Abstract

The measurement of the bonded area of pulp fibers has been an unsolved issue in paper science for more than 40 years. By the use of an established pulp fiber model, and a 4×4 transfer matrix formalism we simulated the optical behavior of pulp fibers in a modified imaging ellipsometer, and we demonstrate that there are rather strong symmetries in the ellipsometric angles Ψ and Δ when comparing single fibers, unbonded fiber crossings, and fiber–fiber bonds. Based on these symmetries we propose and test an algorithm that allows to distinguish the three cases (single fibers, unbonded fiber crossings, and fiber–fiber bonds) in the analysis of ellipsometric data.

© 2012 Optical Society of America

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References

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  1. T. Lindström, L. Wågberg, and T. Larsson, “On the nature of joint strength in paper—a review of dry and wet strength resins used in paper manufacturing,” in 13th Fundamental Research Symposium (Cambridge, 2005), pp. 457–562.
  2. C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).
  3. M. Donoser, M. Wiltsche, and H. Bischof, “A new automated microtomy concept for 3D paper structure analysis,” in Proceedings of the 9th IAPR Conference on Machine Vision Applications (Machine Vision and Application Organization, 2005), pp. 76–79.
  4. M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
    [CrossRef]
  5. G. Jayme and G. Hunger, “Electron microscope 2- and 3-dimensional classification of fibre bonding, formation and structure of paper,” in Transactions of the 2nd Fundamental Research Symposium Oxford (Pulp and Paper Fundamental Research Society, 1961), pp. 135–170.
  6. A. Torgnysdotter, “The link between the fiber contact zone and the physical properties of paper: a way to control paper properties,” J. Compos. Mater. 41, 1619–1633 (2007).
    [CrossRef]
  7. C. I. Thomson, “Probing the nature of cellulosic fiber interfaces with fluorescence resonance energy transfer,” Ph.D. thesis (School of Chemistry and Biochemistry, Georgia Institute of Technology2007).
  8. D. H. Page, “Fibre-to-fibre bonds, part 1—a method for their direct observation,” Paper Technol. 1, 407–411 (1960).
  9. O. Bestsense and C. Ye, “Method and device for determining the orientation angle of the optical axis and the relative phase retardation of a birefingent specimen,” International Patent WO 96/10168 (1995).
  10. E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
    [CrossRef]
  11. L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
    [CrossRef]
  12. L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
    [CrossRef]
  13. H. Sixta, ed., Handbook of Pulp (Wiley-VCH Weinheim, 2006), Vol. 1.
  14. H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).
  15. P. Viitaharju and K. Niskanen, “Chiral curl in thin papers,” J. Pulp Pap. Sci. 20, J148–J152 (1994).
  16. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510(1972).
    [CrossRef]
  17. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
    [CrossRef]
  18. M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures. Phonons, Plasmons and Polaritons (Springer, 2004).
  19. H. Fujiwara, Spectroscopic Ellipsometry—Principles and Applications (Wiley, 2007).
  20. P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express 18, 23095–23103 (2010).
    [CrossRef]

2011 (1)

M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
[CrossRef]

2010 (3)

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express 18, 23095–23103 (2010).
[CrossRef]

2009 (1)

E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
[CrossRef]

2007 (1)

A. Torgnysdotter, “The link between the fiber contact zone and the physical properties of paper: a way to control paper properties,” J. Compos. Mater. 41, 1619–1633 (2007).
[CrossRef]

2002 (1)

H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).

1996 (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

1994 (1)

P. Viitaharju and K. Niskanen, “Chiral curl in thin papers,” J. Pulp Pap. Sci. 20, J148–J152 (1994).

1978 (1)

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

1972 (1)

1960 (1)

D. H. Page, “Fibre-to-fibre bonds, part 1—a method for their direct observation,” Paper Technol. 1, 407–411 (1960).

Aas, L. M. S.

Bauer, W.

M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

Berreman, D. W.

Bestsense, O.

O. Bestsense and C. Ye, “Method and device for determining the orientation angle of the optical axis and the relative phase retardation of a birefingent specimen,” International Patent WO 96/10168 (1995).

Bischof, H.

