Analysis of misalignment-induced measurement error for goniophotometry of light-emitting diode arrays Wentao Cai, Xianming Liu,* Xiaohua Lei, and Weimin Chen Key Laboratory for Optoelectronic Technology & Systems of Ministry of Education, Chongqing University, Chongqing 400044, China *Corresponding author: [email protected] Received 28 August 2013; revised 1 November 2013; accepted 3 November 2013; posted 4 November 2013 (Doc. ID 194471); published 25 November 2013

The luminous distribution characteristics of light sources can be measured by goniophotometry. In this work, the misalignment of luminary-induced measurement errors as the main factor affecting the measurement accuracy is analyzed. A calculation method for measurement error is proposed. Then, the translational and angular misalignment-induced measurement errors are calculated and analyzed. Results show that the measurement errors induced by misalignments may be great in some cases even if the far-field condition is satisfied. For luminaries with different radiation patterns, the acceptable misalignment tolerances corresponding to measurement error of less than 1% are given. © 2013 Optical Society of America OCIS codes: (120.5240) Photometry; (120.0120) Instrumentation, measurement, and metrology; (120.4800) Optical standards and testing; (220.4840) Testing; (230.3670) Light-emitting diodes. http://dx.doi.org/10.1364/AO.52.008381

1. Introduction

All lighting designs are based on the light distribution characteristics of the light sources [1]. It is important to make accurate measurements of the light distribution of light sources for a more precise design [2–4]. Nowadays, light-emitting diode (LED) lamps, which use LEDs as the sources of light, are gaining ground in general lighting due to their advantages of long service life and high energy efficiency. Most LED lamps, especially high-power lamps, consist of several regularly arranged LED clusters. Therefore, LED lamps are always considered as surface light sources. The light distribution for surface light sources is a function of the measurement distance (source-to-detector distance) because the intensity definition only applies to the point source. In order to simplify the lighting design calculation and the 1559-128X/13/348381-07$15.00/0 © 2013 Optical Society of America

light distribution measurement, the light distributions for surface light sources are always obtained in the far-field condition, where the detectors are placed far enough away from the sources. Recently, there were several studies analyzing the far-field condition of LEDs, in which the measurement errors induced by the measurement distance were discussed to determine the far-field distance [5–8]. However, the measurement errors still exist even if the far-field condition is satisfied. Light source installation errors are one of the major factors inducing measurement errors, which are unavoidable because it is difficult to exactly locate the center of the luminous surface of an LED lamp at the rotation center of the sample holder of the goniophotometer [9–11]. The misalignment effects have largely been neglected in previous literature and there is especially a lack of quantitative results. Thus, measurement errors induced by the misalignment of LED lamps should be analyzed to determine whether the measurement results are acceptable. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS

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In this work, light distribution measurement errors induced by the misalignment of luminaries are analyzed. In Section 2, the formula for calculation of measurement error is developed through summarizing and analyzing the calculation methods in previous literature. Four typical radiation patterns of light sources such as Lambertian, concentrated, batwing, and side emitting are taken for examples in the comparison. In Sections 3 and 4, the measurement errors induced by typical translational and angular misalignments are discussed in detail, such as longitudinal and transverse misalignments and rotational and angular tilt misalignments. All calculations are based on LED lamps with different light distributions. Finally, Section 5 concludes the work. 2. Measurement Error Definition and Analysis

The light intensity distribution can be described by a two-dimensional array that provides information about the intensities I at azimuths θ; ϕ. There are several possible systems of measurement planes to define the luminous intensity of a luminaire [12], from which the system of A − α planes is generally applied by most goniophotometer measurement [13]. For a luminary with a cluster of N LEDs as shown in Fig. 1, the angular intensity distribution is contributed to by all the LEDs. We assume each of the LEDs to be a point source with the same light intensity of I s A; α. Hence, the illuminance E at distance d is inversely proportional to the square of the distance from the point source, which is known as the inverse-square law and can be expressed as E  I∕d2 , where d is the measurement distance jrd j. If the detector is placed in the far field of the LEDs, the illuminance of the LEDs is the sum of that of each LED. Then, the intensity of the LEDs can be calculated by using the inverse-square law again [14]: IA; α 

