Abstract

We use transfer matrix formalism to investigate periodic multilayer structures including uniaxial anisotropic materials in the specific case where the optical z axis is normal to the layer interfaces. Having decoupled the problem into two independent states of polarization, we first derive a universal transfer matrix for the unit cell of an arbitrary bilayer (z uniaxial–z uniaxial or z uniaxial–isotropic), and then determine the effective indices in the long-wavelength limit through a homogenization procedure. A graphic representation illustrates the analytical results in terms of generalized filling factors. The procedure is also generalized to n-ary stacks.

© 2008 Optical Society of America

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  1. M. de la Fargue and M. Missous, “Design and fabrication of SiO2/TiO2 dielectric Bragg reflectors by RF sputtering: application to near-infrared InGaAs/GaAs/AlGaAs vertical cavity surface emitting lasers,” in 1999 Symposium on High Performance Electron Devices for Microwave and Optoelectronic Applications (EDMO 1999) (IEEE, 1999), pp. 176-181.
    [CrossRef]
  2. P.S.Zory, Jr., ed., Quantum Well Lasers (Academic, 1993).
  3. J. H. Burroughes, C. A. Jones, and R. H. Friend, “New semiconductor device physics in polymer diodes and transistors,” Nature 335, 137-141 (1988).
    [CrossRef]
  4. C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
    [CrossRef]
  5. S. Visnovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006).
  6. M. Iwanaga, “Effective optical constants in stratified metal-dielectric metamaterial,” Opt. Lett. 32, 1314-1316 (2007).
    [CrossRef] [PubMed]
  7. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  8. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  9. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  10. A. J. Abu El-Haija, “Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique,” J. Appl. Phys. 93, 2590-2594 (2003).
    [CrossRef]
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  12. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  13. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  14. C. Vandenbem, J.-P. Vigneron, and J.-M. Vigoureux, “Tunable band structures in uniaxial multilayer stacks,” J. Opt. Soc. Am. B 23, 2366-2376 (2006).
    [CrossRef]
  15. G. Bastard, Wave Mechanics Applied to Semiconductor Hetero-structures (Les Éditions de Physique, Les Ulis, 1992).
  16. D. J. Griffiths and A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137-154 (2001).
    [CrossRef]
  17. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley-Interscience, 2006), Chap. 1.
  18. M. V. Klein and T. E. Furtak, Optics (Wiley, 1986).
  19. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri, and K. Driss-Khodja, “Effective medium equivalent to a symmetric multilayer at oblique incidence,” Appl. Opt. 29, 2460-2465 (1990).
    [CrossRef] [PubMed]
  20. B. F. Levine, “Quantum-well infrared photodetectors,” J. Appl. Phys. 74, R1-R81 (1993).
    [CrossRef]
  21. E. Rosencher and B. Vinter, Optoelectronics (Cambridge U. Press, 2002).
    [CrossRef]
  22. Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

2007 (1)

2006 (2)

C. Vandenbem, J.-P. Vigneron, and J.-M. Vigoureux, “Tunable band structures in uniaxial multilayer stacks,” J. Opt. Soc. Am. B 23, 2366-2376 (2006).
[CrossRef]

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

2003 (2)

A. J. Abu El-Haija, “Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique,” J. Appl. Phys. 93, 2590-2594 (2003).
[CrossRef]

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

2001 (1)

D. J. Griffiths and A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137-154 (2001).
[CrossRef]

1993 (1)

B. F. Levine, “Quantum-well infrared photodetectors,” J. Appl. Phys. 74, R1-R81 (1993).
[CrossRef]

1990 (1)

1988 (1)

J. H. Burroughes, C. A. Jones, and R. H. Friend, “New semiconductor device physics in polymer diodes and transistors,” Nature 335, 137-141 (1988).
[CrossRef]

1979 (1)

Abu El-Haija, A. J.

A. J. Abu El-Haija, “Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique,” J. Appl. Phys. 93, 2590-2594 (2003).
[CrossRef]

Bastard, G.

G. Bastard, Wave Mechanics Applied to Semiconductor Hetero-structures (Les Éditions de Physique, Les Ulis, 1992).

Bichri, A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Boucher, Y. G.

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

Burroughes, J. H.

J. H. Burroughes, C. A. Jones, and R. H. Friend, “New semiconductor device physics in polymer diodes and transistors,” Nature 335, 137-141 (1988).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley-Interscience, 2006), Chap. 1.

