Abstract

Pupil filters can modify the three dimensional response of an optical system. In this paper, we study different pupil symmetries that produce a predictable image behavior. We show that different pupil-filters that satisfy certain symmetry conditions can produce axial responses which are either identical or mirror reflected. We also establish the differences in the symmetry properties between amplitude-only filters and phase-only filters. In particular, we are interested in phase filters that produce transverse superresolution with axial superresolution or high depth of focus.

©2004 Optical Society of America

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References

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  1. C. S. Chung and H.H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
    [Crossref]
  2. E. W. S. Hee,“Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol.75–79, April (1975).
    [Crossref]
  3. M. J. Yzuel and F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
    [Crossref]
  4. H. B. Chung, K.H. Hong, and S. S. Lee, “Effect of aperture modulation on the MTF of a binocular objective having rotationally symmetric aberrations,” Appl. Opt. 22, 1983, 1812–1814 (1983).
    [Crossref] [PubMed]
  5. I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. A 72, 1287–1291 (1982).
    [Crossref]
  6. Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [Crossref]
  7. H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126–4130 (1992).
    [Crossref]
  8. B.R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966).
    [Crossref]
  9. B.R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [Crossref]
  10. C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [Crossref]
  11. C.J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Optics 43, 525–536 (1996)
    [Crossref]
  12. M. J. Yzuel, J. C. Escalera, and J. Campos, “Polychromatic axial behaviour of axial apodizing and hyperresolving filters,” Appl. Opt. 29, 1631–1641 (1990).
    [Crossref] [PubMed]
  13. J. Ojeda-Castañeda, E. Tepichin, and A. Díaz, “Arbitrarily high focal depth with a quasioptimum real and positive transmitance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
    [Crossref] [PubMed]
  14. R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34, 195 (1997).
    [Crossref]
  15. Z. Hegedus“Pupil filters in confocal imaging,” in Confocal Microscopy,T. Wilson, ed. (Academic Press, London) 171–183 (1990).
  16. M. Gu and C.J.R. Sheppard “Three dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2263 (1991).
    [Crossref]
  17. M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
    [Crossref]
  18. J.C. Escalera, J. Campos, and M. J. Yzuel, “Pupil symmetries for identical axial response,” Microwave Opt. Technol. Lett. 7, 174–178 (1994).
    [Crossref]
  19. J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57–68 (2000).
  20. T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637 (1997).
    [Crossref]
  21. H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
    [Crossref] [PubMed]
  22. H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” App. Opt. 40, 5658–5662 (2001).
    [Crossref]
  23. W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” App. Opt. 41, 6080–6092 (2002)
    [Crossref]
  24. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberration,” Opt. Lett. 28771–773 (2003).
    [Crossref] [PubMed]
  25. D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
    [Crossref]
  26. D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
    [Crossref] [PubMed]
  27. S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to supergaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
    [Crossref] [PubMed]

2004 (1)

2003 (4)

S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberration,” Opt. Lett. 28771–773 (2003).
[Crossref] [PubMed]

D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
[Crossref]

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
[Crossref] [PubMed]

H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
[Crossref] [PubMed]

2002 (1)

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” App. Opt. 41, 6080–6092 (2002)
[Crossref]

2001 (1)

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” App. Opt. 40, 5658–5662 (2001).
[Crossref]

2000 (1)

J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57–68 (2000).

1997 (2)

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637 (1997).
[Crossref]

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34, 195 (1997).
[Crossref]

1996 (1)

C.J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Optics 43, 525–536 (1996)
[Crossref]

1995 (1)

M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

1994 (1)

J.C. Escalera, J. Campos, and M. J. Yzuel, “Pupil symmetries for identical axial response,” Microwave Opt. Technol. Lett. 7, 174–178 (1994).
[Crossref]

1992 (1)

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126–4130 (1992).
[Crossref]

1991 (1)

M. Gu and C.J.R. Sheppard “Three dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2263 (1991).
[Crossref]

1990 (1)

1989 (1)

1988 (2)

C. S. Chung and H.H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[Crossref]

C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
[Crossref]

1986 (1)

1983 (1)

1982 (1)

I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. A 72, 1287–1291 (1982).
[Crossref]

1979 (1)

M. J. Yzuel and F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[Crossref]

1967 (1)

1966 (1)

Andrés, P.

