Abstract

We propose a new synthesis method for the design of multilayer optical filters with intermediate refractive indices, the step method. This method consists in adding infinitesimally small index steps in the index profile at optimal positions and then reoptimizing the thickness and the refractive index of the layers. Application of the method to the design of an antireflective coating, a low-pass edge filter, and an immersed polarizing beam splitter shows that it provides interesting solutions, even in the absence of a proper starting design. The formalism developed for the method also serves to demonstrate that the optimal filter consists of either homogeneous layers that maximize the effective refractive index contrast, or of graded-index layers.

© 2008 Optical Society of America

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  1. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417-5426 (1993).
    [CrossRef] [PubMed]
  2. D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
    [CrossRef]
  3. S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
    [CrossRef]
  4. R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
    [CrossRef]
  5. R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
    [CrossRef] [PubMed]
  6. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc. Am. 57, 1529-1533 (1967).
    [CrossRef]
  7. L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” ENSV Tead. Akad. Toimetised Fuus. Mat. 23, 229-237 (1974), English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).
  8. J. A. Dobrowolski and D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039-3050 (1978).
    [CrossRef] [PubMed]
  9. S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436-7441 (2007).
    [CrossRef] [PubMed]
  10. P. G. Verly, “Modified needle method with simultaneous thickness and refractive-index refinement for the synthesis of inhomogeneous and multilayer optical thin films,” Appl. Opt. 40, 5718-5725 (2001).
    [CrossRef]
  11. P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955-958 (1958).
    [CrossRef]
  12. A. V. Tikhonravov, “A method of synthesis of optical coverings which uses the necessary optimality conditions,” Vestn. Mosk. Univ. 3, Fiz. Astron. 37, 91-93 (1982). Translation of Mosc. Univ. Phys. Bull. (USSR) 37, 108-110 (1982).
  13. B. T. Sullivan and J. A. Dobrowolski, “Implementation of a numerical needle method for thin-film design,” Appl. Opt. 35, 5484-5492 (1996).
    [CrossRef] [PubMed]
  14. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493-5508 (1996).
    [CrossRef] [PubMed]
  15. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  16. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).
  17. C. J. v. d. Laan and H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17, 538-541 (1978).
    [CrossRef] [PubMed]
  18. P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis for inhomogeneous optical coatings,” Appl. Opt. 36, 1487-1495 (1997).
    [CrossRef] [PubMed]
  19. S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
    [CrossRef] [PubMed]
  20. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  21. P. Baumeister, “Rudiments of the design of an immersed polarizing beam divider with a narrow spectral bandwidth and enhanced angular acceptance,” Appl. Opt. 36, 3610-3613(1997).
    [CrossRef] [PubMed]
  22. B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. I. Theoretical description,” Appl. Opt. 31, 3821-3835 (1992).
    [CrossRef] [PubMed]
  23. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46, 704-710 (2007).
    [CrossRef] [PubMed]
  24. P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327-7333 (1998).
    [CrossRef]
  25. S. Larouche and L. Martinu, “A new step method for the synthesis of optical filters with arbitrary indices,” in 49th Annual Technical Conference Proceedings (Society of Vacuum Coaters, 2006), pp. 305-308.

2008

2007

2006

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
[CrossRef]

2004

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
[CrossRef] [PubMed]

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

2001

1998

1997

1996

1993

1992

1982

A. V. Tikhonravov, “A method of synthesis of optical coverings which uses the necessary optimality conditions,” Vestn. Mosk. Univ. 3, Fiz. Astron. 37, 91-93 (1982). Translation of Mosc. Univ. Phys. Bull. (USSR) 37, 108-110 (1982).

1978

1974

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” ENSV Tead. Akad. Toimetised Fuus. Mat. 23, 229-237 (1974), English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

1967

1958

1950

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).

Baumeister, P.

DeBell, G. W.

Delano, E.

Dobrowolski, J. A.