X I s Ai ; αi d2 j⃗ri j2

cos θi ;

(1)

where

Ai  arctanRix ∕Riz ;

(3)

αi  arcsinRiy ∕j⃗ri j:

(4)

Here, xi and yi are the Cartesian coordinates of the ith LED. A and α are the angles of LEDs in A − α coordinates, while Ai and αi are the angles of the ith LED. ni is the optical axis of the ith LED. ri is a vector from the ith LED to the detector. θi is an angle between ri and rd for the detector. The measurement error of light intensity distribution is the difference between a measured value and its true value. Since most of the lighting designs are based on the far-field optical characteristics, the light intensity at infinity (d → ∞) can be treated as the true value for comparison, which is handled in the same way in previous works. Then, the reference data for comparison can be expressed as I ∞ Ai ; αi  and the measurement data as IAi ; αi . The reference I ∞ is obtained by making Ai ; αi  → A; α, jri j → d, and θi → 0: I ∞ A; α  M · I s A; α;

where M is the number of LEDs. For different calculation methods, the difference between these two data should be obtained differently. Recently, Moreno and Sun [6] have presented a method to analyze the measurement error Δ, but considering only the intensity variation on the optical axis. After that, some other error definitions that take into account the variation under different azimuth angles have been introduced, including the normalized cross correlation (NCC) [3,15] and the relative root mean square error (ΔRMS ) [7]. With the definition of NCC, the error between the measurement I and the reference I ∞ is   PM  ¯ ¯ i1 IAi ;αi  − I I ∞ Ai ;αi  − I ∞ ΔNCC  1 − q  ;  P  PM  IAi ;αi  − I¯ 2 M I ∞ Ai ;αi  − I¯ ∞ 2 i1

i1

(6)

r⃗ i  Rix ; Riy ; Riz   d cos α sin A − xi ; d sin α − yi ; d cos α cos A; (2)

i

rs

i

ni

ri

i

rd

where I¯ and I¯ ∞ are the mean values of the measurement and the reference across the angular range, respectively. M is the number of sampling points. However, this definition can distinguish the directions of vectors but not the magnitudes. The error between vectors (1, 2, 3) and (2, 4, 6), for example, is 0.0% calculated by NCC, which suggests that NCC cannot distinguish the vectors with the same directions. As defined in Ref. [7], ΔRMS is obtained by

ΔRMS Fig. 1. Geometry for measuring LED arrays. 8382

(5)

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v u  2 M u1 X I ∞ Ai ; αi  IAi ; αi  − 1 : t M i1 I max I ∞ Ai ; αi  ∞

(7)

0

0

-30

30

-30 60

-60

-90

-90

90

(a)

(b)

0

0

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30

-30 60

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30 60

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90

(c)

ΔRMS

60

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90

-90

30

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90

(d)

Fig. 2. Several radiation patterns: (a) Lambertian; (b) concentrated; (c) side-emitting; and (d) batwing.

However, M X I ∞ Ai ; αi  i1

M · I max ∞



M X I ∞ Ai ; αi  P  1; I∞ i1

(8)

which means the sum of all weighting coefficients is always less than 1. With this weighting calculation method, the influences of small light intensities at some azimuths would be ignored greatly. Therefore, in this work, we put forward an error calculation formula in which the sum of intensity at infinity is set as the denominator in the weighting coefficients:

0.16

v uM 2 uX I ∞ Ai ; αi   IAi ; αi  t P −1 :  I∞ I ∞ Ai ; αi  i1

(9)

