Abstract

Analytical expression of the Airy transform of an arbitrary Hermite-Gaussian beam is derived. The optical field in the x-direction of the Airy transform of Hermite-Gaussian beams with transverse mode number m is the sum of the zero-order derivative to mth-order derivative of the Airy function with different weight coefficients. The analytical expressions of the centre of gravity and the beam spot size of an arbitrary Hermite-Gaussian beam passing through an Airy transform optical system are also presented, which are very concise. Because the Airy transform of a Hermite-Gaussian beam has the same evolution law in the two transverse directions, only the effects of the control parameter α and the transverse mode number m on the normalized intensity distribution, the centre of gravity, and the beam spot size in the x-direction are theoretically investigated, respectively. The Airy transform of Hermite-Gaussian beams is also realized in the experiment. The influence of the control parameters on the normalized intensity distribution, the centre of gravity, and the beam spot size is experimentally investigated, respectively. The experimental results are consistent with the theoretical simulation results. When Hermite-Gaussian beams pass through an Airy transform optical system, the number of lobes may change, and the importance of lobes with the same status in the input plane may become different. By using the Airy transform of Hermite-Gaussian beams, the practical applications of Hermite-Gaussian beams can be extended.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

2019 (12)

M. Yaalou, E. M. El Halba, Z. Hricha, and A. Belafhal, “Transformation of double–half inverse Gaussian hollow beams into superposition of finite Airy beams using an optical Airy transform,” Opt. Quant. Electron. 51(3), 64–75 (2019).
[Crossref]

Y. Zhou, Y. Xu, and G. Zhou, “Beam propagation factor of a cosh-Airy beam,” Appl. Sci. 9(9), 1817 (2019).
[Crossref]

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

S. A. Carvalho and S. De Leo, “The effect of the geometrical optical phase on the propagation of Hermite-Gaussian beams through transversal and parallel dielectric blocks,” J. Mod. Opt. 66(5), 548–556 (2019).
[Crossref]

Z. Ding, X. Li, J. Cao, and X. Ji, “Thermal blooming effect of Hermite-Gaussian beams propagating through the atmosphere,” J. Opt. Soc. Am. A 36(7), 1152–1160 (2019).
[Crossref]

X. Fan, X. Ji, H. Yu, H. Wang, Y. Deng, and L. Chen, “Kerr effect on propagation characteristics of Hermite-Gaussian beams,” Opt. Express 27(16), 23112–23123 (2019).
[Crossref]

L. R. Hofer, L. W. Jones, J. L. Goedert, and R. V. Dragone, “Hermite-Gaussian mode detection via convolution neural networks,” J. Opt. Soc. Am. A 36(6), 936–943 (2019).
[Crossref]

J. Wadhwa and A. Singh, “Generation of second harmonics by a self-focused Hermite-Gaussian laser beam in collisionless plasma,” Phys. Plasmas 26(6), 062118 (2019).
[Crossref]

J. Wadhwa and A. Singh, “Generation of second harmonics of intense Hermite-Gaussian laser beam in relativistic plasma,” Laser Part. Beams 37(01), 79–85 (2019).
[Crossref]

G. Liang and Q. Wang, “Controllable conversion between Hermite Gaussian and Laguerre Gaussian modes due to cross phase,” Opt. Express 27(8), 10684–10691 (2019).
[Crossref]

G. Zhou, R. Chen, and X. Chu, “Propagation of cosh-Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 116, 72–82 (2019).
[Crossref]

Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019).
[Crossref]

2018 (5)

G. Zhou, Z. Ji, and G. Ru, “Complete analytical expression of Lorentz-Hermite-Gauss laser beams,” Laser Eng. 40(1-3), 127–147 (2018).

