Abstract

Monte Carlo methods are an established technique for simulating light transport in biological tissue. Integrating spheres make experimental measurements of the reflectance and transmittance of a sample straightforward and inexpensive. This work presents an extension to existing Monte Carlo photon transport methods to simulate integrating sphere experiments. Crosstalk between spheres in dual-sphere experiments is accounted for in the method. Analytical models, previous works on Monte Carlo photon transport, and experimental measurements of a synthetic tissue phantom validate this method. We present two approaches for using this method to back-calculate the optical properties of samples. Experimental and simulation uncertainties are propagated through both methods. Both back-calculation methods find the optical properties of a sample accurately and precisely. Our model is implemented in standard Python 3 and CUDA C++ [J. Nickolls, I. Buck, M. Garland, and K. Skadron, ACM Queue 6, 40 (2008)] and is publicly available in Code 1.

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References

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  1. C. J. Hourdakis and A. Perris, “A Monte Carlo estimation of tissue optical properties for use in laser dosimetry,” Phys. Med. Biol. 40(3), 351–364 (1995).
    [Crossref]
  2. B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
    [Crossref]
  3. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding–doubling method,” Appl. Opt. 32(4), 559–568 (1993).
    [Crossref]
  4. S. A. Prahl, “Everything I think you should know about inverse adding-doubling,” (2011). [Online; accessed 2018-2019].
  5. P. Lemaillet, C. C. Cooksey, J. Hwang, H. Wabnitz, D. Grosenick, L. Yang, and D. W. Allen, “Correction of an adding-doubling inversion algorithm for the measurement of the optical parameters of turbid media,” Biomed. Opt. Express 9(1), 55–71 (2018).
    [Crossref]
  6. T. P. Moffitt, Y. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms.,” J. Biomed. Opt. 11(4), 041103 (2006).
    [Crossref]
  7. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).
  8. CIE, “CIE S 017/E:2011 ILV: international lighting vocabulary,” (2011).
  9. CIE, “CIE 130-1998 practical methods for the measurement of reflectance and transmittance,” (1998).
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    [Crossref]
  11. J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
    [Crossref]
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    [Crossref]
  14. S. L. Storm, A. Springsteen, and T. M. Ricker, “A discussion of center mount sample holder designs and applications,” Tech. rep., Labsphere (1998).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  23. P. Q. Fiee, “Double integrating sphere characterization of pva-cryogels,” Master’s thesis, McMaster University (2015).
  24. H. C. van de Hulst, Multiple Light Scattering, vol. 2 (Academic Press, 1980).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  29. J. Burkardt, “PRAXIS - scalar function optimization,” (2016). [Online; accessed 2019].
  30. E. Jones, T. Oliphant, and P. Peterson, “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed 2018-2019].
  31. R. Brent, Algorithms for Minimization Without Derivatives (Dover, 2002).
  32. B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. rep., National Institute of Standards and Technology, Gaithersburg, Maryland (1994).
  33. Z. H. Levine, R. H. Streater, A.-M. R. Lieberson, A. L. Pintar, C. C. Cooksey, and P. Lemaillet, “Algorithm for rapid determination of optical scattering parameters,” Opt. Express 25(22), 26728–26746 (2017).
    [Crossref]

2019 (1)

Q. Fang and S. Yan, “Graphics processing unit-accelerated mesh-based Monte Carlo photon transport simulations,” J. Biomed. Opt. 24(11), 1 (2019).
[Crossref]

2018 (1)

2017 (1)

2014 (1)

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

2011 (1)

2008 (1)

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

2006 (1)

T. P. Moffitt, Y. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms.,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref]

1995 (2)

C. J. Hourdakis and A. Perris, “A Monte Carlo estimation of tissue optical properties for use in laser dosimetry,” Phys. Med. Biol. 40(3), 351–364 (1995).
[Crossref]

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of tight transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref]

1993 (1)

1992 (1)

1989 (1)

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” SPIE Institute Series 5, 102–111 (1989).