Frankena, H. J.

Gujrathi, S. C.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
[CrossRef]

Klemberg-Sapieha, J. E.

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
[CrossRef]

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
[CrossRef] [PubMed]

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
[CrossRef]

Laan, C. J. v. d.

Larouche, S.

S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436-7441 (2007).
[CrossRef] [PubMed]

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

S. Larouche and L. Martinu, “A new step method for the synthesis of optical filters with arbitrary indices,” in 49th Annual Technical Conference Proceedings (Society of Vacuum Coaters, 2006), pp. 305-308.

Leroux, P.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
[CrossRef]

Lowe, D.

Macleod, H. A.

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

Martinu, L.

S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436-7441 (2007).
[CrossRef] [PubMed]

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
[CrossRef]

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
[CrossRef] [PubMed]

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
[CrossRef]

S. Larouche and L. Martinu, “A new step method for the synthesis of optical filters with arbitrary indices,” in 49th Annual Technical Conference Proceedings (Society of Vacuum Coaters, 2006), pp. 305-308.

Poitras, D.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
[CrossRef]

Sossi, L.

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” ENSV Tead. Akad. Toimetised Fuus. Mat. 23, 229-237 (1974), English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

Sullivan, B. T.

Szymanowski, H.

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

Tikhonravov, A. V.

Trubetskov, M. K.

Verly, P. G.

Vernhes, R.

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
[CrossRef]

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
[CrossRef] [PubMed]

Zabeida, O.

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
[CrossRef]

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
[CrossRef] [PubMed]

Ann. Phys. (Paris)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596-640, 706-782 (1950).

Appl. Opt.

C. J. v. d. Laan and H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17, 538-541 (1978).
[CrossRef] [PubMed]

P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis for inhomogeneous optical coatings,” Appl. Opt. 36, 1487-1495 (1997).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219-C230 (2008).
[CrossRef] [PubMed]

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97-103 (2004).
[CrossRef] [PubMed]

B. T. Sullivan and J. A. Dobrowolski, “Implementation of a numerical needle method for thin-film design,” Appl. Opt. 35, 5484-5492 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493-5508 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417-5426 (1993).
[CrossRef] [PubMed]

J. A. Dobrowolski and D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039-3050 (1978).
[CrossRef] [PubMed]

S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436-7441 (2007).
[CrossRef] [PubMed]

P. G. Verly, “Modified needle method with simultaneous thickness and refractive-index refinement for the synthesis of inhomogeneous and multilayer optical thin films,” Appl. Opt. 40, 5718-5725 (2001).
[CrossRef]

P. Baumeister, “Rudiments of the design of an immersed polarizing beam divider with a narrow spectral bandwidth and enhanced angular acceptance,” Appl. Opt. 36, 3610-3613(1997).
[CrossRef] [PubMed]

B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. I. Theoretical description,” Appl. Opt. 31, 3821-3835 (1992).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46, 704-710 (2007).
[CrossRef] [PubMed]

P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327-7333 (1998).
[CrossRef]

ENSV Tead. Akad. Toimetised Fuus. Mat.

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” ENSV Tead. Akad. Toimetised Fuus. Mat. 23, 229-237 (1974), English translation available from the Translation Services of the Canada Institute for Scientific and Technical Information (CISTI).

J. Appl. Phys.

R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Pulsed radio frequency plasma deposition of a ─SiNx:H alloys: film properties, growth mechanism, and applications,” J. Appl. Phys. 100, 063308 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Vac. Sci. Technol. A

S. Larouche, H. Szymanowski, J. E. Klemberg-Sapieha, L. Martinu, and S. C. Gujrathi, “Microstructure of plasma-deposited SiO2/TiO2 optical films,” J. Vac. Sci. Technol. A 22, 1200-1207 (2004).
[CrossRef]

Opt. Eng.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, and L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693-2699 (1996).
[CrossRef]

Vestn. Mosk. Univ. 3, Fiz. Astron.

