Abstract

We theoretically and experimentally investigate light diffusion in disordered meso-macroporous materials with a cylindrical shape. High Internal Phase Emulsion (HIPE)-based silica foam samples, exhibiting a polydisperse pore-size distribution centered around 19 μm to resemble certain biological tissues, are realized. To quantify the effect of a finite lateral size on measurable quantities, an analytical model for diffusion in finite cylinders is developed and validated by Monte Carlo random walk simulations. Steady-state and time-resolved transmission experiments are performed and the transport parameters (transport mean free path and material absorption length) are successfully retrieved from fits of the experimental curves with the proposed model. This study reveals that scattering losses on the lateral sides of the samples are responsible for a lowering of the transmission signal and a shortening of the photon lifetime, similar in experimental observables to the effect of material absorption. The recognition of this geometrical effect is essential since its wrong attribution to material absorption could be detrimental in various applications, such as biological tissue diagnosis or conversion efficiency in dye-sensitized solar cells.

© 2014 Optical Society of America

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2014 (1)

C. M. Leroy, C. Olivier, T. Toupance, M. Abbas, L. Hirsch, S. Ravaine, R. Backov, “One-pot easily-processed TiO2 macroporous photoanodes (Ti-HIPE) for dye-sensitized solar cells,” Sol. State Sci. 28, 81–89 (2014).
[CrossRef]

2013 (2)

T. Svensson, K. Vynck, M. Grisi, R. Savo, M. Burresi, D. S. Wiersma, “Holey random walks: Optics of heterogeneous turbid composites,” Phys. Rev. E 87, 022120 (2013).
[CrossRef]

N. Ghofraniha, I. Viola, A. Zacheo, V. Arima, G. Gigli, C. Conti, “Transition from nonresonant to resonant random lasers by the geometrical confinement of disorder,” Opt. Lett. 38, 5043–5046 (2013).
[CrossRef] [PubMed]

2012 (2)

E. Alerstam, T. Svensson, “Observation of anisotropic diffusion of light in compacted granular porous materials,” Phys. Rev. E 89, 040301 (2012).
[CrossRef]

T. van der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
[CrossRef]

2010 (3)

A. B. Davis, A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73, 026801 (2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Z. Shi, C. A. Anderson, “Pharmaceutical applications of separation of absorption and scattering in near-infrared spectroscopy (NIRS),” J. Pharm. Sci. 99, 4766–4783 (2010).
[CrossRef] [PubMed]

2009 (1)

T. Svensson, E. Alerstam, D. Khoptyar, J. Johansson, S. Folestad, S. Andersson-Engels, “Near-infrared photon time-of-flight spectroscopy of turbid materials up to 1400 nm,” Rev. Sci. Instrum. 80, 063105 (2009).
[CrossRef] [PubMed]

2008 (2)

T. Svensson, M. Andersson, L. Rippe, S. Svanberg, S. Andersson-Engels, J. Johansson, S. Folestad, “VCSEL-based oxygen spectroscopy for structural analysis of pharmaceutical solids,” Appl. Phys. B 90, 345–354 (2008).
[CrossRef]

P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

2007 (1)

2005 (3)

M. Schmiedeberg, M. F. Miri, H. Stark, “Photon channelling in foams,” Eur. Phys. J. E 18, 123–131 (2005).
[CrossRef] [PubMed]

M. Xu, R. R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. 95, 213901 (2005).
[CrossRef] [PubMed]

G. Reich, “Near-infrared spectroscopy and imaging: Basic principles and pharmaceutical applications,” Adv. Drug Delivery Rev. 57, 1109 (2005).
[CrossRef]

2004 (3)

F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. 14, 1370–1376 (2004).
[CrossRef]

A. S. Gittings, R. Bandyopadhyay, D. J. Durian, “Photon channelling in foams,” Europhys. Lett. 65, 414–419 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
[CrossRef]

2000 (1)

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

1999 (1)

M. C. W. van Rossum, Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

1997 (1)

1996 (1)

T. G. Mason, J. Bibette, D. A. Weitz, “Yielding and Flow of Monodisperse Emulsions,” J. Colloid. Interface Sci. 179, 439–448 (1996).
[CrossRef]

1995 (1)

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, ”Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine 47, 131–146 (1995).
[CrossRef]

1993 (2)

M.-P. Aronson, M.-F. Petko, “Highly Concentrated Water-in-Oil Emulsions: Influence of Electrolyte on Their Properties and Stability,” J. Colloid Interface Sci. 159, 134–149 (1993).
[CrossRef]

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

1991 (1)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

1990 (1)

A. Z. Genack, J. M. Drake, “Relationship between Optical Intensity, Fluctuations and Pulse Propagation in Random Media,” Europhys. Lett. 11, 4331–4336 (1990).
[CrossRef]

1989 (1)

1951 (1)

M. J. Mooney, “The viscosity of a concentrated suspension of spherical particles,” J. Colloid. Interface Sci. 6, 162–170 (1951).
[CrossRef]

Abbas, M.