1967 (1)

1964 (1)

M. J. D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives,” The Comput. J. 7(2), 155–162 (1964).
[Crossref]

1955 (2)

J. A. Jacquez and H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45(6), 460–470 (1955).
[Crossref]

R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2(4), 153–162 (1955).
[Crossref]

Allen, D. W.

Brent, R.

R. Brent, Algorithms for Minimization Without Derivatives (Dover, 2002).

Buck, I.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Burkardt, J.

J. Burkardt, “PRAXIS - scalar function optimization,” (2016). [Online; accessed 2019].

Chen, Y.

T. P. Moffitt, Y. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms.,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref]

Cooksey, C. C.

Doronin, A.

Fang, Q.

Q. Fang and S. Yan, “Graphics processing unit-accelerated mesh-based Monte Carlo photon transport simulations,” J. Biomed. Opt. 24(11), 1 (2019).
[Crossref]

Fiee, P. Q.

P. Q. Fiee, “Double integrating sphere characterization of pva-cryogels,” Master’s thesis, McMaster University (2015).

Garland, M.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).

Giovanelli, R. G.

R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2(4), 153–162 (1955).
[Crossref]

Goebel, D. G.

Grosenick, D.

Hokr, B. H.

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Hourdakis, C. J.

C. J. Hourdakis and A. Perris, “A Monte Carlo estimation of tissue optical properties for use in laser dosimetry,” Phys. Med. Biol. 40(3), 351–364 (1995).
[Crossref]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).

Hwang, J.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of tight transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” SPIE Institute Series 5, 102–111 (1989).

Jacquez, J. A.

Jones, E.

E. Jones, T. Oliphant, and P. Peterson, “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed 2018-2019].

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” SPIE Institute Series 5, 102–111 (1989).

Kuppenheim, H. F.

Kuyatt, C. E.

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. rep., National Institute of Standards and Technology, Gaithersburg, Maryland (1994).

Lemaillet, P.

Levine, Z. H.

Lieberson, A.-M. R.

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).

M. Sterenborg, H. J. C.

Manzhirov, A. V.

A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Taylor & Francis, 2006).

Meglinski, I.

Moes, C. J. M.

Moffitt, T. P.

T. P. Moffitt, Y. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms.,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref]

Nickolls, J.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).

Oliphant, T.

E. Jones, T. Oliphant, and P. Peterson, “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed 2018-2019].

Perris, A.

C. J. Hourdakis and A. Perris, “A Monte Carlo estimation of tissue optical properties for use in laser dosimetry,” Phys. Med. Biol. 40(3), 351–364 (1995).
[Crossref]

Peterson, P.

E. Jones, T. Oliphant, and P. Peterson, “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed 2018-2019].

Pickering, J. W.

Pintar, A. L.

Polyanin, A. D.

A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Taylor & Francis, 2006).

Powell, M. J. D.

M. J. D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives,” The Comput. J. 7(2), 155–162 (1964).
[Crossref]

Prahl, S. A.

T. P. Moffitt, Y. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms.,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref]

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

J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, and M. J. C. van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9(4), 621–631 (1992).
[Crossref]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” SPIE Institute Series 5, 102–111 (1989).

S. A. Prahl, “Light transport in tissue,” Ph.D. thesis, The University of Texas at Austin (1988).

S. A. Prahl, “Everything I think you should know about inverse adding-doubling,” (2011). [Online; accessed 2018-2019].

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).

Ricker, T. M.

S. L. Storm, A. Springsteen, and T. M. Ricker, “A discussion of center mount sample holder designs and applications,” Tech. rep., Labsphere (1998).

Scully, M. O.

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Skadron, K.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Springsteen, A.

S. L. Storm, A. Springsteen, and T. M. Ricker, “A discussion of center mount sample holder designs and applications,” Tech. rep., Labsphere (1998).

Storm, S. L.

S. L. Storm, A. Springsteen, and T. M. Ricker, “A discussion of center mount sample holder designs and applications,” Tech. rep., Labsphere (1998).

Streater, R. H.

Taylor, B. N.

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. rep., National Institute of Standards and Technology, Gaithersburg, Maryland (1994).

