Abstract

We analyze the time dependent response of strongly scattering media (SSM) to ultra-short pulses of light. A random walk technique is used to model the optical scattering of ultra-short pulses of light propagating through media with random shapes and various packing densities. The pulse spreading was found to be strongly dependent on the average particle size, particle size distribution, and the packing fraction. We also show that the intensity as a function of time-delay can be used to analyze the particle size distribution and packing fraction of an optically thick sample independently of the presence of absorption features. Finally, we propose an all new way to measure the shape of ultra-short pulses that have propagated through a SSM.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. D. J. LeCaptain and K. A. Burglund, "The applicability of second harmonic generation for in situ measurement of induction time of selected crystallization systems," J. Cryst. Growth 203, 564-569 (1999)
    [CrossRef]

2006

L. Taylor, H.Wikstr¨om, A. Gift, and J. Rantanen, "Monitoring and manipulating crystal hydrate formation during high shear wet granulation," European Journal of Pharmaceutical Sciences 28, S7 (2006).

C. C. Sun, "A material-sparing method for simultaneous determination of true density and powder compaction properties—Aspartame as an example," International Journal Pharmaceutics,  326, 94-99, (2006)
[CrossRef]

C. B. Rawle, C. J. Lee, C. J. Strachan, K. Payne, P. J. Manson, and T. Rades, "Towards characterization and identification of solid state pharmaceutical mixtures through second harmonic generation," Journal of Pharmaceutical Sciences 95, 761-768 (2006)
[CrossRef] [PubMed]

2005

I. M. Vellekoop, P. Lodahl, and A. Lagendijk, "Determination of the diffusion constant using phase-sensitive measurements," Phys. Rev. E 71, 056604 (2005).
[CrossRef]

J. Rantanen, H. Wikstr¨om, R. Turner, and L. Taylor, "Use of in-line near-infrared spectroscopy in combination with chemometrics for improved understanding of pharmaceutical processes," Anal. Chem. 77, 556-563 (2005).
[CrossRef] [PubMed]

2004

A. C. Jørgensen, J. Rantanen, P. Luukkonen, S. Laine, and J. Yliruusi, "Visualization of a pharmaceutical unit operation: Wet granulation," Anal. Chem. 76, 5331-5338 (2004).
[CrossRef] [PubMed]

C. J. Strachan, C. J. Lee, and T. Rades, "Partial Characterization of different mixtures of solids by measuring the optical nonlinear response," Journal of Pharmaceutical Sciences 93, 733-742 (2004).
[CrossRef] [PubMed]

2003

L. Liu, M. I. Mishchenko, J. W. Hovenier, H. Volten, and O. Mu˜noz, "Scattering matrix of quartz aerosols: comparison and synthesis of laboratory and Lorenz-Mie results," J. Quant. Spectrosc. Radiat. Transfer 79, 911- 920 (2003).Q2Q3
[CrossRef]

M. I. Mishchenko and A. A. Lacis, "morphology-dependent resonances of nearly spherical particles in random orientation," Appl. Opt. 42, 5551-5556 (2003).
[CrossRef] [PubMed]

2001

L. Mees, G. Gr’ehan, and G. Gouesbet, "Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses," Opt. Commun. 194, 59-65 (2001).
[CrossRef]

M. C. Pasikatan, J. L. Steele, C. K. Spillman, and E. Haque, "Near infrared reflectance spectroscopy for online particle size analysis of powders and ground materials," Journal of Near Infrared Spectroscopy 9, 153-164 (2001).Q1
[CrossRef]

2000

M. Blanco and A. Villar, "Polymorphic analysis of a pharmaceutical preparation by NIR spectroscopy," Analyst 125, 2311-2314 (2000).
[CrossRef]

A. D. Patel, P. E. Luner, and M. S. Kemper, "Quantitative analysis of polymorphs in binary and multi-component powder mixtures by near-infrared reflectance spectroscopy," International Journal of Pharmaceutics 206, 63-74 (2000).
[CrossRef] [PubMed]

