Abstract

We have developed a mathematical model of skin coloration in cephalopods, a class of aquatic animals. Cephalopods utilize neurological and physiological control of various skin components to achieve active camouflage and communication. Specific physical processes of this coloration are identified and modeled, utilizing available biological materials data, to simulate active spectral changes in pigment-bearing organs and structural iridescent cells. Excellent agreement with in vitro measurements of squid skin is obtained. A detailed understanding of the physical principles underlying cephalopod coloration is expected to yield insights into the behavioral ecology of these animals.

© 2008 Optical Society of America

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  1. P. Vukusic and J. R. Sambles, "Photonic structures in biology," Nature 424, 852-855 (2003).
    [CrossRef] [PubMed]
  2. A. R. Parker, "515 million years of structural colour," J. Opt. A, Pure Appl. Opt. 2, R15-R28 (2000).
    [CrossRef]
  3. A. R. Parker and H. E. Townley, "Biomimetics of photonic nanostructures," Nat. Nanotechnol. 2, 347-353 (2007).
    [CrossRef]
  4. R. T. Hanlon and J. B. Messenger, Cephalopod Behaviour (Cambridge U. Press, 1996).
  5. R. T. Hanlon, "Cephalopod dynamic camouflage," Curr. Biol. 17, R400-R404 (2007).
    [CrossRef] [PubMed]
  6. L. Mäthger and R. T. Hanlon, "Malleable skin coloration in cephalopods: selective reflectance, transmission and absorbance of light by chromatophores and iridophores," Cell Tissue Res. 329, 179-186 (2007).
    [CrossRef] [PubMed]
  7. C.-C. Chiao and R. T. Hanlon, "Cuttlefish camouflage: visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration," J. Exp. Biol. 204, 2119-2125 (2001).
    [PubMed]
  8. L. M. Mäthger and R. T. Hanlon, "Anatomical basis for camouflaged polarized light communication in squid," Biol. Lett. 2, 494-496 (2006).
    [CrossRef] [PubMed]
  9. J. B. Messenger, "Cephalopod chromatophores: neurobiology and natural history," Biol. Rev. 76, 473-528 (2001).
    [CrossRef]
  10. M. B. Masthay, "Color changes induced by pigment granule aggregation in chromatophores: a quantitative model based on Beer's law," Photochem. Photobiol. 66, 649-658 (1997).
    [CrossRef]
  11. E. J. Denton and M. F. Land, "Mechanism of reflection in silvery layers of fish and cephalopods," Proc. R. Soc. London, Ser. A 178, 43-61 (1971).
    [CrossRef]
  12. 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, 2103-2118 (2001).
    [PubMed]
  13. W. J. Crookes, L. Ding, Q. L. Huang, J. R. Kimbell, J. Horwitz, and M. J. McFall-Ngai, "Reflectins: the unusual proteins of squid reflective tissues," Science 303, 235-238 (2004).
    [CrossRef] [PubMed]
  14. K. M. Cooper and R. T. Hanlon, "Correlation of iridescence with changes in iridophore platelet ultrastructure in the squid Lolliguncula brevis," J. Exp. Biol. 121, 451-455 (1986).
    [PubMed]
  15. C. D. Mobley, "The optical properties of water," in Handbook of Optics, Vol. II, M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Chap. 43.
  16. B. Maheu, J. N. Letoulouzan, and G. Gouesbet, "Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters," Appl. Opt. 23, 3353-3362 (1984).
    [CrossRef] [PubMed]
  17. R. A. Cloney and S. L. Brocco, "Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods," Am. Zool. 23, 581-592 (1983).
  18. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part I," J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  19. R. Levinson, P. Berdahl, and H. Akbari, "Solar spectral optical properties of pigments--part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements," Sol. Energy Mater. Sol. Cells 89, 319-349 (2005).
    [CrossRef]
  20. R. Levinson, P. Berdahl, and H. Akbari, "Solar spectral optical properties of pigments--part II: survey of common colorants," Sol. Energy Mater. Sol. Cells 89, 351-389 (2005).
    [CrossRef]
  21. W. E. Vargas and G. A. Niklasson, "Forward-scattering ratios and average pathlength parameter in radiative transfer models," J. Phys.: Condens. Matter 9, 9083-9096 (1997).
    [CrossRef]
  22. K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, "Physiological color change in squid iridophores. II. Ultrastructure mechanisms in Lolliguncula brevis," Cell Tissue Res. 259, 15-24 (1990).
    [CrossRef] [PubMed]
  23. R. M. Kramer, W. J. Crookes-Goodson, and R. R. Naik, "The self-organizing properties of squid reflectin protein," Nat. Mater. 6, 533-538 (2007).
    [CrossRef] [PubMed]
  24. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (McGraw-Hill, 1989).
  25. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  26. L. Mäthger, T. F. T. 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, 1759-1769 (2004).
    [CrossRef] [PubMed]
  27. C.-C. Chiao, T. W. Cronin, and D. Osorio, "Color signals in natural scenes: characteristics of reflectance spectra and effects of natural illuminants," J. Opt. Soc. Am. A 17, 218-224 (2000).
    [CrossRef]
  28. H. Kogelnik, "Filter response of nonuniform almost-periodic structures," Bell Syst. Tech. J. 55, 109-126 (1976).

