Abstract

We present a novel way to account for partially coherent interference in multilayer systems via the transfer-matrix method. The novel feature is that there is no need to use modified Fresnel coefficients or the square of their amplitudes to work in the incoherent limit. The transition from coherent to incoherent interference is achieved by introducing a random phase of increasing intensity in the propagating media. This random phase can simulate the effect of defects or impurities. This method provides a general way of dealing with optical multilayer systems, in which coherent and incoherent interference are treated on equal footing.

© 2010 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. Oladeji and L. Chow, “Synthesis and processing of CdS/ZnS multilayer films for solar cell application,” Thin Solid Films 474(1-2), 77–83 (2005).
    [CrossRef]
  2. D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855 (2007).
    [CrossRef]
  3. R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
    [CrossRef]
  4. H. Cho, C. Yun, and S. Yoo, “Multilayer transparent electrode for organic light-emitting diodes: tuning its optical characteristics,” Opt. Express 18(4), 3404–3414 (2010).
    [CrossRef] [PubMed]
  5. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
    [CrossRef] [PubMed]
  6. W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
    [CrossRef] [PubMed]
  7. X. L. Ruan and M. Kaviany, “Photon localization and electromagnetic field enhancement in laser-irradiated, random porous media,” Microscale Thermophys. Eng. 9(1), 63–84 (2005).
    [CrossRef]
  8. C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
    [CrossRef]
  9. X. L. Ruan and M. Kaviany, “Enhanced nonradiative relaxation and photoluminescence quenching in random, doped nanocrystalline powders,” J. Appl. Phys. 97(10), 104331 (2005).
    [CrossRef]
  10. S. Logothetidis and G. Stergioudis, “Studies of density and surface roughness of ultrathin amorphous carbon films with regards to thickness with x-ray reflectometry and spectroscopic ellipsometry,” Appl. Phys. Lett. 71(17), 2463 (1997).
    [CrossRef]
  11. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  12. Z. Knittl, Optics of Thin Films: An Optical Multilayer Theory (Wiley, London, 1976).
  13. O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965).
  14. K. Ohta and H. Ishida, “Matrix formalism for calculation of electric field intensity of light in stratified multilayered films,” Appl. Opt. 29(13), 1952–1959 (1990).
    [CrossRef] [PubMed]
  15. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. 44(35), 7532–7539 (2005).
    [CrossRef] [PubMed]
  16. C. L. Mitsas and D. I. Siapkas, “Generalized matrix method for analysis of coherent and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates,” Appl. Opt. 34(10), 1678 (1995).
    [CrossRef] [PubMed]
  17. J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys. 33(24), 3139–3145 (2000).
    [CrossRef]
  18. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002).
    [CrossRef] [PubMed]
  19. J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
    [CrossRef] [PubMed]
  20. S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).
    [CrossRef]

2010 (3)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

H. Cho, C. Yun, and S. Yoo, “Multilayer transparent electrode for organic light-emitting diodes: tuning its optical characteristics,” Opt. Express 18(4), 3404–3414 (2010).
[CrossRef] [PubMed]

2008 (1)

R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
[CrossRef]

2007 (1)

D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855 (2007).
[CrossRef]

2006 (1)

J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
[CrossRef] [PubMed]

2005 (4)

E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. 44(35), 7532–7539 (2005).
[CrossRef] [PubMed]

O. Oladeji and L. Chow, “Synthesis and processing of CdS/ZnS multilayer films for solar cell application,” Thin Solid Films 474(1-2), 77–83 (2005).
[CrossRef]

X. L. Ruan and M. Kaviany, “Enhanced nonradiative relaxation and photoluminescence quenching in random, doped nanocrystalline powders,” J. Appl. Phys. 97(10), 104331 (2005).
[CrossRef]

X. L. Ruan and M. Kaviany, “Photon localization and electromagnetic field enhancement in laser-irradiated, random porous media,” Microscale Thermophys. Eng. 9(1), 63–84 (2005).
[CrossRef]

2004 (1)

S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).
[CrossRef]

2002 (1)

2000 (1)

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys. 33(24), 3139–3145 (2000).
[CrossRef]

1997 (1)

S. Logothetidis and G. Stergioudis, “Studies of density and surface roughness of ultrathin amorphous carbon films with regards to thickness with x-ray reflectometry and spectroscopic ellipsometry,” Appl. Phys. Lett. 71(17), 2463 (1997).
[CrossRef]

1995 (2)

C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
[CrossRef]

C. L. Mitsas and D. I. Siapkas, “Generalized matrix method for analysis of coherent and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates,” Appl. Opt. 34(10), 1678 (1995).
[CrossRef] [PubMed]

1990 (1)

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Caruso, F.

J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
[CrossRef] [PubMed]

Centurioni, E.

Char, K.