In order to compare the three definitions of error above, we first calculate the error as a function of the measurement distance. The ΔRMS with different weighting methods are expressed as RMSwmax (RMS weight by the max value) and RMSwsum (RMS weight by the sum value) for identification. Four types of radiation patterns are applied in the calculation, which are perfect Lambertian, concentrated, side-emitting, and batwing, with the light distribution shown in Fig. 2. The data of light distribution are taken from the luminaries of a Philips BBG490 spotlight, BGP680 street lamp, and BBP300 ceiling lamp, which are all LED array sources. In order to compare the influences of radiation patterns under the same conditions, we construct four 5 × 5 square LED arrays, of which each LED has the same radiation pattern for one array. The error values with the three different error definitions as functions of measurement distance are plotted in Fig. 3. The relative measurement distance, which is defined by the ratio of measurement distance d to the largest dimension of the source D, is set as abscissa. For the rectangular-shaped LED array, the length–width ratio of the rectangle has little effects

1.0

(a)

(b)

NCC

0.12

NCC

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RMSwmax

RMSwmax RMSwsum

RMSwsum

Error

Error

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Relative measurement distance d/D

Fig. 3. Error values with three different error definitions as functions of relative measurement distance for LED arrays with different radiation patterns: (a) Lambertian; (b) concentrated; (c) side-emitting; and (d) batwing. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS

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on the error if the diagonal is taken as the longest segment [16]. Therefore, the results shown in Fig. 3 can apply to all the rectangular-shaped LED arrays as well. As shown in Fig. 3, the relative error is greatly different from results calculated with different error calculation methods. The RMSwsum is the most sensitive error definition method irrespective of the LED radiation pattern and therefore is more appropriate for precise lighting design and control. With this error definition, the relative far-field distance (RFFD) may be different with other error definitions because the RFFD can be obtained by solving the equation Δ  1%. Figure 4 shows the errors for LED arrays with Lambertian, concentrated, side-emitting, and batwing radiation patterns in the same figure for further comparison. The corresponding RFFDs are 4.1, 15.3, 11.2, and 14.6, respectively. If the RMSwmax and NCC are used as errors, the corresponding RFFDs should be (3.1, 5.6, 5.5, 8.3) and (1.2, 2.8, 1.6, 3.3). For most goniophotometry measurements, the actual measurement distance is often longer than 10 m, which can satisfy the far-field condition for most LED array samples. Therefore, the measurement distance may not be the main factor causing the measurement error. The operation mode for the goniophotometer is to measure the illumination by rotating the sample holder at different azimuths while the illuminance meter keeps still. However, in many systems, the assumption that the motion of sample holder of the goniophotometer is a pure rotation about the center of an LED array’s lighting surface is only an approximation. For instance, the lighting plane of a luminary is typically a few millimeters, sometimes even centimeters below the upper surface of the luminary, which is difficult to calibrate, especially for thick samples. Besides, the optic axis of the LED arrays and the central axis of the sample holder might not be collinear. These kinds of misalignments may induce measurement error for goniophotometry measurements. In next two

sections, the measurement errors induced by several typical misalignments will be discussed in detail based on the error definition of RMSwsum. 3. Translational Misalignment-Induced Measurement Error

There are two possible translational misalignments: the transverse misalignment and the longitudinal misalignment, which are corresponding to the misalignments in the lighting plane and along the optic axis. The illuminance meter is placed at the corresponding far-field distance for each LED array so that measurement errors induced by measurement distance are the same value: 1%. All the luminaries are constructed as 5 × 5 square LED arrays as described in the previous section. Then, the misalignment-induced measurement errors can be compared under the same conditions for LEDs with different radiation patterns. A. Transverse Misalignment

The transverse misalignment is of two cases, along the x-axis translation and along the y-axis translation. For luminaries with rotationally symmetric radiation, such as the Lambertian and concentrated LEDs, the two cases may have equal effects. Here, we take the x-axis translation for an example, as shown in Fig. 5. If the LEDs have offset Δx along the x axis, the actual center of rotation is O but not the center of LEDs O0 . Namely, the z axis is the measurement axis while the z0 axis is the LED array axis. The intensity of the LEDs can be calculated by Eq. (1), in which the corresponding ri may be rewritten as ⃗ri  d cos α sin A − xi − Δx ; d sin α − yi ; d cos α cos A: (10) The measurement errors as functions of relative transverse misalignments of LED arrays are plotted in Fig. 6. Here, the relative transverse misalignment

0.20 Lambertian LEDs spotlight street lamp ceiling lamp

0.15

RMSwsum

0.10

0.05

0.00

-0.05

5

10

15

20

Relative measurement distance d/D Fig. 4. RMSwsum as functions of relative measurement distance for LED arrays with different radiation patterns. 8384

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Fig. 5. Schematic diagram of LED array with transverse misalignment.