M. Donoser, M. Wiltsche, and H. Bischof, “A new automated microtomy concept for 3D paper structure analysis,” in Proceedings of the 9th IAPR Conference on Machine Vision Applications (Machine Vision and Application Organization, 2005), pp. 76–79.

Donoser, M.

M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
[CrossRef]

M. Donoser, M. Wiltsche, and H. Bischof, “A new automated microtomy concept for 3D paper structure analysis,” in Proceedings of the 9th IAPR Conference on Machine Vision Applications (Machine Vision and Application Organization, 2005), pp. 76–79.

Ellingsen, P. G.

Eusufzai, A. R. K.

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry—Principles and Applications (Wiley, 2007).

Gilli, E.

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
[CrossRef]

Hirn, U.

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
[CrossRef]

Hunger, G.

G. Jayme and G. Hunger, “Electron microscope 2- and 3-dimensional classification of fibre bonding, formation and structure of paper,” in Transactions of the 2nd Fundamental Research Symposium Oxford (Pulp and Paper Fundamental Research Society, 1961), pp. 135–170.

Jang, H.

H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).

Jayme, G.

G. Jayme and G. Hunger, “Electron microscope 2- and 3-dimensional classification of fibre bonding, formation and structure of paper,” in Transactions of the 2nd Fundamental Research Symposium Oxford (Pulp and Paper Fundamental Research Society, 1961), pp. 135–170.

Kappel, L.

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
[CrossRef]

Kildemo, M.

Kritzinger, J.

M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
[CrossRef]

Larsson, T.

T. Lindström, L. Wågberg, and T. Larsson, “On the nature of joint strength in paper—a review of dry and wet strength resins used in paper manufacturing,” in 13th Fundamental Research Symposium (Cambridge, 2005), pp. 457–562.

Letnes, P. A.

Lindström, T.

T. Lindström, L. Wågberg, and T. Larsson, “On the nature of joint strength in paper—a review of dry and wet strength resins used in paper manufacturing,” in 13th Fundamental Research Symposium (Cambridge, 2005), pp. 457–562.

Mark, R. E.

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

Nerbø, I. S.

Niskanen, K.

P. Viitaharju and K. Niskanen, “Chiral curl in thin papers,” J. Pulp Pap. Sci. 20, J148–J152 (1994).

Page, D. H.

D. H. Page, “Fibre-to-fibre bonds, part 1—a method for their direct observation,” Paper Technol. 1, 407–411 (1960).

Perkins, R. W.

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

Sankar, R.

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

Schennach, R.

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
[CrossRef]

Schubert, M.

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures. Phonons, Plasmons and Polaritons (Springer, 2004).

Seth, R.

H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).

Thomson, C. I.

C. I. Thomson, “Probing the nature of cellulosic fiber interfaces with fluorescence resonance energy transfer,” Ph.D. thesis (School of Chemistry and Biochemistry, Georgia Institute of Technology2007).

Torgnysdotter, A.

A. Torgnysdotter, “The link between the fiber contact zone and the physical properties of paper: a way to control paper properties,” J. Compos. Mater. 41, 1619–1633 (2007).
[CrossRef]

Viitaharju, P.

P. Viitaharju and K. Niskanen, “Chiral curl in thin papers,” J. Pulp Pap. Sci. 20, J148–J152 (1994).

Wågberg, L.

T. Lindström, L. Wågberg, and T. Larsson, “On the nature of joint strength in paper—a review of dry and wet strength resins used in paper manufacturing,” in 13th Fundamental Research Symposium (Cambridge, 2005), pp. 457–562.

Weigel, G.

H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).

Wiltsche, M.

M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
[CrossRef]

M. Donoser, M. Wiltsche, and H. Bischof, “A new automated microtomy concept for 3D paper structure analysis,” in Proceedings of the 9th IAPR Conference on Machine Vision Applications (Machine Vision and Application Organization, 2005), pp. 76–79.

Wu, C.

H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).

Yang, C.

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

Ye, C.

O. Bestsense and C. Ye, “Method and device for determining the orientation angle of the optical axis and the relative phase retardation of a birefingent specimen,” International Patent WO 96/10168 (1995).

