Abstract

Using a non-local effective-medium approach, we analyze the refraction of light in a colloidal medium. We discuss the theoretical grounds and all the necessary precautions to design and perform experiments to measure the effective refractive index in dilute colloids. As an application, we show that it is possible to retrieve the size of small dielectric particles in a colloid by measuring the complex effective refractive index and the volume fraction occupied by the particles.

© 2008 Optical Society of America

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  1. M. Lax, "Multiple Scattering of Waves II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
    [CrossRef]
  2. W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
    [CrossRef]
  3. C. A. Grimes and D. M. Grimes DM "Permeability and permittivity spectra of granular materials," Phys. Rev. B 43, 10780-10788 (1991).
    [CrossRef]
  4. A. Wachniewski and H. B. McClung, "New Approach to effective medium for composite materials: Application to electromagnetic properties," Phys Rev B 33, 8053-8059 (1986).
    [CrossRef]
  5. A. García-Valenzuela and R. G. Barrera, "Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients," J. Quantum. Spectrosc. Radiat. Transf. 79-80, 627-647 (2003).
    [CrossRef]
  6. L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
    [CrossRef]
  7. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics, (John Wiley and Sons, Inc., New York N.Y., 2001) Chap. 3, pp. 128-130.
    [PubMed]
  8. Y. Kuga, D. Rice, and R. D. West, "Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium," IEEE Trans. Antennas Propag. 44, 326-332 (1996).
    [CrossRef]
  9. V. M. Agranovich and V. L. Ginzburg, Crystal optics with spatial dispersion, and excitons, second corrected and updated edition (Springer Verlag, Berlin, 1984).
  10. M. A. Shapiro, G. Shvets, J. R. Sirigiri, R. J. Temkin, "Spatial dispersion in metamaterials with negative dielectric permittivity and its effects on surface waves," Opt. Lett. 31, 2051-2053 (2006).
    [CrossRef] [PubMed]
  11. R. G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, "Nonlocal nature of the electrodynamic response of colloidal systems," Phys. Rev. B 75, 184202, 1-19, (2007).
    [CrossRef]
  12. A. García-Valenzuela, R. G. Barrera, C. Sánchez-Pérez, A. Reyes Coronado, E. R. Méndez, "Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment," Opt. Express 13, 6723-6737 (2005).
    [CrossRef] [PubMed]
  13. G. H. Meeten and A. N. North, "Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle," Meas. Sci. Technol. 6, 214-221 (1995).
    [CrossRef]
  14. H. C. van de Hulst, Light scattering by small particles, (John Wiley & Sons Inc., New York NY, 1957).
  15. A. Reyes Coronado, A. García-Valenzuela, C. Sánchez-Pérez, and R. G. Barrera, "Measurement of the effective refractive index of a turbid colloidal suspension using light refraction," New J. Phys. 7, 89 1-21 (2005).
    [CrossRef]
  16. A. A. Kokhanovsky and R. Weichert, "Multiple light scattering in laser particle sizing," Appl. Opt. 40, 1507- 1513 (2001).
    [CrossRef]
  17. J. Vargas-Ubera, J. F. Aguilar, and D. M. Gale, "Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods," Appl. Opt. 46, 124-132 (2007).
    [CrossRef]
  18. F. Ferri, A. Bassini, and E. Paganini, "Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing," Appl. Opt. 34, 5829-5839 (1995).
    [CrossRef] [PubMed]
  19. E. Paganini, F. Trespidi, and F. Ferri, "Instrument for long-path spectral extinction measurements in air: application to sizing of airborne particles," Appl. Opt. 40, 4261-4274 (2001).
    [CrossRef]
  20. S. Y. Shchyogolev, "Inverse problems of spectroturbidimetry of biological disperse systems: an overview," J. Biomed. Opt. 4, 490-503 (1999).
    [CrossRef]

2007 (2)

R. G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, "Nonlocal nature of the electrodynamic response of colloidal systems," Phys. Rev. B 75, 184202, 1-19, (2007).
[CrossRef]

J. Vargas-Ubera, J. F. Aguilar, and D. M. Gale, "Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods," Appl. Opt. 46, 124-132 (2007).
[CrossRef]

2006 (1)

2005 (1)

2003 (1)

A. García-Valenzuela and R. G. Barrera, "Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients," J. Quantum. Spectrosc. Radiat. Transf. 79-80, 627-647 (2003).
[CrossRef]

2001 (2)

1999 (1)

