Abstract

The optical properties of transparent single membranes on the wings of the dragonfly Aeshna cyanea have been investigated. These membranes comprise one central thick cuticular layer covered dorsally and ventrally with typical odonatan wax pruinosity. Optical characterisation of individual membranes reveals they can support optical guided modes comprising differential polarisation reflection. We suggest this may offer an intraspecific signalling channel. The guided modes’ characteristics depend on membrane thickness and the nature of the wax pruinosity. We accurately modelled multiple optical data sets simultaneously, thereby inaugurally quantifying the roughness of the pruinosity and the complex refractive indices of the wax and the odonatan cuticle.

© 2006 Optical Society of America

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References

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  1. P. Vukusic and J. R. Sambles, "Photonic structures in Biology," Nature 424, 852-855 (2003).
    [CrossRef] [PubMed]
  2. P. Vukusic, J. R. Sambles, and C. R. Lawrence, "Structurally assisted blackness in butterfly scales," Proc. R. Soc. London Ser. B. (Suppl. i.e. Biology Letters) 271, S237-S239 (2004).
    [CrossRef]
  3. R. J. Wootton, "The functional morphology of the wings of Odonata," Adv. in Odon. 5, 153-169 (1991).
  4. S. N. Gorb, A. Kesel, and J. Berger, "Microsculpture of the wing surface in Odonata: evidence for cuticular wax covering," Arthropod structure and development 29, 129-135 (2000).
    [CrossRef]
  5. T. Wagner, C. Neinhuis, and W. Barthlott, "Wettability and contaminability of insect wings as a function of their surface sculptures," Acta Zoologica 77, 213-225 (1996).
    [CrossRef]
  6. H. Ghiradella and W. Radigan, "Collembolan cuticle: wax layer and anti-wetting properties," J. Insect Physiol. 20, 301-306 (1974).
    [CrossRef] [PubMed]
  7. S. P. Sane, "The aerodynamics of insect flight," J. Exp. Biol. 206, 4191-4208 (2003).
    [CrossRef] [PubMed]
  8. J. M. Wakeling and C. P. Ellington, "Dragonfly flight," J. Exp. Biol. 200, 543-556 (1997).
    [PubMed]
  9. P. S. Corbet, Dragonflies: behaviour and ecology of Odonata, (Comstock Publishing Associates, Cornell University Press. 1999).
  10. P. Vukusic, Natural Photonics, Physics World, Feb. 2004.
  11. P. Vukusic, J. R. Sambles, and R. J. Wootton, "Remarkable iridescence in the hind-wings of the damselfly Neurobasis chinensis (Linnaeus) (Zygoptera Calopterygidae)," Proc. R. Soc. London Ser. B. 271, 595-601 (2004).
    [CrossRef]
  12. T. Hariyama, M. Hironaka, H. Horiguchi, and D. G. Stavenga, "The Leaf Beetle, the Jewel Beetle and the Damselfly; Insects with a multilayered show case" in Structural Colors, S. Yoshioka, S. and Kinoshita, eds. (Osaka University Press, Osaka, 2005), pp. 153-176.
  13. L. Névot and P. Croce, "Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates," Rev..Phys. Appl. 15, 761-779 (1980).
    [CrossRef]
  14. P. K. Tien, "Integrated optics and new wave phenomena in optical waveguides," Rev. Mod. Phys. 49,361-420 (1977).
    [CrossRef]
  15. P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
    [CrossRef]
  16. S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
    [CrossRef]
  17. Z. J. Wang, "Dissecting insect flight," Annu. Rev. Fluid Mech. 37, 183-210 (2005).
    [CrossRef]
  18. J. Kim, "Control of turbulent boundary layers," Phys. Fluids 15, 1093-1105 (2003).
    [CrossRef]
  19. M. Okamoto, K. Yasuda, and A. Azuma, "Aerodynamic characteristics of the wings and body of a dragonfly," J. Exp. Biol. 199, 281-294 (1996).
    [PubMed]
  20. A A. Barybin and V. A. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory: Application to Guided-Wave Optics (Rinton Press, Princeton, New Jersey, 2002).
  21. A. S. Bonnet, G. Caloz, and F. Mahé, "Guided modes of integrated optical guides. A mathematical study," J. Appl. Math. 60, 225-261 (1998).
    [CrossRef]
  22. E. P. Meyer and T. Labhart, "Morphological specializations of dorsal rim ommatidia in the compound eye of dragonflies and damselflies (Odonata)," Cell and Tissue Research,  272, 17-22. (1993).
    [CrossRef]
  23. T. Labhart and E. P. Meyer, "Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye," Microscopy Research and Technique 47, 368-379 (1999).
    [CrossRef] [PubMed]
  24. T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
    [CrossRef]

