Abstract

A supercontinuum bottle beam was successfully produced by focusing a supercontinuum laser beam after passing through an axicon. The supercontinuum radiation was generated from a microstructured fiber pumped by a self-kerr-lens mode-locked femtosecond Ti:sapphire laser. The cross-section intensity distributions of the generated bottle were recorded by a beam profiler. Using the line filters to select different central wavelengths, the different colors of bottle beams show slightly different bottle ranges and diameters due to the dispersion of axicon and focusing lens. The results consist with the theoretical prediction using the Fresnel-Kirchhoff’s formula in considering an incident Gaussian beam.

© 2007 Optical Society of America

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  1. B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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2006 (4)

2005 (4)

2004 (2)

P.-T. Tai, W.-F. Hsieh, and C.-H. Chen, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express. 12, 5827-5833 (2004).
[CrossRef] [PubMed]

D. Yelin, B. E. Bouma, and G. J. Tearney, "Generating an adjustable three-dimensional dark focus," Opt. Lett. 29, 661-663 (2004).
[CrossRef] [PubMed]

2003 (1)

J. C. Gutierrez-Vega, R. Rodriguez-Masegosa, and S. Chavez-Cerda, "Focusing evolution of generalized propagation invariant optical fields," J. Opt. A: Pure Appl. Opt. 5, 276-282 (2003).
[CrossRef]

2001 (1)

2000 (3)

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

J. Arlt and M. J. Padgett, "Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam," Opt. Lett. 25, 191-193 (2000).
[CrossRef]

1999 (2)

J. Yin and Y. Zhu, "LP01-mode output beam from a micro-sized hollow optical fiber: A simple theoretical model and its applications in atom optics," J. Appl. Phys. 85, 2473-2481 (1999).
[CrossRef]

K. Ait-Ameur and F. Sanchez, "Gaussian beam conversion using an axicon," J. Modern Opt,  46, 1537-1548 (1999).

1996 (2)

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

K. T. Gahagan and G. A. Swartzlander, "Optical vortex trapping of particles," Opt. Lett. 21, 827-829 (1996).
[CrossRef] [PubMed]

Ahluwalia, B. P. S.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[CrossRef] [PubMed]

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

Ait-Ameur, K.

K. Ait-Ameur and F. Sanchez, "Gaussian beam conversion using an axicon," J. Modern Opt,  46, 1537-1548 (1999).

Arlt, J.

Bouma, B. E.

Brown, C. T. A.

Brown, C.T.A.

Bu, J.

Carruthers, A. E.

Chai, L.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Chavez-Cerda, S.

J. C. Gutierrez-Vega, R. Rodriguez-Masegosa, and S. Chavez-Cerda, "Focusing evolution of generalized propagation invariant optical fields," J. Opt. A: Pure Appl. Opt. 5, 276-282 (2003).
[CrossRef]

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

Chen, C.-H.

P.-T. Tai, W.-F. Hsieh, and C.-H. Chen, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express. 12, 5827-5833 (2004).
[CrossRef] [PubMed]

Cheong, W. C.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[CrossRef] [PubMed]

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

Chudoba, C.

Cundiff, S. T.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Dholakia, K.

Diddams, S. A.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Dukel’skii, K.V.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Esposito, E.

Fischer, P.

Fujimoto, J. G.

Gahagan, K. T.

Ghanta, R. K.

Gibson, G. M.

Girkin, J. M.

Gutierrez-Vega, J. C.

J. C. Gutierrez-Vega, R. Rodriguez-Masegosa, and S. Chavez-Cerda, "Focusing evolution of generalized propagation invariant optical fields," J. Opt. A: Pure Appl. Opt. 5, 276-282 (2003).
[CrossRef]

Hall, J. L.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Hartl, I.

Hsieh, W.-F.

P.-T. Tai, W.-F. Hsieh, and C.-H. Chen, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express. 12, 5827-5833 (2004).
[CrossRef] [PubMed]

Hu, M.-L.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Huang, W.

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

Jones, D. A.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Khokhlov, A. V.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Ko, T. H.

Kondrat’ev, Y. N.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Kong, F.

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

Leach, J.

Li, P.

Li, X. D.

Li, Y.-F.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Lin, Y. T.

M. D. Wei, W. L. Shiao, and Y. T. Lin, "Adjustable generation of bottle and hollow beams using an axicon," Opt. Commun,  248, 7-14 (2005).
[CrossRef]

Liu, Z.

López-Mariscal, C.

Lu, B.

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

McConnell, G.

Morris, J. E.

New, G. H. C.

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

Padgett, M. J.

Ranka, J. K.

I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001).
[CrossRef]

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Rodriguez-Masegosa, R.

J. C. Gutierrez-Vega, R. Rodriguez-Masegosa, and S. Chavez-Cerda, "Focusing evolution of generalized propagation invariant optical fields," J. Opt. A: Pure Appl. Opt. 5, 276-282 (2003).
[CrossRef]

Sanchez, F.

K. Ait-Ameur and F. Sanchez, "Gaussian beam conversion using an axicon," J. Modern Opt,  46, 1537-1548 (1999).

Serebryannikov, E. E.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Shevandin, V. S.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Shi, K.

Shiao, W. L.

M. D. Wei, W. L. Shiao, and Y. T. Lin, "Adjustable generation of bottle and hollow beams using an axicon," Opt. Commun,  248, 7-14 (2005).
[CrossRef]

Sibbett, W.

Song, Y.-J.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Stentz, A.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tai, P.-T.

P.-T. Tai, W.-F. Hsieh, and C.-H. Chen, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express. 12, 5827-5833 (2004).
[CrossRef] [PubMed]

Tao, S. H.

