Abstract

Theories to design a three-dimensional superresolution filter (TDSF) for confocal microscopy are proposed that can obtain a globally optimal solution through linear programming. The designed TDSF is proved to be a phase-only element introducing a phase delay of 0 or π. Five design examples of the TDSF are presented to demonstrate the validity of these theories. Regardless of transverse superresolution, a curve of S eu(G a ±) defined as the maximum value of Strehl ratio S under the axial resolving power of G a ± is calculated to set the fundamental limits of axial optical superresolution. Finally, what is to our knowledge a novel analytic expression of S eu(G a ±) is deduced.

© 2003 Optical Society of America

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References

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  1. K. Carlsson, P. E. Danielsson, R. Lenz, A. Liljeborg, L. Majlof, N. Aslund, “Three-dimensional microscopy using a confocal laser scanning microscope,” Opt. Lett. 10, 53–55 (1985).
    [Crossref] [PubMed]
  2. T. Wilson, B. R. Masters, “Confocal microscopy,” Appl. Opt. 33, 565–566 (1994).
    [Crossref] [PubMed]
  3. C. J. R. Sheppard, Min. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Optics Commun. 84, 7–13 (1991).
    [Crossref]
  4. M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
    [Crossref]
  5. C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).
  6. T. R. M. Sales, G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
    [Crossref]
  7. M. Martinez-Corral, M. T. Caballero, E. H. K. Stelzer, J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express 10, 98–103 (2002).
    [Crossref] [PubMed]
  8. M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
    [Crossref]
  9. X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).
  10. M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).
  11. C. J. R. Sheppard, G. Calvert, M. Wheatland, “Focal distribution for superresolving Toraldo filters,” J. Opt. Soc. Am. A 15, 849–856 (1998).
    [Crossref]
  12. G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).
  13. M. Martínez-Corral, L. Muñoz-Escrivá, M. Kowalczyk, T. Cichocki, “One-dimensional iterative algorithm for three-dimensional point-spread function engineering,” Opt. Lett. 26, 1861–1863 (2001).
    [Crossref]
  14. J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.
    [Crossref]
  15. T. R. M. Sales, G. M. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
    [Crossref] [PubMed]
  16. H. Liu, Y. Yan, Q. Tan, G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185–2193 (2002).
    [Crossref]
  17. H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for design of hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A.
  18. T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London; New York, 2002), Chap. 1.
    [Crossref]
  19. L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. 1.

2002 (2)

2001 (2)

1999 (1)

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[Crossref]

1998 (3)

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

T. R. M. Sales, G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
[Crossref]

C. J. R. Sheppard, G. Calvert, M. Wheatland, “Focal distribution for superresolving Toraldo filters,” J. Opt. Soc. Am. A 15, 849–856 (1998).
[Crossref]

1997 (1)

1995 (2)

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

1994 (1)

1991 (1)

C. J. R. Sheppard, Min. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Optics Commun. 84, 7–13 (1991).
[Crossref]

1985 (1)

1952 (1)

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Andres, P.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[Crossref]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Aslund, N.

Blyth, T. S.

T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London; New York, 2002), Chap. 1.
[Crossref]

Caballero, M. T.

Calvert, G.

Carlsson, K.

Cichocki, T.

Danielsson, P. E.

Deng, X.

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

Elsgolc, L. E.

L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. 1.

Gu, Min.

C. J. R. Sheppard, Min. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Optics Commun. 84, 7–13 (1991).
[Crossref]

Jin, G.

H. Liu, Y. Yan, Q. Tan, G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185–2193 (2002).
[Crossref]

H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for design of hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A.

Kowalczyk, M.

M. Martínez-Corral, L. Muñoz-Escrivá, M. Kowalczyk, T. Cichocki, “One-dimensional iterative algorithm for three-dimensional point-spread function engineering,” Opt. Lett. 26, 1861–1863 (2001).
[Crossref]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[Crossref]

Lenz, R.

Liljeborg, A.

Liu, H.

H. Liu, Y. Yan, Q. Tan, G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185–2193 (2002).
[Crossref]

H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for design of hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A.

Majlof, L.

Martinez-Corral, M.

M. Martinez-Corral, M. T. Caballero, E. H. K. Stelzer, J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express 10, 98–103 (2002).
[Crossref] [PubMed]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[Crossref]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Martínez-Corral, M.

Masters, B. R.

Morris, G. M.

T. R. M. Sales, G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
[Crossref]

T. R. M. Sales, G. M. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
[Crossref] [PubMed]

Muñoz-Escrivá, L.

Ojeda-Castaneda, J.

