Abstract

The effective medium refractive index of a surface-bound submonolayer of polystyrene nanospheres in water is found to be ill-defined below a rather specific sphere occupied area limit. The submonolayer takes on a recognizable thickness and refractive index only when the average center separation between spheres is at or below the inverse of the wavenumber. An anticipated limit to the Maxwell Garnett theory is therefore confirmed. It is proposed that this reveals a fundamental property of light scattering that is not of the Huygens spherical secondary wave type.

© 2013 Optical Society of America

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References

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  1. G. Brooker, Modern Classical Optics (OUP, 2002).
  2. R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
    [CrossRef]
  3. R. G. Barrera, A. Reyes-Coronado, and A. Garcia-Valenzuela, Phys. Rev. B 75, 84202 (2007).
    [CrossRef]
  4. N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
    [CrossRef]
  5. M. Chen and R. G. Horn, J. Colloid Interface Sci. 315, 814 (2007).
    [CrossRef]
  6. P. Lalanne and D. Lemercier-lalanne, J. Mod. Opt. 43, 2063 (1996).
    [CrossRef]
  7. C. Yang, A. Wax, and M. S. Feld, Opt. Lett. 26, 235 (2001).
    [CrossRef]
  8. R. H. Doremus, J. Appl. Phys. 37, 2775 (1966).
    [CrossRef]
  9. J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
    [CrossRef]
  10. L. A. Girifalco, Statistical Mechanics of Solids (OUP, 2003).
  11. G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
    [CrossRef]
  12. G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
    [CrossRef]
  13. A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
    [CrossRef]
  14. K. Xu, M. M. Ouberai, and M. E. Welland, Biomaterial. 34, 1453 (2012).
  15. G. H. Cross, N. J. Freeman, and M. J. Swann, in Handbook of Biosensors and Biochips, R. S. Marks, D. C. Cullen, I. Karube, C. R. Lowe, and H. H. Weetall, eds. (Wiley, 2007), Chap. 32.
  16. “Resolver” is the CE multi-layer model developed for DPI.
  17. SLAB v.4.0 (1-D waveguide mode solver), BBV Software.
  18. C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
    [CrossRef]
  19. A. F. Ioffe and A. R. Regel, in Progress in Semiconductors, A. F. Gibson, F A. Kroger, and R. E. Burgess, eds. (Heywood, 1960), Vol. 4, p. 237.
  20. M. P. V. Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
    [CrossRef]
  21. S. John, Phys. Today 44(5), 32 (1991).
    [CrossRef]
  22. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
    [CrossRef]
  23. P. Enders, Lat. Am. J. Educ. 3, 19 (2009).
  24. V. K. Varadan, V. N. Bringi, and V. V. Varadan, Phys. Rev. B 19, 2480 (1978).
  25. G. H. Cross, http://arxiv.org/abs/1303.1390 .
  26. R. P. Feynman, R. B. Leighton, and M. Sands, in The Feynman Lectures on Physics (Addison-Wesley1964), Vol. 1, Chap. 26.

2012 (2)

K. Xu, M. M. Ouberai, and M. E. Welland, Biomaterial. 34, 1453 (2012).

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

2010 (1)

R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
[CrossRef]

2009 (1)

P. Enders, Lat. Am. J. Educ. 3, 19 (2009).

2007 (3)

R. G. Barrera, A. Reyes-Coronado, and A. Garcia-Valenzuela, Phys. Rev. B 75, 84202 (2007).
[CrossRef]

M. Chen and R. G. Horn, J. Colloid Interface Sci. 315, 814 (2007).
[CrossRef]

A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
[CrossRef]

2006 (1)

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef]

2004 (1)

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

2003 (1)

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

2001 (1)

1996 (1)

P. Lalanne and D. Lemercier-lalanne, J. Mod. Opt. 43, 2063 (1996).
[CrossRef]

1991 (1)

S. John, Phys. Today 44(5), 32 (1991).
[CrossRef]

1986 (1)

C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

1985 (1)

M. P. V. Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

1978 (1)

V. K. Varadan, V. N. Bringi, and V. V. Varadan, Phys. Rev. B 19, 2480 (1978).

1966 (1)

R. H. Doremus, J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
[CrossRef]

Aegerter, C. M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef]

Albada, M. P. V.

M. P. V. Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

Barrera, R. G.

R. G. Barrera, A. Reyes-Coronado, and A. Garcia-Valenzuela, Phys. Rev. B 75, 84202 (2007).
[CrossRef]

Bohren, C. F.

C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

Brand, S.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

Bringi, V. N.

