## Extended adding-doubling method for fluorescent applications |

Optics Express, Vol. 20, Issue 16, pp. 17856-17872 (2012)

http://dx.doi.org/10.1364/OE.20.017856

Acrobat PDF (1045 KB)

### Abstract

In this paper a fast, yet accurate method to estimate the spectral and angular distribution of the scattered radiation of a fluorescent material is described. The proposed method is an extension of the adding-doubling algorithm for non-fluorescent samples. The method is validated by comparing the spectral and angular transmittance and reflectance characteristics obtained with the extended algorithm with the results obtained using Monte Carlo simulations. The agreement using both methods is within 2%. However, the adding-doubling method achieves a reduction of the calculation time by a factor of 400. Due to the short calculation time, the extended adding-doubling method is very useful when fluorescent layers have to be optimized in an iterative process.

© 2012 OSA

## 1. Introduction

1. J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. **32**(19), 3585–3595 (1993). [CrossRef] [PubMed]

3. A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. **21**(2), 166–178 (1997). [CrossRef] [PubMed]

4. I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci. **38**(5), 1097–1100 (2010). [CrossRef]

6. J. H. Park and J. H. Ko, “Optimization of the emitting structure of flat fluorescent lamps for LCD backlight applications,” J. Opt. Soc. Korea **11**(3), 118–123 (2007). [CrossRef]

7. P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A **25**(1), 61–66 (2007). [CrossRef]

9. E. Klampaftis and B. S. Richards, “Improvement in multi-crystalline silicon solar cell efficiency via addition of luminescent material to EVA encapsulation layer,” Prog. Photovolt. Res. Appl. **19**(3), 345–351 (2011). [CrossRef]

4. I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci. **38**(5), 1097–1100 (2010). [CrossRef]

10. D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol. **6**(12), 645–651 (2010). [CrossRef]

11. R. G. Young and E. G. F. Arnott, “The effect of phosphor coating weight on the lumen output of luorescent lamps,” J. Electrochem. Soc. **112**(10), 982–984 (1965). [CrossRef]

12. W.-T. Chien, C.-C. Sun, and I. Moreno, “Precise optical model of multi-chip white LEDs,” Opt. Express **15**(12), 7572–7577 (2007). [CrossRef] [PubMed]

14. C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys. **44**(8), 6056–6061 (2005). [CrossRef]

15. D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett. **89**(23), 231102 (2006). [CrossRef]

16. J. Y. Chi, J.-S. Chen, C.-Y. Liu, C.-W. Chu, and K.-H. Chiang, “Phosphor converted LEDs with omni-directional-reflector coating,” Opt. Express **17**(26), 23530–23535 (2009). [CrossRef] [PubMed]

17. J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys. **38**(10), 5788–5798 (2011). [CrossRef] [PubMed]

18. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid mediaby using the adding-doubling method,” Appl. Opt. **32**(4), 559–568 (1993). [CrossRef] [PubMed]

19. G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. Lond. **11**(0), 545–556 (1860). [CrossRef]

20. Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf. **105**(2), 243–263 (2007). [CrossRef]

22. P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. **93**(D9), 11037–11050 (1988). [CrossRef]

25. W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron. **26**(12), 2166–2185 (1990). [CrossRef]

*et al.*explored the possibility of the adding-doubling method for describing fluorescent layers using a two flux approach [26

26. A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ. **37**(2), 117–130 (1991). [CrossRef]

*θ*. The optical properties defined in the radiative transfer equation, in particularly absorption and scattering coefficient, are linearly dependent on the concentration [27]. This is advantageous if concentration is an optimization parameter. Furthermore, the use of the n-flux model provides angular information on the scattered and converted light.

## 2. General approach

*θ*. The conical segments are represented in Fig. 1 .

*θ*, channel 2 contains the light travelling in directions lying between

_{1}*θ*and

_{1}*θ*and so forth. The flux in the positive or the negative Z direction can be represented by a vector, where each element represents the amount of light within the conical segment. The vector contains

_{2}*n*elements, equal to the number of channels the flux is divided in.

*I*the incident flux in channel j propagating in the positive Z direction,

^{+}(θ_{j})*I*the reflected flux in channel i, propagating in the negative Z direction and

^{-}(θ_{i})*R(θ*gives the reflection from channel j into channel i. In a similar way a transmission matrix can be composed. The angle

_{i}, θ_{j})*θ*can take values from 0 to 90°.

