asphericsurface and the side part of the lens is a spherical surfacewith the center located at the origin O of the coordinatesThe LED light source is immersed into the primary lens andalso positioned at the origin O of the coordinates [15ndash17]1198741015840 is the center of the spherical reflector and R denotesthe radius of the reflector As can be noted from Figure 1the opening dimension of the spherical reflector equals itsdiameter 120593119888119906119905 represents the cut-off angle of the lighting unitand it is generally given by designers 120593119894 denotes the anglebetween the light ray that will be refracted by the asphericsurface of the lens and z-axis and 1205931015840119894 is the angle between thecorresponding refracted light ray and z-axis One can notethat since the side part of the lens is spherical surface the lightrays will directly pass through the spherical surface of thelens without deflection Thus 120593119903119894 denotes the angle betweenthe light ray that will be reflected by the spherical reflectorand z-axis and 1205931015840119903119894 is the angle between the correspondingreflected light ray and z-axis

x

O

z

ri

ri

ri+1

ri+1

i+1

i

i

cut

d

rmin

rmin

rmax

Figure 2 Diagrammatic sketch of the light energy distribution ofthe lighting unit

From the geometrical relationship illustrated in Figure 1the relationship between the light reflection angle 1205931015840119903119894 andthe light-emitting angle 120593119903119894 of the side rays can be writtenas

1205931015840119903119894 = 120587 minus 120593119903119894 minus 2acrsin [ sin (120593119903119894)tan (120593119888119906119905)] (2)

As illustrated in Figure 1 the light-emitting angle 120593119903119894ranges from 120593119888119906119905 to 1205872 Consider the reflection characteristicof the spherical reflector in this case the minimum value1205931015840119903119898119894119899 and the maximum value 1205931015840119903119898119886119909 of the light reflectionangle 1205931015840119903119894 can be obtained when the light-emitting angle 120593119903119894reaches 1205872 and 120593119888119906119905 respectively Since the reflection angleof the light ray reflected by the spherical reflector is invariantfor a specified emitting angle in this case the uniform lightingon the target surface should be achieved by redistributing thelight energy passing through the aspheric surface of the lensThe right side of Figure 2 shows the diagrammatic sketch ofthe light energy distribution of the developed lighting unitAccording to the geometrical relationship shown in Figure 2one can note that the light reflection angle 1205931015840119903119898119886119909 equals120593119888119906119905 As shown in Figure 2 the light energy emitted fromthe LED light source can be divided into three parts thefirst part contains the light energy in the angle range of 0∘to 120593119889 and the light energy in this part is refracted to theangle range of 0∘ to 1205931015840119903119898119894119899 by the aspheric surface of the lensthe second part contains the light energy in the angle rangeof 120593119889 to 120593119888119906119905 and the third part contains the light energy inthe angle range of 120593119888119906119905 to 1205872 The light energy in these twoparts is modulated to the angle range of 1205931015840119903119898119894119899 to 1205931015840119903119898119886119909 bythe aspheric surface of the lens and the spherical reflectorrespectively



2120587int1205931198890119868119904 (120593) sin (120593) 119889120593 = 2120587int120593

1015840

119903min

0119868119897119904 (120593) sin (120593) 119889120593 (3)

2120587int120593119888119906119905120593119889

119868119904 (120593) sin (120593) 119889120593 + 2120587120588int1205872120593119888119906119905

119868119904 (120593) sin (120593) 119889120593= 2120587int120593119888119906119905

1205931015840119903min

119868119897119904 (120593) sin (120593) 119889120593(4)



2120587int1205931198940119868119904 (120593) sin (120593) 119889120593 = 2120587int120593

1015840

119894

0119868119897119904 (120593) sin (120593) 119889120593 (5)

2120587int120593119894120593119889

119868119904 (120593) sin (120593) 119889120593 + 2120587120588int1205872120593119903119894

119868119904 (120593) sin (120593) 119889120593

= 2120587int1205931015840 1198941205931015840119903min

119868119897119904 (120593) sin (120593) 119889120593(6)


3 Design Examples



119868119904 (120593) = 1198680 cos (120593) (7)



11986801198681198970 =tan2 (1205931015840119903min)sin2 (120593119889) (9)





+ tan2 (1205931015840119903min)) = 0 120593119894 isin (120593119889 120593119888119906119905](11)





