Abstract

The backscattering properties of small layered plates are studied for various size parameter values with respect to the plate thickness, plate aspect ratio, number of layers, incident direction, and polarization states of the incident light. The results are compared with the analytical results for semi-infinite plates. The phase functions and the corresponding backscattering efficiencies of the small plates are computed with the Discrete Dipole Approximation method. The angular width of the reflection peak is found to depend on both the size parameter and the aspect ratio. The criteria for using the reflectance of semi-infinite plates to approximate the backscattering efficiencies of finite size plates are quantified with respect to the number of layers, incident angle, and polarization state.

© 2012 OSA

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References

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  1. M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
    [Crossref] [PubMed]
  2. L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
    [PubMed]
  3. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, 1999).
  4. E. Hecht, Optics (Addison-Wesley, 2002).
  5. A. Huxley, “A theoretical treatment of the reflexion of light by multilayer structures,” J. Exp. Biol. 48, 227–245 (1968).
  6. B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (John Wiley & Sons, 2007).
  7. A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
    [Crossref] [PubMed]
  8. R. L. Sutherland, L. M. Mäthger, R. T. Hanlon, A. M. Urbas, and M. O. Stone, “Cephalopod coloration model. I. Squid chromatophores and iridophores,” J. Opt. Soc. Am. A 25(3), 588–599 (2008).
    [Crossref] [PubMed]
  9. L. M. Mäthger and E. J. Denton, “Reflective properties of iridophores and fluorescent ‘eyespots’ in the loliginid squid Alloteuthis subulata and Loligo vulgaris,” J. Exp. Biol. 204(Pt 12), 2103–2118 (2001).
    [PubMed]
  10. J. Arnold, R. Young, and M. King, “Ultrastructure of a cephalopod photophore. II. Iridophores as reflectors and transmitters,” Biol. Bull. 147(3), 522–534 (1974).
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    [Crossref]
  13. K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, “Physiological color change in squid iridophores. II. Ultrastructural mechanisms in Lolliguncula brevis,” Cell Tissue Res. 259(1), 15–24 (1990).
    [Crossref] [PubMed]
  14. D. Froesch and J. B. Messenger, “On leucophores and the chromatic unit of Octopus vulgaris,” J. Zool. 186(2), 163–173 (1978).
    [Crossref]
  15. L. M. Mäthger, T. F. Collins, and P. A. Lima, “The role of muscarinic receptors and intracellular Ca2+ in the spectral reflectivity changes of squid iridophores,” J. Exp. Biol. 207(11), 1759–1769 (2004).
    [Crossref] [PubMed]
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  21. T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
    [Crossref] [PubMed]
  22. R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, “The self-organizing properties of squid reflectin protein,” Nat. Mater. 6(7), 533–538 (2007).
    [Crossref] [PubMed]
  23. G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1.Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13(11), 1065–1080 (1973).
    [Crossref]
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2011 (2)

A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
[Crossref] [PubMed]

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: Capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

2010 (1)

2009 (1)

L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
[PubMed]

2008 (1)

2007 (2)

T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
[Crossref] [PubMed]

R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, “The self-organizing properties of squid reflectin protein,” Nat. Mater. 6(7), 533–538 (2007).
[Crossref] [PubMed]

2004 (1)

L. M. Mäthger, T. F. Collins, and P. A. Lima, “The role of muscarinic receptors and intracellular Ca2+ in the spectral reflectivity changes of squid iridophores,” J. Exp. Biol. 207(11), 1759–1769 (2004).
[Crossref] [PubMed]

2001 (1)

L. M. Mäthger and E. J. Denton, “Reflective properties of iridophores and fluorescent ‘eyespots’ in the loliginid squid Alloteuthis subulata and Loligo vulgaris,” J. Exp. Biol. 204(Pt 12), 2103–2118 (2001).
[PubMed]

1997 (1)

1996 (1)

1990 (1)

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, “Physiological color change in squid iridophores. II. Ultrastructural mechanisms in Lolliguncula brevis,” Cell Tissue Res. 259(1), 15–24 (1990).
[Crossref] [PubMed]

1983 (1)

R. Cloney and S. Brocco, “Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods,” Integr. Comp. Biol. 23(3), 581–592 (1983).
[Crossref]

1980 (1)

S. L. Brocco and R. A. Cloney, “Reflector cells in the skin of Octopus dofleini,” Cell Tissue Res. 205(2), 167–186 (1980).
[Crossref] [PubMed]

1978 (1)

D. Froesch and J. B. Messenger, “On leucophores and the chromatic unit of Octopus vulgaris,” J. Zool. 186(2), 163–173 (1978).
[Crossref]

1974 (1)

J. Arnold, R. Young, and M. King, “Ultrastructure of a cephalopod photophore. II. Iridophores as reflectors and transmitters,” Biol. Bull. 147(3), 522–534 (1974).
[Crossref]

1973 (2)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1.Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13(11), 1065–1080 (1973).
[Crossref]

1972 (2)

S. Mirow, “Skin color in the squids Loligo pealii and Loligo opalescens. II. Iridophores,” Z. Zellforsch. Mikrosk. Anat. 125(2), 176–190 (1972).
[Crossref] [PubMed]

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[Crossref] [PubMed]

1968 (1)

A. Huxley, “A theoretical treatment of the reflexion of light by multilayer structures,” J. Exp. Biol. 48, 227–245 (1968).

