## An iterative model of diffuse illumination from bidirectional photometric data

Optics Express, Vol. 17, Issue 2, pp. 723-732 (2009)

http://dx.doi.org/10.1364/OE.17.000723

Acrobat PDF (463 KB)

### Abstract

This paper presents a methodology for including the photometric raw data sets into the diffuse illumination design process. The method is based on computing the luminance distribution on the outgoing side of diffusing elements from measured bidirectional scattering distribution functions (BSDFs). The model is limited to specimens that create rotationally symmetric scattering distribution. The calculation procedure includes the linear superposition and the correcting feedback. As an application example, the method is verified by a commercially available diffusing sheet illuminated by a 32-inch backlighting module. Close agreement (correlation coefficient = 98.6%) with the experimental measurement confirmed the validity of the proposed procedure.

© 2009 Optical Society of America

## 1. Introduction

6. A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Waves Random Media **4**, 337–367 (1994). [CrossRef]

8. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surface,” J. Opt. Soc. Am. **57**, 1105–1114 (1967). [CrossRef]

11. L. Tsang, J. A. Kong, K, -H. Ding, and C. O. Ao, *Scattering of Electromagnetic Waves, Numerical Simulations* (Wiley, New York, 2000). [CrossRef]

13. N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough Surfaces,” Pgys. Rev. Lett. **52**, 1798–1801 (1984). [CrossRef]

14. K. Tang, R. Dimenna, and R. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surface,” Int. Heat J. Mass Transfer **40**, 49–59 (1997). [CrossRef]

## 2. BS(T)DFs of Diffusing Components

### 2.1 Bidirectional Transmittance Distribution Function

*q*[17] as following:

*θ*) and (

_{i},ϕ_{i}*θ*) represent the incident and transmitted angle of the light transmitting the specimen. BTDF describes the luminance d

_{t},ϕ_{t}*L*, which is visible under the angles of observation (

_{t}*θ*), induced by the illuminance d

_{t},ϕ_{t}*E*from an incident-side luminance

_{i}*L*for an incident direction (

_{i}*θ*) with a solid angle

_{i},ϕ_{i}*dω*. Since

_{i}*L*is an available functional description, illuminance

_{i}*E*can be decomposed into a linear combination of elementary functions. Equivalently, BTDF can be treated as the two-dimensional impulse response and completely describes the light spreading characteristics of a tested sample. The amount of light transmitted in the outgoing direction can be written as the integral of the BTDF multiplied by the incident flux from each incident direction (

*θ*) [17],

_{i},ϕ_{i}*L*indicates the overall luminance distribution of the transmitted light. Also, the integral can be expressed in a discrete way as following:

_{t}*ω*indicates the solid angle around the specific incident angle (

_{j}*θ*). Based on the linear composition of every

_{i,j},ϕ_{i,j}*j-th*components, the hemispherical luminance distributions over the transmission side can be solved accordingly.

### 2.2 BTDF measurement

18. M. E. Becker, “Evaluation and characterization of display reflectance,” Displays **19**, 35–54 (1998). [CrossRef]

*ϕ*), and the corresponding scattered light can be collected by an objective lens with moderate numerical aperture for observation of imaged hemisphere. Figure 2(b) shows a set of the measured angular spread functions of an arbitrary specimen with different incident polar angle

_{i},θ_{i,j}*θ*along a constant azimuthal angle

_{i}*ϕ*= 90°. Every incident beam illuminating the specimen would lead to specific BTDFs. For the commercial diffusing sheets, the randomly distributed beads are in circular shape. Thus, the scattering fields, which are highly relevant to the geometric natures of the particulates, are circularly symmetric. As the measured BTDFs vary smoothly under different incident angle

_{i}*θ*, the adequate angular sampling are sufficiently complete to represent the diffusing behavior. In this case, the angular interval

_{i,j}*Δθ*was 10 degree.

_{i}### 2.3 Characterizations by BTDFs

*θ*) and the transmitted direction (

_{i},ϕ_{i}*θ*). J. E. Harvey [22

_{t},ϕ_{t}22. J. E. Harvey and C. L. Vernold, “Transfer function characterization of scattering surface,” Proc. SPIE **3141**, 113–127 (1997). [CrossRef]

*ϕ*would be ignored. However, as shown in Fig. 3, the BTDFs exhibit a discrepant scattering shape and peak shift with different inclining illumination. Especially, the phenomena are more observable in larger angles of incidence. Thus, in the incident side, BTDF is a function of pure inclination

_{i}*θ*. A comprehensive model for such shift-variant system is essential for the diffuse illumination design flow.