Compos. Interfaces (1)

E. Gilli, L. Kappel, U. Hirn, and R. Schennach, “An optical model for polarization microscopy analysis of pulp fibre-to-fibre bonds,” Compos. Interfaces 16, 901–922(2009).
[CrossRef]

J. Compos. Mater. (1)

A. Torgnysdotter, “The link between the fiber contact zone and the physical properties of paper: a way to control paper properties,” J. Compos. Mater. 41, 1619–1633 (2007).
[CrossRef]

J. Microsc. (1)

M. Wiltsche, M. Donoser, J. Kritzinger, and W. Bauer, “Automated serial sectioning applied to 3D paper structure analysis,” J. Microsc. 242, 197–205 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Pulp Pap. Sci. (1)

P. Viitaharju and K. Niskanen, “Chiral curl in thin papers,” J. Pulp Pap. Sci. 20, J148–J152 (1994).

Nordic Pulp Pap. Res. J. (2)

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part I: theoretical fundamentals,” Nordic Pulp Pap. Res. J. 25, 65–70 (2010).
[CrossRef]

L. Kappel, U. Hirn, E. Gilli, W. Bauer, and R. Schennach, “Revisiting polarized light microscopy for fiber–fiber bond area measurement—part II: proving the applicability,” Nordic Pulp Pap. Res. J. 25, 71–75 (2010).
[CrossRef]

Opt. Express (1)

Pap. Puu (1)

H. Jang, G. Weigel, R. Seth, and C. Wu, “The effect of fibril angle on the transverse collapse of papermaking fibers,” Pap. Puu 84, 112–115 (2002).

Paper Technol. (1)

D. H. Page, “Fibre-to-fibre bonds, part 1—a method for their direct observation,” Paper Technol. 1, 407–411 (1960).

Phys. Rev. B (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Sven. Papperstidn. (1)

C. Yang, A. R. K. Eusufzai, R. Sankar, R. E. Mark, and R. W. Perkins, “Measurements of geometrical parameters of fiber networks, part 1. bonded surfaces, aspect ratios, fiber moments of inertia, bonding state probabilities,” Sven. Papperstidn. 13, 426–433 (1978).

Other (8)

M. Donoser, M. Wiltsche, and H. Bischof, “A new automated microtomy concept for 3D paper structure analysis,” in Proceedings of the 9th IAPR Conference on Machine Vision Applications (Machine Vision and Application Organization, 2005), pp. 76–79.

G. Jayme and G. Hunger, “Electron microscope 2- and 3-dimensional classification of fibre bonding, formation and structure of paper,” in Transactions of the 2nd Fundamental Research Symposium Oxford (Pulp and Paper Fundamental Research Society, 1961), pp. 135–170.

O. Bestsense and C. Ye, “Method and device for determining the orientation angle of the optical axis and the relative phase retardation of a birefingent specimen,” International Patent WO 96/10168 (1995).

C. I. Thomson, “Probing the nature of cellulosic fiber interfaces with fluorescence resonance energy transfer,” Ph.D. thesis (School of Chemistry and Biochemistry, Georgia Institute of Technology2007).

M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures. Phonons, Plasmons and Polaritons (Springer, 2004).

H. Fujiwara, Spectroscopic Ellipsometry—Principles and Applications (Wiley, 2007).

H. Sixta, ed., Handbook of Pulp (Wiley-VCH Weinheim, 2006), Vol. 1.

T. Lindström, L. Wågberg, and T. Larsson, “On the nature of joint strength in paper—a review of dry and wet strength resins used in paper manufacturing,” in 13th Fundamental Research Symposium (Cambridge, 2005), pp. 457–562.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic drawing of the s2 layer of a pulp fiber (left), and the layer stack used to simulate the fiber. Hatching indicates the direction of the microfibrils, which can be identified with the slow optical axis. The dashed line indicates the principal fiber axis.

Fig. 2.
Fig. 2.

Schematic drawing of the fiber crossing model, with all adjustable parameters of the simulation. ϑ, rotation of the system (both fibers) with respect to the coordinate system; F1, F2, microfibrillar angle of fiber 1 and 2, respectively; C, crossing angle between the two fibers; d1, d2, wall thickness of fiber 1 and 2, respectively; dist, distance between the two fibers; Φa, angle of incidence; and λ0, wavelength of the incident light.

Fig. 3.
Fig. 3.