Arnold, J.

J. Arnold, R. Young, and M. King, “Ultrastructure of a cephalopod photophore. II. Iridophores as reflectors and transmitters,” Biol. Bull. 147(3), 522–534 (1974).
[Crossref]

Bi, L.

Brocco, S.

R. Cloney and S. Brocco, “Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods,” Integr. Comp. Biol. 23(3), 581–592 (1983).
[Crossref]

Brocco, S. L.

S. L. Brocco and R. A. Cloney, “Reflector cells in the skin of Octopus dofleini,” Cell Tissue Res. 205(2), 167–186 (1980).
[Crossref] [PubMed]

Budelmann, B. U.

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, “Physiological color change in squid iridophores. II. Ultrastructural mechanisms in Lolliguncula brevis,” Cell Tissue Res. 259(1), 15–24 (1990).
[Crossref] [PubMed]

Chiou, T. H.

T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
[Crossref] [PubMed]

Cloney, R.

R. Cloney and S. Brocco, “Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods,” Integr. Comp. Biol. 23(3), 581–592 (1983).
[Crossref]

Cloney, R. A.

S. L. Brocco and R. A. Cloney, “Reflector cells in the skin of Octopus dofleini,” Cell Tissue Res. 205(2), 167–186 (1980).
[Crossref] [PubMed]

Collins, T. F.

L. M. Mäthger, T. F. Collins, and P. A. Lima, “The role of muscarinic receptors and intracellular Ca2+ in the spectral reflectivity changes of squid iridophores,” J. Exp. Biol. 207(11), 1759–1769 (2004).
[Crossref] [PubMed]

Cooper, K. M.

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, “Physiological color change in squid iridophores. II. Ultrastructural mechanisms in Lolliguncula brevis,” Cell Tissue Res. 259(1), 15–24 (1990).
[Crossref] [PubMed]

Cronin, T. W.

T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
[Crossref] [PubMed]

Crookes-Goodson, W. J.

R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, “The self-organizing properties of squid reflectin protein,” Nat. Mater. 6(7), 533–538 (2007).
[Crossref] [PubMed]

Denton, E. J.

L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
[PubMed]

L. M. Mäthger and E. J. Denton, “Reflective properties of iridophores and fluorescent ‘eyespots’ in the loliginid squid Alloteuthis subulata and Loligo vulgaris,” J. Exp. Biol. 204(Pt 12), 2103–2118 (2001).
[PubMed]

Froesch, D.

D. Froesch and J. B. Messenger, “On leucophores and the chromatic unit of Octopus vulgaris,” J. Zool. 186(2), 163–173 (1978).
[Crossref]

Hanlon, R. T.

L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
[PubMed]

R. L. Sutherland, L. M. Mäthger, R. T. Hanlon, A. M. Urbas, and M. O. Stone, “Cephalopod coloration model. I. Squid chromatophores and iridophores,” J. Opt. Soc. Am. A 25(3), 588–599 (2008).
[Crossref] [PubMed]

T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
[Crossref] [PubMed]

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, “Physiological color change in squid iridophores. II. Ultrastructural mechanisms in Lolliguncula brevis,” Cell Tissue Res. 259(1), 15–24 (1990).
[Crossref] [PubMed]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: Capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

Holt, A. L.

A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
[Crossref] [PubMed]

Huxley, A.

A. Huxley, “A theoretical treatment of the reflexion of light by multilayer structures,” J. Exp. Biol. 48, 227–245 (1968).

Johnsen, S.

A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
[Crossref] [PubMed]

Kattawar, G. W.

L. Bi, P. Yang, and G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010).
[Crossref] [PubMed]

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1.Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13(11), 1065–1080 (1973).
[Crossref]

King, M.

J. Arnold, R. Young, and M. King, “Ultrastructure of a cephalopod photophore. II. Iridophores as reflectors and transmitters,” Biol. Bull. 147(3), 522–534 (1974).
[Crossref]

Kramer, R. M.