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy "A 32-channel time-resolved instrument for medical optical tomography," Rev. Sci. Instrum. 71, 256-261 (2000).
[CrossRef]

1999

W. E. Vargas, "Diffuse radiation intensity propagating through a particulate slab," J. Opt. Soc. Am. A 16, 1362- 1372 (1999).
[CrossRef]

D. J. LeCaptain and K. A. Burglund, "The applicability of second harmonic generation for in situ measurement of induction time of selected crystallization systems," J. Cryst. Growth 203, 564-569 (1999)
[CrossRef]

1998

M. Blanco, J. Coello, H. Iturriaga, S. Maspoch, and C. de la Pezuela, "Near-infrared spectroscopy in the pharmaceutical industry. Critical review," Analyst 123, 135R-150R (1998).
[CrossRef]

E. Baigar, C. Hauger, and W. Zinth, "Imaging within highly scattering media using time-resolved backscattering of femtosecond pulses," Appl. Phys. B 67, 257-261 (1998).
[CrossRef]

L. S. Taylor and G. Zografi, "The quantitative analysis of crystallinity using FT-Raman spectroscopy," Pharmaceutical Research 15, 755-761 (1998).
[CrossRef] [PubMed]

1997

C. Hauger, E. Baigar, andW. Zinth, "Induced backscattering due to reflecting surfaces in highly scattering media," Opt. Commun. 133, 72-76 (1997).
[CrossRef]

1996

C. Hauger, E. Baigar, T. Wilhelm, and W. Zinth, "Time-resolved backscattering of femtosecond pulses from scattering media—an experimental and numerical investigation," Opt. Commun. 131, 351-358 (1996).
[CrossRef]

1994

1993

1991

1987

Anal. Chem.

A. C. Jørgensen, J. Rantanen, P. Luukkonen, S. Laine, and J. Yliruusi, "Visualization of a pharmaceutical unit operation: Wet granulation," Anal. Chem. 76, 5331-5338 (2004).
[CrossRef] [PubMed]

J. Rantanen, H. Wikstr¨om, R. Turner, and L. Taylor, "Use of in-line near-infrared spectroscopy in combination with chemometrics for improved understanding of pharmaceutical processes," Anal. Chem. 77, 556-563 (2005).
[CrossRef] [PubMed]

Analyst

M. Blanco and A. Villar, "Polymorphic analysis of a pharmaceutical preparation by NIR spectroscopy," Analyst 125, 2311-2314 (2000).
[CrossRef]

M. Blanco, J. Coello, H. Iturriaga, S. Maspoch, and C. de la Pezuela, "Near-infrared spectroscopy in the pharmaceutical industry. Critical review," Analyst 123, 135R-150R (1998).
[CrossRef]

Appl. Opt.

Appl. Phys. B

E. Baigar, C. Hauger, and W. Zinth, "Imaging within highly scattering media using time-resolved backscattering of femtosecond pulses," Appl. Phys. B 67, 257-261 (1998).
[CrossRef]

European Journal of Pharmaceutical Sciences

L. Taylor, H.Wikstr¨om, A. Gift, and J. Rantanen, "Monitoring and manipulating crystal hydrate formation during high shear wet granulation," European Journal of Pharmaceutical Sciences 28, S7 (2006).