300

25

50mm

10mm

minus30 0

(a) (b)


0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)



119868119904 (120593) = 1198680 (1198860 + 11988611205932 + 11988621205934 + 11988631205936 + 11988641205938) (12)






30

210

60

240

90

270

120

300

150

330

180 0



300

25

10mm

50mm

minus30 0

(a) (b)



11989411986801198681198970

120593119889

Valuea0 = 09778 a1 = 005183

489 13∘a2 = -02492 a3 = -005671a4 = 00346





0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)










x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)


x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)



4 Conclusions



Data Availability



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




2120587int1205931198890119868119904 (120593) sin (120593) 119889120593 = 2120587int120593

1015840

119903min

0119868119897119904 (120593) sin (120593) 119889120593 (3)

2120587int120593119888119906119905120593119889

119868119904 (120593) sin (120593) 119889120593 + 2120587120588int1205872120593119888119906119905

119868119904 (120593) sin (120593) 119889120593= 2120587int120593119888119906119905

1205931015840119903min

119868119897119904 (120593) sin (120593) 119889120593(4)



2120587int1205931198940119868119904 (120593) sin (120593) 119889120593 = 2120587int120593

1015840

119894

0119868119897119904 (120593) sin (120593) 119889120593 (5)

2120587int120593119894120593119889

119868119904 (120593) sin (120593) 119889120593 + 2120587120588int1205872120593119903119894

119868119904 (120593) sin (120593) 119889120593

= 2120587int1205931015840 1198941205931015840119903min

119868119897119904 (120593) sin (120593) 119889120593(6)


3 Design Examples



119868119904 (120593) = 1198680 cos (120593) (7)



11986801198681198970 =tan2 (1205931015840119903min)sin2 (120593119889) (9)





+ tan2 (1205931015840119903min)) = 0 120593119894 isin (120593119889 120593119888119906119905](11)





300

25

50mm

10mm

minus30 0

(a) (b)


0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)



119868119904 (120593) = 1198680 (1198860 + 11988611205932 + 11988621205934 + 11988631205936 + 11988641205938) (12)






30

210

60

240

90

270

120

300

150

330

180 0



300

25

10mm

50mm

minus30 0

(a) (b)



11989411986801198681198970

120593119889

Valuea0 = 09778 a1 = 005183

489 13∘a2 = -02492 a3 = -005671a4 = 00346





0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)










x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)


x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)



4 Conclusions



Data Availability



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



300

25

50mm

10mm

minus30 0

(a) (b)


0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)



119868119904 (120593) = 1198680 (1198860 + 11988611205932 + 11988621205934 + 11988631205936 + 11988641205938) (12)






30

210

60

240

90

270

120

300

150

330

180 0



300

25

10mm

50mm

minus30 0

(a) (b)



11989411986801198681198970

120593119889

Valuea0 = 09778 a1 = 005183

489 13∘a2 = -02492 a3 = -005671a4 = 00346





0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)










x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)


x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)



4 Conclusions



Data Availability



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



30

210

60

240

90

270

120

300

150

330

180 0



300

25

10mm

50mm

minus30 0

(a) (b)



11989411986801198681198970

120593119889

Valuea0 = 09778 a1 = 005183

489 13∘a2 = -02492 a3 = -005671a4 = 00346





0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)










x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)


x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)



4 Conclusions



Data Availability



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

0

x (mm)

y (m

m)

Lux

00

05

0

10000minus10000

10000minus10000

1000

0minus10

000

04

03

02

01

00

(a)

30

210

60

240

90

270

120

300

150

330

180 0


(b)










x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)


x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)



4 Conclusions



Data Availability



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



x

z

OΔx

(a)

x (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)


x

O

z

Δz

(a)

z (mm)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

(b)



4 Conclusions



Data Availability



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



x

O

z

Δ

(a)

Nor

mal

ized

vla

ue (

)


1

08

06

04

02

00 05 10 15 20 25 30

Δ (degree)

(b)


150

15

2232mm

minus15 0

(a)15

0

15

2236mm

minus15 0

(b)

(c)



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



10

Lux

0

0

x (mm)

y (m

m)

00

15

0

10000minus10000

10000minus10000

1000

0minus10

000

00


(a)

10Lux

0x (mm)

10000minus10000

00

15

0 10000minus10000

0y

(mm

)10

000

minus10

000

00




Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


a practical method to design reflector-based light...

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