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd ed (University Science Books, 1996).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering, vol. 2 (Academic Press, 1980).

van Gemert, M. J. C.

Wabnitz, H.

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of tight transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref]

Welch, A. J.

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

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” SPIE Institute Series 5, 102–111 (1989).

Yakovlev, V. V.

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Yan, S.

Q. Fang and S. Yan, “Graphics processing unit-accelerated mesh-based Monte Carlo photon transport simulations,” J. Biomed. Opt. 24(11), 1 (2019).
[Crossref]

Yang, L.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of tight transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref]

ACS Photonics (1)

B. H. Hokr, V. V. Yakovlev, and M. O. Scully, “Efficient time-dependent Monte Carlo simulations of stimulated raman scattering in a turbid medium,” ACS Photonics 1(12), 1322–1329 (2014).
[Crossref]

Appl. Opt. (2)

Biomed. Opt. Express (2)

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of tight transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref]

J. Biomed. Opt. (2)

T. P. Moffitt, Y. Chen, and S. A. Prahl, “Preparation and characterization of polyurethane optical phantoms.,” J. Biomed. Opt. 11(4), 041103 (2006).
[Crossref]

Q. Fang and S. Yan, “Graphics processing unit-accelerated mesh-based Monte Carlo photon transport simulations,” J. Biomed. Opt. 24(11), 1 (2019).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2(4), 153–162 (1955).
[Crossref]

Opt. Express (1)

Phys. Med. Biol. (1)

C. J. Hourdakis and A. Perris, “A Monte Carlo estimation of tissue optical properties for use in laser dosimetry,” Phys. Med. Biol. 40(3), 351–364 (1995).
[Crossref]

Queue (1)

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable parallel programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

SPIE Institute Series (1)

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” SPIE Institute Series 5, 102–111 (1989).

The Comput. J. (1)

M. J. D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives,” The Comput. J. 7(2), 155–162 (1964).
[Crossref]

Other (17)

J. Burkardt, “PRAXIS - scalar function optimization,” (2016). [Online; accessed 2019].

E. Jones, T. Oliphant, and P. Peterson, “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed 2018-2019].

R. Brent, Algorithms for Minimization Without Derivatives (Dover, 2002).

B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” Tech. rep., National Institute of Standards and Technology, Gaithersburg, Maryland (1994).

P. Q. Fiee, “Double integrating sphere characterization of pva-cryogels,” Master’s thesis, McMaster University (2015).

H. C. van de Hulst, Multiple Light Scattering, vol. 2 (Academic Press, 1980).

“Report of calibration special photometric tests for four synthetic adult skin samples and one synthetic bulk fat sample,” Tech. rep., National Institute of Standards and Technology, Gaithersburg, Maryland (2017).

S. L. Storm, A. Springsteen, and T. M. Ricker, “A discussion of center mount sample holder designs and applications,” Tech. rep., Labsphere (1998).

S. A. Prahl, “Light transport in tissue,” Ph.D. thesis, The University of Texas at Austin (1988).

“Integrating sphere theory and applications,” Tech. rep., Labsphere (2017). [Online; accessed 2019].

A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Taylor & Francis, 2006).

J. R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd ed (University Science Books, 1996).

P. D. Cook, “ISMC,”, https://github.com/pdcook/ISMC (2019).

S. A. Prahl, “Everything I think you should know about inverse adding-doubling,” (2011). [Online; accessed 2018-2019].

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand.160, 94–145 (1977).

CIE, “CIE S 017/E:2011 ILV: international lighting vocabulary,” (2011).

CIE, “CIE 130-1998 practical methods for the measurement of reflectance and transmittance,” (1998).

Supplementary Material (1)

NameDescription
» Code 1       Integrating Sphere Monte Carlo (ISMC) - Implementation of the model and inverse solvers.