_{i}## 3. Algorithm for modeling diffusers

### 3.1 Rotational Construction by BTDFs

*q*, which is the

_{j}*j-th*inclination set as shown in Fig. 2, and a comb function along the azimuthal direction:

*j*and Δ

*θ*represent the shifts along

*θ*direction with the interval Δ

*θ*.

*m*and Δ

_{j}*ϕ*indicate the shifts along

_{j}*ϕ*direction of the

*j-th*inclination.

*q*is the

^{R}_{j}*j-th*rotationally-constructed BTDF (R-BTDF). In our case, the width of the horizontal band Δ

*θ*is 10 degrees, which is mentioned in the measured results in section

*2.2*. Here the sampling number

*M*, which means 2π is equally divided into

_{j}*M*divisions, is increased with the outer band, so the sampling interval Δ

_{j}*ϕ*is varied with different

_{j}*j*. Fig. 5 shows an numerical calculation of convolution of the 40°-inclination (

*j*=4 the fourth ring) BTDF. Because of the nature of the integrated solid angle, the arrangement of

*M*should be directly proportional to the zonal constant [23], which is a convenient factor to calculate the luminous flux emitted into a narrow band and multiplying the summation by a solid angle factor. After a number of straight manipulations, eight donut-like R-BTDF

_{j}*q*can be obtained accordingly.

^{R}_{j}### 3.2. Weighting & Superposition of R-BTDFs

*j*=0~7) is weighted by the factor

*W*in accordance with the Lambertian reference source subject to the corresponding inclinations. The weights

_{j}*W*is the luminance at

_{j}*j-th*inclination

*θ*of the reference incident function

_{j}*L*over the value in normal direction

_{i,r}*θ*

_{0}under a constant azimuth

*ϕ*

_{0}.

*A*is the normalized coefficient, and calculated by

### 3.3 Correlation Coefficient

*CC*is defined as:

## 4. Luminance Calculation

*L*. The weighting function

_{i}*W*(

*j,m*) corresponding to the known sampling grid, can be obtained by:

*θ*) as following:

_{t}and Φ

_{i}are the overall luminous flux from the incident and transmitted sides of the diffusing sheet. In most case of diffusing sheets, the transmittances

*τ*(

*θ*) are a constant with respect to different incident angles, so the absolute value of the transmittance can be applied on light sources with variant angular distributions. The transmittances

_{i}*τ*can correct the normalized luminance distribution to absolute luminance value. The calculated result for the sample we mentioned is shown in Fig 7(a), where the transmitted luminance distribution was from a commercially available 32”-TV backlighting source. Comparing with the experimental results in Fig 7(b), the close agreement with the measurement (

*CC*=98.6%) demonstrates the validity of the proposed training process and corresponding diffusing model. The negligible deviations at large angles were mainly resulted from the measurement errors attributed by the conoscope distortion.

## 5. Conclusions

*M*to characterize one diffusing sheet and it is convenient for designers or factories to build a diffuser database. We successfully demonstrate the validity by using a general backlight source, where calculated emergent luminance distribution is 98.6% close to the measurement. In most case,

_{j}*CC*can achieve the value larger than 98%. The algorithm provides a relatively effective way for diffusing simulation, and is useful for the lighting development in display or luminance application.

## Acknowledgment

## References and links

1. | J. C. Stover, |

2. | M. Nieto-Vesperinas, |

3. | L. Tsang, J. A. Kong, and K. -H. Ding, |

4. | L. Tsang and J. A. Kong, |

5. | A. K. Fung, |

6. | A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces,” Waves Random Media |

7. | A. Voronovich, |

8. | K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surface,” J. Opt. Soc. Am. |

9. | B. van Ginneken, M. Staveridi, and J. J. Koendrik, “Diffuse and specular reflectance from rough surface,” Appl. Opt. |

10. | K. Tang and R. O. Buckius, “A statistical model of wave scattering from random rough surfaces,” Int. J. Heat Mass Transfer |

11. | L. Tsang, J. A. Kong, K, -H. Ding, and C. O. Ao, |

12. | F. D. Hastings, J. B. Schneider, and S. L. Broschat, “A Monte Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. |

13. | N. Garcia and E. Stoll, “Monte Carlo calculation for electromagnetic-wave scattering from random rough Surfaces,” Pgys. Rev. Lett. |

