Abstract

We propose a new method to generate Bessel-like beam using cross-phase modulation. The hot rubidium atomic sample is shown to have the ability to vary a Gaussian beam (probe beam) into a Bessel-like beam when the sample is illuminated with a counter-propagating Gaussian beam (pump beam) tuned close to the atomic resonances. It is demonstrated that the Bessel-like beam exhibits self-healing after encountering an obstruction on the beam path. The parameters of the Bessel-like beam are found to be easily adjusted by the pump beam power and sample temperature. Moreover, this method is even applicable to the probe beam of low power, having more practical value than the method using self-phase modulation which needs high input beam power. The merits of variable parameter, no requirement for input beam power, simple setup, and low cost would make this method of significance in a variety of applications, especially in those areas where Bessel beam power is needed to be low and parameter to be adjusted easily without changing the setup.

© 2017 Optical Society of America

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References

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  21. R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
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2017 (2)

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

2015 (1)

2014 (1)

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

2012 (1)

2011 (1)

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

2006 (4)

Y. Matsuoka and M. Hirasawa, “Micro grooving of metallic material using a Bessel beam,” Rev. Lser. Eng. 34, 842–847 (2006).

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84, 423–430 (2006).

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

E. McLeod, A. B. Hopkins, and C. B. Arnold, “Multiscale Bessel beams generated by a tunable acoustic gradient index of refraction lens,” Opt. Lett. 31(21), 3155–3157 (2006).
[PubMed]

2003 (1)

2001 (2)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).

W. Lee, Y. Noh, J. Jeon, J. Lee, J. Chang, K. Ko, and J. Lee, “Conical emission as a result of self-phase modulation in samarium vapor under the near-resonant condition,” J. Opt. Soc. Am. B 18, 101–105 (2001).

2000 (2)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).

1997 (1)

F. Castaldo, D. Paparo, and E. Santamato, “Chaotic and hexagonal spontaneous pattern formation in the cross section of a laser beam in a defocusing Kerr-like film with single feedback mirror,” Opt. Commun. 143, 57–61 (1997).

1991 (1)

1990 (1)

1987 (1)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[PubMed]

1981 (1)

1978 (1)

1970 (1)

D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).

Arakelian, S. M.

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).

Arnold, C. B.

Bai, J.

Bai, J.-T.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Bai, S.-J.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Bai, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Bélanger, P. A.

Bian, F.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Castaldo, F.

F. Castaldo, D. Paparo, and E. Santamato, “Chaotic and hexagonal spontaneous pattern formation in the cross section of a laser beam in a defocusing Kerr-like film with single feedback mirror,” Opt. Commun. 143, 57–61 (1997).

Chang, J.

Chattrapiban, N.

Chen, H.-W.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Cheng, X.

Cheng, X.-M.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Cofield, D.

Danaci, O.

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

David, K. N.

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

Denschlag, J. H.

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

Dholakia, K.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).

Du, X.-L.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Durbin, S. D.

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[PubMed]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[PubMed]

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).

Glasser, R. T.

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

Grischkowsky, D.

D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).

Guo, C.-X.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Herman, R. M.

Hesselink, L.

Hill, W. T.

Hirasawa, M.

Y. Matsuoka and M. Hirasawa, “Micro grooving of metallic material using a Bessel beam,” Rev. Lser. Eng. 34, 842–847 (2006).

Hopkins, A. B.

Inoue, T.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84, 423–430 (2006).

Jeon, J.

Jiang, M.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Kizuka, Y.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84, 423–430 (2006).

Knutson, E. M.

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

Ko, K.

Lang, F.

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

Lee, J.

Lee, W.

Lu, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Matsuoka, Y.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84, 423–430 (2006).

Y. Matsuoka and M. Hirasawa, “Micro grooving of metallic material using a Bessel beam,” Rev. Lser. Eng. 34, 842–847 (2006).

McLeod, E.

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[PubMed]

Mishra, S. R.

Nemoto, S.

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).

Noh, Y.

Paparo, D.

F. Castaldo, D. Paparo, and E. Santamato, “Chaotic and hexagonal spontaneous pattern formation in the cross section of a laser beam in a defocusing Kerr-like film with single feedback mirror,” Opt. Commun. 143, 57–61 (1997).