S. Y. Shchyogolev, "Inverse problems of spectroturbidimetry of biological disperse systems: an overview," J. Biomed. Opt. 4, 490-503 (1999).
[CrossRef]

1996 (1)

Y. Kuga, D. Rice, and R. D. West, "Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium," IEEE Trans. Antennas Propag. 44, 326-332 (1996).
[CrossRef]

1995 (2)

G. H. Meeten and A. N. North, "Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle," Meas. Sci. Technol. 6, 214-221 (1995).
[CrossRef]

F. Ferri, A. Bassini, and E. Paganini, "Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing," Appl. Opt. 34, 5829-5839 (1995).
[CrossRef] [PubMed]

1991 (1)

C. A. Grimes and D. M. Grimes DM "Permeability and permittivity spectra of granular materials," Phys. Rev. B 43, 10780-10788 (1991).
[CrossRef]

1989 (1)

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

1986 (1)

A. Wachniewski and H. B. McClung, "New Approach to effective medium for composite materials: Application to electromagnetic properties," Phys Rev B 33, 8053-8059 (1986).
[CrossRef]

1982 (1)

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

1952 (1)

M. Lax, "Multiple Scattering of Waves II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
[CrossRef]

Aguilar, J. F.

Barrera, R. G.

R. G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, "Nonlocal nature of the electrodynamic response of colloidal systems," Phys. Rev. B 75, 184202, 1-19, (2007).
[CrossRef]

A. García-Valenzuela, R. G. Barrera, C. Sánchez-Pérez, A. Reyes Coronado, E. R. Méndez, "Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment," Opt. Express 13, 6723-6737 (2005).
[CrossRef] [PubMed]

A. García-Valenzuela and R. G. Barrera, "Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients," J. Quantum. Spectrosc. Radiat. Transf. 79-80, 627-647 (2003).
[CrossRef]

A. Reyes Coronado, A. García-Valenzuela, C. Sánchez-Pérez, and R. G. Barrera, "Measurement of the effective refractive index of a turbid colloidal suspension using light refraction," New J. Phys. 7, 89 1-21 (2005).
[CrossRef]

Bassini, A.

Doyle, W. T.

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

Ferri, F.

Gale, D. M.

García-Valenzuela, A.

R. G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, "Nonlocal nature of the electrodynamic response of colloidal systems," Phys. Rev. B 75, 184202, 1-19, (2007).
[CrossRef]

A. García-Valenzuela, R. G. Barrera, C. Sánchez-Pérez, A. Reyes Coronado, E. R. Méndez, "Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment," Opt. Express 13, 6723-6737 (2005).
[CrossRef] [PubMed]

A. García-Valenzuela and R. G. Barrera, "Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients," J. Quantum. Spectrosc. Radiat. Transf. 79-80, 627-647 (2003).
[CrossRef]

A. Reyes Coronado, A. García-Valenzuela, C. Sánchez-Pérez, and R. G. Barrera, "Measurement of the effective refractive index of a turbid colloidal suspension using light refraction," New J. Phys. 7, 89 1-21 (2005).
[CrossRef]

Grimes, C. A.

C. A. Grimes and D. M. Grimes DM "Permeability and permittivity spectra of granular materials," Phys. Rev. B 43, 10780-10788 (1991).
[CrossRef]

Kokhanovsky, A. A.

Kong, J. A.

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

Kuga, Y.

Y. Kuga, D. Rice, and R. D. West, "Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium," IEEE Trans. Antennas Propag. 44, 326-332 (1996).
[CrossRef]

Lax, M.

M. Lax, "Multiple Scattering of Waves II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
[CrossRef]

McClung, H. B.

A. Wachniewski and H. B. McClung, "New Approach to effective medium for composite materials: Application to electromagnetic properties," Phys Rev B 33, 8053-8059 (1986).
[CrossRef]

Meeten, G. H.

G. H. Meeten and A. N. North, "Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle," Meas. Sci. Technol. 6, 214-221 (1995).
[CrossRef]

Méndez, E. R.

North, A. N.

G. H. Meeten and A. N. North, "Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle," Meas. Sci. Technol. 6, 214-221 (1995).
[CrossRef]

Paganini, E.

Reyes Coronado, A.

A. García-Valenzuela, R. G. Barrera, C. Sánchez-Pérez, A. Reyes Coronado, E. R. Méndez, "Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment," Opt. Express 13, 6723-6737 (2005).
[CrossRef] [PubMed]

A. Reyes Coronado, A. García-Valenzuela, C. Sánchez-Pérez, and R. G. Barrera, "Measurement of the effective refractive index of a turbid colloidal suspension using light refraction," New J. Phys. 7, 89 1-21 (2005).
[CrossRef]

Reyes-Coronado, A.