International Journal of Pharmaceutics

A. D. Patel, P. E. Luner, and M. S. Kemper, "Quantitative analysis of polymorphs in binary and multi-component powder mixtures by near-infrared reflectance spectroscopy," International Journal of Pharmaceutics 206, 63-74 (2000).
[CrossRef] [PubMed]

International Journal Pharmaceutics

C. C. Sun, "A material-sparing method for simultaneous determination of true density and powder compaction properties—Aspartame as an example," International Journal Pharmaceutics,  326, 94-99, (2006)
[CrossRef]

J. Cryst. Growth

D. J. LeCaptain and K. A. Burglund, "The applicability of second harmonic generation for in situ measurement of induction time of selected crystallization systems," J. Cryst. Growth 203, 564-569 (1999)
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

L. Liu, M. I. Mishchenko, J. W. Hovenier, H. Volten, and O. Mu˜noz, "Scattering matrix of quartz aerosols: comparison and synthesis of laboratory and Lorenz-Mie results," J. Quant. Spectrosc. Radiat. Transfer 79, 911- 920 (2003).Q2Q3
[CrossRef]

Journal of Near Infrared Spectroscopy

M. C. Pasikatan, J. L. Steele, C. K. Spillman, and E. Haque, "Near infrared reflectance spectroscopy for online particle size analysis of powders and ground materials," Journal of Near Infrared Spectroscopy 9, 153-164 (2001).Q1
[CrossRef]

Journal of Pharmaceutical Sciences

C. B. Rawle, C. J. Lee, C. J. Strachan, K. Payne, P. J. Manson, and T. Rades, "Towards characterization and identification of solid state pharmaceutical mixtures through second harmonic generation," Journal of Pharmaceutical Sciences 95, 761-768 (2006)
[CrossRef] [PubMed]

C. J. Strachan, C. J. Lee, and T. Rades, "Partial Characterization of different mixtures of solids by measuring the optical nonlinear response," Journal of Pharmaceutical Sciences 93, 733-742 (2004).
[CrossRef] [PubMed]

Opt. Commun.

C. Hauger, E. Baigar, andW. Zinth, "Induced backscattering due to reflecting surfaces in highly scattering media," Opt. Commun. 133, 72-76 (1997).
[CrossRef]

C. Hauger, E. Baigar, T. Wilhelm, and W. Zinth, "Time-resolved backscattering of femtosecond pulses from scattering media—an experimental and numerical investigation," Opt. Commun. 131, 351-358 (1996).
[CrossRef]

L. Mees, G. Gr’ehan, and G. Gouesbet, "Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses," Opt. Commun. 194, 59-65 (2001).
[CrossRef]

Pharmaceutical Research

L. S. Taylor and G. Zografi, "The quantitative analysis of crystallinity using FT-Raman spectroscopy," Pharmaceutical Research 15, 755-761 (1998).
[CrossRef] [PubMed]

Phys. Rev. E

I. M. Vellekoop, P. Lodahl, and A. Lagendijk, "Determination of the diffusion constant using phase-sensitive measurements," Phys. Rev. E 71, 056604 (2005).
[CrossRef]

Rev. Sci. Instrum.

F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delpy "A 32-channel time-resolved instrument for medical optical tomography," Rev. Sci. Instrum. 71, 256-261 (2000).
[CrossRef]

STP Pharma Sciences

S. Pederson,and H. G. Kristensen, "Change in crystal density of acetylsalicylic acid during compaction," STP Pharma Sciences,  4, 201-206 (1994).Q4

Other

Schott GlassAG , "Optical glass catalog," URL http://www.schott.com/opticsdevices/english/download/index.htm, pp 11 (2006)

USFDA, "Process and Analytical Technology (PAT) Initiative," (2006). URL http://www.fda.gov/Cder/OPS/PAT.htm.

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Figures (13)

Fig. 1.
Fig. 1.

The probability of a ray traversing a sphere along a particular chord length (a). The calculated data from Snell’s law is given by the black line, while the red line is equation 1 with α=2.7 and β=1.7. Subfigure (b): equation 1 for α=2, β=2 (black), α=5, β=2 (red), α=2, β=10 (magenta), and α=2, β=5 (blue).

Fig. 2.
Fig. 2.

Exit pulse from a random media with a packing fraction of 0.5. The Gaussian input pulse is centered around 0 ps and is 50 fs in duration (FWHM). The solid line represents the response from a system of particles, uniformly distributed in size from 85 to 135 µm. The dotted line is the response from a system of particles with a Gaussian distribution centered on 110 µm, with the half-width of the distribution at 85 and 135 µm.