K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018).
[Crossref]

D. Liu, Y. Wang, and H. Zhong, “Average intensity of radial phase-locked partially coherent standard Hermite-Gaussian beam in oceanic turbulence,” Opt. Laser Technol. 106, 495–505 (2018).
[Crossref]

Y. Shen, Y. Meng, X. Fu, and M. Gong, “Wavelength-tunable Hermite-Gaussian modes and an orbital-angular-momentum-tunable vortex beam in a dual-off-axis pumped Yb: CALGO laser,” Opt. Lett. 43(2), 291–294 (2018).
[Crossref]

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “A conversion of the hyperbolic-cosine Gaussian beam to a novel finite Airy-related beam using an optical Airy transform system,” Optik 171, 501–506 (2018).
[Crossref]

2017 (4)

2016 (2)

S. Restuccia, D. Giovannini, G. Gibson, and M. Padgett, “Comparing the information capacity of Laguerre-Gaussian and Hermite-Gaussian modal sets in a finite-aperture system,” Opt. Express 24(24), 27127–27136 (2016).
[Crossref]

H. S. Ghotra and N. Kant, “TEM modes influenced electron acceleration by Hermite-Gaussian laser beam in plasma,” Laser Part. Beams 34(3), 385–393 (2016).
[Crossref]

2015 (1)

2014 (4)

A. A. Kovalev, V. V. Kotlyar, and S. G. Zaskanov, “Diffraction integral and propagation of Hermite-Gaussian modes in a linear refractive index medium,” J. Opt. Soc. Am. A 31(5), 914–919 (2014).
[Crossref]

J. Rogel-Salazar, H. Jiménez-Romero, and A. S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89(2), 023807 (2014).
[Crossref]

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11(10), 105001 (2014).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self- bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

2013 (4)

2012 (6)

G. Zhou, “Vectorial structure of the far field of an elegant Hermite-Gaussian beam,” Opt. Laser Technol. 44(1), 218–225 (2012).
[Crossref]

Y. Jiang, K. Huang, and X. Lu, “The optical Airy transform and its application in generating and controlling the Airy beam,” Opt. Commun. 285(24), 4840–4843 (2012).
[Crossref]

Y. Jiang, K. Huang, and X. Lu, “Airy related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29(7), 1412–1416 (2012).
[Crossref]

P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012).
[Crossref]

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
[Crossref]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

2011 (4)

L. Li, T. Li, S. Wang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref]

G. Porat, I. Dolev, O. Barlev, and A. Arie, “Airy beam laser,” Opt. Lett. 36(20), 4119–4121 (2011).
[Crossref]

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Y. Ni, Z. Ji, G. Ru, and G. Zhou, “Propagation properties of controllable dark hollow laser beams in uniaxial crystals along the optical axis,” Laser Eng. 31(3-6), 0826001 (2011).
[Crossref]

2010 (2)

G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

2009 (3)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

H. Dai, X. Sun, D. Luo, and Y. Liu, “Airy beams generated by a binary phase element made of polymer-dispersed liquid crystals,” Opt. Express 17(22), 19365–19370 (2009).
[Crossref]

2008 (3)

2007 (3)

2005 (2)

2001 (1)

1995 (1)

1993 (1)

1985 (1)

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Acebal, P.

P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012).
[Crossref]

Aire, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Aire, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Arie, A.

G. Porat, I. Dolev, O. Barlev, and A. Arie, “Airy beam laser,” Opt. Lett. 36(20), 4119–4121 (2011).
[Crossref]

T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Arrizon, V.

Barlev, O.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Belafhal, A.

M. Yaalou, E. M. El Halba, Z. Hricha, and A. Belafhal, “Transformation of double–half inverse Gaussian hollow beams into superposition of finite Airy beams using an optical Airy transform,” Opt. Quant. Electron. 51(3), 64–75 (2019).
[Crossref]

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “A conversion of the hyperbolic-cosine Gaussian beam to a novel finite Airy-related beam using an optical Airy transform system,” Optik 171, 501–506 (2018).
[Crossref]

Bencheikh, A.

K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018).
[Crossref]

Bexter, T.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Blaya, S.

P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012).
[Crossref]

Boufalah, F.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “A conversion of the hyperbolic-cosine Gaussian beam to a novel finite Airy-related beam using an optical Airy transform system,” Optik 171, 501–506 (2018).
[Crossref]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Bu, J.

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Cai, Y.

Cao, J.

Cao, R.

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Carrada, R.

Carretero, L.

P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012).
[Crossref]

Carvalho, S. A.

S. A. Carvalho and S. De Leo, “The effect of the geometrical optical phase on the propagation of Hermite-Gaussian beams through transversal and parallel dielectric blocks,” J. Mod. Opt. 66(5), 548–556 (2019).
[Crossref]

Chávez-Cerda, A. S.