2007

A. R. Parker and H. E. Townley, "Biomimetics of photonic nanostructures," Nat. Nanotechnol. 2, 347-353 (2007).
[CrossRef]

R. T. Hanlon, "Cephalopod dynamic camouflage," Curr. Biol. 17, R400-R404 (2007).
[CrossRef] [PubMed]

L. Mäthger and R. T. Hanlon, "Malleable skin coloration in cephalopods: selective reflectance, transmission and absorbance of light by chromatophores and iridophores," Cell Tissue Res. 329, 179-186 (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, 533-538 (2007).
[CrossRef] [PubMed]

2006

L. M. Mäthger and R. T. Hanlon, "Anatomical basis for camouflaged polarized light communication in squid," Biol. Lett. 2, 494-496 (2006).
[CrossRef] [PubMed]

2005

R. Levinson, P. Berdahl, and H. Akbari, "Solar spectral optical properties of pigments--part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements," Sol. Energy Mater. Sol. Cells 89, 319-349 (2005).
[CrossRef]

R. Levinson, P. Berdahl, and H. Akbari, "Solar spectral optical properties of pigments--part II: survey of common colorants," Sol. Energy Mater. Sol. Cells 89, 351-389 (2005).
[CrossRef]

2004

W. J. Crookes, L. Ding, Q. L. Huang, J. R. Kimbell, J. Horwitz, and M. J. McFall-Ngai, "Reflectins: the unusual proteins of squid reflective tissues," Science 303, 235-238 (2004).
[CrossRef] [PubMed]

L. Mäthger, T. F. T. 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, 1759-1769 (2004).
[CrossRef] [PubMed]

2003

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

2001

J. B. Messenger, "Cephalopod chromatophores: neurobiology and natural history," Biol. Rev. 76, 473-528 (2001).
[CrossRef]

C.-C. Chiao and R. T. Hanlon, "Cuttlefish camouflage: visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration," J. Exp. Biol. 204, 2119-2125 (2001).
[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, 2103-2118 (2001).
[PubMed]

2000

1997

W. E. Vargas and G. A. Niklasson, "Forward-scattering ratios and average pathlength parameter in radiative transfer models," J. Phys.: Condens. Matter 9, 9083-9096 (1997).
[CrossRef]

M. B. Masthay, "Color changes induced by pigment granule aggregation in chromatophores: a quantitative model based on Beer's law," Photochem. Photobiol. 66, 649-658 (1997).
[CrossRef]

1990

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, "Physiological color change in squid iridophores. II. Ultrastructure mechanisms in Lolliguncula brevis," Cell Tissue Res. 259, 15-24 (1990).
[CrossRef] [PubMed]

1986

K. M. Cooper and R. T. Hanlon, "Correlation of iridescence with changes in iridophore platelet ultrastructure in the squid Lolliguncula brevis," J. Exp. Biol. 121, 451-455 (1986).
[PubMed]

1984

1983

R. A. Cloney and S. L. Brocco, "Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods," Am. Zool. 23, 581-592 (1983).

1976

H. Kogelnik, "Filter response of nonuniform almost-periodic structures," Bell Syst. Tech. J. 55, 109-126 (1976).

1971

E. J. Denton and M. F. Land, "Mechanism of reflection in silvery layers of fish and cephalopods," Proc. R. Soc. London, Ser. A 178, 43-61 (1971).
[CrossRef]

1948

Am. Zool.

R. A. Cloney and S. L. Brocco, "Chromatophore organs, reflector cells, iridocytes and leucophores in cephalopods," Am. Zool. 23, 581-592 (1983).

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, "Filter response of nonuniform almost-periodic structures," Bell Syst. Tech. J. 55, 109-126 (1976).

