Abstract

A promising technique has been proposed recently [Opt. Commun. 284, 1331 (2011), Opt. Commun. 284, 4107 (2011)] for breaking the diffraction limit of light. This technique consists of transforming a symmetrical Laguerre–Gaussian LGp0 beam into a near-Gaussian beam at the focal plane of a thin converging lens thanks to a binary diffractive optical element (DOE) having a transmittance alternatively equal to 1 or +1, transversely. The effect of the DOE is to convert the alternately out-of-phase rings of the LGp0 beam into a unified phase front. The benefits of the rectified beam at the lens focal plane are a short Rayleigh range, which is very useful for many laser applications, and a focal volume much smaller than that obtained with a Gaussian beam. In this paper, we demonstrate numerically that the central lobe’s radius of the rectified beam at the lens focal plane depends exclusively on the dimensionless radial intensity vanishing factor of the incident beam. Consequently, this value can be easily predicted.

© 2011 Optical Society of America

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References

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  1. W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
    [CrossRef]
  2. R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
    [CrossRef]
  3. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
    [CrossRef] [PubMed]
  4. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
    [CrossRef]
  8. A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. L. Ingber, “Very fast simulated re-annealing, Math. Comput. Model. 12, 967–973 (1989).
    [CrossRef]
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    [CrossRef]
  13. L. Ingber, “Adaptive simulated annealing (ASA): Lessons learned,” Contr. Cybernet. 25, 33–54 (1996).
  14. W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516–3516(2008).
    [CrossRef]

2011

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

2008

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef] [PubMed]

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516–3516(2008).
[CrossRef]

2007

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

2004

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

2003

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef] [PubMed]

1996

L. Ingber, “Adaptive simulated annealing (ASA): Lessons learned,” Contr. Cybernet. 25, 33–54 (1996).

1993

L. Ingber, “Simulated annealing: Practice versus theory,” Math. Comput. Model. 18, 29–57 (1993).
[CrossRef]

1990

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

1989

L. Ingber, “Very fast simulated re-annealing, Math. Comput. Model. 12, 967–973 (1989).
[CrossRef]

1988

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1987

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

1970

Aït-Ameur, K.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

Ashkin, A.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Bencheikh, A.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef] [PubMed]

Cagniot, E.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef] [PubMed]

Collins, S. A.

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Drechsler, U.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Durig, U.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Fromager, M.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

Guntherodt, H. J.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Haase, A.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Harbers, R.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Hasnaoui, A.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

Heller, C.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Infuehr, R.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Ingber, L.

L. Ingber, “Adaptive simulated annealing (ASA): Lessons learned,” Contr. Cybernet. 25, 33–54 (1996).

L. Ingber, “Simulated annealing: Practice versus theory,” Math. Comput. Model. 18, 29–57 (1993).
[CrossRef]

L. Ingber, “Very fast simulated re-annealing, Math. Comput. Model. 12, 967–973 (1989).
[CrossRef]

Ji, P.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516–3516(2008).
[CrossRef]

Kuna, L.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Lichtenegger, H.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Liska, R.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Liu, W.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516–3516(2008).
[CrossRef]

Mahrt, R. F.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef] [PubMed]

Pucher, N.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Salis, G.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Schmidt, V.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Smith, C. G.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef] [PubMed]

Stampfl, J.

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Teh, W. H.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef] [PubMed]

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Yang, J.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516–3516(2008).
[CrossRef]

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef] [PubMed]

Annu. Rev. Biochem.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef] [PubMed]

Appl. Phys. Lett.

W. H. Teh, U. Durig, G. Salis, R. Harbers, U. Drechsler, R. F. Mahrt, C. G. Smith, and H. J. Guntherodt, “SU-8 for real three-dimensional subdiffraction-limit two-photon microfabrication,” Appl. Phys. Lett. 84, 4095–4097 (2004).
[CrossRef]

Appl. Surf. Science

R. Infuehr, N. Pucher, C. Heller, H. Lichtenegger, R. Liska, V. Schmidt, L. Kuna, A. Haase, and J. Stampfl, “Functional polymers by two-photon 3D lithography,” Appl. Surf. Science 254, 836–840 (2007).
[CrossRef]

Contr. Cybernet.