J. Rogel-Salazar, H. Jiménez-Romero, and A. S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89(2), 023807 (2014).
[Crossref]

Chavez-Rivas, F.

Chen, C.

Chen, L.

Chen, R.

G. Zhou, R. Chen, and X. Chu, “Propagation of cosh-Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 116, 72–82 (2019).
[Crossref]

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11(10), 105001 (2014).
[Crossref]

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
[Crossref]

Chen, X.

Cheng, M.

Christodoulides, D. N.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Chu, X.

G. Zhou, R. Chen, and X. Chu, “Propagation of cosh-Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 116, 72–82 (2019).
[Crossref]

Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019).
[Crossref]

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
[Crossref]

Chuu, C.-S.

Courvoisier, F.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Cruz, S. C. Y.

S. C. Y. Cruz and Z. Gress, “Group approach to the paraxial propagation of Hermite-Gaussian modes in a parabolic medium,” Ann. Phys. 383, 257–277 (2017).
[Crossref]

Dai, H.

Dalil-Essakali, L.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “A conversion of the hyperbolic-cosine Gaussian beam to a novel finite Airy-related beam using an optical Airy transform system,” Optik 171, 501–506 (2018).
[Crossref]

De Leo, S.

S. A. Carvalho and S. De Leo, “The effect of the geometrical optical phase on the propagation of Hermite-Gaussian beams through transversal and parallel dielectric blocks,” J. Mod. Opt. 66(5), 548–556 (2019).
[Crossref]

Deng, Y.

Dennis, M. R.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Ding, Z.

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Dolev, I.

Dragone, R. V.

Duan, K.

Dudley, J. M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

El Halba, E. M.

M. Yaalou, E. M. El Halba, Z. Hricha, and A. Belafhal, “Transformation of double–half inverse Gaussian hollow beams into superposition of finite Airy beams using an optical Airy transform,” Opt. Quant. Electron. 51(3), 64–75 (2019).
[Crossref]

Ellenbogen, T.

T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Ez-zariy, L.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “A conversion of the hyperbolic-cosine Gaussian beam to a novel finite Airy-related beam using an optical Airy transform system,” Optik 171, 501–506 (2018).
[Crossref]

Fallnich, C.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Fan, X.

Feng, S.

Froehly, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Fu, X.

Furfaro, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Ghotra, H. S.

H. S. Ghotra and N. Kant, “TEM modes influenced electron acceleration by Hermite-Gaussian laser beam in plasma,” Laser Part. Beams 34(3), 385–393 (2016).
[Crossref]

Gibson, G.

Giovannini, D.

Goedert, J. L.

Gong, M.

Gonzalez, L. A.

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Aire, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Gress, Z.

S. C. Y. Cruz and Z. Gress, “Group approach to the paraxial propagation of Hermite-Gaussian modes in a parabolic medium,” Ann. Phys. 383, 257–277 (2017).
[Crossref]

Guo, J.

Guo, L.

Hanssen, J. L.

Hellwig, T.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Hofer, L. R.

Howls, C. J.

Hricha, Z.

M. Yaalou, E. M. El Halba, Z. Hricha, and A. Belafhal, “Transformation of double–half inverse Gaussian hollow beams into superposition of finite Airy beams using an optical Airy transform,” Opt. Quant. Electron. 51(3), 64–75 (2019).
[Crossref]

Huang, K.

Y. Jiang, K. Huang, and X. Lu, “Airy related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29(7), 1412–1416 (2012).
[Crossref]

Y. Jiang, K. Huang, and X. Lu, “The optical Airy transform and its application in generating and controlling the Airy beam,” Opt. Commun. 285(24), 4840–4843 (2012).
[Crossref]

Huang, Q.

Jacquot, M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Ji, X.

Ji, Z.

G. Zhou, Z. Ji, and G. Ru, “Complete analytical expression of Lorentz-Hermite-Gauss laser beams,” Laser Eng. 40(1-3), 127–147 (2018).

Y. Ni, Z. Ji, G. Ru, and G. Zhou, “Propagation properties of controllable dark hollow laser beams in uniaxial crystals along the optical axis,” Laser Eng. 31(3-6), 0826001 (2011).
[Crossref]

Jia, S.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self- bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

Jiang, Y.

