Abstract

In this study, a simple method to determine the reflectance and transmittance of turbid media is proposed. The method is based on a single integrating sphere system. The integrating sphere theory is used to calculate the reflectance and transmittance under diffuse and directional illumination, and an experimental setup is proposed to measure these quantities. This method was used to determine the reflectance and transmittance of poly(vinyl alcohol) (PVA) hydrogels. Monte Carlo calculations were also implemented to validate our results. The comparison shows that the optical properties obtained through this method propagate low values of uncertainty, and a setup simpler than that of the double integrating sphere system can be realized. However, our system is limited to the diffuse–diffuse transmittance. The results suggest that the single sphere system and Monte Carlo method can be a strong combination to calculate the inherent optical properties of PVA samples.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

16 May 2019: Typographical corrections were made to the author listing and author affiliations.


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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  33. S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
    [Crossref]
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    [Crossref]
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    [Crossref]

2014 (1)

S. L. Jacques, “Coupling 3D Monte Carlo light transport in optically heterogeneous tissues to photoacoustic signal generation,” J. Photoacoust. 2(4), 137–142 (2014).
[Crossref]

2012 (2)

2011 (2)

Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011).
[Crossref]

A. M. Nilsson, A. Jonsson, J. C. Jonsson, and A. Roos, “Method for more accurate transmittance measurements of low-angle scattering samples using an integrating sphere with an entry port beam diffuser,” Appl. Opt. 50(7), 999–1006 (2011).
[Crossref]

2010 (1)

2008 (2)

H. S. Mansur and H. S. Costa, “Nanostructured poly (vinyl alcohol)/bioactive glass and poly (vinyl alcohol)/chitosan/bioactive glass hybrid scaffolds for biomedical applications,” Chem. Eng. J. 137(1), 72–83 (2008).
[Crossref]

T. Nakaoki and H. Yamashita, “Bound states of water in poly (vinyl alcohol) hydrogel prepared by repeated freezing and melting method,” J. Mol. Struct. 875(1-3), 282–287 (2008).
[Crossref]

2007 (2)

2006 (1)

2004 (2)

S. Nevas, F. Manoocheri, and E. Ikonen, “Gonioreflectometer for measuring spectral diffuse reflectance,” Appl. Opt. 43(35), 6391–6399 (2004).
[Crossref]

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

2002 (1)

2001 (1)

2000 (1)

C. M. Hassan and N. A. Peppas, “Structure and applications of poly(vinyl alcohol) hydrogels produced by conventional crosslinking or by freezing/thawing methods,” Adv. Polym. Sci. 153, 37–65 (2000).
[Crossref]

1998 (1)

1995 (2)

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

M. Hammer, A. Roggan, D. Schweitzer, and G. Muller, “Optical properties of ocular fundus tissues-an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40(6), 963–978 (1995).
[Crossref]

1993 (2)

1992 (1)

1991 (1)

1990 (1)

G. D. Boreman, A. B. Centore, and Y. Sun, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29(4), 339–342 (1990).
[Crossref]

1983 (1)

1965 (2)

J. M. Davies and W. Zagieboylo, “An integrating sphere system for measuring average reflectance and transmittance,” Appl. Opt. 4(2), 167–174 (1965).
[Crossref]

B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. A 55(9), 1122–1125 (1965).
[Crossref]

1938 (1)

1935 (1)

Ahtee, V.

AL-Rubaiee, M.

Beek, J. F.

Boreman, G. D.

G. D. Boreman, A. B. Centore, and Y. Sun, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29(4), 339–342 (1990).
[Crossref]

Brown, S. W.

Butler, J. J.

Carr, K. F.

K. F. Carr, A Guide to Integrating Sphere Radiometry & Photometry (Labsphere Technical Guide, Labsphere Inc.1997).

Centore, A. B.

G. D. Boreman, A. B. Centore, and Y. Sun, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29(4), 339–342 (1990).
[Crossref]

Clare, J. F.

Cooper, J. W.

Costa, H. S.