## 3. Adding-doubling

### 3.1 Adding-doubling for non-fluorescent slabs

20. Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf. **105**(2), 243–263 (2007). [CrossRef]

22. P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. **93**(D9), 11037–11050 (1988). [CrossRef]

**) and transmission (**

*R***) matrices of two given slabs at a particular wavelength are known, the new**

*T***and**

*R***matrices of a slab consisting of the two slabs joined together can be calculated with the adding method. The doubling method is similar as the adding method for the combination of two identical slabs. For one slab the following equations can be written for each wavelength with Eq. (2) and Eq. (3):**

*T*

*I**is the flux within the channels in vector notation, the plus and minus sign denote the propagation direction along the positive and negative Z-axis respectively. The subindex represents the location of an interface starting from 0.*

_{x}^{±}

*R**respectively*

_{xy}

*T**is the reflection matrix and transmission matrix for a flux incident on interface x of the slab defined by interface x and y. This is schematically shown in Fig. 2 .*

_{xy}**the unity matrix. A more extensive description can be found in literature [20**

*E*20. Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf. **105**(2), 243–263 (2007). [CrossRef]

22. P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. **93**(D9), 11037–11050 (1988). [CrossRef]

### 3.2 Adding-doubling for fluorescent slabs

*Δλ*and L wavelength intervals

^{X}_{i}*Δλ*, with i going from 1 to N and j going from 1 to L. Each wavelength region

^{M}_{j}*Δλ*can have a contribution to the flux at emission wavelength interval

^{X}_{i}*Δλ*, as schematically shown in Fig. 3 .

^{M}_{j}*Δλ*must be adapted and are given by Eq. (8) and Eq. (9):

^{M}_{j}

*R**and*

_{xy}

*T**are the reflection and transmission matrices at the selected emission wavelength, representing scattering and absorption. The dependence on wavelength has been omitted from the equations for readability reasons. The matrices*

_{xy}

*R**and*

^{c}_{xy}(Δλ^{X}_{i}, Δλ^{M}_{j})

*T**represent the conversion of light from an excitation wavelength interval*

^{c}_{xy}(Δλ^{X}_{i}, Δλ^{M}_{j})*Δλ*towards a particular emission wavelength interval

^{X}_{i}*Δλ*for each incident and scattered conical segment.

^{M}_{j}

*R**and*

^{c}_{xy}

*T**are identical to zero and Eqs. (8) and (9) become equal to Eqs. (2) and (3).*

^{c}_{xy}## 4. Initial matrices

### 4.1 Initial matrices for non-fluorescent slabs

**and**

*R***matrices are known for each layer. In this section we explain how the initial**

*T***and**

*R***matrices can be determined. For non-fluorescent applications this is already extensively described in literature [28**

*T*28. W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf. **16**(8), 637–658 (1976). [CrossRef]

28. W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf. **16**(8), 637–658 (1976). [CrossRef]

*I*the flux within a selected channel,

*ν*the cosine between the average propagation direction in the selected channel and the normal on the slab (

*ν*can take values from −1 to 1).

*P(ν,ν’)*is the azimuthally averaged phase function (normalized to 1),

*µ*and

_{a}*µ*are respectively the absorption and scattering coefficient. The phase function models the spatial redistribution of scattered light.

_{s}29. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. **93**, 70–83 (1941). [CrossRef]

30. J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. **42**(11), 2255–2261 (1997). [CrossRef] [PubMed]

31. W. Saeys, M. A. Velazco-Roa, S. N. Thennadil, H. Ramon, and B. M. Nicolaï, “Optical properties of apple skin and flesh in the wavelength range from 350 to 2200 nm,” Appl. Opt. **47**(7), 908–919 (2008). [CrossRef] [PubMed]

*P*is given in Eq. (15), where

_{HG}(Υ)*Υ*is the cosine of the angle between incident and scattered direction.

*g*represents the anisotropy factor. Using the Henyey-Greenstein phase function as a fixed phase function for all materials, a non-fluorescent material can be described with three optical parameters:

*µ*,

_{a}*µ*,

_{s}*g*.