Dal Negro, L.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

de la Fargue, M.

M. de la Fargue and M. Missous, “Design and fabrication of SiO2/TiO2 dielectric Bragg reflectors by RF sputtering: application to near-infrared InGaAs/GaAs/AlGaAs vertical cavity surface emitting lasers,” in 1999 Symposium on High Performance Electron Devices for Microwave and Optoelectronic Applications (EDMO 1999) (IEEE, 1999), pp. 176-181.
[CrossRef]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley-Interscience, 2006), Chap. 1.

Driss-Khodja, K.

Friend, R. H.

J. H. Burroughes, C. A. Jones, and R. H. Friend, “New semiconductor device physics in polymer diodes and transistors,” Nature 335, 137-141 (1988).
[CrossRef]

Frigerio, J. M.

Furtak, T. E.

M. V. Klein and T. E. Furtak, Optics (Wiley, 1986).

Gaburro, Z.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths and A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137-154 (2001).
[CrossRef]

Guérineau, N.

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

Haïdar, R.

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

Iwanaga, M.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Johnson, P. J.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Jones, C. A.

J. H. Burroughes, C. A. Jones, and R. H. Friend, “New semiconductor device physics in polymer diodes and transistors,” Nature 335, 137-141 (1988).
[CrossRef]

Klein, M. V.

M. V. Klein and T. E. Furtak, Optics (Wiley, 1986).

Lafait, J.

Lagendijk, A.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Laloë, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley-Interscience, 2006), Chap. 1.

Le Rouzo, J.

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

Levine, B. F.

B. F. Levine, “Quantum-well infrared photodetectors,” J. Appl. Phys. 74, R1-R81 (1993).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Missous, M.

M. de la Fargue and M. Missous, “Design and fabrication of SiO2/TiO2 dielectric Bragg reflectors by RF sputtering: application to near-infrared InGaAs/GaAs/AlGaAs vertical cavity surface emitting lasers,” in 1999 Symposium on High Performance Electron Devices for Microwave and Optoelectronic Applications (EDMO 1999) (IEEE, 1999), pp. 176-181.
[CrossRef]

Oton, C. J.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Pavesi, L.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Ribet, I.

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

Rosencher, E.

E. Rosencher and B. Vinter, Optoelectronics (Cambridge U. Press, 2002).
[CrossRef]

Steinke, A.

D. J. Griffiths and A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137-154 (2001).
[CrossRef]

Vandenbem, C.

Vigneron, J.-P.

Vigoureux, J.-M.

Vinter, B.

E. Rosencher and B. Vinter, Optoelectronics (Cambridge U. Press, 2002).
[CrossRef]

Visnovský, S.

S. Visnovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006).

Wiersma, D. S.

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yamaguchi, T.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742-756 (1979).
[CrossRef]

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

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Am. J. Phys. (1)

D. J. Griffiths and A. Steinke, “Waves in locally periodic media,” Am. J. Phys. 69, 137-154 (2001).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (2)

B. F. Levine, “Quantum-well infrared photodetectors,” J. Appl. Phys. 74, R1-R81 (1993).
[CrossRef]

A. J. Abu El-Haija, “Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique,” J. Appl. Phys. 93, 2590-2594 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

J. Phys. IV (1)

Y. G. Boucher, J. Le Rouzo, I. Ribet, R. Haïdar, and N. Guérineau, “Description matricielle de l'anisotropie de la transition inter-sous-bande d'une structure à multi-puits quantiques,” J. Phys. IV 135, 99-102 (2006) (in French).

Nature (1)

J. H. Burroughes, C. A. Jones, and R. H. Friend, “New semiconductor device physics in polymer diodes and transistors,” Nature 335, 137-141 (1988).
[CrossRef]

Opt. Lett. (1)

Phys. Status Solidi A (1)

C. J. Oton, L. Dal Negro, Z. Gaburro, L. Pavesi, P. J. Johnson, A. Lagendijk, and D. S. Wiersma, “Light propagation in one-dimensional porous silicon complex systems,” Phys. Status Solidi A 197, 298-302 (2003).
[CrossRef]

Other (12)

S. Visnovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006).

M. de la Fargue and M. Missous, “Design and fabrication of SiO2/TiO2 dielectric Bragg reflectors by RF sputtering: application to near-infrared InGaAs/GaAs/AlGaAs vertical cavity surface emitting lasers,” in 1999 Symposium on High Performance Electron Devices for Microwave and Optoelectronic Applications (EDMO 1999) (IEEE, 1999), pp. 176-181.
[CrossRef]

P.S.Zory, Jr., ed., Quantum Well Lasers (Academic, 1993).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

G. Bastard, Wave Mechanics Applied to Semiconductor Hetero-structures (Les Éditions de Physique, Les Ulis, 1992).

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley-Interscience, 2006), Chap. 1.