A. V. Tikhonravov, “A method of synthesis of optical coverings which uses the necessary optimality conditions,” Vestn. Mosk. Univ. 3, Fiz. Astron. 37, 91-93 (1982). Translation of Mosc. Univ. Phys. Bull. (USSR) 37, 108-110 (1982).

Other

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

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

S. Larouche and L. Martinu, “A new step method for the synthesis of optical filters with arbitrary indices,” in 49th Annual Technical Conference Proceedings (Society of Vacuum Coaters, 2006), pp. 305-308.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the step method: (a) parameters can be added to the refinement by adding steps in existing layers; (b) to determine where to add steps, the derivative of the MF is calculated for rising (blue/black) and descending (red/gray) steps as a function of the position; and (c) one or a few steps are added at the positions where the derivative of the MF is minimal.

Fig. 2
Fig. 2

Index profile (black) and derivative of the MF with respect to the addition of rising (blue/black curve) or descending (red/black curve) step (left column), and reflection spectrum (right column) at progressive stages (from top to bottom) of the design of an AR coating on glass for the visible.

Fig. 3
Fig. 3

Comparison of the index profile (left) and transmission spectra (right) of an edge filter designed using a QW stack as the starting design (top) by the step method (middle) or the needle method (bottom).

Fig. 4
Fig. 4

Index profile and worst case transmission spectrum between 51 ° and 71 ° of a polarizing beam splitter.

Fig. 5
Fig. 5

Effective refractive indices for s and p polarizations and multiple angles of incidence in glass ( N = 1.5 ).

Equations (39)