0.20

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Lambertian LEDs Spotlight Street Lamp Ceiling Lamp

0.15

RMSwsum

RMSwsum

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Lambertian LEDs-x Spotlight-x Street lamp-x Ceiling lamp-x Street lamp-y Ceiling lamp-y

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0.10 1% error line

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0.00 0.00

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Relative transverse misalignment Fig. 6. RMSwsum as functions of relative transverse misalignments for LED arrays with different radiation patterns.

is defined as the ratio of the misalignment to the largest dimension of the LED array. It is shown that the measurement error is approximately proportional to the transverse misalignment for all the samples. If the measurement error Δ  1% is set as the acceptable value as defined in the far-field distance, the corresponding relative transverse misalignments for LED arrays with Lambertian, concentrated, side-emitting (along the x and y directions), and batwing radiation patterns (along the x and y directions) are 3.9%, 4.4%, (6.4%, 5.6%), and (5.8%, 3.9%), respectively. Taking a 10 cm × 10 cm LED array with Lambertian radiation, for example, the largest dimension is 14.1 cm and the maximum permissible transverse misalignment is 0.0055 cm, which can be satisfied in most measurements. B.

Longitudinal Misalignment

There is longitudinal misalignment if the rotation center of the sample holder is not on the lighting plane. As shown in Fig. 7, when the offset along the z axis is Δz , which could be positive or negative, the corresponding ri are

0.00

-0.2

-0.1

0.0

0.1

0.2

Relative longitudinal misalignment Fig. 8. RMSwsum as functions of relative longitudinal misalignments for LED arrays with different radiation patterns.

⃗ri  d cos α sin A − xi ; d sin α − yi ; d cos α cos A  Δz : (11) The measurement errors as functions of relative longitudinal misalignments of LED arrays are plotted in Fig. 8. The relative longitudinal misalignment is defined as the ratio of the misalignment to the largest dimension of LED array. Corresponding to the measurement error Δ  1%, the relative longitudinal misalignment along the positive and negative directions are (−0.9%, −10.6%, −1.9%, −3.7%) and (0.8%, 10.6%, 1.8%, 3.7%), respectively, for Lambertian, concentrated, side-emitting, and batwing radiation patterns LED arrays. For 10 cm × 10 cm LED arrays with these four radiation patterns, the maximum permissible longitudinal misalignments are 0.0012 cm, 0.015 cm, 0.0026 cm, and 0.0052 cm. We can see that the LED array with the Lambertain is more sensitive to longitudinal misalignment. This happens due to the Lambertain LEDs having the shortest farfield distance, while RMSwsum is inversely related with the far-field distance of LEDs because maxΔθ  ≈ arcsinΔx ∕d. 4. Angular Misalignment-Induced Measurement Error

The angular misalignment would induce measurement errors. There are two kinds of angular misalignment, corresponding deflection around the x axis (or y axis) and z axis (optical axis), which are named as the angular tilt misalignment and the rotational misalignment. They are very common misalignments for goniophotometry measurement. In this section, we will analyze the measurement error introduced by these two kinds of angular misalignment. A. Angular Tilt Misalignment

Fig. 7. Schematic diagram of LED array with longitudinal misalignment.