Y. Jiang, K. Huang, and X. Lu, “The optical Airy transform and its application in generating and controlling the Airy beam,” Opt. Commun. 285(24), 4840–4843 (2012).
[Crossref]

Y. Jiang, K. Huang, and X. Lu, “Airy related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29(7), 1412–1416 (2012).
[Crossref]

Jiménez-Romero, H.

J. Rogel-Salazar, H. Jiménez-Romero, and A. S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89(2), 023807 (2014).
[Crossref]

Jones, L. W.

Kant, N.

H. S. Ghotra and N. Kant, “TEM modes influenced electron acceleration by Hermite-Gaussian laser beam in plasma,” Laser Part. Beams 34(3), 385–393 (2016).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Kotlyar, V. V.

Kovalev, A. A.

Lacourt, P. A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Aire, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Li, J.

Li, L.

L. Li, T. Li, S. Wang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref]

Li, T.

L. Li, T. Li, S. Wang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref]

Li, X.

Liang, G.

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Aire, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Liu, D.

D. Liu, Y. Wang, and H. Zhong, “Average intensity of radial phase-locked partially coherent standard Hermite-Gaussian beam in oceanic turbulence,” Opt. Laser Technol. 106, 495–505 (2018).
[Crossref]

Liu, H.

Liu, S.

Liu, Y.

Lu, X.

Y. Jiang, K. Huang, and X. Lu, “Airy related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29(7), 1412–1416 (2012).
[Crossref]

Y. Jiang, K. Huang, and X. Lu, “The optical Airy transform and its application in generating and controlling the Airy beam,” Opt. Commun. 285(24), 4840–4843 (2012).
[Crossref]

Lu, Y.

Lü, B.

Luk, K. M.

Luo, D.

Manallah, A.

K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018).
[Crossref]

Manuel, S.

O. Vallée and S. Manuel, Airy Functions and Applications to Physics (Imperial College Press, 2010).

Martínez-Herrero, R.

Mata-Mendez, O.

Mathis, A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Mejías, P. M.

Meng, Y.

Meyrath, T. P.

Mihoubi, K.

K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018).
[Crossref]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Murciano, A.

P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012).
[Crossref]

Ni, Y.

Y. Ni, Z. Ji, G. Ru, and G. Zhou, “Propagation properties of controllable dark hollow laser beams in uniaxial crystals along the optical axis,” Laser Eng. 31(3-6), 0826001 (2011).
[Crossref]

Padgett, M.

Papazoglou, D.

D. Abdollahpour, S. Suntsov, D. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Porat, G.

Porfirev, A. P.

Raizen, M. G.

Restuccia, S.

Ring, J. D.

Rogel-Salazar, J.

J. Rogel-Salazar, H. Jiménez-Romero, and A. S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89(2), 023807 (2014).
[Crossref]

Ru, G.

G. Zhou, Z. Ji, and G. Ru, “Complete analytical expression of Lorentz-Hermite-Gauss laser beams,” Laser Eng. 40(1-3), 127–147 (2018).

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11(10), 105001 (2014).
[Crossref]

Y. Ni, Z. Ji, G. Ru, and G. Zhou, “Propagation properties of controllable dark hollow laser beams in uniaxial crystals along the optical axis,” Laser Eng. 31(3-6), 0826001 (2011).
[Crossref]

Ruiz, U.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Schepers, F.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Schreck, F.

Shen, Y.

Singh, A.

J. Wadhwa and A. Singh, “Generation of second harmonics by a self-focused Hermite-Gaussian laser beam in collisionless plasma,” Phys. Plasmas 26(6), 062118 (2019).
[Crossref]

J. Wadhwa and A. Singh, “Generation of second harmonics of intense Hermite-Gaussian laser beam in relativistic plasma,” Laser Part. Beams 37(01), 79–85 (2019).
[Crossref]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Skidanov, R. V.

Sun, P.

Sun, X.

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Tanaka, T.

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref]

Vallée, O.

O. Vallée and S. Manuel, Airy Functions and Applications to Physics (Imperial College Press, 2010).

Vaughan, J. C.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self- bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

Voloch, N.

T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Aire, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013).
[Crossref]

Wadhwa, J.

