Abstract

We discuss the use of Mueller matrices for characterizing the reflecting properties of beetles, including matching patterns of significant elements to specific cuticular architectures. In the case of illumination by natural light, the parameters of the reflected light are related to the elements of the first column of the matrix. The green and red beetle Stephanorrhina guttata is shown to be a narrowband polarization- preserving reflector apart from depolarizing white patches, the green Calloodes grayanus a narrowband left-circular reflector, and the gold Anoplognathus parvulus behaves as a broadband left-circular reflector. Comparison of experimental and simulated matrices confirms that the beetle reflectors are natural analogs of all-dielectric thin-film reflectors. However, the gold Chrysina resplendens, which was formerly known as Plusiotis resplendens and reflects both right-handed and left-handed light, is represented by an ensemble of laterally incoherent chiral thin-film reflectors.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  7. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  24. A. R. Parker, D. R. McKenzie and M. C. J. Large, “Multilayer reflectors in animals using green and gold beetles as contrasting examples,” J. Exp. Biol. 201, 1307–1313(1998).
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    [CrossRef]
  27. M. Srinivasarao, “Nano-optics in the biological world: beetles, butterflies, birds and moths,” Chem. Rev. 99, 1935–1961 (1999).
    [CrossRef]

2010

J. Guo, F. Liu, F. Chen, J. Wei, and H. Yang, “Realisation of cholesteric liquid-crystalline materials reflecting both right- and left-circularly polarised light using the wash-out/refill technique,” Liq. Cryst. 37, 171–178 (2010).
[CrossRef]

2009

M. McCall and I. J. Hodgkinson, “Properties of partially polarized light remitted from lossless polarizing elements,” Eur. J. Phys. 30, S63–S80 (2009).
[CrossRef]

V. Sharma, M. Crne, J. O. Park, and M. Srinivasarao, “Structural origin of circularly polarized iridescence in jeweled beetles,” Science 325, 449–451 (2009).
[CrossRef] [PubMed]

2008

2007

D. J. Brink, N. G. van der Berg, L. C. Prinsloo, and I. J. Hodgkinson, “Unusual coloration in scarabaeid beetles,” J. Phys. D 40, 2189–2196 (2007).
[CrossRef]

S. Lowrey, L. De Silva, I. J. Hodgkinson, and J. Leader, “Observation and modeling of polarized light from scarab beetles,” J. Opt. Soc. Am. A 24, 2418–2425 (2007).
[CrossRef]

2006

D. H. Goldstein, “Polarization properties of Scarabaeidae,” Appl. Opt. 45, 7944–7950 (2006).
[CrossRef] [PubMed]

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364(2006).
[CrossRef] [PubMed]

2005

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

2004

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

2000

A. R. Parker, “515 million years of structural colour,” J. Opt. A Pure Appl. Opt. 2, R15–R28 (2000).
[CrossRef]

1999

M. Srinivasarao, “Nano-optics in the biological world: beetles, butterflies, birds and moths,” Chem. Rev. 99, 1935–1961 (1999).
[CrossRef]

1998

A. R. Parker, D. R. McKenzie and M. C. J. Large, “Multilayer reflectors in animals using green and gold beetles as contrasting examples,” J. Exp. Biol. 201, 1307–1313(1998).

1989

T. D. Schultz and G. D. Bernard, “Pointillistic mixing of interference colours in cryptic tiger beetles,” Nature 337, 72–73(1989).
[CrossRef]

1971

S. Caveney, “Cuticle reflectivity and optical activity in scarab beetles: the role of uric acid,” Proc. R. Soc. B 178, 205–225(1971).
[CrossRef]

1969

A. C. Neville and S. Caveney, “Scarabaeid beetle exocuticle as an optical analogue of cholesteric liquid crystals,” Biol. Rev. Camb. Philos. Soc. 44, 531–562 (1969).
[CrossRef] [PubMed]

1911

A. A. Michelson, “On metallic colouring in birds and insects,” Philos. Mag. 21, 554–567 (1911).

Arnold, M.

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

Bernard, G. D.

T. D. Schultz and G. D. Bernard, “Pointillistic mixing of interference colours in cryptic tiger beetles,” Nature 337, 72–73(1989).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Bourke, L.