The angular tilt misalignment exists when the optical axis deflects, as shown in Fig. 9. For luminaries with rotational symmetric radiation patterns such as the Lambertian and concentrated LED arrays, the 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS

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z x

Fig. 9. Schematic diagram of LED array with angular tilt misalignment. Fig. 11. Schematic diagram of LED array with rotational misalignment.

deflecting direction has the same influence on the measurement error, while for others the influences may be different. Here, we consider only the situations for deflection around the x axis and y axis for nonrotational symmetric radiation LED arrays. If there is angular tilt misalignment as shown in Fig. 9, the A–α coordinates need to be transformed using the rotational matrix. Geometric transformation formulas for the x axis are tan Ax 

cos α sin A ; sin α sin φx  cos α cos A cos φx

sin αx  sin α cos φx − cos α cos A sin φx ;

(12)

(13)

and for the y axis are Ay  A  φ y ;

(14)

αy  α:

(15)

Here, φx and φy are deflection angles around the x axis and y axis, respectively. The coordinates A; α can be replaced by Ax ; αx  or Ay ; αy  in Eq (2). Figure 10 shows the measurement errors as functions of angular tilt misalignment for the above four LED arrays with different radiation

patterns. It is clear that the angular tilt misalignment would introduce very large measurement error for all samples, especially for spotlight with concentrated radiation. Even if the tilt angle is only 1°, the measurement error may be larger than 5%. The tilt angle around the x axis should be less than 0.26°, 0.10°, 0.23°, and 0.26° for Lambertian, concentrated, side-emitting and batwing radiation LED arrays, corresponding to a measurement error of less than 1%. Therefore, the angular tilt misalignment may be the main factor introducing the measurement error for goniophotometry measurement. B. Rotational Misalignment

There is rotational misalignment if the LED array deflects around the optical axis, as shown in Fig. 11. For LED arrays with a rotational symmetric radiation pattern, the rotational misalignment has no effect on the measurement results. However, for those nonrotational symmetric radiation LED arrays, there must be measurement error induced by rotational misalignment. If the deflection angle is φz , as shown in Fig. 11, the light intensity can be simply expressed with a system of C–γ planes: ICi ; γ i   ICi  φz ; γ i . Then the intensity IAi ; αi  can be obtained by transforming A; α into C; γ with [13]

0.4

Street Lamp Ceiling Lamp

0.12

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0.3

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Lambertian LEDs-x Spotlight-x Street lamp-x Ceiling lamp-x Street lamp-y Ceiling lamp-y

0.2

0.1

0.08

0.04 1% error line

1% error line

0.0

0

1

2

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Angular tilt misalignment (degree) Fig. 10. RMSwsum as functions of angular tilt misalignments for LED arrays with different radiation patterns. 8386

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0.00

0

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2

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Rotational misalignment (degree) Fig. 12. RMSwsum as functions of rotational misalignments for LED arrays with different radiation patterns.

tan A  cos C × tan γ;

(16)

The support of the Fundamental Research Funds for the Central Universities (CDJZ11120002) is gratefully acknowledged.

sin α  sin C × sin γ:

(17)

References

and

The measurement errors as functions of rotational misalignment for nonrotational symmetric radiation LED arrays such as the street lamp and ceiling lamp are introduced above, as shown in Fig. 12. For radiation patterns with a large degree of asymmetry, the rotational misalignment may induce large measurement error. The corresponding rotation angle should be less than 0.32° and 0.39° for side-emitting and batwing radiation LED arrays. 5. Conclusion

In summary, we have analyzed the goniophotometry errors induced by misalignment of LED luminaries. Calculation results show that the misalignment of LED luminaries has a great impact on the measurement error. Even on the premise of meeting the far-field condition, the mounting accuracy of the luminary must be strictly controlled to minimize the errors for goniophotometry. For rotational symmetric radiation LED arrays, the translational misalignments are the main factors inducing measurement errors. The relative longitudinal misalignments for Lambertian radiation LED arrays should be less than 0.9% of the maximum luminary dimension, which should attract particular attention in goniophotometry. The angular misalignment is the most sensitive mounting error affecting the measurement error because the goniophotometry results are highly dependent on the accuracy of goniometry. Small tilt or rotational angle may induce large measurement error. Therefore, the calibration of the mounting direction of the luminary is the most important step before goniophotometry measurement.

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Analysis of misalignment-induced measurement error for goniophotometry of light-emitting diode arrays.

The luminous distribution characteristics of light sources can be measured by goniophotometry. In this work, the misalignment of luminary-induced meas...
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