J. Wadhwa and A. Singh, “Generation of second harmonics by a self-focused Hermite-Gaussian laser beam in collisionless plasma,” Phys. Plasmas 26(6), 062118 (2019).
[Crossref]

J. Wadhwa and A. Singh, “Generation of second harmonics of intense Hermite-Gaussian laser beam in relativistic plasma,” Laser Part. Beams 37(01), 79–85 (2019).
[Crossref]

Wang, B.

Wang, F.

Wang, H.

Wang, J.

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Wang, M.

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Wang, Q.

Wang, S.

L. Li, T. Li, S. Wang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref]

Wang, T.

Wang, Y.

D. Liu, Y. Wang, and H. Zhong, “Average intensity of radial phase-locked partially coherent standard Hermite-Gaussian beam in oceanic turbulence,” Opt. Laser Technol. 106, 495–505 (2018).
[Crossref]

Xia, J.

Xu, Y.

Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019).
[Crossref]

Y. Zhou, Y. Xu, and G. Zhou, “Beam propagation factor of a cosh-Airy beam,” Appl. Sci. 9(9), 1817 (2019).
[Crossref]

Yaalou, M.

M. Yaalou, E. M. El Halba, Z. Hricha, and A. Belafhal, “Transformation of double–half inverse Gaussian hollow beams into superposition of finite Airy beams using an optical Airy transform,” Opt. Quant. Electron. 51(3), 64–75 (2019).
[Crossref]

Yan, X.

Yang, Y.

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Yu, H.

Yu, P. K.

Zaskanov, S. G.

Zhong, H.

D. Liu, Y. Wang, and H. Zhong, “Average intensity of radial phase-locked partially coherent standard Hermite-Gaussian beam in oceanic turbulence,” Opt. Laser Technol. 106, 495–505 (2018).
[Crossref]

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G. Zhou, F. Wang, and S. Feng, “Airy transform of Laguerre-Gaussian beams,” Opt. Express 28(13), 19683–19699 (2020).
[Crossref]

Y. Zhou, Y. Xu, and G. Zhou, “Beam propagation factor of a cosh-Airy beam,” Appl. Sci. 9(9), 1817 (2019).
[Crossref]

Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019).
[Crossref]

G. Zhou, R. Chen, and X. Chu, “Propagation of cosh-Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 116, 72–82 (2019).
[Crossref]

G. Zhou, Z. Ji, and G. Ru, “Complete analytical expression of Lorentz-Hermite-Gauss laser beams,” Laser Eng. 40(1-3), 127–147 (2018).

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11(10), 105001 (2014).
[Crossref]

G. Zhou, “Far field structural properties of a Gaussian vortex beam,” Laser Eng. 26(1-2), 1–17 (2013).

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
[Crossref]

G. Zhou, “Vectorial structure of the far field of an elegant Hermite-Gaussian beam,” Opt. Laser Technol. 44(1), 218–225 (2012).
[Crossref]

Y. Ni, Z. Ji, G. Ru, and G. Zhou, “Propagation properties of controllable dark hollow laser beams in uniaxial crystals along the optical axis,” Laser Eng. 31(3-6), 0826001 (2011).
[Crossref]

G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010).
[Crossref]

Zhou, Y.

Y. Zhou, Y. Xu, and G. Zhou, “Beam propagation factor of a cosh-Airy beam,” Appl. Sci. 9(9), 1817 (2019).
[Crossref]

Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019).
[Crossref]

Zhu, S.

L. Li, T. Li, S. Wang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011).
[Crossref]

Zhuang, X.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self- bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

Ann. Phys. (1)

S. C. Y. Cruz and Z. Gress, “Group approach to the paraxial propagation of Hermite-Gaussian modes in a parabolic medium,” Ann. Phys. 383, 257–277 (2017).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Appl. Phys. Lett. (2)

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011).
[Crossref]

Appl. Sci. (2)

Y. Zhou, Y. Xu, and G. Zhou, “Beam propagation factor of a cosh-Airy beam,” Appl. Sci. 9(9), 1817 (2019).
[Crossref]

Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019).
[Crossref]

IEEE Photonics J. (1)

P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012).
[Crossref]

J. Mod. Opt. (1)

S. A. Carvalho and S. De Leo, “The effect of the geometrical optical phase on the propagation of Hermite-Gaussian beams through transversal and parallel dielectric blocks,” J. Mod. Opt. 66(5), 548–556 (2019).
[Crossref]

J. Opt. Soc. Am. A (13)

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Figures (17)
Fig. 1.
Fig. 1. A schematic diagram of the Airy transform optical system.