L. Bourke, I. J. Hodgkinson, L. De Silva, and J. Leader, “Chiral photonic film and flake,” Opt. Express 16, 16889–16894(2008).
[CrossRef] [PubMed]

I. J. Hodgkinson and L. Bourke, “Chiral photonic media,” in Tutorials in Complex Photonic Media, M.A.Noginov, M.W.McCall, G.Dewar, and N.I.Zheludev, eds. (SPIE, 2009), Chap. 6, pp. 131–156.
[CrossRef]

Brink, D. J.

D. J. Brink, N. G. van der Berg, L. C. Prinsloo, and I. J. Hodgkinson, “Unusual coloration in scarabaeid beetles,” J. Phys. D 40, 2189–2196 (2007).
[CrossRef]

Caveney, S.

S. Caveney, “Cuticle reflectivity and optical activity in scarab beetles: the role of uric acid,” Proc. R. Soc. B 178, 205–225(1971).
[CrossRef]

A. C. Neville and S. Caveney, “Scarabaeid beetle exocuticle as an optical analogue of cholesteric liquid crystals,” Biol. Rev. Camb. Philos. Soc. 44, 531–562 (1969).
[CrossRef] [PubMed]

Chen, F.

J. Guo, F. Liu, F. Chen, J. Wei, and H. Yang, “Realisation of cholesteric liquid-crystalline materials reflecting both right- and left-circularly polarised light using the wash-out/refill technique,” Liq. Cryst. 37, 171–178 (2010).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics—Devices, Measurements and Properties, M.Bass, ed. (McGraw-Hill, 1995) Vol. 2, Chap. 22.

Crne, M.

V. Sharma, M. Crne, J. O. Park, and M. Srinivasarao, “Structural origin of circularly polarized iridescence in jeweled beetles,” Science 325, 449–451 (2009).
[CrossRef] [PubMed]

De Silva, L.

L. Bourke, I. J. Hodgkinson, L. De Silva, and J. Leader, “Chiral photonic film and flake,” Opt. Express 16, 16889–16894(2008).
[CrossRef] [PubMed]

S. Lowrey, L. De Silva, I. J. Hodgkinson, and J. Leader, “Observation and modeling of polarized light from scarab beetles,” J. Opt. Soc. Am. A 24, 2418–2425 (2007).
[CrossRef]

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

Dessaud, N.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364(2006).
[CrossRef] [PubMed]

Goldstein, D. H.

Guo, J.

J. Guo, F. Liu, F. Chen, J. Wei, and H. Yang, “Realisation of cholesteric liquid-crystalline materials reflecting both right- and left-circularly polarised light using the wash-out/refill technique,” Liq. Cryst. 37, 171–178 (2010).
[CrossRef]

Hecht, E.

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

Hodgkinson, I. J.

M. McCall and I. J. Hodgkinson, “Properties of partially polarized light remitted from lossless polarizing elements,” Eur. J. Phys. 30, S63–S80 (2009).
[CrossRef]

L. Bourke, I. J. Hodgkinson, L. De Silva, and J. Leader, “Chiral photonic film and flake,” Opt. Express 16, 16889–16894(2008).
[CrossRef] [PubMed]

D. J. Brink, N. G. van der Berg, L. C. Prinsloo, and I. J. Hodgkinson, “Unusual coloration in scarabaeid beetles,” J. Phys. D 40, 2189–2196 (2007).
[CrossRef]

S. Lowrey, L. De Silva, I. J. Hodgkinson, and J. Leader, “Observation and modeling of polarized light from scarab beetles,” J. Opt. Soc. Am. A 24, 2418–2425 (2007).
[CrossRef]

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

I. J. Hodgkinson and L. Bourke, “Chiral photonic media,” in Tutorials in Complex Photonic Media, M.A.Noginov, M.W.McCall, G.Dewar, and N.I.Zheludev, eds. (SPIE, 2009), Chap. 6, pp. 131–156.
[CrossRef]

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1998).

Hwang, J.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Ishikawa, K.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Large, M. C. J.

A. R. Parker, D. R. McKenzie and M. C. J. Large, “Multilayer reflectors in animals using green and gold beetles as contrasting examples,” J. Exp. Biol. 201, 1307–1313(1998).

Leader, J.

L. Bourke, I. J. Hodgkinson, L. De Silva, and J. Leader, “Chiral photonic film and flake,” Opt. Express 16, 16889–16894(2008).
[CrossRef] [PubMed]

S. Lowrey, L. De Silva, I. J. Hodgkinson, and J. Leader, “Observation and modeling of polarized light from scarab beetles,” J. Opt. Soc. Am. A 24, 2418–2425 (2007).
[CrossRef]

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

Liu, F.