R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, “The self-organizing properties of squid reflectin protein,” Nat. Mater. 6(7), 533–538 (2007).
[Crossref] [PubMed]

Land, M. F.

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[Crossref] [PubMed]

Lima, P. A.

L. M. Mäthger, T. F. Collins, and P. A. Lima, “The role of muscarinic receptors and intracellular Ca2+ in the spectral reflectivity changes of squid iridophores,” J. Exp. Biol. 207(11), 1759–1769 (2004).
[Crossref] [PubMed]

Liou, K. N.

Marshall, N. J.

L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
[PubMed]

Mäthger, L. M.

L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
[PubMed]

R. L. Sutherland, L. M. Mäthger, R. T. Hanlon, A. M. Urbas, and M. O. Stone, “Cephalopod coloration model. I. Squid chromatophores and iridophores,” J. Opt. Soc. Am. A 25(3), 588–599 (2008).
[Crossref] [PubMed]

T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
[Crossref] [PubMed]

L. M. Mäthger, T. F. Collins, and P. A. Lima, “The role of muscarinic receptors and intracellular Ca2+ in the spectral reflectivity changes of squid iridophores,” J. Exp. Biol. 207(11), 1759–1769 (2004).
[Crossref] [PubMed]

L. M. Mäthger and E. J. Denton, “Reflective properties of iridophores and fluorescent ‘eyespots’ in the loliginid squid Alloteuthis subulata and Loligo vulgaris,” J. Exp. Biol. 204(Pt 12), 2103–2118 (2001).
[PubMed]

Messenger, J. B.

D. Froesch and J. B. Messenger, “On leucophores and the chromatic unit of Octopus vulgaris,” J. Zool. 186(2), 163–173 (1978).
[Crossref]

Mirow, S.

S. Mirow, “Skin color in the squids Loligo pealii and Loligo opalescens. II. Iridophores,” Z. Zellforsch. Mikrosk. Anat. 125(2), 176–190 (1972).
[Crossref] [PubMed]

Morse, D. E.

A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
[Crossref] [PubMed]

Naik, R. R.

R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, “The self-organizing properties of squid reflectin protein,” Nat. Mater. 6(7), 533–538 (2007).
[Crossref] [PubMed]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Plass, G. N.

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1.Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13(11), 1065–1080 (1973).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Stone, M. O.

Sutherland, R. L.

Sweeney, A. M.

A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
[Crossref] [PubMed]

Urbas, A. M.

Yang, P.

Young, R.

J. Arnold, R. Young, and M. King, “Ultrastructure of a cephalopod photophore. II. Iridophores as reflectors and transmitters,” Biol. Bull. 147(3), 522–534 (1974).
[Crossref]

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: Capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

Appl. Opt. (2)

Astrophys. J. (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Biol. Bull. (1)

J. Arnold, R. Young, and M. King, “Ultrastructure of a cephalopod photophore. II. Iridophores as reflectors and transmitters,” Biol. Bull. 147(3), 522–534 (1974).
[Crossref]

Cell Tissue Res. (2)

S. L. Brocco and R. A. Cloney, “Reflector cells in the skin of Octopus dofleini,” Cell Tissue Res. 205(2), 167–186 (1980).
[Crossref] [PubMed]

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, “Physiological color change in squid iridophores. II. Ultrastructural mechanisms in Lolliguncula brevis,” Cell Tissue Res. 259(1), 15–24 (1990).
[Crossref] [PubMed]

Integr. Comp. Biol. (1)

R. Cloney and S. Brocco, “Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods,” Integr. Comp. Biol. 23(3), 581–592 (1983).
[Crossref]

J. Exp. Biol. (4)

L. M. Mäthger, T. F. Collins, and P. A. Lima, “The role of muscarinic receptors and intracellular Ca2+ in the spectral reflectivity changes of squid iridophores,” J. Exp. Biol. 207(11), 1759–1769 (2004).
[Crossref] [PubMed]

A. Huxley, “A theoretical treatment of the reflexion of light by multilayer structures,” J. Exp. Biol. 48, 227–245 (1968).