J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
[CrossRef] [PubMed]

Chen, S.

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

Cho, H.

Cho, J.

J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
[CrossRef] [PubMed]

Chow, L.

O. Oladeji and L. Chow, “Synthesis and processing of CdS/ZnS multilayer films for solar cell application,” Thin Solid Films 474(1-2), 77–83 (2005).
[CrossRef]

Ghosh, K.

R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
[CrossRef]

Gupta, R. K.

R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
[CrossRef]

Hatzopoulos, N.

C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
[CrossRef]

Hong, J.

J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
[CrossRef] [PubMed]

Huang, J. L.

D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855 (2007).
[CrossRef]

Ishida, H.

Kahol, P. K.

R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
[CrossRef]

Katsidis, C. C.

C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002).
[CrossRef] [PubMed]

C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
[CrossRef]

Kaviany, M.

X. L. Ruan and M. Kaviany, “Enhanced nonradiative relaxation and photoluminescence quenching in random, doped nanocrystalline powders,” J. Appl. Phys. 97(10), 104331 (2005).
[CrossRef]

X. L. Ruan and M. Kaviany, “Photon localization and electromagnetic field enhancement in laser-irradiated, random porous media,” Microscale Thermophys. Eng. 9(1), 63–84 (2005).
[CrossRef]

Lin, S. Y.

D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855 (2007).
[CrossRef]

Livingstone, J.

S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).
[CrossRef]

Logothetidis, S.

S. Logothetidis and G. Stergioudis, “Studies of density and surface roughness of ultrathin amorphous carbon films with regards to thickness with x-ray reflectometry and spectroscopic ellipsometry,” Appl. Phys. Lett. 71(17), 2463 (1997).
[CrossRef]

Lu, Y.

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

Lund, C. P.

S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).
[CrossRef]

Mitsas, C. L.

Ohta, K.

Oladeji, O.

O. Oladeji and L. Chow, “Synthesis and processing of CdS/ZnS multilayer films for solar cell application,” Thin Solid Films 474(1-2), 77–83 (2005).
[CrossRef]

Panknin, D.

C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
[CrossRef]

Patel, R.

R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
[CrossRef]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Prentice, J. S. C.

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys. 33(24), 3139–3145 (2000).
[CrossRef]

Reinhardt, K.

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

Rowlands, S. F.

S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).
[CrossRef]

Ruan, X. L.

X. L. Ruan and M. Kaviany, “Photon localization and electromagnetic field enhancement in laser-irradiated, random porous media,” Microscale Thermophys. Eng. 9(1), 63–84 (2005).
[CrossRef]

X. L. Ruan and M. Kaviany, “Enhanced nonradiative relaxation and photoluminescence quenching in random, doped nanocrystalline powders,” J. Appl. Phys. 97(10), 104331 (2005).
[CrossRef]

Sahu, D. R.

D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855 (2007).
[CrossRef]

Siapkas, D. I.

Skorupa, W.

C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
[CrossRef]

Stergioudis, G.

S. Logothetidis and G. Stergioudis, “Studies of density and surface roughness of ultrathin amorphous carbon films with regards to thickness with x-ray reflectometry and spectroscopic ellipsometry,” Appl. Phys. Lett. 71(17), 2463 (1997).
[CrossRef]

Wang, W.

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

Wu, S.

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

Yoo, S.

Yun, C.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

S. Logothetidis and G. Stergioudis, “Studies of density and surface roughness of ultrathin amorphous carbon films with regards to thickness with x-ray reflectometry and spectroscopic ellipsometry,” Appl. Phys. Lett. 71(17), 2463 (1997).
[CrossRef]

J. Am. Chem. Soc. (1)

J. Cho, J. Hong, K. Char, and F. Caruso, “Nanoporous block copolymer micelle/micelle multilayer films with dual optical properties,” J. Am. Chem. Soc. 128(30), 9935–9942 (2006).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

X. L. Ruan and M. Kaviany, “Enhanced nonradiative relaxation and photoluminescence quenching in random, doped nanocrystalline powders,” J. Appl. Phys. 97(10), 104331 (2005).
[CrossRef]

J. Phys. D Appl. Phys. (2)

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys. 33(24), 3139–3145 (2000).
[CrossRef]

R. K. Gupta, K. Ghosh, R. Patel, and P. K. Kahol, “Properties of ZnO/W-doped In2O3/ZnO multilayer thin films deposited at different growth temperatures,” J. Phys. D Appl. Phys. 41(21), 215309 (2008).
[CrossRef]

Microelectron. Eng. (1)

C. C. Katsidis, D. I. Siapkas, D. Panknin, N. Hatzopoulos, and W. Skorupa, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent,and incoherent interference,” Microelectron. Eng. 28, 439 (1995).
[CrossRef]

Microscale Thermophys. Eng. (1)