Biol. Lett.

L. M. Mäthger and R. T. Hanlon, "Anatomical basis for camouflaged polarized light communication in squid," Biol. Lett. 2, 494-496 (2006).
[CrossRef] [PubMed]

Biol. Rev.

J. B. Messenger, "Cephalopod chromatophores: neurobiology and natural history," Biol. Rev. 76, 473-528 (2001).
[CrossRef]

Cell Tissue Res.

L. Mäthger and R. T. Hanlon, "Malleable skin coloration in cephalopods: selective reflectance, transmission and absorbance of light by chromatophores and iridophores," Cell Tissue Res. 329, 179-186 (2007).
[CrossRef] [PubMed]

K. M. Cooper, R. T. Hanlon, and B. U. Budelmann, "Physiological color change in squid iridophores. II. Ultrastructure mechanisms in Lolliguncula brevis," Cell Tissue Res. 259, 15-24 (1990).
[CrossRef] [PubMed]

Curr. Biol.

R. T. Hanlon, "Cephalopod dynamic camouflage," Curr. Biol. 17, R400-R404 (2007).
[CrossRef] [PubMed]

J. Exp. Biol.

C.-C. Chiao and R. T. Hanlon, "Cuttlefish camouflage: visual perception of size, contrast and number of white squares on artificial checkerboard substrata initiates disruptive coloration," J. Exp. Biol. 204, 2119-2125 (2001).
[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, 2103-2118 (2001).
[PubMed]

K. M. Cooper and R. T. Hanlon, "Correlation of iridescence with changes in iridophore platelet ultrastructure in the squid Lolliguncula brevis," J. Exp. Biol. 121, 451-455 (1986).
[PubMed]

L. Mäthger, T. F. T. 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, 1759-1769 (2004).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys.: Condens. Matter

W. E. Vargas and G. A. Niklasson, "Forward-scattering ratios and average pathlength parameter in radiative transfer models," J. Phys.: Condens. Matter 9, 9083-9096 (1997).
[CrossRef]

Nat. Mater.

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

Nat. Nanotechnol.

A. R. Parker and H. E. Townley, "Biomimetics of photonic nanostructures," Nat. Nanotechnol. 2, 347-353 (2007).
[CrossRef]

Nature

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

Photochem. Photobiol.

M. B. Masthay, "Color changes induced by pigment granule aggregation in chromatophores: a quantitative model based on Beer's law," Photochem. Photobiol. 66, 649-658 (1997).
[CrossRef]

Proc. R. Soc. London, Ser. A

E. J. Denton and M. F. Land, "Mechanism of reflection in silvery layers of fish and cephalopods," Proc. R. Soc. London, Ser. A 178, 43-61 (1971).
[CrossRef]

Science

W. J. Crookes, L. Ding, Q. L. Huang, J. R. Kimbell, J. Horwitz, and M. J. McFall-Ngai, "Reflectins: the unusual proteins of squid reflective tissues," Science 303, 235-238 (2004).
[CrossRef] [PubMed]

Sol. Energy Mater. Sol. Cells

R. Levinson, P. Berdahl, and H. Akbari, "Solar spectral optical properties of pigments--part I: model for deriving scattering and absorption coefficients from transmittance and reflectance measurements," Sol. Energy Mater. Sol. Cells 89, 319-349 (2005).
[CrossRef]

R. Levinson, P. Berdahl, and H. Akbari, "Solar spectral optical properties of pigments--part II: survey of common colorants," Sol. Energy Mater. Sol. Cells 89, 351-389 (2005).
[CrossRef]

Other

R. T. Hanlon and J. B. Messenger, Cephalopod Behaviour (Cambridge U. Press, 1996).

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (McGraw-Hill, 1989).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

C. D. Mobley, "The optical properties of water," in Handbook of Optics, Vol. II, M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Chap. 43.

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Figures (10)

Fig. 1
Fig. 1

Three-flux model for computing transmittance and reflectance of a chromatophore.

Fig. 2
Fig. 2

Schematic illustration of the random spacing of platelets in iridophores. The platelets (dark) have a uniform thickness h H , while the spacer thickness varies as h L + ζ k for layer k, where ζ k ( x ) is a random variable. A sensor will integrate over the spatial variable x within a pixel and yield a response related to the reflectance (or transmittance) averaged over the random variables. The mean spacer thickness is h L .