*et al.*extensively described the solution of the radiative transfer equation given in Eq. (14) [28

28. W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf. **16**(8), 637–658 (1976). [CrossRef]

*ν*the value of

_{i}*ν*in conical segment i. The weights in matrix

**are determined by the used quadrature scheme. Prahl et al. described the use of a Radau and Gaussian quadrature scheme for the discretization [18**

*c*18. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid mediaby using the adding-doubling method,” Appl. Opt. **32**(4), 559–568 (1993). [CrossRef] [PubMed]

**16**(8), 637–658 (1976). [CrossRef]

**defined by Eq. (25):**

*Γ*### 4.2 Initial matrices for fluorescent slabs

32. D. Yudovsky and L. Pilon, “Modeling the local excitation fluence rate and fluorescence emission in absorbing and strongly scattering multilayered media,” Appl. Opt. **49**(31), 6072–6084 (2010). [CrossRef]

*Δλ*with central wavelength

^{M}_{j}*λ*

^{M}_{j}.*I(ν)*is the channel flux at the selected emission wavelength interval,

*I(ν’, Δλ*represents the channel flux at the excitation wavelength interval

^{X}_{i})*Δλ*The third term on the right hand side of Eq. (26) represents the contribution of each excitation wavelength interval

^{X}_{i}.*Δλ*(with central wavelength

^{X}_{i}*λ*) to the selected emission wavelength interval.

^{X}_{i}*µ*and

_{a}*µ*are respectively the absorption and scattering coefficient for the selected emission wavelength interval,

_{s}*µ*is the absorption coefficient at excitation wavelength interval

_{a}(Δλ^{X}_{i})*Δλ*.

^{X}_{i}*QE(Δλ*expresses the quantum efficiency the excitation wavelength interval. The wavelength ratio is used to convert the number of photons to powerflux.

^{X}_{i})32. D. Yudovsky and L. Pilon, “Modeling the local excitation fluence rate and fluorescence emission in absorbing and strongly scattering multilayered media,” Appl. Opt. **49**(31), 6072–6084 (2010). [CrossRef]

*Δλ*will be emitted within the selected wavelength interval

^{X}_{i}*Δλ*(schematically shown in Fig. 5 ). A weight factor

^{M}_{j}*w(Δλ*has been introduced to describe the fraction of light generated by an excitation at

^{M}_{j})*Δλ*and emitted within the selected emission wavelength region. The coefficient

^{X}_{i}*w(Δλ*is normalized so that the integration over the entire emission wavelength region is 1 with Eq. (27):

^{M}_{j})**and**

*α***are still defined by Eq. (18) and Eq. (19). The matrix**

*β***γ**

_{i}is defined by Eq. (30):

*T**(Δλ*and

^{X}_{i})

*R**(Δλ*are the reflection and transmission matrices calculated at the excitation wavelength region

^{X}_{i})*Δλ*, calculated with the method for non-fluorescent slabs. The matrix

^{X}_{i}**is still defined by Eq. (25). A more extensive description of the derivation of the equations can be found in appendix B.**

*Γ*

*R**,*

_{01}

*T**,*

_{10}

*R**and*

^{c}_{01}(Δλ^{X}_{i}, Δλ^{M}_{j})

*T**, the results will be the same as Eqs. (32-34) since the slab will behave identical for light entering the slab in the opposite direction.*

^{c}_{10}(Δλ^{X}_{i}, Δλ^{M}_{j})

*R**and*

_{10}

*T**, describing the behavior of incident light at the emission wavelength, are identical to the matrices in defined Eqs. (23) and (24). Indeed, fluorescence is independent on the absorption and scattering characteristics at the emission wavelengths. The*

_{01}

*R**and*

^{c}

*T**matrices describing the contribution due to fluorescence are equal for transmission and reflection. This meets the expectations since the emission due to fluorescence is assumed to be isotropic in the single-scatter layer. If there is no fluorescence (QE = 0), the matrices with superindex c become zero and the equations become equal to the equations valid for the non-fluorescent case. The formulas for the non-fluorescent case can thus be considered as special cases of the general formulas taking fluorescence into account.*

^{c}## 5. Validation

*σ*= 12 and 20 nm have been used.

*I(θ*for three wavelengths: one excitation wavelength (450 nm), one emission wavelength (600 nm), and a wavelength not participating in the fluorescence (750 nm).

_{s})*θ*= 0° obtained with the Monte Carlo simulations and with the adding- doubling method are shown for the three selected wavelengths.