X. L. Ruan and M. Kaviany, “Photon localization and electromagnetic field enhancement in laser-irradiated, random porous media,” Microscale Thermophys. Eng. 9(1), 63–84 (2005).
[CrossRef]

Nano Lett. (1)

W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10(6), 2012–2018 (2010).
[CrossRef] [PubMed]

Nat. Mater. (1)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef] [PubMed]

Opt. Express (1)

Sol. Energy (1)

S. F. Rowlands, J. Livingstone, and C. P. Lund, “Optical modelling of thin film solar cells with textured interfaces using the effective medium approximation,” Sol. Energy 76(1-3), 301–307 (2004).
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

D. R. Sahu, S. Y. Lin, and J. L. Huang, “Deposition of Ag-based Al-doped ZnO multilayer coatings for the transparent conductive electrodes by electron beam evaporation,” Sol. Energy Mater. Sol. Cells 91(9), 851–855 (2007).
[CrossRef]

Thin Solid Films (1)

O. Oladeji and L. Chow, “Synthesis and processing of CdS/ZnS multilayer films for solar cell application,” Thin Solid Films 474(1-2), 77–83 (2005).
[CrossRef]

Other (3)

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Z. Knittl, Optics of Thin Films: An Optical Multilayer Theory (Wiley, London, 1976).

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Multilayer system composed of N layers with complex refractive indices, ni, and N + 1 interfaces. The sign + and - on the electric field amplitudes, Ei, indicate left- and right-going waves, respectively. The prime is used for waves at the right hand side of an interface.

Fig. 2
Fig. 2

Transmittance vs. wavelength for a crystalline Si film of 150 nm thickness in the coherent and incoherent limits. The solid red line corresponds to the coherent limit; the solid green line corresponds to the incoherent limit calculated using the squares of the amplitudes of the transmission coefficients; and the dashed blue line corresponds to the incoherent limit calculated using the added random phase with β = π. A filter of 10 moving averages was used to smooth the blue curve.

Fig. 3
Fig. 3

Transmittance vs. wavelength for a crystalline Si film of 150 nm thickness in the coherent and partially coherent regimes. The dash-dotted red curve corresponds to the coherent limit. The solid blue and green curves, which correspond to the partially coherent regime, were calculated using the random phase method with β = π/4 and β = π/3 respectively. As a reference the incoherent limit is also plotted (dashed green curve). The curves corresponding to the partial coherence limit were smoothed using a filter of 5 moving averages.

Fig. 4
Fig. 4

Transmittance vs. wavelength for a two layer system consisting of a ZnO film of 150 nm thickness and a crystalline Si film of 150 nm thickness. The black curve represents the complete incoherent limit; the (dash-dotted) green curve represents a coherent ZnO layer on an incoherent Si layer; the (dashed) blue curve represents a coherent ZnO layer on a partially coherent Si layer; and the solid red curve represents the complete coherent limit. The curves corresponding to the partial coherence and incoherent limit were smoothed using a filter of 5 moving averages (for wavenumbers > 0.6 μm).

Equations (14)

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

( E m 1 + E m 1 ) = I m 1 , m ( E m + E m )
I m 1 , m = 1 t m 1 , m ( 1 r m 1 , m r m 1 , m 1 )
( E m ' + E m ' ) = P m ( E m + E m ) ,
P m = ( e i δ m 0 0 e i δ m ) .
δ m = 2 π σ n m d m cos φ m ,
( E 0 + E 0 ) = I 01 P 1 I 12 P 2 I 23 P N I N ( N + 1 ) ( E N + 1 + E N + 1 ) = ( T 11 T 12 T 21 T 22 ) ( E N + 1 + E N + 1 ) .
T 0 , ( N + 1 ) = I 01 P 1 I 12 P 2 I 23 P N I N ( N + 1 ) ,
r = E 0 E 0 + = T 21 T 11 ,
t = E N + 1 ' + E 0 + = 1 T 11 .
T 0 , ( N + 1 ) = 1 t 0 , N + 1 [ 1 r N + 1 , 0 r 0 , N + 1 t 0 , N + 1 t N + 1 , 0 r 0 , N + 1 r N + 1 , 0 ] .
T 0 , ( N + 1 ) = 1 | t 0 , N + 1 | 2 [ 1 | r N + 1 , 0 | 2 | r 0 , N + 1 | 2 ( | t 0 , N + 1 t N + 1 , 0 | 2 | r 0 , N + 1 r N + 1 , 0 | 2 ) ] .
δ m = 2 π σ n m d m cos φ m + β R a n d ,
δ 1 = 2 π σ n 1 d 1 cos φ 1 + β 1 R a n d 1 ,
δ 2 = 2 π σ n 2 d 2 cos φ 2 + β 2 R a n d 2 .

Metrics