14. | K. Tang, R. Dimenna, and R. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surface,” Int. Heat J. Mass Transfer |

15. | M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, |

16. | E. Kreyszig, |

17. | J. de Boer, “Modelling indoor illumination by complex fenestration systems based on bidirectional : Basics, Measurement, and Rating,” J. Society for Information Display |

18. | M. E. Becker, “Evaluation and characterization of display reflectance,” Displays |

19. | J. W. Goodman, |

20. | K. Iizuka, |

21. | A. M. Nuijs and J. J. L. Horikx, “Diffraction and scattering at antiglare structures for display devices,” Appl. Opt. |

22. | J. E. Harvey and C. L. Vernold, “Transfer function characterization of scattering surface,” Proc. SPIE |

23. | M. W. Hodapp, ”Applications for High-Brightness Light-Emitting Diodes,” in |

**OCIS Codes**

(120.5240) Instrumentation, measurement, and metrology : Photometry

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(290.1990) Scattering : Diffusion

(290.1483) Scattering : BSDF, BRDF, and BTDF

**ToC Category:**

Illumination and nonimaging optics

**History**

Original Manuscript: October 30, 2008

Revised Manuscript: December 31, 2008

Manuscript Accepted: January 6, 2009

Published: January 8, 2009

**Citation**

Chung-Hao Tien and Chien-Hsiang Hung, "An iterative model of diffuse illumination from bidirectional photometric data," Opt. Express **17**, 723-732 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-2-723

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### References

- J. C. Stover, Optical Scattering: Measurement and Analysis (Mc Graw-Hill, New York, 1990).
- M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
- L. Tsang, J. A. Kong, and K. -H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, New York, 2000). [CrossRef]
- L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topiics s (Wiley, New York, 2001).
- A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Boston, 1994).
- A. Voronovich, "Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces," Waves Random Media 4, 337-367 (1994). [CrossRef]
- A. Voronovich, Wave Scattering from Rough Surfaces, 2nd Edition (Springer-Verlag, Berlin Heidelberg, 1994).
- K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surface," J. Opt. Soc. Am. 57, 1105-1114 (1967). [CrossRef]
- B. van Ginneken, M. Staveridi and J. J. Koendrik, "Diffuse and specular reflectance from rough surface," Appl. Opt. 37, 130-139 (1998). [CrossRef]
- K. Tang and R. O. Buckius, "A statistical model of wave scattering from random rough surfaces," Int. J. Heat Mass Transfer 44, 4059-4073 (2001). [CrossRef]
- L. Tsang, J. A. Kong, K, -H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, New York, 2000). [CrossRef]
- F. D. Hastings, J. B. Schneider, and S. L. Broschat, "A Monte Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).
- N. Garcia and E. Stoll, "Monte Carlo calculation for electromagnetic-wave scattering from random rough Surfaces," Phys. Rev. Lett. 52, 1798-1801 (1984). [CrossRef]
- K. Tang, R. Dimenna, and R. Buckius, "Regions of validity of the geometric optics approximation for angular scattering from very rough surface," Int. Heat J. Mass Transfer 40, 49-59 (1997). [CrossRef]
- M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, Handbook of Optics, Volume II (McGraw-Hill, New York, 1991).
- E. Kreyszig, Introductory Mathematical Statistics (Wiley, New York, 1970)
- J. de Boer, "Modelling indoor illumination by complex fenestration systems based on bidirectional : Basics, Measurement, and Rating," J. Soc. Info. Dis. 14/11, 1003-1017 (2006).
- M. E. Becker, "Evaluation and characterization of display reflectance," Displays 19, 35-54 (1998). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 2004)
- K. Iizuka, Elements of Photonics I (Willey, New York, 2002)
- A. M. Nuijs and J. J. L. Horikx, "Diffraction and scattering at antiglare structures for display devices," Appl. Opt. 33, 4058-4068 (1994). [CrossRef] [PubMed]
- J. E. Harvey and C. L. Vernold, "Transfer function characterization of scattering surface," Proc. SPIE 3141, 113-127 (1997). [CrossRef]
- M. W. Hodapp, Applications for High-Brightness Light-Emitting Diodes, in Semiconductors and Semimetals Vol. 48, G. B. Stringfellow and M. G. Craford ed., (Academic Press, San Diego, 1997) Semiconductors and Semimetals Vol. 48, Chap. 6, p. 227.

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