V. K. Varadan, V. N. Bringi, and V. V. Varadan, Phys. Rev. B 19, 2480 (1978).

Brismar, H.

A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
[CrossRef]

Brooker, G.

G. Brooker, Modern Classical Optics (OUP, 2002).

Callison, T. H.

A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
[CrossRef]

Chen, M.

M. Chen and R. G. Horn, J. Colloid Interface Sci. 315, 814 (2007).
[CrossRef]

Cross, G. H.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

G. H. Cross, N. J. Freeman, and M. J. Swann, in Handbook of Biosensors and Biochips, R. S. Marks, D. C. Cullen, I. Karube, C. R. Lowe, and H. H. Weetall, eds. (Wiley, 2007), Chap. 32.

Doremus, R. H.

R. H. Doremus, J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

Elofsson, U. M.

A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
[CrossRef]

Enders, P.

P. Enders, Lat. Am. J. Educ. 3, 19 (2009).

Feld, M. S.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, in The Feynman Lectures on Physics (Addison-Wesley1964), Vol. 1, Chap. 26.

Freeman, N. J.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

G. H. Cross, N. J. Freeman, and M. J. Swann, in Handbook of Biosensors and Biochips, R. S. Marks, D. C. Cullen, I. Karube, C. R. Lowe, and H. H. Weetall, eds. (Wiley, 2007), Chap. 32.

Garcia-Valenzuela, A.

R. G. Barrera, A. Reyes-Coronado, and A. Garcia-Valenzuela, Phys. Rev. B 75, 84202 (2007).
[CrossRef]

Girifalco, L. A.

L. A. Girifalco, Statistical Mechanics of Solids (OUP, 2003).

Godovsky, D.

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

Gross, P.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef]

Horn, R. G.

M. Chen and R. G. Horn, J. Colloid Interface Sci. 315, 814 (2007).
[CrossRef]

Ioffe, A. F.

A. F. Ioffe and A. R. Regel, in Progress in Semiconductors, A. F. Gibson, F A. Kroger, and R. E. Burgess, eds. (Heywood, 1960), Vol. 4, p. 237.

John, S.

S. John, Phys. Today 44(5), 32 (1991).
[CrossRef]

Ko, M.

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

Lagendijk, A.

M. P. V. Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

Lalanne, P.

P. Lalanne and D. Lemercier-lalanne, J. Mod. Opt. 43, 2063 (1996).
[CrossRef]

Lee, M.

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, in The Feynman Lectures on Physics (Addison-Wesley1964), Vol. 1, Chap. 26.

Lemercier-lalanne, D.

P. Lalanne and D. Lemercier-lalanne, J. Mod. Opt. 43, 2063 (1996).
[CrossRef]

Maret, G.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
[CrossRef]

Miles, R. E.

R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
[CrossRef]

Orr-Ewing, A. J.

R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
[CrossRef]

Ouberai, M. M.

K. Xu, M. M. Ouberai, and M. E. Welland, Biomaterial. 34, 1453 (2012).

Peel, L. L.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

Popplewell, J. F.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

Reeves, A. A.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

Regel, A. R.

A. F. Ioffe and A. R. Regel, in Progress in Semiconductors, A. F. Gibson, F A. Kroger, and R. E. Burgess, eds. (Heywood, 1960), Vol. 4, p. 237.

Reid, J. P.

R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
[CrossRef]

Reyes-Coronado, A.

R. G. Barrera, A. Reyes-Coronado, and A. Garcia-Valenzuela, Phys. Rev. B 75, 84202 (2007).
[CrossRef]

Rudic, S.

R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, in The Feynman Lectures on Physics (Addison-Wesley1964), Vol. 1, Chap. 26.

Sergienko, N.

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

Sonesson, A. W.

A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
[CrossRef]

Störzer, M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef]

Swann, M. J.

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

G. H. Cross, N. J. Freeman, and M. J. Swann, in Handbook of Biosensors and Biochips, R. S. Marks, D. C. Cullen, I. Karube, C. R. Lowe, and H. H. Weetall, eds. (Wiley, 2007), Chap. 32.

Varadan, V. K.

V. K. Varadan, V. N. Bringi, and V. V. Varadan, Phys. Rev. B 19, 2480 (1978).

Varadan, V. V.

V. K. Varadan, V. N. Bringi, and V. V. Varadan, Phys. Rev. B 19, 2480 (1978).

Wax, A.

Welland, M. E.

K. Xu, M. M. Ouberai, and M. E. Welland, Biomaterial. 34, 1453 (2012).

Xu, K.

K. Xu, M. M. Ouberai, and M. E. Welland, Biomaterial. 34, 1453 (2012).

Yang, C.