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Fig. 2.
Fig. 2. Normalized intensity distribution in the x0-direction of Hermite-Gaussian beams in the input plane z=0. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

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Fig. 3.
Fig. 3. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.7 mm. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

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Fig. 4.
Fig. 4. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.4 mm. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

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Fig. 5.
Fig. 5. The centre of gravity (a) and the beam spot size (b) in the x-direction of the Airy transform of Hermite-Gaussian beams as a function of the control parameter α.

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Fig. 6.
Fig. 6. Normalized intensity distribution of Hermite-Gaussian beams in the input plane z=0. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 7.
Fig. 7. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.5 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 8.
Fig. 8. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.4 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 9.
Fig. 9. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.3 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 10.
Fig. 10. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.2 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 11.
Fig. 11. Schematic diagram of the experimental setup for generation of the Hermite-Gaussian beam as well as for the measurement of its spectral density after the Airy transform. BE: beam expander; SLM1 and SLM2: spatial light modulator; BS: beam splitter; L1, L2, L3, L4: thin lenses; BPA: beam profile analysis.

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Fig. 12.
Fig. 12. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams in the input plane z=0. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 13.
Fig. 13. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.5 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 14.
Fig. 14. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.4 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 15.
Fig. 15. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.3 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 16.
Fig. 16. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.2 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

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Fig. 17.
Fig. 17. Experimental measurement of the centre of gravity (a) and the beam spot size (b) in the x-direction of the Airy transform of Hermite-Gaussian beams as a function of the control parameter α.

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Equations (25)