L. M. Mäthger and E. J. Denton, “Reflective properties of iridophores and fluorescent ‘eyespots’ in the loliginid squid Alloteuthis subulata and Loligo vulgaris,” J. Exp. Biol. 204(Pt 12), 2103–2118 (2001).
[PubMed]

T. H. Chiou, L. M. Mäthger, R. T. Hanlon, and T. W. Cronin, “Spectral and spatial properties of polarized light reflections from the arms of squid (Loligo pealeii) and cuttlefish (Sepia officinalis L.),” J. Exp. Biol. 210(20), 3624–3635 (2007).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (2)

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: Capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1.Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transf. 13(11), 1065–1080 (1973).
[Crossref]

J. R. Soc. Interface (2)

A. L. Holt, A. M. Sweeney, S. Johnsen, and D. E. Morse, “A highly distributed Bragg stack with unique geometry provides effective camouflage for Loliginid squid eyes,” J. R. Soc. Interface 8(63), 1386–1399 (2011).
[Crossref] [PubMed]

L. M. Mäthger, E. J. Denton, N. J. Marshall, and R. T. Hanlon, “Mechanisms and behavioural functions of structural coloration in cephalopods,” J. R. Soc. Interface 6(2), S149–S163 (2009).
[PubMed]

J. Zool. (1)

D. Froesch and J. B. Messenger, “On leucophores and the chromatic unit of Octopus vulgaris,” J. Zool. 186(2), 163–173 (1978).
[Crossref]

Nat. Mater. (1)

R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, “The self-organizing properties of squid reflectin protein,” Nat. Mater. 6(7), 533–538 (2007).
[Crossref] [PubMed]

Prog. Biophys. Mol. Biol. (1)

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[Crossref] [PubMed]

Z. Zellforsch. Mikrosk. Anat. (1)

S. Mirow, “Skin color in the squids Loligo pealii and Loligo opalescens. II. Iridophores,” Z. Zellforsch. Mikrosk. Anat. 125(2), 176–190 (1972).
[Crossref] [PubMed]

Other (5)

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, 1999).

E. Hecht, Optics (Addison-Wesley, 2002).

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (John Wiley & Sons, 2007).

C. Bohren and E. Clothiaux, Fundamentals of atmospheric radiation (Wiley-VCH, 2006).

H. C. van de Hulst, Light scattering by small particles (Dover Publications, 1981).

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Figures (9)

Fig. 1
Fig. 1

(a) A single layer plate (b) A 5-layer plate.

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Fig. 2
Fig. 2

The projected geometric areas ( C g ) which are perpendicular to the incident direction are shown for (a) a single-layer plate and (b) a 5-layer plate. Incident angle β is from 0° to 90°.

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Fig. 3
Fig. 3

Illustration of the interference of the light reflected from the top and bottom surfaces of a semi-infinite plate with thickness d and relative refractive index n r in the principal plane.

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Fig. 4
Fig. 4

Phase function P( θ s ) with aspect ratio a ranging from 1 to 40 for a single-layer plate under normal incidence and size parameter x = 1.3. The FWHM of the reflection peaks ( Δθ ) is indicated as a solid horizontal red bar. The aspect ratio, a, is shown close to its corresponding curve.

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Fig. 5
Fig. 5

The FWHM of the reflection peak ( Δθ ) versus the plate thickness size parameter for a single-layer plate. The aspect ratio is a. Vertical dashed lines are for size parameters x = 1.3, 2.6, and 3.9.

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Fig. 6
Fig. 6

Backscattering efficiency Qb versus number of layers for a size parameter of the plate thickness x=1.3 with different aspect ratios and under normal incidence. The reflection for semi-infinite plates is indicated by asterisks.

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Fig. 7
Fig. 7

Backscattering efficiency Qb versus size parameter x=2πd/λ of the plate thickness at different aspect ratios under normal incidence. The reflection for semi-infinite plates is indicated by solid red lines. (a) Results for a single-layer plate. Each curve is successively moved upward by 0.05. Maximum reflection of the semi-infinite plates is 0.033 at x = 1.3 and 3.9. (b) Results for 5-layer plate. Each curve is successively moved upward by 1.0. Maximum reflection of the semi-infinite plates is 0.52 at x = 1.3 and 3.9. The step for the size parameter in the calculation is 0.1. Vertical dashed lines are for the interference maxima with size parameter x = 1.3, 2.6, and 3.9

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Fig. 8
Fig. 8

Same as Fig. 7 but for a single-layer plate with an aspect ratio 10 and varying incident angles β. Incident light is polarized in (a) parallel direction and (b) perpendicular direction relative to the principal plane. Each curve is successively moved upward by 0.1 except forβ = 70°, 80°, and 90° which are plotted in the upper panels. The solid red lines indicate the reflection for semi-infinite plates forβ = 10° to 90° from bottom to top.

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Fig. 9
Fig. 9

Same as Fig. 8 but for a five-layer plate. Each curve is successively moved upward by 1.0.

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Equations (4)

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Q b = 1 k 2 C g ∬ Ω + P(Ω)dΩ ,
R 1s,p = 4 r s,p 2 sin 2 α (1− r s,p 2 ) 2 +4 r s,p 2 sin 2 α ,
r s = cosβ− n r cosθ cosβ+ n r cosθ ,
r p = n r cosβ−cosθ n r cosβ+cosθ ,

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