Zavin, B.

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

Biomaterial. (1)

K. Xu, M. M. Ouberai, and M. E. Welland, Biomaterial. 34, 1453 (2012).

Biosens. Bioelectron. (1)

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, M. J. Swann, and N. J. Freeman, Biosens. Bioelectron. 19, 383 (2003).
[CrossRef]

Colloids Surf. B (1)

A. W. Sonesson, T. H. Callison, H. Brismar, and U. M. Elofsson, Colloids Surf. B 54, 236 (2007).
[CrossRef]

J. Appl. Phys. (1)

R. H. Doremus, J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

J. Atmos. Sci. (1)

C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

J. Colloid Interface Sci. (1)

M. Chen and R. G. Horn, J. Colloid Interface Sci. 315, 814 (2007).
[CrossRef]

J. Mod. Opt. (1)

P. Lalanne and D. Lemercier-lalanne, J. Mod. Opt. 43, 2063 (1996).
[CrossRef]

J. Phys. Chem. A (1)

R. E. Miles, S. Rudić, A. J. Orr-Ewing, and J. P. Reid, J. Phys. Chem. A 114, 7077 (2010).
[CrossRef]

J. Phys. D (1)

G. H. Cross, A. A. Reeves, S. Brand, J. F. Popplewell, L. L. Peel, M. J. Swann, and N. J. Freeman, J. Phys. D 37, 74 (2004).
[CrossRef]

Lat. Am. J. Educ. (1)

P. Enders, Lat. Am. J. Educ. 3, 19 (2009).

Nanoscale Res. Lett. (1)

N. Sergienko, D. Godovsky, B. Zavin, M. Lee, and M. Ko, Nanoscale Res. Lett. 7, 181 (2012).
[CrossRef]

Opt. Lett. (1)

Phil. Trans. R. Soc. A (1)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
[CrossRef]

Phys. Rev. B (2)

V. K. Varadan, V. N. Bringi, and V. V. Varadan, Phys. Rev. B 19, 2480 (1978).

R. G. Barrera, A. Reyes-Coronado, and A. Garcia-Valenzuela, Phys. Rev. B 75, 84202 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef]

M. P. V. Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

Phys. Today (1)

S. John, Phys. Today 44(5), 32 (1991).
[CrossRef]

Other (8)

A. F. Ioffe and A. R. Regel, in Progress in Semiconductors, A. F. Gibson, F A. Kroger, and R. E. Burgess, eds. (Heywood, 1960), Vol. 4, p. 237.

G. H. Cross, http://arxiv.org/abs/1303.1390 .

R. P. Feynman, R. B. Leighton, and M. Sands, in The Feynman Lectures on Physics (Addison-Wesley1964), Vol. 1, Chap. 26.

G. Brooker, Modern Classical Optics (OUP, 2002).

L. A. Girifalco, Statistical Mechanics of Solids (OUP, 2003).

G. H. Cross, N. J. Freeman, and M. J. Swann, in Handbook of Biosensors and Biochips, R. S. Marks, D. C. Cullen, I. Karube, C. R. Lowe, and H. H. Weetall, eds. (Wiley, 2007), Chap. 32.

“Resolver” is the CE multi-layer model developed for DPI.

SLAB v.4.0 (1-D waveguide mode solver), BBV Software.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Deduced layer thickness (circles) and average separation (squares) versus TE mode phase retardation for the deposition of (nominally) 30 nm diameter polystyrene spheres, MG model S shown for comparison (solid line). Data points represent sample time intervals of 1 s. Error bars are calculated using the phase retardation random noise 5 σ uncertainty of ± 5 mrad and ± 3 mrad , for Δ ϕ TE and Δ ϕ TM , respectively averaged over a total pre-deposition time of 120 s. (b) Data from a sphere deposition experiment using (nominally) 21 nm diameter spheres [symbols and error methods as for (a)].

Fig. 2.
Fig. 2.

(a) SEM micrograph of the surface bound 100 nm spheres. Coverage is almost uniform along the full interferometer path (see supplementary information in [25]). (b)  R versus Δ ϕ TE with error bars removed for clarity. The CE model R versus Δ ϕ TE (for slope values, see text) is shown as the dashed line. The data range indicated by the dashed arrow and labelled “wash” represents the behavior when pure water is reintroduced to the interferometer chip.

Equations (2)

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( ε eff ε m ) ( ε eff + 2 ε m ) = f A ( ε sp ε m ) ( ε sp + 2 ε m ) .
1 / k = λ eff / 2 π = / p M = S C ,

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