H. S. Mansur and H. S. Costa, “Nanostructured poly (vinyl alcohol)/bioactive glass and poly (vinyl alcohol)/chitosan/bioactive glass hybrid scaffolds for biomedical applications,” Chem. Eng. J. 137(1), 72–83 (2008).
[Crossref]

D’Sa, E. J.

Davies, J. M.

Ducoste, J. J.

Enríque, S.

Fukshansky, L.

Fukshansky-Kazarinova, N.

Gatebe, C. K.

Gayen, S. K.

Hammer, M.

M. Hammer, A. Roggan, D. Schweitzer, and G. Muller, “Optical properties of ocular fundus tissues-an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40(6), 963–978 (1995).
[Crossref]

Hanssen, L.

Hardy, A. C.

Hassan, C. M.

C. M. Hassan and N. A. Peppas, “Structure and applications of poly(vinyl alcohol) hydrogels produced by conventional crosslinking or by freezing/thawing methods,” Adv. Polym. Sci. 153, 37–65 (2000).
[Crossref]

Hisdal, B. J.

B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. A 55(9), 1122–1125 (1965).
[Crossref]

Iglesias- Prieto, R.

Ikonen, E.

Jacques, S. L.

S. L. Jacques, “Coupling 3D Monte Carlo light transport in optically heterogeneous tissues to photoacoustic signal generation,” J. Photoacoust. 2(4), 137–142 (2014).
[Crossref]

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

Jia, G.-x.

Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011).
[Crossref]

Jonsson, A.

Jonsson, J. C.

Kharine, A.

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

King, M. D.

Kortoum, G.

G. Kortoum, Reflectance Spectroscopy (Springer-Verlag, 1969).

Kowalewski, M.

Larason, T. C.

Linden, K. G.

Lykke, K. R.

Mamane, H.

Manohar, S.

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

Manoocheri, F.

Mansur, H. S.

H. S. Mansur and H. S. Costa, “Nanostructured poly (vinyl alcohol)/bioactive glass and poly (vinyl alcohol)/chitosan/bioactive glass hybrid scaffolds for biomedical applications,” Chem. Eng. J. 137(1), 72–83 (2008).
[Crossref]

Méndez, E. R.

Moes, C. J.

Moffitt, T. P.

T. P. Moffitt, “Compact fiber-optic diffuse reflection probes for medical diagnostics,” Ph.D. dissertation (Oregon Health & Science University, 2007).
[Crossref]

Muller, G.

M. Hammer, A. Roggan, D. Schweitzer, and G. Muller, “Optical properties of ocular fundus tissues-an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40(6), 963–978 (1995).
[Crossref]

Naik, P.

Nakaoki, T.

T. Nakaoki and H. Yamashita, “Bound states of water in poly (vinyl alcohol) hydrogel prepared by repeated freezing and melting method,” J. Mol. Struct. 875(1-3), 282–287 (2008).
[Crossref]

Nevas, S.

Nilsson, A. M.

Noorma, M.

Okada, E.

S. Takano and E. Okada, “Analysis of the diffuse reflectance spectra of skin due to detection system,” in European Conferences on Biomedical Optics, International Society for Optics and Photonics, 80881A (2011).

Ouyang, J.-f. J.

Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011).
[Crossref]

Peppas, N. A.

C. M. Hassan and N. A. Peppas, “Structure and applications of poly(vinyl alcohol) hydrogels produced by conventional crosslinking or by freezing/thawing methods,” Adv. Polym. Sci. 153, 37–65 (2000).
[Crossref]

Pickering, J. W.

Prahl, S.

S. Prahl, Everything I think you should know about inverse adding-doubling. Oregon Medical Laser Center, Manual of the Inverse Adding-Doubling Program (2011). http://omlc.ogi.edu/software/iad/ (retrieved 01.10.12).

Prahl, S. A.

Pu, Y.

Qu, X.-h.

Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011).
[Crossref]

Remisowsky, A. M. V.

Roggan, A.

M. Hammer, A. Roggan, D. Schweitzer, and G. Muller, “Optical properties of ocular fundus tissues-an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40(6), 963–978 (1995).
[Crossref]

Roos, A.

Schafer, E.

Schweitzer, D.