2005 (1)

Z. J. Wang, "Dissecting insect flight," Annu. Rev. Fluid Mech. 37, 183-210 (2005).
[CrossRef]

2004 (1)

P. Vukusic, J. R. Sambles, and R. J. Wootton, "Remarkable iridescence in the hind-wings of the damselfly Neurobasis chinensis (Linnaeus) (Zygoptera Calopterygidae)," Proc. R. Soc. London Ser. B. 271, 595-601 (2004).
[CrossRef]

2003 (4)

P. Vukusic and J. R. Sambles, "Photonic structures in Biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

S. P. Sane, "The aerodynamics of insect flight," J. Exp. Biol. 206, 4191-4208 (2003).
[CrossRef] [PubMed]

J. Kim, "Control of turbulent boundary layers," Phys. Fluids 15, 1093-1105 (2003).
[CrossRef]

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

1999 (2)

T. Labhart and E. P. Meyer, "Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye," Microscopy Research and Technique 47, 368-379 (1999).
[CrossRef] [PubMed]

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
[CrossRef]

1998 (1)

A. S. Bonnet, G. Caloz, and F. Mahé, "Guided modes of integrated optical guides. A mathematical study," J. Appl. Math. 60, 225-261 (1998).
[CrossRef]

1997 (2)

S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
[CrossRef]

J. M. Wakeling and C. P. Ellington, "Dragonfly flight," J. Exp. Biol. 200, 543-556 (1997).
[PubMed]

1996 (2)

T. Wagner, C. Neinhuis, and W. Barthlott, "Wettability and contaminability of insect wings as a function of their surface sculptures," Acta Zoologica 77, 213-225 (1996).
[CrossRef]

M. Okamoto, K. Yasuda, and A. Azuma, "Aerodynamic characteristics of the wings and body of a dragonfly," J. Exp. Biol. 199, 281-294 (1996).
[PubMed]

1993 (1)

E. P. Meyer and T. Labhart, "Morphological specializations of dorsal rim ommatidia in the compound eye of dragonflies and damselflies (Odonata)," Cell and Tissue Research,  272, 17-22. (1993).
[CrossRef]

1980 (1)

L. Névot and P. Croce, "Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates," Rev..Phys. Appl. 15, 761-779 (1980).
[CrossRef]

1977 (1)

P. K. Tien, "Integrated optics and new wave phenomena in optical waveguides," Rev. Mod. Phys. 49,361-420 (1977).
[CrossRef]

1974 (1)

H. Ghiradella and W. Radigan, "Collembolan cuticle: wax layer and anti-wetting properties," J. Insect Physiol. 20, 301-306 (1974).
[CrossRef] [PubMed]

Azuma, A.

M. Okamoto, K. Yasuda, and A. Azuma, "Aerodynamic characteristics of the wings and body of a dragonfly," J. Exp. Biol. 199, 281-294 (1996).
[PubMed]

Barthlott, W.

T. Wagner, C. Neinhuis, and W. Barthlott, "Wettability and contaminability of insect wings as a function of their surface sculptures," Acta Zoologica 77, 213-225 (1996).
[CrossRef]

Bonnet, A. S.