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

MF = χ 2 = i = 1 m ( B i B ¯ i Δ B i ) 2 ,
M i = [ cos ϕ i ( i / η i ) sin ϕ i i η i sin ϕ i cos ϕ i ] ,
η i = { N i 2 α 2 s   polarization N i 2 / N i 2 α 2 p   polarization
ϕ i = 2 π λ N i 2 α 2 d i
α = N i sin θ i ,
M = i = q 1 M i ,
d M d a k = i = q j + 1 M i d M j d a k i = j 1 1 M i .
d M j d N j = d M j d ϕ j d ϕ j d N j + d M j d η j d η j d N j ,
d M j d ϕ j = [ sin ϕ j ( i / η j ) cos ϕ j i η j cos ϕ j sin ϕ j , ] ,
d M j d η j = [ 0 ( i / η j 2 ) sin ϕ j i sin ϕ j 0 ] ,
d η j d N j = { N j N j 2 α 2 s   polarization N j N j 2 α 2 ( 2 N j 2 N j 2 α 2 ) p   polarization ,
d ϕ j d N j = 2 π λ d j N j N j 2 α 2 .
d M j d n j , 0 = d M j d N j d N j d n j , 0 ,
d R d a k = 2 r * d r d a k ,
d r d a k = Tr ( d M d a k ψ r )
ψ r = t 2 η inc [ η inc ( 1 r ) ( 1 + r ) η inc η ex ( 1 r ) η ex ( 1 + r ) ] ,
d MF d a k = 2 i = 1 m ( B i B ¯ i ( Δ B i ) 2 d B i d a k ) .
ϕ j , 1 = 2 π λ N j 2 α 2 z ,
ϕ j , 2 = 2 π λ N j 2 α 2 ( d j z ) .
d M j d Δ N | Δ N = 0 = M j , 2 d M j , 1 d Δ N | Δ N = 0 + d M j , 2 d Δ N | Δ N = 0 M j , 1 .
d M j , 1 d Δ N | Δ N = 0 = 1 2 d M j , 1 d N j ,
d M j , 2 d Δ N | Δ N = 0 = 1 2 d M j , 2 d N j .
d M d Δ N | Δ N = 0 = 1 2 d M j , 2 d N j M j , 1 M j , 2 1 2 d M j , 1 d N j ,
d M j d Δ N | Δ N = 0 = 1 2 ( d M j d ϕ j d Δ ϕ j d N j + ( d M j ( Δ ϕ j ) d η j + 1 η j ( cos ϕ j cos Δ ϕ j ) [ 1 0 0 1 ] ) d η j d N j ) ,
d M j d Δ N | Δ N = 0 , N min = d M j d Δ N | Δ N = 0 + 1 2 d M j d N j
d M j d Δ N | Δ N = 0 , N max = d M j d Δ N | Δ N = 0 1 2 d M j d N j .
N = 2 α .
d MF d Δ N | Δ N = 0 , rising = d MF d Δ N | Δ N = 0 , descending .
d M j d Δ N | Δ N = 0 = 1 2 d M j , 2 d N j M j , 1 M j , 2 1 2 d M j , 1 d N j ,
d M j d Δ N | Δ N = 0 = 1 2 ( d M j , 2 d ϕ j , 2 d ϕ j , 2 d N j + d M j , 2 d η j d η j d N j ) M j , 1 1 2 M j , 2 ( d M j , 1 d ϕ j , 1 d ϕ j , 1 d N j + d M j , 1 d η j d η j d N j ) ,
d M j d Δ N | Δ N = 0 = 1 2 ( d M j , 2 d ϕ j , 2 d ϕ j , 2 d N j M j , 1 M j , 2 d M j , 1 d ϕ j , 1 d ϕ j , 1 d N j + ( d M j , 2 d η j M j , 1 M j , 2 d M j , 1 d η j ) d η j d N j ) .
d M j d Δ N | Δ N = 0 = 1 2 ( [ sin ϕ j , 2 ( i / η j ) cos ϕ j , 2 i η j cos ϕ j , 2 sin ϕ j , 2 ] [ cos ϕ j , 1 ( i / η j ) sin ϕ j , 1 i η j sin ϕ j , 1 cos ϕ j , 1 ] d ϕ j , 2 d N j [ cos ϕ j , 2 ( i / η j ) sin ϕ j , 2 i η j sin ϕ j , 2 cos ϕ j , 2 ] [ sin ϕ j , 1 ( i / η j ) cos ϕ j , 1 i η j cos ϕ j , 1 sin ϕ j , 1 ] d ϕ j , 1 d N j + ( [ 0 ( i / η j 2 ) sin ϕ j , 2 i sin ϕ j , 2 0 ] [ cos ϕ j , 1 ( i / η j ) sin ϕ j , 1 i η j sin ϕ j , 1 cos ϕ j , 1 ] [ cos ϕ j , 2 ( i / η j ) sin ϕ j , 2 i η j sin ϕ j , 2 cos ϕ j , 2 ] [ 0 ( i / η j 2 ) sin ϕ j , 1 i sin ϕ j , 1 0 ] ) d η j d N j ) ,