M. V. Klein and T. E. Furtak, Optics (Wiley, 1986).

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

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

E. Rosencher and B. Vinter, Optoelectronics (Cambridge U. Press, 2002).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Symmetric unit cell made of a medium (2) sandwiched between two layers made of medium (1). The total length is Λ = d 1 + d 2 . The optical axes of the uniaxial layers, if any, are along the z axis.

Fig. 2
Fig. 2

Scheme of a typical isotropic–uniaxial interface, and corresponding notations.

Fig. 3
Fig. 3

Scheme of the index surfaces of a z-uniaxial medium, and corresponding notations. The e-index ε e i ( θ e i ) is defined by the semiaxis values ε o i and ε e i .

Fig. 4
Fig. 4

Direction of the propagation vector k through an isotropic/z-uniaxial stack with the angles corresponding to each medium.

Fig. 5
Fig. 5

Case of an isotropic–negative z-uniaxial stack (a) Behavior of ε o and ε e as functions of the external angle θ (see Fig. 4). (b) Schematic view of the index surface (o sphere and e-quasi-ellipsoid). Values of parameters: f 1 = 0.70 , n 1 = 3.4 , n o 2 = 3.5 , n e 2 = 3 .

Fig. 6
Fig. 6

Case of an isotropic–positive z-uniaxial stack. (a) Behavior of ε o and ε e as functions of the external angle θ (see Fig. 4). (b) Schematic view of the index surface (o sphere and e-quasi-ellipsoid).Values of parameters: f 1 = 0.75 , n 1 = 3.75 , n o 2 = 3 , n e 2 = 3.5 .

Fig. 7
Fig. 7

Scheme of a quaternary structure.

Tables (3)

Tables Icon

Table 1 Generic Notations of the Permittivity in Medium (1)

Tables Icon

Table 2 Summary of the Various Parameters Used in Transfer Matrix Formalism, Along With Correspondence Algebra

Tables Icon

Table 3 Summary of the Various Homogenized Parameters in the Case of Binary Stacks

Equations (50)