C. M. Leroy, C. Olivier, T. Toupance, M. Abbas, L. Hirsch, S. Ravaine, R. Backov, “One-pot easily-processed TiO2 macroporous photoanodes (Ti-HIPE) for dye-sensitized solar cells,” Sol. State Sci. 28, 81–89 (2014).
[CrossRef]

Achard, M-.F.

F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. 14, 1370–1376 (2004).
[CrossRef]

Akkermans, E.

E. Akkermans, G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

Alerstam, E.

E. Alerstam, T. Svensson, “Observation of anisotropic diffusion of light in compacted granular porous materials,” Phys. Rev. E 89, 040301 (2012).
[CrossRef]

T. Svensson, E. Alerstam, D. Khoptyar, J. Johansson, S. Folestad, S. Andersson-Engels, “Near-infrared photon time-of-flight spectroscopy of turbid materials up to 1400 nm,” Rev. Sci. Instrum. 80, 063105 (2009).
[CrossRef] [PubMed]

Alfano, R. R.

M. Xu, R. R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. 95, 213901 (2005).
[CrossRef] [PubMed]

Altube, A.

P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Anderson, C. A.

Z. Shi, C. A. Anderson, “Pharmaceutical applications of separation of absorption and scattering in near-infrared spectroscopy (NIRS),” J. Pharm. Sci. 99, 4766–4783 (2010).
[CrossRef] [PubMed]

Andersson, M.

T. Svensson, M. Andersson, L. Rippe, S. Svanberg, S. Andersson-Engels, J. Johansson, S. Folestad, “VCSEL-based oxygen spectroscopy for structural analysis of pharmaceutical solids,” Appl. Phys. B 90, 345–354 (2008).
[CrossRef]

Andersson-Engels, S.

T. Svensson, E. Alerstam, D. Khoptyar, J. Johansson, S. Folestad, S. Andersson-Engels, “Near-infrared photon time-of-flight spectroscopy of turbid materials up to 1400 nm,” Rev. Sci. Instrum. 80, 063105 (2009).
[CrossRef] [PubMed]

T. Svensson, M. Andersson, L. Rippe, S. Svanberg, S. Andersson-Engels, J. Johansson, S. Folestad, “VCSEL-based oxygen spectroscopy for structural analysis of pharmaceutical solids,” Appl. Phys. B 90, 345–354 (2008).
[CrossRef]

Arendt, J. T.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Arima, V.

Aronson, M.-P.

M.-P. Aronson, M.-F. Petko, “Highly Concentrated Water-in-Oil Emulsions: Influence of Electrolyte on Their Properties and Stability,” J. Colloid Interface Sci. 159, 134–149 (1993).
[CrossRef]

Backman, V.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Backov, R.

C. M. Leroy, C. Olivier, T. Toupance, M. Abbas, L. Hirsch, S. Ravaine, R. Backov, “One-pot easily-processed TiO2 macroporous photoanodes (Ti-HIPE) for dye-sensitized solar cells,” Sol. State Sci. 28, 81–89 (2014).
[CrossRef]

F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. 14, 1370–1376 (2004).
[CrossRef]

Badizadegan, K.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Bandyopadhyay, R.

A. S. Gittings, R. Bandyopadhyay, D. J. Durian, “Photon channelling in foams,” Europhys. Lett. 65, 414–419 (2004).
[CrossRef]

Barby, D.

D. Barby, Z. Haq, “Low density porous cross-linked polymeric materials and their preparation,” Eur. Patent Appl.60138 (1982).

Barthelemy, P.

T. van der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B 85, 115401 (2012).
[CrossRef]

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Bertolotti, J.

P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Bibette, J.

T. G. Mason, J. Bibette, D. A. Weitz, “Yielding and Flow of Monodisperse Emulsions,” J. Colloid. Interface Sci. 179, 439–448 (1996).
[CrossRef]

Blanco, Á.

P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A 78, 023823 (2008).
[CrossRef]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

Brinker, C. J.

C. J. Brinker, G. W. Scherer, in Sol-Gel Science: the Physics and Chemistry of Sol-Gel Processing (Academic Press, San Diego, 1990).

Burresi, M.

T. Svensson, K. Vynck, M. Grisi, R. Savo, M. Burresi, D. S. Wiersma, “Holey random walks: Optics of heterogeneous turbid composites,” Phys. Rev. E 87, 022120 (2013).
[CrossRef]

R. Savo, M. Burresi, T. Svensson, K. Vynck, D. S. Wiersma, “Measuring the fractal dimension of an optical random walk,” arXiv:1312.5962.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104, 100601 (2010).
[CrossRef] [PubMed]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
[CrossRef]

Carn, F.

F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. 14, 1370–1376 (2004).
[CrossRef]

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, 2011).

Colin, A.

F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. 14, 1370–1376 (2004).
[CrossRef]

Conti, C.

Contini, D.

Crawford, J. M.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Dasari, R. R.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Davis, A. B.

A. B. Davis, A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73, 026801 (2010).
[CrossRef]

Deleuze, H.

F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. 14, 1370–1376 (2004).
[CrossRef]

Drake, J. M.