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E m n ( x 0 , y 0 ) = E m ( x 0 , 0 ) E n ( y 0 , 0 ) = H m ( 2 x 0 w 0 ) H n ( 2 y 0 w 0 ) exp ⁡ ( − x 0 2 + y 0 2 w 0 2 ) ,
E m n ( x , y ) = E m ( x ) E n ( y ) = 1 | α β | ∫ − ∞ ∞ ∫ − ∞ ∞ E m n ( x 0 , y 0 ) A i ( x − x 0 α ) A i ( y − y 0 β ) d x 0 d y 0 ,
H m ( x ) = ∑ l = 0 [ m / 2 ] ( − 1 ) l 2 m − 2 l m ! l ! ( m − 2 l ) ! x m − 2 l ,
∫ − ∞ ∞ exp ⁡ ( i u 3 3 + i p u 2 + i q u ) d u  =  2 π exp ⁡ [ i p ( 2 p 2 3 − q ) ] A i ( q − p 2 ) ,
E m ( x ) = i m w 0 α 2 π | α | ∫ − ∞ ∞ H m ( − w 0 ξ 2 ) exp ⁡ ( i α 3 ξ 3 3 − w 0 2 ξ 2 4 + i ξ x ) d ξ  =  i m τ π ∫ − ∞ ∞ ∑ l = 0 [ m / 2 ] ( − 1 ) m − l m ! l ! ( m − 2 l ) ! ( α | α | 2 2 τ ) m − 2 l u m − 2 l exp ⁡ ( i u 3 3 − τ u 2 + i x α u ) d u  =  i m 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) ∑ l = 0 [ m / 2 ] ( − 1 ) m − l m ! l ! ( m − 2 l ) ! ( α | α | 2 2 τ ) m − 2 l ∑ s = 0 m − 2 l C m − 2 l , s A i ( s ) ( x 1 ) ,
C 0 , 0  =  1 ,
C 1 , 0  =  − i τ C 0 , 0 , C 1 , 1  =  − i C 0 , 0 ,
C t , 0 = − i τ C t − 1 , 0 , C t , s = ( − i τ C t − 1 , s − i C t − 1 , s − 1 ) , C t , t = − i C t − 1 , t − 1 , t > 1 , 0 < s < t − 1.
E 0 ( x ) = 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) A i ( x 1 ) ,
E 1 ( x ) = − 4 2 π α τ | α | exp ⁡ ( 2 τ 3 3 + τ x α ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] ,
E 2 ( x ) = 4 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 1 + 4 τ 3 ) A i ( x 1 )  +  8 τ 2 A i ′ ( x 1 ) + 4 τ A i ( 2 ) ( x 1 ) ] = 4 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 1 + 8 τ 3 + 4 τ x α ) A i ( x 1 )  +  8 τ 2 A i ′ ( x 1 ) ] ,
E 3 ( x )  =  − 4 2 π α τ | α | exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 6 τ + 8 τ 4 ) A i ( x 1 ) + ( 6 + 24 τ 3 ) A i ′ ( x 1 ) + 24 τ 2 A i ( 2 ) ( x 1 ) + 8 τ A i ( 3 ) ( x 1 ) ]  =  − 4 2 π α τ | α | exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 14 τ + 32 τ 4 + 24 τ 2 x α ) A i ( x 1 ) + ( 6 + 32 τ 3 + 8 τ x α ) A i ′ ( x 1 ) ] ,
E 4 ( x ) = 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 12  +  96 τ 3 + 64 τ 6 ) A i ( x 1 ) + ( 192 τ 2 + 256 τ 5 ) A i ′ ( x 1 ) + ( 96 τ + 384 τ 4 ) A i ( 2 ) ( x 1 ) + 256 τ 3 A i ( 3 ) ( x 1 ) + 64 τ 2 A i ( 4 ) ( x 1 ) ] = 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 12  +  448 τ 3 + 512 τ 6  +  96 τ x α + 512 τ 4 x α + 64 τ 2 x 2 α 2 ) A i ( x 1 ) + ( 320 τ 2 + 512 τ 5 + 256 τ 3 x α ) A i ′ ( x 1 ) ] ,
I m n ( x , y ) = I m ( x ) I n ( y ) = | E m ( x ) | 2 | E n ( y ) | 2 .
X m , c = ∫ − ∞ ∞ x | E m ( x ) | 2 d x ∫ − ∞ ∞ | E m ( x ) | 2 d x .
X 1 , c = α ( ∫ − ∞ ∞ x 1 exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 ∫ − ∞ ∞ exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 − τ 2 ) = − 3 α 3 w 0 2 .
X 0 , c = − α 3 w 0 2 , X 2 , c = − 5 α 3 w 0 2 , X 3 , c = − 7 α 3 w 0 2 , X 4 , c = − 9 α 3 w 0 2 .
W m , x = [ ∫ − ∞ ∞ x 2 | E m ( x ) | 2 d x ∫ − ∞ ∞ | E m ( x ) | 2 d x − X m , c 2 ] 1 / 2 .
W 1 , x = { α 2 ∫ − ∞ ∞ x 1 2 exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 ∫ − ∞ ∞ exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 − 2 α τ 2 X 1 , c − α 2 τ 4 − X 1 , c 2 } 1 / 2 = | α | 2 2 τ ( 24 τ 3 + 3 ) 1 / 2 .
W 0 , x = | α | 2 2 τ ( 8 τ 3 + 1 ) 1 / 2 , W 2 , x = | α | 2 2 τ ( 40 τ 3 + 7 ) 1 / 2 ,
W 3 , x = | α | 2 2 τ ( 56 τ 3 + 13 ) 1 / 2 , W 4 , x = | α | 2 2 τ ( 72 τ 3 + 21 ) 1 / 2 .
X m , c = − ( 2 m + 1 ) α 3 w 0 2 , Y n , c = − ( 2 n + 1 ) β 3 w 0 2 ,
W m , x = | α | 2 2 τ [ 8 ( 2 m + 1 ) τ 3 + m ( m + 1 ) + 1 ] 1 / 2 , W n , y = | β | 2 2 γ [ 8 ( 2 n + 1 ) γ 3 + n ( n + 1 ) + 1 ] 1 / 2 ,
X m , c = ∑ i N 1 ∑ j N 2 x i I ( x i , y j ) / ∑ i N 1 ∑ j N 2 I ( x i , y j ) ,
W m , x 2 = ∑ i N 1 ∑ j N 2 ( x i − X m , c ) 2 I ( x i , y j ) / ∑ i N 1 ∑ j N 2 I ( x i , y j ) ,

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