A. S. Bonnet, G. Caloz, and F. Mahé, "Guided modes of integrated optical guides. A mathematical study," J. Appl. Math. 60, 225-261 (1998).
[CrossRef]

Caldwell, R. L.

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

Caloz, G.

A. S. Bonnet, G. Caloz, and F. Mahé, "Guided modes of integrated optical guides. A mathematical study," J. Appl. Math. 60, 225-261 (1998).
[CrossRef]

Cheroske, A. G.

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

Chiou, T.

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

Croce, P.

L. Névot and P. Croce, "Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates," Rev..Phys. Appl. 15, 761-779 (1980).
[CrossRef]

Cronin, T.

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

Ellington, C. P.

J. M. Wakeling and C. P. Ellington, "Dragonfly flight," J. Exp. Biol. 200, 543-556 (1997).
[PubMed]

Ghiradella, H.

H. Ghiradella and W. Radigan, "Collembolan cuticle: wax layer and anti-wetting properties," J. Insect Physiol. 20, 301-306 (1974).
[CrossRef] [PubMed]

Jin, X.

S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
[CrossRef]

Kim, J.

J. Kim, "Control of turbulent boundary layers," Phys. Fluids 15, 1093-1105 (2003).
[CrossRef]

Labhart, T.

T. Labhart and E. P. Meyer, "Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye," Microscopy Research and Technique 47, 368-379 (1999).
[CrossRef] [PubMed]

E. P. Meyer and T. Labhart, "Morphological specializations of dorsal rim ommatidia in the compound eye of dragonflies and damselflies (Odonata)," Cell and Tissue Research,  272, 17-22. (1993).
[CrossRef]

Lawrence, C. R.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
[CrossRef]

Liang, S.

S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
[CrossRef]

Mahé, F.

A. S. Bonnet, G. Caloz, and F. Mahé, "Guided modes of integrated optical guides. A mathematical study," J. Appl. Math. 60, 225-261 (1998).
[CrossRef]

Marshall, J.

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

Meyer, E. P.

T. Labhart and E. P. Meyer, "Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye," Microscopy Research and Technique 47, 368-379 (1999).
[CrossRef] [PubMed]

E. P. Meyer and T. Labhart, "Morphological specializations of dorsal rim ommatidia in the compound eye of dragonflies and damselflies (Odonata)," Cell and Tissue Research,  272, 17-22. (1993).
[CrossRef]

Neinhuis, C.

T. Wagner, C. Neinhuis, and W. Barthlott, "Wettability and contaminability of insect wings as a function of their surface sculptures," Acta Zoologica 77, 213-225 (1996).
[CrossRef]

Névot, L.

L. Névot and P. Croce, "Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates," Rev..Phys. Appl. 15, 761-779 (1980).
[CrossRef]

Okamoto, M.

M. Okamoto, K. Yasuda, and A. Azuma, "Aerodynamic characteristics of the wings and body of a dragonfly," J. Exp. Biol. 199, 281-294 (1996).
[PubMed]

Radigan, W.

H. Ghiradella and W. Radigan, "Collembolan cuticle: wax layer and anti-wetting properties," J. Insect Physiol. 20, 301-306 (1974).
[CrossRef] [PubMed]

Sambles, J. R.

P. Vukusic, J. R. Sambles, and R. J. Wootton, "Remarkable iridescence in the hind-wings of the damselfly Neurobasis chinensis (Linnaeus) (Zygoptera Calopterygidae)," Proc. R. Soc. London Ser. B. 271, 595-601 (2004).
[CrossRef]

P. Vukusic and J. R. Sambles, "Photonic structures in Biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
[CrossRef]

Sane, S. P.

S. P. Sane, "The aerodynamics of insect flight," J. Exp. Biol. 206, 4191-4208 (2003).
[CrossRef] [PubMed]

Shashar, N.

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

Strahler, A. H.

S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
[CrossRef]

Tien, P. K.

P. K. Tien, "Integrated optics and new wave phenomena in optical waveguides," Rev. Mod. Phys. 49,361-420 (1977).
[CrossRef]

Vukusic, P.