M. Hammer, A. Roggan, D. Schweitzer, and G. Muller, “Optical properties of ocular fundus tissues-an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40(6), 963–978 (1995).
[Crossref]

Seyfried, M.

Sjostrand, J.

Steenbergen, W.

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

Sterenborg, H. J.

Sterenborg, H. J. C.

Sun, Y.

G. D. Boreman, A. B. Centore, and Y. Sun, “Generation of laser speckle with an integrating sphere,” Opt. Eng. 29(4), 339–342 (1990).
[Crossref]

Takano, S.

S. Takano and E. Okada, “Analysis of the diffuse reflectance spectra of skin due to detection system,” in European Conferences on Biomedical Optics, International Society for Optics and Photonics, 80881A (2011).

Taylor, A. H.

Terán, E.

Terán-Bobadilla, E.

E. Terán-Bobadilla, “Consecuencias del esparcimiento múltiple en la absorción de algunos sistemas biológicos,” PhD thesis, Centro de Investigación en Científica y de Educación Superior de Ensenada, Carretera Ensenada-Tijuana No. 3918 Zona Playitas Código Postal 22860 Apdo. Postal 360 Ensenada, B.C. México, August 2010.

Thaung, J.

Van Gemert, M. J.

van Hespen, J. C. G.

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

van Leeuwen, T. G.

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

Van Wieringen, N.

Wang, L.

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

Wang, W.

Welch, A. J.

Xu, M.

Yamashita, H.

T. Nakaoki and H. Yamashita, “Bound states of water in poly (vinyl alcohol) hydrogel prepared by repeated freezing and melting method,” J. Mol. Struct. 875(1-3), 282–287 (2008).
[Crossref]

Zagieboylo, W.

Zheng, L.

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

Zhu, Z.-m.

Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011).
[Crossref]

Adv. Polym. Sci. (1)

C. M. Hassan and N. A. Peppas, “Structure and applications of poly(vinyl alcohol) hydrogels produced by conventional crosslinking or by freezing/thawing methods,” Adv. Polym. Sci. 153, 37–65 (2000).
[Crossref]

Appl. Opt. (12)

A. M. Nilsson, A. Jonsson, J. C. Jonsson, and A. Roos, “Method for more accurate transmittance measurements of low-angle scattering samples using an integrating sphere with an entry port beam diffuser,” Appl. Opt. 50(7), 999–1006 (2011).
[Crossref]

E. Terán, E. R. Méndez, S. Enríque, and R. Iglesias- Prieto, “Multiple light scattering and absorption in reef-building corals,” Appl. Opt. 49(27), 5032–5042 (2010).
[Crossref]

J. M. Davies and W. Zagieboylo, “An integrating sphere system for measuring average reflectance and transmittance,” Appl. Opt. 4(2), 167–174 (1965).
[Crossref]

L. Hanssen, “Integrating-sphere system and method for absolute measurement of transmittance, reflectance, and absorptance of specular samples,” Appl. Opt. 40(19), 3196–3204 (2001).
[Crossref]

S. Nevas, F. Manoocheri, and E. Ikonen, “Gonioreflectometer for measuring spectral diffuse reflectance,” Appl. Opt. 43(35), 6391–6399 (2004).
[Crossref]

V. Ahtee, S. W. Brown, T. C. Larason, K. R. Lykke, E. Ikonen, and M. Noorma, “Comparison of absolute spectral irradiance responsivity measurement techniques using wavelength-tunable lasers,” Appl. Opt. 46(20), 4228–4236 (2007).
[Crossref]

M. Seyfried, L. Fukshansky, and E. Schafer, “Correcting remission and transmission spectra of plant tissue measured in glass cuvettes: a technique,” Appl. Opt. 22(3), 492–496 (1983).
[Crossref]

L. Fukshansky, N. Fukshansky-Kazarinova, and A. M. V. Remisowsky, “Absorption spectra of leaves corrected for scattering and distributional error: a radiative transfer and absorption statistics treatment,” Appl. Opt. 30(22), 3145–3153 (1991).
[Crossref]