J. Guo, F. Liu, F. Chen, J. Wei, and H. Yang, “Realisation of cholesteric liquid-crystalline materials reflecting both right- and left-circularly polarised light using the wash-out/refill technique,” Liq. Cryst. 37, 171–178 (2010).
[CrossRef]

Lowrey, S.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001).
[CrossRef]

McCall, M.

M. McCall and I. J. Hodgkinson, “Properties of partially polarized light remitted from lossless polarizing elements,” Eur. J. Phys. 30, S63–S80 (2009).
[CrossRef]

McKenzie, D. R.

A. R. Parker, D. R. McKenzie and M. C. J. Large, “Multilayer reflectors in animals using green and gold beetles as contrasting examples,” J. Exp. Biol. 201, 1307–1313(1998).

McNaughton, A.

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

Michelson, A. A.

A. A. Michelson, “On metallic colouring in birds and insects,” Philos. Mag. 21, 554–567 (1911).

Mitov, M.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364(2006).
[CrossRef] [PubMed]

Murray, P.

L. De Silva, I. J. Hodgkinson, P. Murray, Q. H. Wu, M. Arnold, J. Leader, and A. McNaughton, “Natural and Nanoengineered chiral reflectors: structural color of Manuka beetles and titania coatings,” Electromagnetics 25, 391–408 (2005).
[CrossRef]

Neville, A. C.

A. C. Neville and S. Caveney, “Scarabaeid beetle exocuticle as an optical analogue of cholesteric liquid crystals,” Biol. Rev. Camb. Philos. Soc. 44, 531–562 (1969).
[CrossRef] [PubMed]

A. C. Neville, Biology of Fibrous Composites: Development Beyond the Cell Membrane (Cambridge University, 1993).
[CrossRef]

Nishimura, S.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Park, B.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Park, J. O.

V. Sharma, M. Crne, J. O. Park, and M. Srinivasarao, “Structural origin of circularly polarized iridescence in jeweled beetles,” Science 325, 449–451 (2009).
[CrossRef] [PubMed]

Parker, A. R.

A. R. Parker, “515 million years of structural colour,” J. Opt. A Pure Appl. Opt. 2, R15–R28 (2000).
[CrossRef]

A. R. Parker, D. R. McKenzie and M. C. J. Large, “Multilayer reflectors in animals using green and gold beetles as contrasting examples,” J. Exp. Biol. 201, 1307–1313(1998).

Prinsloo, L. C.

D. J. Brink, N. G. van der Berg, L. C. Prinsloo, and I. J. Hodgkinson, “Unusual coloration in scarabaeid beetles,” J. Phys. D 40, 2189–2196 (2007).
[CrossRef]

Schultz, T. D.

T. D. Schultz and G. D. Bernard, “Pointillistic mixing of interference colours in cryptic tiger beetles,” Nature 337, 72–73(1989).
[CrossRef]

Sharma, V.

V. Sharma, M. Crne, J. O. Park, and M. Srinivasarao, “Structural origin of circularly polarized iridescence in jeweled beetles,” Science 325, 449–451 (2009).
[CrossRef] [PubMed]

Shin, K. C.

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Song, M. H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Srinivasarao, M.

V. Sharma, M. Crne, J. O. Park, and M. Srinivasarao, “Structural origin of circularly polarized iridescence in jeweled beetles,” Science 325, 449–451 (2009).
[CrossRef] [PubMed]

M. Srinivasarao, “Nano-optics in the biological world: beetles, butterflies, birds and moths,” Chem. Rev. 99, 1935–1961 (1999).
[CrossRef]

Swager, T.

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Takanishi, Y.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Takezoe, H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Toyooka, T.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

van der Berg, N. G.

D. J. Brink, N. G. van der Berg, L. C. Prinsloo, and I. J. Hodgkinson, “Unusual coloration in scarabaeid beetles,” J. Phys. D 40, 2189–2196 (2007).
[CrossRef]

Watanabe, J.

M. H. Song, K. C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437–441 (2004).
[CrossRef]

Wei, J.