Fig. 3
Fig. 3

Spectral absorption and backscattering coefficients (normalized to thickness d) derived from reflectance and transmittance measurements of the brown, red, and yellow chromatophores of the squid Loligo pealeii. (a) Absorption ( K ) spectra. (b) Backscattering ( S ) spectra.

Fig. 4
Fig. 4

Examples of model fits to specific chromatophore reflectance measurements of the squid Loligo pealeii obtained by using K and S values in Fig. 3 and adjusting the scaling parameter ρ.

Fig. 5
Fig. 5

Examples of model fits to specific chromatophore transmittance measurements of the squid Loligo pealeii obtained by using K and S values in Fig. 3 and adjusting the scaling parameter ρ. These data are not from the same samples given in Fig. 4.

Fig. 6
Fig. 6

Fit of iridophore model to reflectance data from the red dorsal iridophore of the squid Loligo pealeii. The dotted curve gives the calculated spectrum of an ideal thin-film stack (Bragg reflector), while the dashed curve illustrates the effect of averaging over an ensemble of Bragg reflectors with randomly variable mean spacer thickness. The solid curve shows the effects of additionally averaging over individual random spacing of each spacer layer and provides a good fit to the data (circles). The inset shows the dependence of bandwidth Δ λ on σ 0 and the ratio of out-of-band to in-band integrated reflectance on σ ζ .

Fig. 7
Fig. 7

Theoretical chromaticity coordinates for the iridophore of Fig. 6 for various values of the statistical parameters σ 0 and σ ζ . Also shown is the CIE 1931 chromaticity diagram. CIE standard daylight illuminant D65 was used for these calculations. Circles represent various σ 0 values, ranging from 0 to 12 nm , for a constant σ ζ = 2.4 nm . Squares represent various σ ζ values, ranging from 0 to 4.8 nm , for a constant σ 0 = 7.6 nm .

Fig. 8
Fig. 8

Theoretical reflectance plots and experimental data for an active iridophore treated with acetylcholine (ACh). The points show the measured spectra at various times after application of ACh. The curves are fits to the data obtained from a dynamic iridophore model by adjusting the volume fraction f of water in the platelet regions of the iridophore.

Fig. 9
Fig. 9

Calculated reflection spectra for a red chromatophore, a blue iridophore, and the combination of a red chromatophore over a blue iridophore. The chromatophore and total reflectance spectra have been multiplied by a factor of 5 for ease of viewing.

Fig. 10
Fig. 10

Theoretical chromaticity coordinates for a blue iridophore overlaid with a red chromatophore illustrating color changes for various values of ρ and f. Also shown is the CIE 1931 chromaticity diagram. CIE standard daylight illuminant D65 was used for these calculations. Circles represent various ρ values, ranging from 0.5 to 3, for a constant f = 0.3 . The leftmost circle represents a bare iridophore. Squares represent various f values, ranging from 0.3 to 0.9, for a constant ρ = 1.8 .

Equations (48)