Fig. 3.
Fig. 3.

Pulse spread as a function of particle size and size range. For all simulations, the packing fraction was set to 0.5. Pulse spread as a function of particle size (a). The particles were uniformly distributed over a 2 µm range centered on the marker. Pulse spread as a function of size range (b). The particle sizes were uniformly distributed about 110 µm.

Fig. 4.
Fig. 4.

Delay time between input pulse and main return pulse as a function of particle size at a constant packing fraction (0.5) (a). Delay time between input pulse and main return pulse as a function of packing fraction at constant particle size (110 µm) (b). The inset shows the dependence on the inverse cube root of packing fraction, which is expected to be linear. For both simulations the input pulse width was 50 fs.

Fig. 5.
Fig. 5.

The delay time between input pulse and main return pulse as a function of particle size (a). The packing fraction was 0.5. The delay time between input pulse and main return pulse as a function of packing fraction (b). The particle size was 110 µm. For both simulations the input pulse width was 50 fs.

Fig. 6.
Fig. 6.

The intensity as a function of frequency and time delay in a dispersive medium for 0.125 (a), 0.25 (b), 0.5 (d) and 0.75 (e) packing fractions. For comparison with Fig. 2, line plots for 686 nm (black), 800 nm (red), and 961 nm (blue) for packing fractions 0.125 (c) and 0.25 (d) are also shown. The particle system was 110±1 µm and the input pulse duration 50 fs for all simulations.

Fig. 7.
Fig. 7.

The intensity as a function of frequency and time delay in a dispersive medium for 2.5±1 (a), 25±7.5 (b), 110±25 (c) and 325±70 (d) µm particles. The packing fraction was 0.5 and the input pulse duration 50 fs for all simulations.

Fig. 8.
Fig. 8.

The intensity as a function of frequency and time delay in a dispersive, absorbing medium. 110 µm, 0.125 packing fraction (a) and 25±7.5 µm, 0.5 packing fraction (b). The input pulse duration 50 fs for both simulations.

Fig. 9.
Fig. 9.

The exit pulse from a random media with a packing fraction of 0.5. The particles are uniformly distributed between 85–135 µm. The free path distribution is given by equation 1, with α=2, β=2 (black), α=2, β=5 (blue), and α=5, β=2 (red).

Fig. 10.
Fig. 10.

The time delay as a function of particle size (a) and particle size range (b). Open circles are for a packing fraction of 0.5, while closed circles are for a packing fraction of 0.9. Three different free path distributions were used: α=2, β=2 (black), α=5, β=2 (red), and α=2, β=5 (blue). The lines are to guide the eye.

Fig. 11.
Fig. 11.

The ratio of the rising and falling edge slopes (R/F) as a function of the ratio of α and β (a) for 110 µm (triangles), 200 µm (circles), and 325 µm (squares) particles. The size range was 2 µm. The ratio of the rising and falling edge slopes as a function of particle size range (b) for α=5, β=2, and an average particle size of 110 µm.

Fig. 12.
Fig. 12.

The dependence of the pulse width on the particle size range (a) and the average particle size (b) for packing fractions of 0.5 (open circles) and 0.9 (closed circles). The free path distribution is given by α=2, β=2 (black), α=5, β=2 (red), and α=2, β=5 (blue). The lines are to guide the eye

Fig. 13.
Fig. 13.

The time delay between the first and second peaks of the scattered intensity for 110 µm±1 µm (closed circles) and ±25 µm (open circles). The free path distribution parameters are α=5 and β=2.

Equations (2)

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f ( x ; α , β ) = 1 B ( α , β ) x α 1 ( 1 x ) β 1
B ( α , β ) = 0 1 t α 1 ( 1 t ) β 1 dt

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