M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Cagigal, M. P.

D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
[Crossref]

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
[Crossref] [PubMed]

Calvo, F.

M. J. Yzuel and F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[Crossref]

Campos, J.

S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to supergaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
[Crossref] [PubMed]

J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57–68 (2000).

J.C. Escalera, J. Campos, and M. J. Yzuel, “Pupil symmetries for identical axial response,” Microwave Opt. Technol. Lett. 7, 174–178 (1994).
[Crossref]

M. J. Yzuel, J. C. Escalera, and J. Campos, “Polychromatic axial behaviour of axial apodizing and hyperresolving filters,” Appl. Opt. 29, 1631–1641 (1990).
[Crossref] [PubMed]

Canales, V. F.

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
[Crossref] [PubMed]

D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
[Crossref]

Cathey, W. T.

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” App. Opt. 41, 6080–6092 (2002)
[Crossref]

Chung, C. S.

C. S. Chung and H.H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[Crossref]

Chung, H. B.

Cox, I. J.

I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. A 72, 1287–1291 (1982).
[Crossref]

de Juana, D. M.

D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
[Crossref]

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
[Crossref] [PubMed]

Díaz, A.

Dowski, E. R.

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” App. Opt. 41, 6080–6092 (2002)
[Crossref]

Escalera, J. C.

Escalera, J.C.

J.C. Escalera, J. Campos, and M. J. Yzuel, “Pupil symmetries for identical axial response,” Microwave Opt. Technol. Lett. 7, 174–178 (1994).
[Crossref]

Frieden, B.R.

Fukuda, H.

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126–4130 (1992).
[Crossref]

Gan, F.

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” App. Opt. 40, 5658–5662 (2001).
[Crossref]

Gu, M.

M. Gu and C.J.R. Sheppard “Three dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2263 (1991).
[Crossref]

Harvey, A. R.

Hee, E. W. S.

E. W. S. Hee,“Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol.75–79, April (1975).
[Crossref]

Hegedus, Z.

Z. Hegedus“Pupil filters in confocal imaging,” in Confocal Microscopy,T. Wilson, ed. (Academic Press, London) 171–183 (1990).

Hegedus, Z. S.

Hild, R.

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34, 195 (1997).
[Crossref]

Hong, K.H.

Hopkins, H.H.

C. S. Chung and H.H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[Crossref]

Jin, G.

H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
[Crossref] [PubMed]

Ledesma, S.

Lee, S. S.

Liu, H.

H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
[Crossref] [PubMed]

Martínez-Corral, M

M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Mezouari, S.

Morris, G. M.

Ojeda-Castañeda, J.

M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

J. Ojeda-Castañeda, E. Tepichin, and A. Díaz, “Arbitrarily high focal depth with a quasioptimum real and positive transmitance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
[Crossref] [PubMed]

Oti, J. E.

Saavedra, G.

M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Sales, T. R. M.

Sarafis, V.

Sheppard, C. J. R.

J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57–68 (2000).

C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
[Crossref]

I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. A 72, 1287–1291 (1982).
[Crossref]

Sheppard, C.J. R.

C.J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Optics 43, 525–536 (1996)
[Crossref]

Sheppard, C.J.R.

M. Gu and C.J.R. Sheppard “Three dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2263 (1991).
[Crossref]

Tepichin, E.

Valle, P.J.

D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
[Crossref]

Wang, H.

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” App. Opt. 40, 5658–5662 (2001).
[Crossref]

Wilson, T.

I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. A 72, 1287–1291 (1982).
[Crossref]

Yamanaka, R.

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126–4130 (1992).
[Crossref]

Yan, Y.

H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
[Crossref] [PubMed]

Yi, D.

H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
[Crossref] [PubMed]

Yzuel, M. J.

S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to supergaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
[Crossref] [PubMed]

J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57–68 (2000).