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I c = ( α + σ ) I c ,
I d = η α I d η ( 1 ξ ) σ I d + η ( 1 ξ ) σ J d + ξ σ I c ,
J d = η α J d + η ( 1 ξ ) σ J d η ( 1 ξ ) σ I d ( 1 ξ ) σ I c ,
T cc = exp [ ( α + σ ) d ] ,
T dd = κ κ cosh η κ d + [ α + ( 1 ξ ) σ ] sinh η κ d ,
R dd = ( 1 ξ ) σ sinh η κ d κ cosh η κ d + [ α + ( 1 ξ ) σ ] sinh η κ d ,
κ = α 2 + 2 ( 1 ξ ) σ α .
T cd κ κ cosh κ d + [ α + ( 1 ξ ) σ ] sinh κ d exp [ ( α + σ ) d ] ,
R cd ( 1 ξ ) σ sinh κ d κ cosh κ d + [ α + ( 1 ξ ) σ ] sinh κ d .
a = 1 + R cd 2 T 2 2 R cd ,
b = a 2 1 .
S ( 1 ξ ) σ = 1 b d sinh 1 ( b R cd T ) ,
K α = ( 1 a ) S .
( E 0 E r ) = P ( E t 0 ) .
R mf = E r E 0 2 = P 21 P 11 2 ,
T mf = n s cos θ s n 0 cos θ 0 E t E 0 2 = n s cos θ s n 0 cos θ 0 1 P 11 2 ,
M = Φ H F H L Φ L F L H .
Φ k = [ exp ( i 2 π n k h k cos θ k λ ) 0 0 exp ( + i 2 π n k h k cos θ k λ ) ] ,
F j k = 1 t j k [ 1 r j k r j k 1 ] ,
n H ( λ ) = f n w ( λ ) + ( 1 f ) n r ( λ ) ,
h H = h 0 + d h d f ( f f 0 ) .
D k = [ exp ( i δ φ k ) 0 0 exp ( + i δ φ k ) ] ,
T cc = T mf ( ζ 0 , { ζ k } ) p ( ζ 0 ) p ( ζ 1 ) p ( ζ 2 ) p ( ζ N ) d ζ 0 d ζ 1 d ζ 2 d ζ N ,
T dd ( λ ) = I λ , t I λ , 0 = 0 π 2 T cc , u ( λ , θ ) sin 2 θ d θ .
I c ( z ) = I c , 0 exp [ ( α + σ ) z ] .
I d η 2 κ 2 I d = β I I c ,
β I = σ { ξ ( α + σ ) + η [ ξ α + ( 1 ξ ) σ ] } .
J d η 2 κ 2 J d = β J I c ,
β J = σ ( η 1 ) ( 1 ξ ) ( α + σ ) .
I d = A 1 e η κ z + A 2 e η κ z + β I I c ( η κ ) 2 ( α + σ ) 2 ,
J d = B 1 e η κ z + B 2 e η κ z + β J I c ( η κ ) 2 ( α + σ ) 2 ,
R d = J d , 0 I c , 0 + I d , 0 ( 1 q ) R dd + q R cd ,
R cd = b 1 cosh η κ d b 2 sinh η κ d η [ ( η κ ) 2 ( α + σ ) 2 ] { κ cosh η κ d + [ α + ( 1 ξ ) σ ] sinh η κ d } ,
b 1 = η β J { κ [ κ + α + ( 1 ξ ) σ ] e ( α + σ η κ ) d } ,
b 2 = { β J [ α + σ η ( κ + α + ( 1 ξ ) σ ) e ( α + σ η κ ) d ] ( 1 ξ ) σ [ ( η κ ) 2 ( α + σ ) 2 ] } .
T d = I d ( d ) I c , 0 + I d , 0 ( 1 q ) T dd + q T cd ,
T cd = a 1 sinh η κ d a 2 cosh η κ d + a 3 sinh 2 η κ d + β I e ( α + σ ) d ( η κ ) 2 ( α + σ ) 2 ,
a 1 = ξ σ [ ( η κ ) 2 ( α + σ ) 2 ] + β I ( α + σ ) η κ [ ( η κ ) 2 ( α + σ ) 2 ] + β J ( 1 ξ ) σ κ [ ( η κ ) 2 ( α + σ ) 2 ] × { 1 e ( α + σ η κ ) d [ { α + σ η κ e ( α + σ η κ ) d + η [ α + ( 1 ξ ) σ ] ( 1 e ( α + σ η κ ) d ) } sinh η κ d η κ cosh η κ d + η [ α + ( 1 ξ ) σ ] sinh η κ d ] } ,
a 2 = β I ( η κ ) 2 ( α + σ ) 2 ,
a 3 = ( 1 ξ ) 2 σ 2 η κ cosh η κ d + η [ α + ( 1 ξ ) σ ] sinh η κ d .
T mf ( ζ 0 , { ζ k } ) = T 0 + 2 Re ( τ 0 * Δ τ + Δ τ * Δ τ ) ,
Δ τ ( ζ 0 , { ζ k } ) = k = 1 N m = 1 1 m ! ( m Δ τ ζ k m ) { ζ k } = 0 ζ k m .
m ζ k m P ( { ζ k } ) = F L H Φ H F H L D ( ζ 1 ) Φ L F L H Φ H F H L m ζ k m D ( ζ k ) Φ L F L H Φ H F H L D ( ζ N ) Φ L
m D ζ k m ζ k = 0 = [ ( i 2 π n L λ ) m 0 0 ( + i 2 π n L λ ) m ] .
T cc = T mf = T 0 ¯ + 2 Re ( τ 0 * Δ τ ¯ ) ,
T 0 ¯ = T ( ζ 0 ) p ( ζ 0 ) d ζ 0 ,
τ 0 * Δ τ ¯ = 1 2 k = 1 N τ 0 * ( ζ 0 ) 2 Δ τ ( ζ 0 , { ζ k } ) ζ k 2 { ζ k } = 0 σ ζ 2 p ( ζ 0 ) d ζ 0 ,
p ( ζ 0 ) = 1 2 π σ 0 exp ( ζ 0 2 2 σ 0 2 ) .

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