A. Z. Genack, J. M. Drake, “Relationship between Optical Intensity, Fluctuations and Pulse Propagation in Random Media,” Europhys. Lett. 11, 4331–4336 (1990).
[CrossRef]

Durian, D. J.

A. S. Gittings, R. Bandyopadhyay, D. J. Durian, “Photon channelling in foams,” Europhys. Lett. 65, 414–419 (2004).
[CrossRef]

Elaloufi, R.

Feld, M. S.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, M. S. Feld, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

(a) SEM micrograph of the SiO2(HIPE) and (b) normalized distribution (green histogram) of pore diameters. The normal probability distribution function determined from the arithmetic mean and standard deviation of the pore sizes is also shown (red dashed line). The inset of (a) shows 2 monoliths of different thicknesses, evidencing the equivalence of the lateral and longitudinal dimensions.

Fig. 2
Fig. 2

(a) Sketch of the Monte Carlo simulations. Random walkers are launched on a side of the cylinder (here, top side). Those that cross the other (bottom) side are counted in the transmission measurements. (b) Time-resolved transmission curves through a cylinder with thickness L = 100 μm, radius R = 100 μm and transport mean free path t = 5 μm, obtained from Monte Carlo simulations (hollow dots), the analytical expression derived for slabs [16] (gray dashed line), and the analytical expression in Eq. (13) derived for finite cylinders (blue solid line). The agreement between the Monte Carlo simulations and the model for diffusion in finite cylinders is excellent, while a clear deviation is observed with the model for diffusion in slabs. The faster decay observed in the case of the cylinder could be wrongly interpreted as material absorption.

Fig. 3
Fig. 3

(a) White light all-angle integrated transmittance of the SiO2(HIPE) monoliths for 25 thicknesses (from top to bottom: the shortest to the longest) as a function of wavelength. The intersections of these plots with the vertical line drawn at λ = 515 nm provide the transmittance as a function of thickness (red stars) at this wavelength (b). The black solid (dashed) line shows the best fit of T (L) performed by the standard diffusion equation, without (with) absorption, while the red dashed and solid lines show the best fit of T (L) by Eq. (14) (including absorption) and the curve when absorption is neglected, respectively. (c) Time of flight experiments performed on 3.2 mm(red stars) and 6.9 mm (blue circles) thick samples. The corresponding dashed lines show the predicted trends of T (L, t) by Eq. (13), according to the values t = 48 μm and a = 0.954 m determined in (b). (d) Exponential decay rates of the tail of T (L, t) versus L (red stars). The red dashed and solid lines show the best fit of T (L) by Eq. (15) (including absorption) and the curve when absorption is neglected, respectively.

Tables (1)

Tables Icon

Table 1 Oil volumic fraction of the starting emulsion and Mercury intrusion porosimetry results.

Equations (15)

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D 2 u ( r , t ) = u ( r , t ) t + μ a v u ( r , t ) ,
2 u ( ρ , z , t ) = 1 ρ u ( ρ , z , t ) ρ + 2 u ( ρ , z , t ) ρ 2 + u ( ρ , z , t ) z .
T ( t ) = exp [ ( λ ρ + λ z + μ a v ) t ] ,
Z ( z ) = C 1 cos [ λ z D z ] + C 2 sin [ λ z D z ] ,
P ( ρ ) = C 3 J 0 [ λ ρ D z ] + C 4 Y 0 [ λ ρ D z ] .
u ( ρ , z = 0 , t ) = 0 ,
u ( ρ , z = L e , t ) = 0 ,
u ( ρ = 0 , z , t ) ± ,
u ( ρ = R e , z , t ) = 0 .
u ( ρ , z , t ) = 2 π L e R e 2 m = 1 1 J 1 2 ( α m ) J 0 [ α m R e ρ ] exp [ α m 2 R e 2 D t ] × n = 1 sin [ n π L e z 0 ] sin [ n π L e z ] exp [ n 2 π 2 L e 2 D t ] exp [ μ a ν t ] .
J = D u ( ρ , z , t ) = D [ u ( ρ , z , t ) ρ ρ ^ + u ( ρ , z , t ) z z ^ ] .
J z ( ρ , z , t ) = 2 D L e 2 R e 2 m = 1 1 J 1 2 ( α m ) J 0 [ α m R e ρ ] exp [ α m 2 R e 2 D t ] × n = 1 n sin [ n π L e z 0 ] cos [ n π L e z ] exp [ n 2 π 2 L e 2 D t ] exp [ μ a v t ] .
T ( z , t ) = 4 π D R L e 2 R e m = 1 1 α m J 1 2 ( α m ) J 1 [ α m R e R ] exp [ α m 2 R e 2 D t ] × n = 1 n sin [ n π L e z 0 ] cos [ n π L e z ] exp [ n 2 π 2 L e 2 D t ] exp [ μ a v t ] .
T ( z ) = 4 π R L e 2 R e m = 1 n = 1 J 1 [ α m R e R ] n sin [ n π L e z 0 ] cos [ n π L e z ] α m J 1 2 ( α m ) ( α m 2 R e 2 + n 2 π 2 R e 2 + μ a v D ) .
Γ = D ( α 1 2 R e 2 + π 2 L e 2 + 3 t a ) .

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