R. G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, "Nonlocal nature of the electrodynamic response of colloidal systems," Phys. Rev. B 75, 184202, 1-19, (2007).
[CrossRef]

Rice, D.

Y. Kuga, D. Rice, and R. D. West, "Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium," IEEE Trans. Antennas Propag. 44, 326-332 (1996).
[CrossRef]

Sánchez-Pérez, C.

A. García-Valenzuela, R. G. Barrera, C. Sánchez-Pérez, A. Reyes Coronado, E. R. Méndez, "Coherent reflection of light from a turbid suspension of particles in an internal-reflection configuration: Theory versus experiment," Opt. Express 13, 6723-6737 (2005).
[CrossRef] [PubMed]

A. Reyes Coronado, A. García-Valenzuela, C. Sánchez-Pérez, and R. G. Barrera, "Measurement of the effective refractive index of a turbid colloidal suspension using light refraction," New J. Phys. 7, 89 1-21 (2005).
[CrossRef]

Shapiro, M. A.

Shchyogolev, S. Y.

S. Y. Shchyogolev, "Inverse problems of spectroturbidimetry of biological disperse systems: an overview," J. Biomed. Opt. 4, 490-503 (1999).
[CrossRef]

Shvets, G.

Sirigiri, J. R.

Temkin, R. J.

Trespidi, F.

Tsang, L.

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

Vargas-Ubera, J.

Wachniewski, A.

A. Wachniewski and H. B. McClung, "New Approach to effective medium for composite materials: Application to electromagnetic properties," Phys Rev B 33, 8053-8059 (1986).
[CrossRef]

Weichert, R.

West, R. D.

Y. Kuga, D. Rice, and R. D. West, "Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium," IEEE Trans. Antennas Propag. 44, 326-332 (1996).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

Y. Kuga, D. Rice, and R. D. West, "Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium," IEEE Trans. Antennas Propag. 44, 326-332 (1996).
[CrossRef]

J. Appl. Phys. (1)

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

J. Biomed. Opt. (1)

S. Y. Shchyogolev, "Inverse problems of spectroturbidimetry of biological disperse systems: an overview," J. Biomed. Opt. 4, 490-503 (1999).
[CrossRef]

J. Quantum. Spectrosc. Radiat. Transf. (1)

A. García-Valenzuela and R. G. Barrera, "Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients," J. Quantum. Spectrosc. Radiat. Transf. 79-80, 627-647 (2003).
[CrossRef]

Meas. Sci. Technol. (1)

G. H. Meeten and A. N. North, "Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle," Meas. Sci. Technol. 6, 214-221 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys Rev B (1)

A. Wachniewski and H. B. McClung, "New Approach to effective medium for composite materials: Application to electromagnetic properties," Phys Rev B 33, 8053-8059 (1986).
[CrossRef]

Phys. Rev. (1)

M. Lax, "Multiple Scattering of Waves II. The effective field in dense systems," Phys. Rev. 85, 621-629 (1952).
[CrossRef]

Phys. Rev. B (3)

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

C. A. Grimes and D. M. Grimes DM "Permeability and permittivity spectra of granular materials," Phys. Rev. B 43, 10780-10788 (1991).
[CrossRef]

R. G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, "Nonlocal nature of the electrodynamic response of colloidal systems," Phys. Rev. B 75, 184202, 1-19, (2007).
[CrossRef]

Other (4)

H. C. van de Hulst, Light scattering by small particles, (John Wiley & Sons Inc., New York NY, 1957).

A. Reyes Coronado, A. García-Valenzuela, C. Sánchez-Pérez, and R. G. Barrera, "Measurement of the effective refractive index of a turbid colloidal suspension using light refraction," New J. Phys. 7, 89 1-21 (2005).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics, (John Wiley and Sons, Inc., New York N.Y., 2001) Chap. 3, pp. 128-130.
[PubMed]

V. M. Agranovich and V. L. Ginzburg, Crystal optics with spatial dispersion, and excitons, second corrected and updated edition (Springer Verlag, Berlin, 1984).

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Figures (8)

Fig. 1.
Fig. 1.

Schematic illustrating the bulk and surface region in the effective medium of a half-space.

Fig. 2.
Fig. 2.