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E ( x , y , z , t ) = E ( z ) exp [ j ( ω t k x x k y y ) ] .
E i ( z ) = ( E a i + E a i E b i + E b i ) ,
( E a 1 + E a 1 E b 1 + E b 1 ) = ( [ I a a 1 2 ] [ I a b 1 2 ] [ I b a 1 2 ] [ I b b 1 2 ] ) ( E a 2 + E a 2 E b 2 + E b 2 ) ,
( E u i + ( z ) E u i ( z ) ) = [ P u i ( d ) ] ( E u i + ( z + d ) E u i ( z + d ) ) ,
[ P u i ( d ) ] = ( exp ( + j γ u i d ) 0 0 exp ( j γ u i d ) ) ,
γ s i = γ p i = [ ε i k 0 2 k 2 ] 1 2 ,
γ o i = [ ε o i k 0 2 k 2 ] 1 2 ,
γ e i = [ ε o i k 0 2 ( ε o i ε e i ) k 2 ] 1 2 .
( E u 1 + E u 1 ) z 0 = [ I u u 1 2 ] ( E u 2 + E u 2 ) z 0 + ,
[ I u u 1 2 ] = η 12 2 ( 1 + q 12 σ ( 1 q 12 ) σ ( 1 q 12 ) 1 + q 12 ) .
α 12 = ε e 1 ( θ e 1 ) ε e 2 ( θ e 2 ) 1 + δ 2 1 + δ 1 ,
η 12 = ε e 2 ( θ e 2 ) ε e 1 ( θ e 1 ) cos ( θ e 2 φ e 2 ) cos ( θ e 1 φ e 1 ) ,
1 + δ i = cos φ e i cos θ e i cos ( θ e i φ e i ) .
[ m ] = [ P u 1 ( d 1 2 ) ] [ I u u 1 2 ] [ P u 2 ( d 2 ) ] [ I u u 2 1 ] [ P u 1 ( d 1 2 ) ] ,
m 11 = [ cos ψ u 2 + j 2 ( q 12 + q 21 ) sin ψ u 2 ] exp ( j ψ u 1 ) ,
m 12 = j σ 2 ( q 12 q 21 ) sin ψ u 2 ,
m 21 = j σ 2 ( q 12 q 21 ) sin ψ u 2 ,
m 22 = [ cos ψ u 2 j 2 ( q 12 + q 21 ) sin ψ u 2 ] exp ( j ψ u 1 ) ,
cos ( K u u Λ ) = Tr [ m ] 2 ,
cos ( K u u Λ ) = cos ψ u 1 cos ψ u 2 ( 1 2 ) ( q 12 + q 21 ) sin ψ u 1 sin ψ u 2 .
K u u 2 = f 1 2 γ u 1 2 + f 2 2 γ u 2 2 + f 1 f 2 ( 1 α 12 γ u 1 2 + α 12 γ u 2 2 ) ,
K s s 2 + k 2 = ε ̱ o k 0 2 ,
ε ̱ o = f 1 ε 1 + f 2 ε 2 .
K p p 2 ε ̱ o + k 2 ε ̱ e = k 0 2 ,
ε ̱ e ( θ e ) cos 2 θ e ε ̱ o + ε ̱ e ( θ e ) sin 2 θ e ε ̱ e = 1 ,
( K p p k 0 ) 2 = ε ̱ e ( θ e ) cos 2 ( θ e ) ,
( k k 0 ) 2 = ε ̱ e ( θ e ) sin 2 ( θ e ) ,
( K p p k ) 2 = cotan 2 ( θ e ) .
ε ̱ o = f 1 ε 1 + f 2 ε 2 = ε ̱ o ,
1 ε ̱ e = f 1 ε 1 + f 2 ε 2 .
K o o 2 + k 2 = ε ̱ o k 0 2 ,
ε ̱ o = f 1 ε o 1 + f 2 ε o 2 .
α 12 = ε e 1 ( θ e 1 ) ε e 2 ( θ e 2 ) 1 + δ 2 1 + δ 1 .
K e e 2 ε ̱ o ( θ e ) + k 2 ε ̱ e ( θ e ) = k 0 2 ,
ε ̱ e ( θ e ) cos 2 θ e ε ̱ o ( θ e ) + ε ̱ e ( θ e ) sin 2 θ e ε ̱ e ( θ e ) = 1 .
ε ̱ o ( θ e ) = F 1 ε e 1 ( θ e 1 ) + F 2 ε e 2 ( θ e 2 ) ,
1 ε ̱ e ( θ e ) = G 1 ε e 1 ( θ e 1 ) + G 2 ε e 2 ( θ e 2 ) ,
F i = f i 1 + δ i ( 1 + δ 1 f 1 + δ 2 f 2 ) ,
G i = f i ( 1 + δ i ) 1 + δ 1 f 1 + δ 2 f 2 .
ε ̱ e ( θ e ) < ε ̱ o ( θ e ) .
ε ̱ o = f 1 ε 1 + f 2 ε o 2 ,
ε ̱ o ( θ e ) = F 1 ε 1 + F 2 ε e 2 ( θ e 2 ) ,
1 ε ̱ e ( θ e ) = G 1 ε 1 + G 2 ε e 2 ( θ e 2 ) ,
[ m ] = [ P u 1 ( d 1 2 ) ] { m = 1 n 1 [ I u u m m + 1 ] [ P u ( m + 1 ) ( d m + 1 ) ] } [ I u u n 1 ] [ P u 1 ( d 1 2 ) ] .
K 2 = m = 1 n f m 2 γ u m m 2 + k , l [ 1 , n ] k < l f k f l ( 1 α k l γ u k k 2 + α k l γ u l l 2 ) ,
ε ̱ o = i = 1 n f i ε o i ,
ε ̱ o ( θ e ) = i = 1 n F i ε e i ( θ e i ) ,
1 ε ̱ e ( θ e ) = i = 1 n G i ε e i ( θ e i ) ,
F i = f i 1 + δ i ( 1 + m = 1 n δ m f m ) ,
G i = f i ( 1 + δ i ) 1 + m = 1 n δ m f m .

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