_{i}## 6. Conclusion

## Appendix A

*R**and*

_{xy}

*T**are the reflection and transmission matrices at the selected emission wavelength, representing scattering and absorption. The dependence on wavelength has been omitted from the equations for readability reasons.*

_{xy}

*R**and*

^{c}_{xy}(Δλ^{X}_{i}, Δλ^{M}_{j})

*T**represent the conversion of light from an excitation wavelength interval*

^{c}_{xy}(Δλ^{X}_{i}, Δλ^{M}_{j})*Δλ*towards a particular emission wavelength interval

^{X}_{i}*Δλ*for each incident and scattered conical segment. Similar as with the adding-doubling method at non-emission wavelengths, the same equations can be written for a slab 1-2, given by Eq. (37) and Eq. (38):

^{M}_{j}

*I**can be written in function of*

_{1}^{-}(Δλ^{X}_{i})

*I**and*

_{2}^{-}(Δλ^{X}_{i})

*I**given by Eq. (40):*

_{0}^{+}(Δλ^{X}_{i})

*R**and*

_{xy}(Δλ^{X}_{i})

*T**are the reflection and transmission matrices at excitation wavelength interval*

_{xy}(Δλ^{X}_{i})*Δλ*. In a similar way,

^{X}_{i}

*I**can be written in function of*

_{1}^{+}(Δλ^{X}_{i})

*I**and*

_{2}^{-}(Δλ^{X}_{i})

*I**, given by Eq. (41):*

_{0}^{+}(Δλ^{X}_{i})**and**

*R***matrices at the selected emission wavelength for a layer consisting of two joined layers are finally given by Eqs. (44)-(47):**

*T*## Appendix B

*Δλ*with central wavelength

^{M}_{j}*λ*given by Eq. (48):

^{M}_{j}*I*

*(ν)*is the channel flux at the selected emission wavelength interval,

*I(ν’, Δλ*represents the channel flux at the excitation wavelength interval

^{X}_{i})*Δλ*(with central wavelength

^{X}_{i}*λ*). In Eq. (48)

^{X}_{i}*µ*and

_{a}, µ_{s}*P(ν,ν’)*are respectively the absorption and scattering coefficient and phase function for the selected emission wavelength interval,

*µ*is the absorption coefficient at excitation wavelength interval

_{a}(Δλ^{X}_{i})*Δλ*.

^{X}_{i}*QE(Δλ*expresses the quantum efficiency the excitation wavelength interval. The wavelength ratio is used to convert the number of photons to powerflux. A weight factor

^{X}_{i})*w(Δλ*has been to describe the fraction of light generated by an excitation at

^{M}_{j})*Δλ*and emitted within the selected emission wavelength region.

^{X}_{i}*µ*and corresponding weights

_{M}*c*, which results in Eq. (49):

_{i}*ν*can take values from 0 to 1,

*p*is the phase function in matrix notation defined by Eq. (50):

*I**and*

^{+}

*I**represent a channel flux at the selected emission wavelength interval in vector notation with the angle between the propagation direction and the positive Z-axis respectively smaller and larger than 90°.*

^{-}

*I**and*

^{+}(Δλ^{X}_{i})

*I**represent the fluxes at excitation wavelength interval*

^{-}(Δλ^{X}_{i})*Δλ*

^{X}_{i}.**16**(8), 637–658 (1976). [CrossRef]

**represents the unity matrix and**

*E***,**

*α***and**

*β***γ**

_{i}are defined by respectively Eq. (60), Eq. (61) and Eq. (62):

*R**=*

_{10}

*R**and*

_{01}

*T**=*

_{10}

*T**. Using this identities in Eq. (63), Eq. (64) is obtained:*

_{01}**defined by Eq. (65):**

*Γ*## Acknowledgment

## References and links

1. | J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. |

2. | A. Liebert, H. Wabnitz, N. Zołek, and R. Macdonald, “Monte Carlo algorithm for efficient simulation of time-resolved fluorescence in layered turbid media,” Opt. Express |

3. | A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. |

4. | I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci. |

5. | J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater. |

6. | J. H. Park and J. H. Ko, “Optimization of the emitting structure of flat fluorescent lamps for LCD backlight applications,” J. Opt. Soc. Korea |

7. | P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A |

8. | W. G. J. H. M. van Sark, “Enhancement of solar cell performance by employing planar spectral converters,” Appl. Phys. Lett. |

9. | E. Klampaftis and B. S. Richards, “Improvement in multi-crystalline silicon solar cell efficiency via addition of luminescent material to EVA encapsulation layer,” Prog. Photovolt. Res. Appl. |

10. | D. Bera, S. Maslov, L. Qian, J. S. Yoo, and P. H. Holloway, “Optimization of the yellow phosphor concentration and layer thickness for down-conversion of blue to white light,” J. Disp. Technol. |

11. | R. G. Young and E. G. F. Arnott, “The effect of phosphor coating weight on the lumen output of luorescent lamps,” J. Electrochem. Soc. |

12. | W.-T. Chien, C.-C. Sun, and I. Moreno, “Precise optical model of multi-chip white LEDs,” Opt. Express |