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34, 195 (1997).
[Crossref]

J.C. Escalera, J. Campos, and M. J. Yzuel, “Pupil symmetries for identical axial response,” Microwave Opt. Technol. Lett. 7, 174–178 (1994).
[Crossref]

M. J. Yzuel, J. C. Escalera, and J. Campos, “Polychromatic axial behaviour of axial apodizing and hyperresolving filters,” Appl. Opt. 29, 1631–1641 (1990).
[Crossref] [PubMed]

M. J. Yzuel and F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[Crossref]

App. Opt. (2)

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” App. Opt. 40, 5658–5662 (2001).
[Crossref]

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” App. Opt. 41, 6080–6092 (2002)
[Crossref]

Appl Opt. (1)

H. Liu, Y. Yan, D. Yi, and G. Jin, “Design of three-dimensional superresolution filtres and limits of axial superresolution,” Appl Opt. 42, 1463–1476 (2003).
[Crossref] [PubMed]

Appl. Opt. (3)

J. Mod. Opt. (3)

J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57–68 (2000).

C. S. Chung and H.H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485–1511 (1988).
[Crossref]

M. Gu and C.J.R. Sheppard “Three dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2263 (1991).
[Crossref]

J. Mod. Optics (1)

C.J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Optics 43, 525–536 (1996)
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Jpn. J. Appl. Phys. (1)

H. Fukuda and R. Yamanaka, “A new pupil filter for annular illumination in optical lithography,” Jpn. J. Appl. Phys. 31, 4126–4130 (1992).
[Crossref]

Microelectron. Eng. (1)

R. Hild, M. J. Yzuel, and J. C. Escalera, “High focal depth imaging of small structures,” Microelectron. Eng. 34, 195 (1997).
[Crossref]

Microwave Opt. Technol. Lett. (1)

J.C. Escalera, J. Campos, and M. J. Yzuel, “Pupil symmetries for identical axial response,” Microwave Opt. Technol. Lett. 7, 174–178 (1994).
[Crossref]

Opt. Acta (1)

M. J. Yzuel and F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[Crossref]

Opt. Commun. (2)

D. M. de Juana, V. F. Canales, P.J. Valle, and M. P. Cagigal, “Focusing properties of annular binary phase filters,” Opt. Commun. 229, 71–77 (2003).
[Crossref]

M Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

Opt. Lett. (3)

Other (2)

E. W. S. Hee,“Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol.75–79, April (1975).
[Crossref]

Z. Hegedus“Pupil filters in confocal imaging,” in Confocal Microscopy,T. Wilson, ed. (Academic Press, London) 171–183 (1990).

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Figures (5)

Fig. 1.
Fig. 1. Transmission of two supergaussian filters, centered at t 0=-0.3 (green), and 0.3 (blue). In the right axis amplitude transmission corresponding to the amplitude supergaussian filter (Eq. (29), Ω=0.1, α=2) and in the left axis phase transmission ϕ corresponding to the phase supergaussian filter (Eq. (31), Ω=0.1, α=2).
Fig. 2.
Fig. 2. (a) Transverse gain, GT (dashed line) and axial gain, GA (filled line) for an amplitude supergaussian filter as a function of t0 , with Ω=0.1, α=2. (b) Strehl ratio as a function of t0 for amplitude supergaussian filter and the same parameters as (a).
Fig. 3.
Fig. 3. (a) and c) Axial intensities for the amplitude supergaussian filters shown in Fig. 1 centered at t0 =-0.3 and t0 =0.3 respectively. (b) and (d) Transverse intensities for these two cases. In dashed line the response for the pupil without filter.
Fig. 4.
Fig. 4. (a) Transverse gain, GT (dashed line) and axial gain, GA (filled line) for a phase supergaussian filter as a function of t0 , for Ω=0.1, α=2 and a=3. (b) Strehl ratio as a function of t0 for a phase supergaussian filter with the same parameters as (a).
Fig. 5.
Fig. 5. (a) and c) Axial intensities for the phase supergaussian filters shown in Fig. 1 centered at t0 =-0.3 and t0 =0.3 respectively. (b) and (d) Transverse intensities for these two cases. In dashed line the response for the pupil without filter.