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Robertson, E. F.

T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London; New York, 2002), Chap. 1.
[Crossref]

Saavedra, G.

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Sales, T. R. M.

T. R. M. Sales, G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
[Crossref]

T. R. M. Sales, G. M. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
[Crossref] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, G. Calvert, M. Wheatland, “Focal distribution for superresolving Toraldo filters,” J. Opt. Soc. Am. A 15, 849–856 (1998).
[Crossref]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

C. J. R. Sheppard, Min. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Optics Commun. 84, 7–13 (1991).
[Crossref]

Stelzer, E. H. K.

Strayer, J. K.

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.
[Crossref]

Swoger, J.

Tan, Q.

Toraldo di Francia, G.

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Wang, G.

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

Wheatland, M.

Wilson, T.

Xu, Z.

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

Yan, Y.

H. Liu, Y. Yan, Q. Tan, G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185–2193 (2002).
[Crossref]

H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for design of hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A.

Yi, D.

H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for design of hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A.

Zapata-Rodriguez, C. J.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[Crossref]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

Appl. Opt. (1)

Atti Fond. Giorgio Ronchi (1)

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Chin. J. Lasers (1)

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

T. R. M. Sales, G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
[Crossref]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Optics Commun. (1)

C. J. R. Sheppard, Min. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Optics Commun. 84, 7–13 (1991).
[Crossref]

Optik (Stuttgart) (2)

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

Other (4)

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, New York, 1989), Chap. 2.
[Crossref]

H. Liu, Y. Yan, D. Yi, G. Jin, “Theories for design of hybrid refractive-diffractive superresolution lens with high numerical aperture,” J. Opt. Soc. Am. A.

T. S. Blyth, E. F. Robertson, Further Linear Algebra (Springer, London; New York, 2002), Chap. 1.
[Crossref]

L. E. Elsgolc, Calculus of Variations (Pergamon, Oxford, UK, 1961), Chap. 1.

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Figures (9)

Fig. 1
Fig. 1

Three-dimensional superresolution filter in a confocal scanning microscope.

Fig. 2
Fig. 2

Constraint (19c) corresponding to the area surrounded by a circle can be approximated to a set of linear constraints corresponding to the area surrounded by a regular polygon with 4P sides. Here k (p) = (p - 1)π/(2P), p = 1, …, P.

Fig. 3
Fig. 3

Intensity distribution on the focal plane for design examples 1–3 of the TDSF.

Fig. 4
Fig. 4

Axial intensity distribution of design examples 1–3 of the TDSF.

Fig. 5
Fig. 5

Intensity distribution on the focal plane for design examples 1, 4, and 5 of the TDSF.

Fig. 6
Fig. 6

Axial intensity distribution of design examples 1, 4, and 5 of the TDSF.

Fig. 7
Fig. 7

Intensity distribution in the image space for design example 1 of the TDSF.

Fig. 8
Fig. 8

Curves of S eu (G a ±). The inserted curves are of a logarithmic coordinate used to magnify those of a linear coordinate.

Fig. 9
Fig. 9

Curves of μ ma (G a ±).

Tables (2)

Tables Icon

Table 1 Design Examples of the Three-Dimensional Superresolution Filter

Tables Icon

Table 2 Design Examples of the Axial Superresolution Filter

Equations (105)