P. Vukusic, J. R. Sambles, and R. J. Wootton, "Remarkable iridescence in the hind-wings of the damselfly Neurobasis chinensis (Linnaeus) (Zygoptera Calopterygidae)," Proc. R. Soc. London Ser. B. 271, 595-601 (2004).
[CrossRef]

P. Vukusic and J. R. Sambles, "Photonic structures in Biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
[CrossRef]

Wagner, T.

T. Wagner, C. Neinhuis, and W. Barthlott, "Wettability and contaminability of insect wings as a function of their surface sculptures," Acta Zoologica 77, 213-225 (1996).
[CrossRef]

Wakeling, J. M.

J. M. Wakeling and C. P. Ellington, "Dragonfly flight," J. Exp. Biol. 200, 543-556 (1997).
[PubMed]

Wang, Z. J.

Z. J. Wang, "Dissecting insect flight," Annu. Rev. Fluid Mech. 37, 183-210 (2005).
[CrossRef]

Wootton, R. J.

P. Vukusic, J. R. Sambles, and R. J. Wootton, "Remarkable iridescence in the hind-wings of the damselfly Neurobasis chinensis (Linnaeus) (Zygoptera Calopterygidae)," Proc. R. Soc. London Ser. B. 271, 595-601 (2004).
[CrossRef]

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
[CrossRef]

Yasuda, K.

M. Okamoto, K. Yasuda, and A. Azuma, "Aerodynamic characteristics of the wings and body of a dragonfly," J. Exp. Biol. 199, 281-294 (1996).
[PubMed]

Zhu, Q.

S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
[CrossRef]

Acta Zoologica (1)

T. Wagner, C. Neinhuis, and W. Barthlott, "Wettability and contaminability of insect wings as a function of their surface sculptures," Acta Zoologica 77, 213-225 (1996).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

Z. J. Wang, "Dissecting insect flight," Annu. Rev. Fluid Mech. 37, 183-210 (2005).
[CrossRef]

Cell and Tissue Research (1)

E. P. Meyer and T. Labhart, "Morphological specializations of dorsal rim ommatidia in the compound eye of dragonflies and damselflies (Odonata)," Cell and Tissue Research,  272, 17-22. (1993).
[CrossRef]

Integr. Comp Biol. (1)

T. Cronin, N. Shashar, R. L. Caldwell, J. Marshall, A. G. Cheroske, and T. Chiou, "Polarization vision and its role in Biological signaling," Integr. Comp Biol. 43, 549-558 (2003).
[CrossRef]

J. Appl. Math. (1)

A. S. Bonnet, G. Caloz, and F. Mahé, "Guided modes of integrated optical guides. A mathematical study," J. Appl. Math. 60, 225-261 (1998).
[CrossRef]

J. Exp. Biol. (3)

M. Okamoto, K. Yasuda, and A. Azuma, "Aerodynamic characteristics of the wings and body of a dragonfly," J. Exp. Biol. 199, 281-294 (1996).
[PubMed]

S. P. Sane, "The aerodynamics of insect flight," J. Exp. Biol. 206, 4191-4208 (2003).
[CrossRef] [PubMed]

J. M. Wakeling and C. P. Ellington, "Dragonfly flight," J. Exp. Biol. 200, 543-556 (1997).
[PubMed]

J. Insect Physiol. (1)

H. Ghiradella and W. Radigan, "Collembolan cuticle: wax layer and anti-wetting properties," J. Insect Physiol. 20, 301-306 (1974).
[CrossRef] [PubMed]

Microscopy Research and Technique (1)

T. Labhart and E. P. Meyer, "Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye," Microscopy Research and Technique 47, 368-379 (1999).
[CrossRef] [PubMed]

Nature (1)

P. Vukusic and J. R. Sambles, "Photonic structures in Biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

Phys. Appl. (1)

L. Névot and P. Croce, "Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates," Rev..Phys. Appl. 15, 761-779 (1980).
[CrossRef]

Phys. Fluids (1)