L. Ingber, “Adaptive simulated annealing (ASA): Lessons learned,” Contr. Cybernet. 25, 33–54 (1996).

J. Acoust. Soc. Am.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516–3516(2008).
[CrossRef]

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Opt. Soc. Am.

Math. Comput. Model.

L. Ingber, “Very fast simulated re-annealing, Math. Comput. Model. 12, 967–973 (1989).
[CrossRef]

L. Ingber, “Simulated annealing: Practice versus theory,” Math. Comput. Model. 18, 29–57 (1993).
[CrossRef]

Nat. Biotechnol.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef] [PubMed]

Nature

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Opt. Commun.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011).
[CrossRef]

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Erratum to ‘Creation of a sharper focus by using a rectified TEMp0 beam’,” Opt. Commun. 284, 4107–4107 (2011).
[CrossRef]

Science

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Optical layout.

Fig. 2
Fig. 2

Incident LG 3 0 beam and the corresponding binary structure.

Fig. 3
Fig. 3

Incident LG 4 0 beam and the corresponding binary structure.

Fig. 4
Fig. 4

Incident LG 8 0 beam and the corresponding source Gaussian beam.

Fig. 5
Fig. 5

Rectified LG 8 0 beam at the focal plane of the lens and the corresponding target Gaussian beam.

Tables (2)

Tables Icon

Table 1 Results of the Optimization Process

Tables Icon

Table 2 α as a Function of p

Equations (35)