d M j d Δ N | Δ N = 0 = 1 2 ( [ sin ϕ j , 2 cos ϕ j , 1 cos ϕ j , 2 sin ϕ j , 1 ( i / η j ) ( sin ϕ j , 2 sin ϕ j , 1 + cos ϕ j , 2 cos ϕ j , 1 ) i η j ( cos ϕ j , 2 cos ϕ j , 1 sin ϕ j , 2 sin ϕ j , 1 ) cos ϕ j , 2 sin ϕ j , 1 sin ϕ j , 2 cos ϕ j , 1 ] d ϕ j , 2 d N [ cos ϕ j , 2 sin ϕ j , 1 sin ϕ j , 2 cos ϕ j , 1 ( i / η j ) ( cos ϕ j , 2 cos ϕ j , 1 sin ϕ j , 2 sin ϕ j , 1 ) i η j ( sin ϕ j , 2 sin ϕ j , 1 + cos ϕ j , 2 cos ϕ j , 1 ) sin ϕ j , 2 cos ϕ j , 1 cos ϕ j , 2 sin ϕ j , 1 ] d ϕ j , 1 d N + ( [ ( 1 / η j ) sin ϕ j , 2 sin ϕ j , 1 ( i / η j 2 ) sin ϕ j , 2 cos ϕ j , 1 i sin ϕ j , 2 cos ϕ j , 1 ( 1 / η j ) sin ϕ j , 2 sin ϕ j , 1 ] [ ( 1 / η j ) sin ϕ j , 2 sin ϕ j , 1 ( i / η j 2 ) cos ϕ j , 2 sin ϕ j , 1 i cos ϕ j , 2 sin ϕ j , 1 ( 1 / η j ) sin ϕ j , 2 sin ϕ j , 1 ] ) d η j d N ) .
d M j d Δ N | Δ N = 0 = 1 2 ( [ ( sin ϕ j , 2 cos ϕ j , 1 + cos ϕ j , 2 sin ϕ j , 1 ) i η j ( cos ϕ j , 2 cos ϕ j , 1 sin ϕ j , 2 sin ϕ j , 1 ) ( i / η j ) ( cos ϕ j , 2 cos ϕ j , 1 sin ϕ j , 2 sin ϕ j , 1 ) ( cos ϕ j , 2 sin ϕ j , 1 + sin ϕ j , 2 cos ϕ j , 1 ) ] ( d ϕ j , 2 d N j d ϕ j , 1 d N j ) + [ ( 2 / η j ) sin ϕ j , 2 sin ϕ j , 1 ( i / η j 2 ) ( sin ϕ j , 2 cos ϕ j , 1 cos ϕ j , 2 sin ϕ j , 1 ) i ( sin ϕ j , 2 cos ϕ j , 1 cos ϕ j , 2 sin ϕ j , 1 ) ( 2 / η j ) sin ϕ j , 2 sin ϕ j , 1 ] d η j d N j ) .
sin ϕ j , 2 cos ϕ j , 1 + cos ϕ j , 2 sin ϕ j , 1 = sin ( ϕ j , 2 + ϕ j , 1 ) , sin ϕ j , 2 cos ϕ j , 1 cos ϕ j , 2 sin ϕ j , 1 = sin ( ϕ j , 2 ϕ j , 1 ) , cos ϕ j , 2 cos ϕ j , 1 sin ϕ j , 2 sin ϕ j , 1 = cos ( ϕ j , 2 + ϕ j , 1 ) ,
sin ϕ j , 2 sin ϕ j , 1 = 1 2 ( cos ( ϕ j , 2 ϕ j , 1 ) cos ( ϕ j , 2 + ϕ j , 1 ) ) ,
d M j d Δ N | Δ N j = 0 = 1 2 ( [ sin ϕ j ( i / η j ) cos ϕ j i η j cos ϕ j sin ϕ j ] d Δ ϕ j d N j + [ ( 1 / η j ) ( cos Δ ϕ j cos ϕ j ) ( i / η j 2 ) sin Δ ϕ j i     sin Δ ϕ j ( 1 / η j ) ( cos Δ ϕ j cos ϕ j ) ] d η j d N j ) ,
d M j d Δ N | Δ N j = 0 = 1 2 ( [ sin ϕ j ( i / η j ) cos ϕ j i η j cos ϕ j sin ϕ j ] d Δ ϕ j d N j + ( [ 0 ( i / η j 2 ) sin Δ ϕ j i sin Δ ϕ j ] + 1 η j [ cos Δ ϕ j cos ϕ j 0 0 cos Δ ϕ j + cos ϕ j ] ) d η j d N j ) ,
d M j d Δ N | Δ N = 0 = 1 2 ( d M j d ϕ j d Δ ϕ j d N j + ( d M j ( Δ ϕ j ) d η j + 1 η j ( cos ϕ j cos Δ ϕ j ) [ 1 0 0 1 ] ) d η j d N j ) ,

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