Pender, J.

Pu, J.

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).

Ram, S. P.

Rawat, H. S.

Ren, Z.

Rios, C.

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

Rioux, M.

Rogers, E. A.

Roy, R.

Santamato, E.

F. Castaldo, D. Paparo, and E. Santamato, “Chaotic and hexagonal spontaneous pattern formation in the cross section of a laser beam in a defocusing Kerr-like film with single feedback mirror,” Opt. Commun. 143, 57–61 (1997).

Schmid, S.

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

Shen, Y. R.

Sibbett, W.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).

Sun, Y.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Swaim, J. D.

J. D. Swaim, K. N. David, E. M. Knutson, C. Rios, O. Danaci, and R. T. Glasser, “Atomic vapor as a source of tunable, non-Gaussian self-reconstructing optical modes,” Sci. Rep. 7, 42311 (2017).
[PubMed]

Thalhammer, G.

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

Tiwari, S. K.

Tremblay, R.

Wang, E.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Wang, H.-L.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Wang, W.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Wiggins, T. A.

Winkler, K.

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

Wu, R.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Yan, S.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Yin, J.-P.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Yin, X.

Yin, X.-L.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Yin, Y.-L.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Yong, X.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Zhang, H.

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).

Zhang, Q.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Zhang, Y.

Q. Zhang, X.-M. Cheng, Y. Zhang, X.-L. Yin, M. Jiang, H.-W. Chen, and J.-T. Bai, “Optical limiting using spatial self-phase modulation in hot atomic sample,” Opt. Laser Technol. 88, 54–60 (2017).

Y. Zhang, X. Cheng, X. Yin, J. Bai, P. Zhao, and Z. Ren, “Research of far-field diffraction intensity pattern in hot atomic Rb sample,” Opt. Express 23(5), 5468–5476 (2015).
[PubMed]

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Zhao, J.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

Zhao, P.

Zheng, G.-J.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Zhou, Z.-N.

X.-L. Du, Y.-L. Yin, G.-J. Zheng, C.-X. Guo, Y. Sun, Z.-N. Zhou, S.-J. Bai, H.-L. Wang, X. Yong, and J.-P. Yin, “Generation of a dark hollow beam by a nonlinear ZnSe crystal and its propagation properties in free space: Theoretical analysis,” Opt. Commun. 322, 179–182 (2014).

Appl. Opt. (2)

Appl. Phys., A Mater. Sci. Process. (1)

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys., A Mater. Sci. Process. 84, 423–430 (2006).

J. Opt. Soc. Am. A (1)

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

Nano Lett. (1)

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely Coherent Nonlinear Optical Response in Solution Dispersions of Graphene Sheets,” Nano Lett. 11(12), 5159–5164 (2011).
[PubMed]

New J. Phys. (1)

S. Schmid, G. Thalhammer, K. Winkler, F. Lang, and J. H. Denschlag, “Long distance transport of ultracold atoms using a 1D optical lattice,” New J. Phys. 8, 73–197 (2006).

Opt. Commun. (4)

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[PubMed]

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Figures (6)

Fig. 1
Fig. 1 Scheme of the experimental setup. Ti: S laser: Ti: sapphire laser; HWP: half-wave plate; HR: high reflective mirror; PBS: polarization beam splitter.
Fig. 2
Fig. 2 Non-Gaussian Bessel-like beam profiles. (a) Image of the output pump beam; (b) Image of the output probe beam; (c) The lateral intensity distribution of the beam and the theoretical simulation obtained by Eq. (9) with n 2 =1.97× 10 16 cm 2 /W; (d) Interferometric image of the output probe beam and a Gaussian beam.
Fig. 3
Fig. 3 Self-reconstruction of the Bessel-like beam generated using XPM in atomic Rb vapor. Images of unobstructed (top) and obstructed (bottom) modes at various positions along the propagation coordinate. The right side of the sample cell is taken as the origin point of z, and the obstruction is inserted at z = 18 cm.
Fig. 4
Fig. 4 Tunability of the size of Bessel-like beam obtained from atomic vapor by varying pump power. (a) Beam images with various pump power, keeping the cell temperature fixed at T = 140 °C and probe power at 0.1 mW; (b) Central spot radius and intensity as a function of pump power; (c) Rings number with respect to pump power.
Fig. 5
Fig. 5 Tunability of the size of Bessel-like beam obtained from atomic vapor by varying sample cell temperature. (a) Beam images with various cell temperature, keeping the pump power fixed at T = 120 mW and probe power at 0.1 mW; (b) Central spot radius as a function of cell temperature; (c) Number of the rings with respect to cell temperature.
Fig. 6
Fig. 6 Cross-induced transparency in atomic vapor. (a) Transmission of the probe beam from sample cell with respect to the pump power, keeping the cell temperature fixed at 140 °C; (b) Transmission of the probe beam from sample cell with respect to the cell temperature, keeping the pump power fixed at power fixed at 3mW and 120 mW.