Schematic image of the optical setup in two versions: A: The sample is rotated, B: The relevant optical components are rotated (see text).

Fig. 4.
Fig. 4.

Contour plot of a two-dimensional histogram of ellipsometric angles for single fibers (left-hand frames), unbonded fiber crossings (center frames), and fiber bonds (right-hand frames). Top frames: Ellipsometric angle Ψ, bottom frames: ellipsometric angle Δ.

Fig. 5.
Fig. 5.

Contour plot of a two-dimensional histogram of ellipsometric angles Δ for single fibers (left-hand frames), unbonded fiber crossings (center frames), and fiber bonds (right-hand frames).

Tables (5)

Tables Icon

Table 1. Simulation Parameters Used to Study the Symmetries of the System

Tables Icon

Table 2. Simulation Parameters Used to Simulate the Measurement with the Algorithm, Using Noise-Free Data

Tables Icon

Table 3. Detection Rate for n=3112960 Simulated Measurements, Using Noise-Free Simulation Data

Tables Icon

Table 4. Simulation Parameters as Used, to Simulate the Measurement with the Algorithm, Using Noisy Data with a Noise Level of 0.5% in Ψ and 3.5% in Δ

Tables Icon

Table 5. Detection Rates for n=4515840 Simulated Fiber Configurations, Using Noisy Simulation Data

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

ϵs2(0,0,0)=(nfast2000nslow2000nfast2),
nfast=1.53092,
nslow=1.63985.
Rz(α)=(cosα-sinα0sinαcosα0001)
Rx(α)=(1000cosαsinα0sinαcosα),
REuler(Ψ,Θ,Φ)=Rz(Ψ)Rx(Θ)Rz(Φ),
ϵs2(Ψ,Θ,Φ)=REuler(Ψ,Θ,Φ)ϵs2(0,0,0)REuler1(Ψ,Θ,Φ).
(AsBsApBp)=T(CsDsCpDp)=(T11T12T13T14T21T22T23T24T31T32T33T34T41T42T43T44)(Cs0Cp0),
JPr=(rpprpsrsprss),
rpp=T11T43-T13T41T11T33-T13T31,
rsp=T11T23-T13T21T11T33-T13T31,
rps=T33T41-T31T43T11T33-T13T31,
rss=T33T21-T31T23T11T33-T13T31.
JA=Jλ/4(0)·JBS(0)·JPr(ϑ)·JP(0),
JB=Jλ/4(ϑ)·JM(0)·JBS(0)·JPr(0)·JP(ϑ),
JP=(1000)polarizer,
Jλ/4=(i001)λ/4-phase retarder,
JM=(1001)nonpolarizing mirror,
JBS=(0.5000.5)=0.5·JMnonpolarizing beam splitter.
JA=(i001)(0.5000.5)(rpprpsrsprss)(1000)=0.5(irpp0rsp0),
JB=R(ϑ)Jλ/4(0)R(ϑ)(1001)(0.5000.5)=0.5·1JPr(0)R(ϑ)JP(0)R(ϑ)=0.5R(ϑ)Jλ/4(0)R(ϑ)JPr(0)R(-ϑ)JP(0)R(ϑ)=0.5R(-ϑ)Jλ/4(0)JPr(ϑ)JP(0)R(ϑ)=0.5R(-ϑ)(irpp0rsp0)R(ϑ),
R(α)=(cosα-sinαsinαcosα).
ρ^=rprs=tanΨeiΔ,
ρ^A=irpprsp,ρ^B(ϑ)=irpprsp,
Ψ=arctan|ρ^|,
Δ=-argρ^=ln(Im(ρ^)).
ρ^B(ϑ)=-ρ^A(0)=ρ^A(0)e±iπ=tanΨei(Δ±π),ΨB=ΨA,ΔB=ΔA±180°.
Ψcrit=85.5°.
Δcrit+=106.5°,
Δcrit=73.5°.
min[Ψ(ϑ)]=Ψ(ϑ0),
Ψ(ϑ0){<Ψcritfiberfiber bond>Ψcrit{Δ(ϑ0){>0Δcrit=Δcrit+<0Δcrit=ΔcritΔ(ϑ0){>Δcritsingle fiber<Δcritunbonded crossing.
δΔδΨ6.75,

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