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

Tao, S.-H.

Tearney, G. J.

Volke-Sepulveda, K.

Wang, C.-Y.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Wang, H.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[CrossRef] [PubMed]

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

Wei, M. D.

M. D. Wei, W. L. Shiao, and Y. T. Lin, "Adjustable generation of bottle and hollow beams using an axicon," Opt. Commun,  248, 7-14 (2005).
[CrossRef]

Windeler, R. S.

I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001).
[CrossRef]

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Wright, A. J.

Wright, E. M.

Yelin, D.

Yin, J.

J. Yin and Y. Zhu, "LP01-mode output beam from a micro-sized hollow optical fiber: A simple theoretical model and its applications in atom optics," J. Appl. Phys. 85, 2473-2481 (1999).
[CrossRef]

Yuan, X.-C.

B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006).
[CrossRef] [PubMed]

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

Zhai, Q.

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

Zhang, B.

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

Zhang, L. S.

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

Zhang, L.-S.

Zheltikov, A. M.

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Zhu, Y.

J. Yin and Y. Zhu, "LP01-mode output beam from a micro-sized hollow optical fiber: A simple theoretical model and its applications in atom optics," J. Appl. Phys. 85, 2473-2481 (1999).
[CrossRef]

J. Appl. Phys. (1)

J. Yin and Y. Zhu, "LP01-mode output beam from a micro-sized hollow optical fiber: A simple theoretical model and its applications in atom optics," J. Appl. Phys. 85, 2473-2481 (1999).
[CrossRef]

J. Modern Opt (1)

K. Ait-Ameur and F. Sanchez, "Gaussian beam conversion using an axicon," J. Modern Opt,  46, 1537-1548 (1999).

J. Opt. A: Pure Appl. Opt. (1)

J. C. Gutierrez-Vega, R. Rodriguez-Masegosa, and S. Chavez-Cerda, "Focusing evolution of generalized propagation invariant optical fields," J. Opt. A: Pure Appl. Opt. 5, 276-282 (2003).
[CrossRef]

Laser Phys. Lett. (1)

M.-L. Hu, C.-Y. Wang, E. E. Serebryannikov, Y.-J. Song, Y.-F. Li, L. Chai, K.V. Dukel’skii, A. V. Khokhlov, V. S. Shevandin, Y. N. Kondrat’ev, and A. M. Zheltikov, "Wavelength-tunable hollow-beam generation by a photonic-crystal fiber," Laser Phys. Lett. 3, 306-309 (2006).
[CrossRef]

Opt. Commun (1)

M. D. Wei, W. L. Shiao, and Y. T. Lin, "Adjustable generation of bottle and hollow beams using an axicon," Opt. Commun,  248, 7-14 (2005).
[CrossRef]

Opt. Commun. (2)

B. Lu, W. Huang, B. Zhang, F. Kong, and Q. Zhai, "Focusing properties of Bessel beams," Opt. Commun. 131, 223-228 (1996).
[CrossRef]

S. Chavez-Cerda and G. H. C. New, "Evolution of focused Hankel waves and Bessel beams," Opt. Commun. 181, 369-377 (2000).
[CrossRef]

Opt. Express (4)

Opt. Express. (1)

P.-T. Tai, W.-F. Hsieh, and C.-H. Chen, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express. 12, 5827-5833 (2004).
[CrossRef] [PubMed]

Opt. Lett. (6)

Science (1)

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-649 (2000).
[CrossRef] [PubMed]

Other (2)

B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W. C. Cheong, L. S. Zhang, and H. Wang, "Micromanipulation of high and low indices microparticles using a microfabricated double axicon," J. Appl. Phys. 99, 113104-1-113104-6 (2006).
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integral, Series, and Products", (Academic Press, New York, 1980).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup for generating supercontinuum bottle beam.

Fig. 2.
Fig. 2.

(a). The spectra of the supercontinuum (black) and the selected band centered at 530 nm (green), 670 nm (red), and 780 nm (pink), respectively; inset are the photography of spanning visible spectra from the prism (upper left) and the output pattern from the MF (upper right). (b) The 3D intensity distribution (top) and its 1D radial intensity profile (bottom).

Fig. 3.
Fig. 3.

The normalized 3D and 1D intensity distributions at (a) z = 26.20 mm, (b) z = 28.23 mm, (c) z = 32.09 mm, (d) z = 34.58 mm, (e) z = 40.27 mm, (f) z = 54.29 mm, (g) z = 59.12 mm, and (h) z = 64.50 mm.

Fig. 4.
Fig. 4.

Diameter variations of the dark region of the doughnut versus the distance z for the red light (red circles) and green light (green triangles).

Fig. 5.
Fig. 5.

The on-axis intensity of the optical bottle as a function of z.

Tables (1)

Tables Icon

Table.1 The measured (black) and calculated (red) range of bottle beam for the white-light SC, green light, red light, and IR.

Equations (4)

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E in ( r ) = E 0 exp [ r 2 w 2 ] .
E ( r , z ) = 2 πi λ B E 0 exp ( ikL ) exp ( i kD r 2 2 B ) 0 exp ( r 2 W 2 ) J 0 ( krr B ) exp ( i kA r 2 2 B ibr ) rdr ,
E ( o , z ) = 2 πi λB E 0 exp ( ikL ) 0 exp ( r 2 w 2 ) exp ( i kA r 2 2 B ibr ) rdr .
E ( o , z ) = 2 πi λB E 0 exp ( ikL ) { 1 2 μ ib 4 μ π μ exp ( b 2 4 μ ) [ 1 Φ ( ib 2 μ ) ] } .

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