Schematic illustration of refraction in (a) a colloidal slab and (b) a colloidal prism.

Fig. 3.
Fig. 3.

Graphs of Im(neff)/f versus Re(neff)/f for a colloid with a matrix of refractive index nm =1.36 and particles of refractive index np =1.45, 1.47, …1.55. The symbols in each plot in (a) are for a particles of radius a=0, 30nm, 60nm, …1500 nm. In (b) are for a=0, 5nm, 10nm, 15nm, …50nm.

Fig. 4.
Fig. 4.

Graphs of Im(neff)/f versus Re(neff)/f for a colloid with a matrix of refractive index nm =1.36 and particles of refractive index np =1.80, 1.85, …2.0. The symbols in each plot in (a) are for a particles of radius a=0, 10nm, 20nm, …800 nm. In (b) are for a=0, 5nm, 10nm, 15nm, …50nm.

Fig. 5.
Fig. 5.

Particle radius at which the imaginary part of the effective refractive is maximum as a function of the refractive index of the particles.

Fig. 6.
Fig. 6.

Two curves for two different values of the particles’ refractive index, (a) np =1.50, 1.55 and (b) 1.80, 1.90 at steps of (a) Δa=80 nm and (b) Δa=80 nm. Points with the same value a and np but with an imaginary part added to np of Im(np )=10-3 (squares) and Im(np )=10-2 (triangles) are also plotted.

Fig. 7.
Fig. 7.

Difference between the effective refractive index calculated by solving exactly the non-local dispersion equation for transverse modes [Eq. (19)] and that calculated with the van de Hulst expression [Eq. (21)] normalized by the particle’s volume fraction f. The curves shown are for a volume fraction of f=1%, 2%, 3%, 4% and 5% and a particles’ refractive index of 1.53 immersed on a matrix of refractive index 1.36 and a vacuum wavelength of 635 nm.

Fig. 8.
Fig. 8.

Difference between the effective refractive index calculated by solving exactly the non-local dispersion equation for transverse modes [Eq. (19)] and that calculated with the van de Hulst expression [Eq. (21)] normalized by the particle’s volume fraction f. The curves shown are for a volume fraction of f=1%, 2%, 3%, 4% and 5% and a particles’ refractive index of 2.00 immersed on a matrix of refractive index 1.36 and a vacuum wavelength of 635 nm.

Equations (21)

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J ind , j ( r ) = σ s NL ( r r j , r r j ) · E exc ( r ) d 3 r
J ind , j ( r ) = σ s ( r ) · E exc ( r ) ,
σ s ( r ) = σ s NL ( r r j , r r j ) d 3 r
J ind ( r ) = j J ind , j ( r ) .
J ind ( r ) σ eff ( r , r ) · E ( r ) d 3 r
σ eff ( r , r ) = ρ 0 V σ s NL ( r r j , r r j ) d 3 r j .
σ eff ( r , r ) = σ eff B ( r r )
σ eff B ( k , ω ) = σ eff B , L ( k , ω ) k ̂ k ̂ + σ eff B , T ( k , ω ) [ 1 k ̂ k ̂ ] ,
1 + i ω ε 0 σ eff B , L ( k , ω ) = 0
k 2 = ω 2 ε 0 μ 0 + i ω μ 0 σ eff B , T ( k , ω ) .
n eff ( ω ) = k T ( ω ) ω ε 0 μ 0 ,
p ( r j ) = { d 3 r j V if z j > 0 0 if z j < 0 .
σ eff ( r , r ) = { σ eff B ( r r ) if z bulk σ eff ( r r ; z , z ) if z surface region 0 otherwise .
E inc ( r ) = E inc exp ( i k i · r ) exp ( i k z i z ) e ̂ i ,
E ( r ) = E 0 exp ( i k i · r ) exp ( i k z z ) e ̂
σ eff B ( k , ω ) = σ eff B ( r r ) exp [ i k · ( r r ) ] d 3 r ,
k z ( ω ) = k T ( ω ) k i · k i
k z ( ω ) = k 0 2 n eff ( ω ) k i · k i .
k 2 = ω 2 ε m μ 0 + i ω μ 0 σ eff , m B , T ( k , ω ) ,
σ eff , m B ; L ( k , ω ) = i ω ε m ( ω ) ,
n eff ( ω ) = n m ( ω ) 1 + 2 i 3 f 2 x m 3 S ( 0 ) n m ( ω ) [ 1 + i 3 2 f x m 3 S ( 0 ) + ]

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