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

w ( z ) = w 0 [ 1 + ( z z R ) 2 ] 1 2 ,
V G = z R + z R S ( z ) d z = 4 π 2 w 0 4 3 λ .
w 0 = λ f π w 0 [ 1 + ( f z R ) 2 ] 1 2 .
w 0 λ f π w 0 .
Ψ p ( r , z ) = 2 π 1 w ( z ) exp [ r 2 w ( z ) 2 ] L p [ 2 r 2 w ( z ) 2 ] exp [ j k r 2 2 R ( z ) ] × exp [ j ( 2 p + 1 ) γ ( z ) ] exp [ j k ( z z 0 ) ] = Γ ( z ) L p [ 2 r 2 w ( z ) 2 ] exp [ ( 1 w ( z ) 2 + j k 2 R ( z ) ) r 2 ] ,
w ( z ) = w 0 [ 1 + ( z z 0 z R ) 2 ] 1 2 ,
R ( z ) = ( z z 0 ) [ 1 + ( z R z z 0 ) 2 ] ,
γ ( z ) = arctan ( z z 0 z R ) ,
( A B C D ) = ( 1 f 0 1 ) × ( 1 0 1 f 1 ) .
E ( r , z ) = j k B exp ( j k z ) exp ( j k D 2 B r 2 ) × 0 + τ ( ρ ) Ψ p ( ρ , 0 ) exp ( j k A 2 B ρ 2 ) J 0 ( k B r ρ ) ρ d ρ ,
E ( r , z ) = Γ ( z ) j k B exp ( j k z ) exp ( j k D 2 B r 2 ) × { l = 0 L 1 Δ l 0 a l L p ( α ρ 2 ) exp ( Q ρ 2 ) J 0 ( k B r ρ ) ρ d ρ + exp ( j θ L ) 0 + L p ( α ρ 2 ) exp ( Q ρ 2 ) J 0 ( k B r ρ ) ρ d ρ } ,
{ a 1 = 0 , l 0 , a l = a l 1 + s l ,
{ α = 2 w ( 0 ) 2 , Q = 1 w ( 0 ) 2 + j k 2 ( A B + 1 R ( 0 ) ) , Δ l = exp ( j θ l ) exp ( j θ l + 1 ) .
0 + L p ( α ρ 2 ) exp ( Q ρ 2 ) J 0 ( ρ x ) ρ d ρ = ( Q α ) p 2 Q p + 1 exp ( x 2 4 Q ) L p [ α x 2 4 Q ( α Q ) ] ,
E ( r , z ) = Γ ( z ) j k B exp ( j k z ) exp ( j k D 2 B r 2 ) × { l = 0 L 1 Δ l 0 a l L p ( α ρ 2 ) exp ( Q ρ 2 ) J 0 ( k r B ρ ) ρ d ρ + exp ( j θ L ) ( Q α ) p 2 Q p + 1 exp ( k 2 r 2 4 Q B 2 ) L p [ α k 2 r 2 4 Q ( α Q ) B 2 ] } .
circ ( ρ a ) = { 1 , 0 ρ a , 0 , ρ > a ,
E ( r , z ) = Γ ( z ) j k B exp ( j k z ) exp ( j k D 2 B r 2 ) × { l = 0 L 1 Δ l 0 + circ ( ρ a l ) L p ( α ρ 2 ) exp ( Q ρ 2 ) J 0 ( k r B ρ ) ρ d ρ + exp ( j θ L ) ( Q α ) p 2 Q p + 1 exp ( k 2 r 2 4 Q B 2 ) L p [ α k 2 r 2 4 Q ( α Q ) B 2 ] } .
circ ( ρ a ) n = 0 N 1 F n exp ( G n a 2 ρ 2 ) ,
E ( r , z ) Γ ( z ) j k B exp ( j k z ) exp ( j k D 2 B r 2 ) × { l = 0 L 1 Δ l n = 0 N 1 F n 0 + L p ( α ρ 2 ) exp ( Q n ( l ) ρ 2 ) J 0 ( k r B ρ ) ρ d ρ + exp ( j θ L ) ( Q α ) p 2 Q p + 1 exp ( k 2 r 2 4 Q B 2 ) L p [ α k 2 r 2 4 Q ( α Q ) B 2 ] } ,
n < N , Q n ( l ) = Q + G n a l 2 .
E ( r , z ) Γ ( z ) 2 j k B exp ( j k z ) exp ( j k D 2 B r 2 ) × { l = 0 L 1 Δ l n = 0 N 1 F n ( Q n ( l ) α ) p ( Q n ( l ) ) p + 1 exp ( k 2 r 2 4 Q n ( l ) B 2 ) L p [ α k 2 r 2 4 Q n ( l ) ( α Q n l ) B 2 ] + exp ( j θ L ) ( Q α ) p Q p + 1 exp ( k 2 r 2 4 Q B 2 ) L p [ α k 2 r 2 4 Q ( α Q ) B 2 ] } .
w 0 ( TG ) = α × λ f π w ( 0 ) [ 1 + ( f z R ) 2 ] 1 2 .
err = i I 1 ( i ) I 2 ( i ) i I 1 ( i ) × 100.
w 0 ( TG ) λ f π β w 0 ,
I ( r ) = I 0 [ L p ( 2 r 2 w 2 ) ] 2 exp ( 2 r 2 w 2 ) ,
P = 2 π 0 + I ( r ) r d r .
P = π w 2 I 0 2 0 + L p ( u ) L p ( u ) exp ( u ) d u = π w 2 I 0 2 ,
P t = π w 2 I 0 2 0 a P 2 p ( u ) exp ( u ) d u ,
P m ( x ) exp ( a x ) d x = exp ( a x ) a k = 0 m ( 1 ) k P ( k ) ( x ) a k ,
P t = π w 2 I 0 2 { k = 0 2 p P ( k ) ( 0 ) exp ( 2 κ 2 ) k = 0 2 p P ( k ) ( 2 κ 2 ) } ,
T ( κ ) = P t P = k = 0 2 p P ( k ) ( 0 ) exp ( 2 κ 2 ) k = 0 2 p P ( k ) ( 2 κ 2 ) .
k = 0 2 p P ( k ) ( 0 ) exp ( 2 κ 2 ) k = 0 2 p P ( k ) ( 2 κ 2 ) = 0.99 .
exp ( 2 κ 0 2 ) = 0.01 ,
κ 0 = a β w 0 = κ p β ,
α = κ 0 κ p .

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