J. W. Pickering, S. A. Prahl, N. Van Wieringen, J. F. Beek, H. J. Sterenborg, and M. J. Van Gemert, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32(4), 399–410 (1993).
[Crossref]

S. A. Prahl, M. J. Van Gemert, and A. J. Welch, “Double-integrating-sphere system for measuring the optical properties of tissue,” Appl. Opt. 32(4), 559–568 (1993).
[Crossref]

H. Mamane, J. J. Ducoste, and K. G. Linden, “Effect of particles on ultraviolet light penetration in natural and engineered systems,” Appl. Opt. 45(8), 1844–1856 (2006).
[Crossref]

C. K. Gatebe, J. J. Butler, J. W. Cooper, M. Kowalewski, and M. D. King, “Characterization of errors in the use of integrating-sphere systems in the calibration of scanning radiometers,” Appl. Opt. 46(31), 7640–7651 (2007).
[Crossref]

Appl. Spectrosc. (1)

Chem. Eng. J. (1)

H. S. Mansur and H. S. Costa, “Nanostructured poly (vinyl alcohol)/bioactive glass and poly (vinyl alcohol)/chitosan/bioactive glass hybrid scaffolds for biomedical applications,” Chem. Eng. J. 137(1), 72–83 (2008).
[Crossref]

Comput. Methods Programs Biomed. (1)

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

J. Biomed. Opt. (1)

S. Manohar, A. Kharine, J. C. G. van Hespen, W. Steenbergen, and T. G. van Leeuwen, “Photoacoustic mammography laboratory prototype: imaging of breast tissue phantoms,” J. Biomed. Opt. 9(6), 1172–1181 (2004).
[Crossref]

J. Disp. Technol. (1)

Z.-m. Zhu, X.-h. Qu, G.-x. Jia, and J.-f. J. Ouyang, “Uniform illumination design by configuration of LED array and diffuse reflection surface for color vision application,” J. Disp. Technol. 7(2), 84–89 (2011).
[Crossref]

J. Mol. Struct. (1)

T. Nakaoki and H. Yamashita, “Bound states of water in poly (vinyl alcohol) hydrogel prepared by repeated freezing and melting method,” J. Mol. Struct. 875(1-3), 282–287 (2008).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Photoacoust. (1)

S. L. Jacques, “Coupling 3D Monte Carlo light transport in optically heterogeneous tissues to photoacoustic signal generation,” J. Photoacoust. 2(4), 137–142 (2014).
[Crossref]

Opt. Eng. (1)

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[Crossref]

Opt. Express (1)

Phys. Med. Biol. (1)

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[Crossref]

Other (6)

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[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagrams of an integrating sphere used to illuminate a sample with diffuse or directional light. The reflection geometry is shown in (a) and (b), and the transmittance geometry is shown in (c) and (d).
Fig. 2.
Fig. 2. Scheme to evaluate the diffuse–diffuse reflectance $R_d$ . (a) Sphere without a sample, (b) sphere with a diffuse standard, and (c) with a sample.
Fig. 3.
Fig. 3. Schematic of the experimental setup used to measure the optical properties of a turbid medium.
Fig. 4.
Fig. 4. Comparison of the uncertainties for the three ways of expressing diffuse reflectance. The horizontal axis denotes the relative uncertainty and the vertical axis denotes the total magnitude of uncertainty propagated.The Eq. (10) is represented by solid line. It is considered that $R_d=0.41$ and a power detected ratio of ${\rho _d}=1.122$ . (a) Evaluating the uncertainty propagation in Eq. (14). Effect of the change in each variable for a given uncertainty in that variable at $a_s=0.015$ , $a_{\delta }=1.5\times 10^{{-}4}$ , and $m= 0.98$ . The contribution of the wall reflectance $m$ is denoted by diamonds. The contribution of the sample aperture $a_s$ is denoted by dashed line. The contribution of the detector aperture $a_d$ is denoted by circles. (b) Evaluating the uncertainty propagation in Eq. (15). Effect of the change in each variable for a given uncertainty in that variable at $b_1=0.8$ and $b_2=0.25$ . Contribution of the power detected $\rho _d$ is represented by diamonds, the constant of the sphere $b_1$ by dashed line and constant of the sphere $b_2$ by circles. (c) Evaluating the uncertainty propagation in Eq. (16). Effect of the change in each variable for a given uncertainty in that variable at $R_d^{(std)}=0.98$ , $\rho _D=0.9$ , and $\rho _{(std)} =0.75$ . Contribution of the power detected ratio $\rho _D$ is represented by dashed line, the reflectance standard $R_d^{(std)}$ by diamonds and power detected ratio with the standard $\rho _{std}$ by circles.
Fig. 5.
Fig. 5. Diffuse–diffuse reflectance $R_d$ as a function of the PVA content.
Fig. 6.
Fig. 6. Monte Carlo simulation of a PVA layer as a function of the percentage increment $n$ . (a) Directional-diffuse transmittance $T_{MC}$ , collimated transmittance $T_c$ , and total transmittance $T_{tot}=T_{MC}+T_c$ of the layer. (b) Directional-diffuse reflectance $R_{MC}$ and collimated or specular reflectance $R_{sp}$ of the layer. (c) Absorption of the layer ${\cal A}_{MC}$ .
Fig. 7.
Fig. 7. Integrating sphere measurements of a PVA layer with increasing PVA content. (a) Directional–diffuse transmittance $T_{cd}$ and collimated transmittance $T_c$ . (b) Directional–diffuse reflectance $R_{cd}$ and (c) absorption ${\cal A}=1- (R_{cd}+T_{cd} )$ of the PVA.