Equations (32)

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

U ( v , u ) = 2 0 1 P ( ρ ) J 0 ( v ρ ) exp [ iu ρ 2 2 ] ρ d ρ ,
U ( 0 , u ) = 2 0 1 P ( ρ ) exp [ iu ρ 2 2 ] ρ d ρ .
ρ 2 = t + 0.5 ,
U ( 0 , u ) = 0.5 0.5 Q ( t ) exp [ iu t + 0.5 2 ] dt ,
U ( 0 , u ) = exp ( i 0.5 u 2 ) 0.5 0.5 Q ( t ) exp [ iut 2 ] dt .
I ( 0 , u ) = 0.5 0 . 5 Q ( t ) exp [ iut 2 ] dt 2 .
I 1 ( 0 , u ) = 0.5 0 . 5 Q 1 ( t ) exp [ iut 2 ] dt 2 ,
and I 2 ( 0 , u ) = 0.5 0 . 5 Q 2 ( t ) exp [ iut 2 ] dt 2 = 0.5 0 . 5 Q 1 ( t ) exp [ iut 2 ] dt 2 .
U ( 0 , u ) exp ( iu max 0.5 2 ) 0.5 0.5 Q ( t ) exp [ iu max t 2 ] { 1 + [ i ( t + 0.5 ) 2 ] ( u u max ) [ ( t + 0.5 ) 2 8 ] ( u u max ) 2 } dt .
I n = 0.5 0.5 Q ( t ) ( t + 0.5 ) n exp [ iu max t 2 ] dt .
U ( 0 , u ) exp ( iu max 0.5 2 ) { I 0 + I 1 i 2 ( u u max ) I 2 8 ( u u max ) 2 } .
I ( 0 , u ) | I 0 | 2 Im ( I 0 * I 1 ) u + 1 4 ( | I 1 | 2 Re ( I 0 I 2 * ) ) u 2 ,
G A = 12 Re ( I 0 I 2 * ) I 1 2 I 0 2 1 2 u 0 Im ( I 0 * I 1 ) .
U ( v , u max ) 0.5 0.5 Q ( t ) [ 1 1 4 v 2 ( t + 0.5 ) ] exp ( iu max t ) dt .
I ( v , u max ) I 0 2 1 2 Re ( I 0 I 1 * ) v 2 .
G T = 2 Re ( I 0 I 1 * ) I 0 2 .
S = I 0 2 u 0 Im ( I 0 * I 1 ) .
I 1 = 0 . 5 0.5 exp [ iu max t 2 ] Q ( t ) t dt + 0.5 I 0 .
I 1 = 0.5 0.5 Q ( t ) t dt + 0.5 I 0 ,
G T = 1 + I 0 I 0 2 0.5 0.5 Q ( t ) t dt ,
u max 2 = u max 1 .
I n ( Q 2 ) = 05 0.5 Q 1 ( t ) ( t + 0.5 ) n exp [ iu max t 2 ] dt .
I 0 ( Q 2 ) = 05 0.5 Q 1 ( t ) exp [ iu max t 2 ] dt = I 0 ( Q 1 ) .
I 1 ( Q 2 ) = 05 0.5 Q 1 ( t ) ( t + 0.5 ) exp [ iu max t 2 ] dt = I 1 ( Q 1 ) + I 0 ( Q 1 ) .
G T ( Q 2 ) = 2 Re ( I 0 ( Q 1 ) . I 1 ( Q 1 ) + 0.5 I 0 ( Q 1 ) 2 ) I 0 ( Q 1 2
= 2 Re ( I 0 ( Q 1 ) I 1 ( Q 1 ) ) I 0 2 + 2 .
G T ( Q 1 ) = 2 Re ( I 0 ( Q 1 ) I 1 ( Q 1 ) ) I 0 2 .
G T ( Q 1 ) + G T ( Q 2 ) = 2 .
S ( Q 1 ) = S ( Q 2 ) .
Q ( t ) = exp { [ ( t t 0 ) Ω ] } 2 α ,
Q ( t ) = exp [ i ϕ ] ,
ϕ = a exp { [ ( t t 0 ) Ω ] } 2 α ,

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