Equations (15)

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P (3) =3 ε 0 χ (3) ( E ˜ 1 + E ˜ 2 ) ( E ˜ 1 + E ˜ 2 ) * ( E ˜ 1 + E ˜ 2 )+c.c = 3 8 ε 0 χ (3) {[| A ˜ 1 | 2 +2| A ˜ 2 | 2 + A ˜ 1 A ˜ 2 * exp(2i k 0 z)] × A ˜ 1 exp[i(ωt k 0 z)]+[2| A ˜ 1 | 2 +| A ˜ 2 | 2 + A ˜ 1 * A ˜ 2 exp(2i k 0 z)] A ˜ 2 exp[i(ωt+ k 0 z)]}+c.c.
P ˜ backward (3) = 1 2 P ˜ 2 (3) exp[i(ωt+ k 0 z)] = 3 8 ε 0 χ (3) [2| A ˜ 1 | 2 +| A ˜ 2 | 2 ] A ˜ 2 ×exp[i(ωt+ k 0 z)]
A ˜ 2 z =i μωc 2 n 0 P ˜ 2 (3)
A ˜ 2 z =i k 0 3 8 χ (3) n 0 [2| A ˜ 1 | 2 +| A ˜ 2 | 2 ] A ˜ 2
A ˜ 2 z =i k 0 n 2 [2| A ˜ 1 | 2 +| A ˜ 2 | 2 ] A ˜ 2 =i k NL A ˜ 2
φ NL =2 k 0 0 l n 2 I 1 (0,0) r 0 2 r p 2 (z) exp(2 r 2 / r p 2 (z)) dz
I 2 = | 1 iλD | 2 | 0 0 2π E ˜ 2 (0,l)exp(ikrθcosϕ)exp[i( k r 2 2R(z) + ϕ NL (r)]rdrdφ | 2
J 0 (x)= 1 2π 0 2π exp(ixcosϕ)dϕ = 1 2π 0 2π exp(ixcosϕ)dϕ
I 2 = I 0 | 0 J 0 (krθ)exp[i( k r 2 2R( z 0 ) + ϕ NL (r)]rdr | 2
E 2 (r,z)= E 20 exp[ r 2 r p 2 (z) ]exp{i k 0 [z+ r 2 2R( z ) ]+ ϕ NL (r)]}
E g (r, z g )= E g0 exp[ r g 2 r p 2 ( z g ) ]×exp{i k 0 [ z g + r g 2 2R( z g ) ]}
I t =[ E 2 (r,z)+ E g (r, z g )] [ E 2 (r,z)+ E g (r, z g )] * = E 20 2 exp[ 2 r 2 r p 2 (z) ]+ E g0 2 exp[ 2 r 2 r p 2 ( z g ) ] +2 E 20 E g0 exp[ r 2 r p 2 (z) r 2 r p 2 ( z g ) ] ×cos{ k 0 (z z g )+ k 0 [ r 2 2R(z) r 2 2R( z g ) ]+ ϕ NL (r)]}}
I t = I 2 + I g +2 I 2 I g cos[ k 0 (z z g )+ ϕ NL (r)]
ϕ NL (0)exp(2 r 2 / r 0 2 )=2mπ(m=0,1,2)
r m+1 2 r m 2 r 0 2 =ln m m+1

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