13. | Y. Shuai, N. T. Tran, and F. G. Shi, “Nonmonotonic phosphor size dependence of luminous efficacy for typical white LED emitters,” IEEE Photon. Technol. Lett. |

14. | C. C. Chang, R.-L. Chern, C. C. Chang, C.-C. Chu, J. Y. Chi, J.-C. Su, I.-M. Chan, and J.-F. T. Wang, “Monte Carlo simulation of optical properties of phosphor-screened ultraviolet light in a white light-emitting device,” Jpn. J. Appl. Phys. |

15. | D.-Y. Kang, E. Wu, and D.-M. Wang, “Modeling white light-emitting diodes with phosphor layers,” Appl. Phys. Lett. |

16. | J. Y. Chi, J.-S. Chen, C.-Y. Liu, C.-W. Chu, and K.-H. Chiang, “Phosphor converted LEDs with omni-directional-reflector coating,” Opt. Express |

17. | J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography-computational efficiency,” Med. Phys. |

18. | S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid mediaby using the adding-doubling method,” Appl. Opt. |

19. | G. G. Stokes, “On the intensity of the light reflected from or transmitted through a pile of plates,” Proc. R. Soc. Lond. |

20. | Z. Zhang, P. Yang, G. Kattawar, H.-L. Huang, T. Greenwald, J. Li, B. A. Baum, D. K. Zhou, and Y. Hu, “A fast infrared radiative transfer model based on the adding–doubling method for hyperspectral remote-sensing applications,” J. Quant. Spectrosc. Radiat. Transf. |

21. | G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. |

22. | P. J. Flatau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. |

23. | H. G. Völz, |

24. | S. N. Thennadil, “Relationship between the Kubelka-Munk scattering and radiative transfer coefficients,” J. Opt. Soc. Am. A |

25. | W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review on the optical properties of biological tissues,” IEEE J. Quantum Electron. |

26. | A. Rosema, W. Verhoef, J. Schroote, and J. F. H. Snel, “Simulating fluorescence light-canopy interaction in support of laser-induced fluorescence measurements,” Remote Sens. Environ. |

27. | C. F. Bohren and D. Huffman, |

28. | W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transf. |

29. | L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. |

30. | J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, and M. J. C. Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. |

31. | W. Saeys, M. A. Velazco-Roa, S. N. Thennadil, H. Ramon, and B. M. Nicolaï, “Optical properties of apple skin and flesh in the wavelength range from 350 to 2200 nm,” Appl. Opt. |

32. | D. Yudovsky and L. Pilon, “Modeling the local excitation fluence rate and fluorescence emission in absorbing and strongly scattering multilayered media,” Appl. Opt. |

**OCIS Codes**

(260.2510) Physical optics : Fluorescence

(290.0290) Scattering : Scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: May 30, 2012

Revised Manuscript: July 16, 2012

Manuscript Accepted: July 16, 2012

Published: July 20, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Sven Leyre, Guy Durinck, Bart Van Giel, Wouter Saeys, Johan Hofkens, Geert Deconinck, and Peter Hanselaer, "Extended adding-doubling method for fluorescent applications," Opt. Express **20**, 17856-17872 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17856

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### References

- J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt.32(19), 3585–3595 (1993). [CrossRef] [PubMed]
- A. Liebert, H. Wabnitz, N. Zołek, and R. Macdonald, “Monte Carlo algorithm for efficient simulation of time-resolved fluorescence in layered turbid media,” Opt. Express16(17), 13188–13202 (2008). [CrossRef] [PubMed]
- A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med.21(2), 166–178 (1997). [CrossRef] [PubMed]
- I. Seo, J. Jung, B. J. Oh, and K. Whang, “Improvement of luminance and luminous efficacy of mercury-free, flat fluorescent lamp by optimizing phosphor profile,” IEEE Trans. Plasma Sci.38(5), 1097–1100 (2010). [CrossRef]
- J. P. You, N. T. Tran, Y. Lin, Y. He, and F. G. Shi, “Phosphor-concentration-dependent characteristics of white LEDs in different current regulation modes,” J. Electron. Mater.38(6), 761–766 (2009). [CrossRef]
- J. H. Park and J. H. Ko, “Optimization of the emitting structure of flat fluorescent lamps for LCD backlight applications,” J. Opt. Soc. Korea11(3), 118–123 (2007). [CrossRef]
- P. Chung, H. Chung, and P. H. Holloway, “Phosphor coatings to enhance Si photovoltaic cell performance,” J. Vac. Sci. Technol. A25(1), 61–66 (2007). [CrossRef]
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