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i2r2, z=2πλz20R ur1expiπr12λ1z-1f×J02πr1r2λzr1dr12,
NA  1.
r2λNAor r2<λNA
|δ|λNA2or |δ|<λNA2
|δ|  f  NA  λR,
i2r2, z=2πλf20R ur1exp-iπr12λδf2×J02πr1r2λfr1dr12.
λR  NA  1,
Iη, μ=401 Uαexp-i2πμα2J0xJηααdα2,
maxUα I0, 0
Iηi, 0εiI0, 0, εi0, 1,i=1, 2, , Nr, 0<ηi<ηi+1
I0, -μjωjI0, 0
I0, μjωjI0, 0, ωj0, 1,j=1, 2, , Na, 0<μj<μj+1
|Uα|1,
maxAα,Bα01 Aααdα2+01 Bααdα2
01 AαJ0xJηiααdα2+01 BαJ0xJηiααdα2εi01 Aααdα2+01 Bααdα2
01 Aαcos2πμjα2αdα-01 Bαsin2πμjα2αdα2+01 Bαcos2πμjα2αdα+01 Aαsin2πμjα2αdα2ωj01 Aααdα2+01 Bααdα2
01 Aαcos2πμjα2αdα+01 Bαsin2πμjα2αdα2+01 Bαcos2πμjα2αdα-01 Aαsin2πμjα2αdα2ωj01 Aααdα2+01 Bααdα2
Aα2+Bα21.
minAα,Bα01 Aααdα
01 Aααdα=01 Bααdα
01 AαJ0xJηiααdα2+01 BαJ0xJηiααdα22εi01 Aααdα2
01 Aαcos2πμjα2αdα-01 Bαsin2πμjα2αdα2+01 Bαcos2πμjα2αdα+01 Aαsin2πμjα2αdα22ωj01 Aααdα2
01 Aαcos2πμjα2αdα+01 Bαsin2πμjα2αdα2+01 Bαcos2πμjα2αdα-01 Aαsin2πμjα2αdα22ωj01 Aααdα2
Aα2+Bα21.
Aα=A*α Bα=B*α
Aα=Bα=A*α+B*α2
Fmin=01 A*ααdα,
01 A*ααdα=01 B*ααdα
01 A*αJ0xJηiααdα2+01 B*αJ0xJηiααdα22εi01 A*ααdα2
01 A*αcos2πμjα2αdα-01 B*αsin2πμjα2αdα2+01 B*αcos2πμjα2αdα+01 A*αsin2πμjα2αdα22ωj01 A*ααdα2
01 A*αcos2πμjα2αdα+01 B*αsin2πμjα2αdα2+01 B*αcos2πμjα2αdα-01 A*αsin2πμjα2αdα22ωj01 A*ααdα2
A*α2+B*α21.
01A*α+B*α2αdα=01 A*ααdα=Fmin,
01A*α+B*α2 J0xJηiααdα2+01A*α+B*α2 J0xJηiααdα2=121×01 A*αJ0xJηiααdα+1×01 B*αJ0xJηiααdα21212+1201 A*αJ0xJηiααdα2+01 B*αJ0xJηiααdα22εi01 A*ααdα2=2εi01A*α+B*α2 αdα2,
01 A*αcos2πμjα2αdα2+01 B*αcos2πμjα2αdα2+01 A*αsin2πμjα2αdα2+01 B*αsin2πμjα2αdα22ωj01 A*ααdα2.
01A*α+B*α2cos2πμjα2αdα-01A*α+B*α2sin2πμjα2αdα2+01A*α+B*α2cos2πμjα2αdα+01A*α+B*α2sin2πμjα2αdα2=121×01 A*αcos2πμjα2αdα+1×01 B*αcos2πμjα2αdα2+121×01 A*αsin2πμjα2αdα+1×01 B*αsin2πμjα2αdα21212+1201 A*αcos2πμjα2αdα2+01 B*αcos2πμjα2αdα2+1212+1201 A*αsin2πμjα2αdα2+01 B*αsin2πμjα2αdα22ωj01 A*ααdα2=2ωj01A*α+B*α2 αdα2,
A*α+B*α22+A*α+B*α22=121×A*α+1×B*α21212+12A*α2+B*α21,
minAα01 Aααdα
01 AαJ0xJηiααdα2εi01 Aααdα2
01 Aαcos2πμjα2αdα2+01 Aαsin2πμjα2αdα2ωj01 Aααdα2
-12Aα12
Bα=Aα.
Aα=Amα
Fmin=01 Amααdα
01 AmαJ0xJηiααdα2εi01 Amααdα2
01 Amαcos2πμjα2αdα2+01 Amαsin2πμjα2αdα2ωj01 Amααdα2
-12Amα12.
01-AmαJ0xJηiααdα2εi01-Amααdα2
01-Amαcos2πμjα2αdα2+01-Amαsin2πμjα2αdα2ωj01-Amααdα2
-12-Amα12.
Aα=-Amα
01 Amααdα=Fmin01-Amααdα,
01 Amααdα0.
εi01 Aααdα01 AαJ0xJηiααdα-εi01 Aααdα.
01 Aαcos2πμjα2αdαcos γp+01 Aαsin2πμjα2αdαsin γp=-ωj01 Aααdαcosπ4P
-01 Aαcos2πμjα2αdαcos γp+01 Aαsin2πμjα2αdαsin γp=-ωj01 Aααdαcosπ4P