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Figures (10)

Fig. 1.
Fig. 1. Depictions of how integrating spheres would be used in optical experiments measuring (a) reflectance and (b) transmittance as well as (c) a dual-sphere setup measuring both reflectance and transmittance.
Fig. 2.
Fig. 2. Simulated effect of integrating spheres—with properties listed in Table 4—on the measured reflectance and transmittance of samples (a) with $\mu _a=1\, \textrm{cm}^{-1}$ and varying $\mu _s^\prime$ and (b) with varying $\mu _a$ and $\mu _s^\prime=10\, \textrm{cm}^{-1}$ . All samples had refractive index $n=1.4$ , anisotropy factor $g=0.5$ , and thickness $t=0.135\, \textrm{cm}$ .
Fig. 3.
Fig. 3. Configurations necessary to make the measurements required for the calculation of $M_R$ .
Fig. 4.
Fig. 4. Configurations necessary to make the measurements required for the calculation of $M_T$ .
Fig. 5.
Fig. 5. Configurations necessary to make the measurements required for the calculation of $M_U$ .
Fig. 6.
Fig. 6. A flowchart of our integrating sphere method.
Fig. 7.
Fig. 7. Lookup tables for the (a) absorption coefficient and (b) reduced scattering coefficient for a sample with $n=1.4$ , $g=0.9$ , and $t=1\, \textrm{cm}$ measured with a zero-sphere setup. Marked points represent values obtained directly from simulation that were interpolated from.
Fig. 8.
Fig. 8. A flowchart showing the propagation of uncertainties through the lookup table method.
Fig. 9.
Fig. 9. A flowchart showing the propagation of uncertainties through the minimization method.
Fig. 10.
Fig. 10. Lookup tables for the (a) absorption coefficient and (c) reduced scattering coefficient for our tissue phantom measured with a single-sphere setup as well as the percent uncertainty from simulation parameters only in the (b) absorption coefficient and (d) reduced scattering coefficient Marked points represent values obtained directly from simulation that were interpolated from.

Tables (8)

Tables Icon

Table 1. Verification for a sample with a refractive-index-matched boundary.

Tables Icon

Table 2. Verification for a sample with a refractive-index-mismatched boundary.

Tables Icon

Table 3. Optical Properties of our tissue phantom at λ = 532 nm according to a National Institute of Standards and Technology (NIST) report [27].

Tables Icon

Table 4. Properties of the integrating sphere used to verify the model in each orientation.

Tables Icon

Table 5. Raw measurement values for the integrating sphere measurements of our tissue phantom.

Tables Icon

Table 6. Reflectance and transmittance values found by experiment and simulation.

Tables Icon

Table 7. Input parameters and associated uncertainties used to solve for the optical coefficients of our tissue phantom as well as the method each parameter is used in.

Tables Icon

Table 8. Inversely calculated absorption and reduced scattering coefficients of our tissue phantom at λ = 532 nm from integrating sphere measurements using both the lookup table method, the minimization method, and a NIST calibration report [27].

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

M R = r s t d R ( r s d i r e c t , r s ) R ( 0 , 0 ) R ( r s t d , r s t d ) R ( 0 , 0 ) .
M T = T ( r s d i r e c t , r s ) T d a r k T ( 0 , 0 ) T d a r k .
M U = U s U 0 U 100 U 0 .
θ U = tan 1 A 2 l .
f X = 1 1 ( d X D ) 2 2 ,
| μ z | 1 f p .
P = ρ w f s 1 ρ w ( 1 f ) .
x = 0 , y = 0 , z = 0   o r   t , μ x = cos ( 2 π ξ 2 ) 1 ξ 1 2 , μ y = sin ( 2 π ξ 2 ) 1 ξ 1 2 , μ z = ± ξ 1 .
P = n = 1 ρ w n ( 1 f ) n 1 f s .
P = ρ w f s 1 ρ w ( 1 f ) .
A c a p = 2 π r 2 ( 1 cos α ) .
P ( θ , ϕ ) = { C for   0 θ π 2 0 otherwise .
0 2 π 0 θ C sin θ d θ d ϕ = 0 ξ 1 d x , 0 ϕ 0 π 2 C sin θ d θ d ϕ = 0 ξ 2 d x ,
cos θ = ξ 1 , ϕ = 2 π ξ 2 ,
μ x = cos ϕ 1 cos 2 θ , μ y = sin ϕ 1 cos 2 θ , μ z = cos θ ,

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