Tables (3)

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Table 1. Optical properties as functions of the detected power obtained through the integrating sphere method.

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Table 2. Monte Carlo simulation results of a PVA layer for $\mu _a=0.0342$ mm $^{{-}1}$ , $\mu '_s=0.5$ mm $^{{-}1}$ , and $n = 1.36$ for $\lambda = 1064$ nm over a transparent substrate with a refractive index of 1.48. Note that, owing to energy conservation, ${\cal A}+ (R_{MC}+R_{sp} )+T_{tot}=1$ .

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Table 3. Detected power of a set of samples with different PVA contents and a width of 100 $\mu$ m.

Equations (17)

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P d = A δ A m P o 1 ( m α + R d A s / A ) ,
R d = A A s [ 1 m α m P o P d A δ A ] .
b 1 = A δ A m 1 m α and b 2 = A s A 1 1 m α .
R d = 1 b 2 [ 1 b 1 P d / P o ] .
b 1 = P d ( o ) P o and b 2 = 1 R d ( s t d ) [ 1 b 1 P d ( s t d ) / P o ] ,
R d = R d ( s t d ) [ 1 ρ d ] [ 1 ρ s t d ] .
P r = A δ A ( R c d + R s p ) P 1 1 ( m α + R d A s / A ) .
R c d = m P r P d P o P 1 .
R s p = m P r P d P o P 1 m P r P d P o P 1 .
( δ X ) 2 = i [ ( X / x i ) δ x i ] 2 ,
T M C = T t o t e [ ( μ s + μ a ) d z ] .
μ s ( n ) = ρ ( n ) C s = μ s ( 1 + n / 100 ) ,
μ a ( n ) = μ a ( 1 + n / 100 ) .
[ δ R d R d ] 2 = [ m ( α + a δ ρ d ) ( 1 m α m a δ ρ d ) ] 2 ( δ m m ) 2 ,
( δ R d R d ) 2 = [ b 1 ρ d ( 1 b 1 ρ d ) ( δ b 1 b 1 ) ] 2 + [ δ b 2 b 2 ] 2 + [ b 1 ρ d ( 1 b 1 ρ d ) ( δ ρ d ρ d ) ] 2 ,
[ δ R d R d ] 2 = [ δ R d ( s t d ) R d ( s t d ) ] 2 + [ ρ D ( 1 ρ D ) ( δ ρ D ρ D ) ] 2 + [ ρ s t d ( 1 ρ s t d ) ( δ ρ s t d ρ s t d ) ] 2 .
[ δ R c d / R c d ] 2 = [ δ m / m ] 2 + [ δ P r / P r ] 2 + [ δ P d / P d ] 2 .