-01 Aαcos2πμjα2αdαcos γp-01 Aαsin2πμjα2αdαsin γp=-ωj01 Aααdαcosπ4P
01 Aαcos2πμjα2αdαcos γp-01 Aαsin2πμjα2αdαsin γp=-ωj01 Aααdαcosπ4P,
-1m01 Aαcos2πμjα2αdαcos γp+-1n01 Aαsin2πμjα2αdαsin γp-ωj01 Aααdαcosπ4P, m, n1, 2, p=1,, P.
minAkk=1K Akαk2-αk-12
εik=1K Akαk2-αk2k=1K AkJ1xJηiαkαk-J1xJηiαk-1αk-1xJηi2-εik=1K Akαk2-αk2
-1mk=1K Ak×sin2πμjαk2-sin2πμjαk-122πμjcos γp+-1nk=1K Ak×cos2πμjαk2-cos2πμjαk-12-2πμjsin γp
-ωjk=1K Akαk2-αk-12cosπ4P, m, n1, 2, p=1,, P-12Ak12, k=1,, K.
Aα=12sin θα,
minθα01sin θααdα
01sin θαJ0xJηiααdα2εi01sin θααdα2
01sin θαcos2πμjα2αdα2+01sin θαsin2πμjα2αdα2ωj01sin θααdα2.
θα=θmα,
Fθα, σie, τje=01sin θααdα+ieIe σie01sin θα×J0xJηieααdα2-εie01sin θααdα2+jeJe τje01sin θα×cos2πμjeα2αdα2+01sin θαsin2πμjeα2αdα2-ωje01sin θααdα2.
δθαFθα, σie, τje=01cos θααδθαdα+ieIe σie2cieJ01 J0xJηieαcos θααδθαdα-2c0εie01cos θααδθαdα+jeJe τje2cjec01cos2πμjeα2cos θααδθαdα+2cjes01sin2πμjeα2cos θααδθαdα-2c0ωje01cos θααδθαdα=011-2c0ieIe σieεie-2c0jeJe τjeωje+2 ieIe σiecieJJ0xJηieα+2 jeJe τjecjeccos2πμjeα2+2 jeJe τjecjessin2πμjeα2cos θααδθαdα=0,
c0=01sin θααdα
cieJ=01sin θαJ0xJηieααdα
cjec=01sin θαcos2πμjeα2αdα
cjes=01sin θαsin2πμjeα2αdα
1-2c0ieIe σieεie-2c0jeJe τjeωje+2 ieIe σiecieJJ0xJηieα+2 jeJe τjecjeccos2πμjeα2+2 jeJe τjecjessin2πμjeα2cos θαα=0, α0, 1.
1-2c0ieIe σieεie-2c0jeJe τjeωje+2 ieIe σiecieJJ0xJηieα+2 jeJe τjecjeccos2πμjeα2+2 jeJe τjecjessin2πμjeα2=0αa, b, a, b0, 1.
Ω=1, J0xJηieα, cos2πμjeα2,sin2πμjeα2|ieIe, jeJe,
1-2c0ieIe σieεie-2c0jeJe τjeωje=0
σiecieJ=0
τjecjec=τjecjes=0.
εie=0  cieJ=0
ωje=0  cjec=cjes=0
σieεie=0,
τjeωje=0.
1=0,
1-2c0ieIe σieεie-2c0jeJe τjeωje+2 ieIe σiecieJJ0xJηieα+2 jeJe τjecjeccos2πμjeα2+2 jeJe τjecjessin2πμjeα20αa, b, a, b0, 1.
cos θα=0, αa, b, a, b0, 1.
cos θα=0,  α0, 1.
Aα-12, 12, α0, 1.
ε1=ω1=0.5 and η1<η0 and μ1<μ0,
Gr=η1η0 and Ga±=μ1μ0.
8a dependent on 8b8e, 47, 48
8a dependent on 8b with i2, 8c and 8d with j2, 8e, 48.
ω1=0.5 and μ1<μ0.
Ga±=μ1μ0.
8a dependent on 8c and 8d with j2, 8e, 52.
Uα=1,if α0, αi-1,if ααi, αo1,if ααo, 1,
I0, -μma=I0, μma=0.
cosπμmaαi2sinπμmaαi2-cosπμmaαo2+αi2×sinπμmaαo2-αi2+cosπμma1+αo2×sinπμma1-αo2=0
sinπμmaαi2sinπμmaαi2-sinπμmaαo2+αi2×sinπμmaαo2-αi2+sinπμma1+αo2×sinπμma1-αo2=0.
αiμma=0.5-0.5 arcsin0.5 sinπμmaπμma1/2
αoμma=0.5+0.5 arcsin0.5 sinπμmaπμma1/2,
Seu=1-2 arcsin0.5 sinπμmaπμma2,
μmaGa±n=19 anGa±n+1-n=19 anGa±10,
minani=1imn=19 anGa,i±n+1-n=19 anGa,i±10-μma,i2,

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