J. Kim, "Control of turbulent boundary layers," Phys. Fluids 15, 1093-1105 (2003).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, "Quantified interference and diffraction in single Morpho butterfly scales," Proc. R. Soc. London Ser. B 266, 1403-1411 (1999).
[CrossRef]

Proc. R. Soc. London Ser. B. (1)

P. Vukusic, J. R. Sambles, and R. J. Wootton, "Remarkable iridescence in the hind-wings of the damselfly Neurobasis chinensis (Linnaeus) (Zygoptera Calopterygidae)," Proc. R. Soc. London Ser. B. 271, 595-601 (2004).
[CrossRef]

Remote Sens. Environ. (1)

S. Liang, A. H. Strahler, X. Jin, and Q. Zhu, "Comparisons of Radiative transfer models of vegetation canopies and laboratory measurements," Remote Sens. Environ. 61, 129-138 (1997).
[CrossRef]

Rev. Mod. Phys. (1)

P. K. Tien, "Integrated optics and new wave phenomena in optical waveguides," Rev. Mod. Phys. 49,361-420 (1977).
[CrossRef]

Other (7)

T. Hariyama, M. Hironaka, H. Horiguchi, and D. G. Stavenga, "The Leaf Beetle, the Jewel Beetle and the Damselfly; Insects with a multilayered show case" in Structural Colors, S. Yoshioka, S. and Kinoshita, eds. (Osaka University Press, Osaka, 2005), pp. 153-176.

P. Vukusic, J. R. Sambles, and C. R. Lawrence, "Structurally assisted blackness in butterfly scales," Proc. R. Soc. London Ser. B. (Suppl. i.e. Biology Letters) 271, S237-S239 (2004).
[CrossRef]

R. J. Wootton, "The functional morphology of the wings of Odonata," Adv. in Odon. 5, 153-169 (1991).

S. N. Gorb, A. Kesel, and J. Berger, "Microsculpture of the wing surface in Odonata: evidence for cuticular wax covering," Arthropod structure and development 29, 129-135 (2000).
[CrossRef]

P. S. Corbet, Dragonflies: behaviour and ecology of Odonata, (Comstock Publishing Associates, Cornell University Press. 1999).

P. Vukusic, Natural Photonics, Physics World, Feb. 2004.

A A. Barybin and V. A. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory: Application to Guided-Wave Optics (Rinton Press, Princeton, New Jersey, 2002).

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Figures (3)

Fig. 1.
Fig. 1.

An SEM image showing a cleaved section of the single wing membrane from the fore-wing of Aeshna cyanea. The thin wax pruinosity is visible as the rough overlayer on the top surface (scale bar: 2 µm).

Fig. 2.
Fig. 2.

A selection of SEM images, at the same magnification, of the surface cuticular wax layer from the Aeshna cyanea dragonfly wing shown on the right. (a) dorsal surface from region 1. (b) dorsal surface from region 2. (c) ventral surface from region 3. (d) dorsal surface from region 4. Although (c) is the only image which corresponds to the ventral surface, most other regions on the ventral surface resemble those of images (a) and (d).

Fig. 3.
Fig. 3.

Angle-dependent TE polarised transmission data and theoretical fits for the four regions investigated. These regions correspond to the windows in the wing shown in the photograph in Fig 2. d1 is the thickness of the cuticular wax layer on the incident side of the structure, d2 is the thickness of the bulk cuticle layer, and d3 is the thickness of the cuticular wax layer on the exit side of the structure. σ1 and σ3 are the diminution constants describing the scattering due to the rough surfaces of the incident and exit interfaces respectively.

Equations (5)

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r ij = k zi k zj k zi + k zj
t ij = 2 k zi k zi + k zj
r ij = r ij e 2 k zi k zj σ 2 and t ij = t ij e ( k zi k zj ) 2 σ 2 2
t i , n = t ij t i + 1 , n e 2 i β i 1 + r ij r i + 1 , n e 2 i β i
T = Re [ k zn k z 1 ] tt *

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