J. Guo, F. Liu, F. Chen, J. Wei, and H. Yang, “Realisation of cholesteric liquid-crystalline materials reflecting both right- and left-circularly polarised light using the wash-out/refill technique,” Liq. Cryst. 37, 171–178 (2010).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Wu, J.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005).
[CrossRef] [PubMed]

Wu, Q. H.

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Figures (10)

Fig. 1
Fig. 1

Patterns of elements in standard Mueller matrices: a, depolarizer; b, polarization-preserving reflector; c, left- circular polarizer; and d, right-circular polarizer. The sign of nonzero matrix elements are represented as schematic spectra (see later results and simulations).

Fig. 2
Fig. 2

Ellipsometer used for recording Mueller matrices of beetle cuticle.

Fig. 3
Fig. 3

Beetles used in the study: a, Stephanorrhina guttata; b, Calloodes grayanus; c, Anoplognathus parvulus; and d, Chrysina resplendens. The arrows show both the measurement sites and the reference direction for ellipsometric measurements.

Fig. 4
Fig. 4

Mueller-matrix spectra (solid lines) and reflected unpolarized light spectra R u (shaded regions) recorded from a green area of S. guttata (upper image) and for a simulation of the green reflector based on a quarter-wave stack of isotropic materials (lower image). In each square, the matrix elements are negative in the lower half and the wavelength range is 400 to 900 nm .

Fig. 5
Fig. 5

Mueller-matrix spectra (solid lines) and reflected unpolarized light spectrum R u (shaded region) recorded from a white area of S. guttata. In each square, the matrix elements are negative in the lower half and the wavelength range is 400 to 900 nm .

Fig. 6
Fig. 6

Mueller-matrix spectra (solid lines) and reflected unpolarized light spectra R u (shaded regions) recorded for a green area of C. grayanus (upper image) and for a simulation of the green reflector based on a stack of locally birefringent layers twisted in a left-handed sense and with a constant pitch (lower image). In each square, the matrix elements are negative in the lower half and the wavelength range is 400 to 900 nm .

Fig. 7
Fig. 7

Mueller-matrix spectra (solid lines) and reflected unpolarized light spectra R u (shaded regions) recorded for A. parvulus (upper image) and for a simulation of the gold reflector based on a stack of locally birefringent layers twisted in a left-handed sense and with a variable pitch (lower image). In each square, the matrix elements are negative in the lower half and the wavelength range is 400 to 900 nm .

Fig. 8
Fig. 8

Mueller-matrix spectra (solid lines) and reflected unpolarized light spectra R u (shaded regions) recorded for C. resplendens (upper image) and computed for Caveney’s model for the beetle (lower image). In each square, the matrix elements are negative in the lower half and the wavelength range is 400 to 900 nm .

Fig. 9
Fig. 9

Stokes matrices for remittance from C. resplendens illuminated in turn by the six basis polarizations (upper image) and from an implementation of Caveney’s model. The shaded regions show the unpolarized component of each reflected beam. In each square, the matrix elements are negative in the lower half and the wavelength range is 400 to 900 nm .

Fig. 10
Fig. 10

Domain of R p for light remitted from lossless thin-film models representing the beetles used in the study. Caveney’s model for C. resplendens is the only structure that can roam the entire domain.

Equations (11)

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M = [ M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 22 M 23 M 30 M 31 M 32 M 33 ] ,
s = M S .
R t = M 00 ,
R p = ( M 10 2 + M 20 2 + M 30 2 ) 1 / 2 ,
R u = M 00 ( M 10 2 + M 20 2 + M 30 2 ) 1 / 2 .
V = ( M 10 2 + M 20 2 + M 30 2 ) 1 / 2 / M 00 .
b / a = tan { 1 2 arcsin [ M 30 / ( M 10 2 + M 20 2 + M 30 2 ) 1 / 2 ] } ,
θ = 1 2 arctan ( M 20 / M 10 ) .
M = [ M 00 0 0 M 03 0 M 11 M 12 0 0 M 21 M 22 0 M 30 0 0 M 33 ] .
S = [ 1 1 1 1 1 1 1 1 0 0 cos Δ cos Δ 0 0 1 1 0 0 0 0 0 0 sin Δ sin Δ ] .
s = [ s 00 s 01 s 02 s 03 s 04 s 05 s 10 s 11 s 12 s 13 s 14 s 15 s 20 s 21 s 22 s 23 s 24 s 25 s 30 s 31 s 32 s 33 s 34 s 35 ] .

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