Abstract

A virtual source that yields a family of a Pearcey wave is demonstrated. A closed-form expression is derived for the Pearcey wave that simplifies to the paraxial Pearcey beam (PB) in the appropriate limit. From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a PB is obtained. The infinite series expression of a PB can give accuracy up to any order of the diffraction angle. By applying the integral representation of the Pearcey wave, the first three terms in the nonparaxial correction series to the paraxial PB are provided.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. S. Barwick, Opt. Lett. 35, 4118 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, Opt. Express 20, 18955 (2012).
[CrossRef]

S. H. Yan, B. L. Yao, M. Lei, D. Dan, Y. L. Yang, and P. Gao, Opt. Lett. 37, 4774 (2012).

2011

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

2010

S. Barwick, Opt. Lett. 35, 4118 (2010).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

2009

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

D. M. Deng and Q. Guo, J. Opt. Soc. Am. B 26, 2044 (2009).
[CrossRef]

2008

2007

2004

2003

2002

1979

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

1977

1976

1971

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

1946

T. Pearcey, Philos. Mag. Ser. 37(268), 311 (1946).

Aleahmad, P.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Ament, C.

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bandres, M. A.

Barwick, S.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Cannan, D.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

Chen, Z. G.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Chen, Z. R.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Christodoulides, D. N.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

Dan, D.

Deng, D. M.

Dennis, M. R.

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Dholakia, K.

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Felsen, L. B.

Gao, P.

Guo, Q.

Gutiérrez-Vega, J. C.

Hu, Y.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Ji, J. H.

Kaminer, I.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Koleskik, M.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Lei, M.

Li, T. C.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

Lindberg, J.

Mazilu, M.

Mills, M. S.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Miri, M. A.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Moloney, J. V.

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Morandotti, R.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

Mourka, A.

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Pearcey, T.

T. Pearcey, Philos. Mag. Ser. 37(268), 311 (1946).

Polynkin, P.

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Ring, J. D.

Salandrino, A.

Segev, M.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Seshadri, S. R.

Shi, Z. X.

Shin, S. Y.

Siviloglou, G. A.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

Song, Y. J.

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Yan, S. H.

Yang, Y. L.

Yao, B. L.

Yin, X. B.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zhang, P.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

Zhang, X.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. C. Li, R. Morandotti, X. Zhang, and Z. G. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zhang, Y. C.

Am. J. Phys.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Electron. Lett.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Nat. Photonics

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Philos. Mag. Ser.

T. Pearcey, Philos. Mag. Ser. 37(268), 311 (1946).

Phys. Rev. Lett.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

Science

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef]

Other

M. V. Berry and C. J. Howls, “Catastrophes and Canonical Integrals,” Digital Library of Mathematical Functions, National Institute of Standards and Technology, 2012, http://dlmf.nist.gov/36.2 .

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Figures (2)

Fig. 1.
Fig. 1.

Behavior of the normalized amplitude of the (1+1)-dimensional electric field as a function of x/w0 for a PB with a spot size. Solid curves show the paraxial approximation solution; dotted curves with triangle show the second-order nonparaxial correction solution; dash-dotted curves show the fourth-order nonparaxial correction solution; dotted curves with square show the sixth-order nonparaxial correction solution. The parameters are chosen as c0=0, (a) σ=0.1, z=1zR; (b) σ=0.1, z=2zR; (c) σ=0.3, z=1zR; and (d) σ=0.3, z=2zR.

Fig. 2.
Fig. 2.

Transverse normalized intensity distributions of the (1+2)-dimensional electric field for (a) and (e) the paraxial approximation solution, (b) and (f) the second-order nonparaxial correction solution, (c) and (g) the fourth-order nonparaxial correction solution, and (d) and (h) the sixth-order nonparaxial correction solution of the PBs with c0=0, σ=0.3, z=1zR in (a)–(d) and z=2zR in (e)–(h). Solid curves, dotted curves with triangle, dash-dotted curves, and dotted curves with squares in (i) and (j) are the cross-section normalized intensity distributions at y=0 corresponding to (a)–(d) and (e)–(h), respectively.

Equations (20)

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E(x,0)=Pe(xw0,c0),
(2x2+2z2+k2)E(x,z)=Scs×exp[D(x)]δ(x)δ(zzcs),
E(x,z)=12π+E˜(fx,z)exp(ifxx)dfx,
E˜(fx,z)=+E(x,z)exp(ifxx)dx,
E(x,z)=iScs4π+dfxexp(ifxx)exp(iw04fx4)×1/ζexp[iζ(zzcs)],
E(x,z)=exp[ik(zzcs)]iScs4πk×+dfxexp(ifxx)exp(iw14fx4)×exp[ifx2(zzcs)/(2k)].
Pe(s,t)=+dqexp(iq4+iq2t+iqs),
E(x,z)=iScs4πkw0exp[ik(zzcs)]×Pe(x/w0,i(zzcs)/(2kw02)).
E(x,0)=iScs4πkw0exp(ikzcs)×Pe(x/w0,izcs/(2kw02)).
zcs=2ic0kw02=4ic0zR,
Scs=4iπkw0exp(4kc0zR),
E(x,z)=kw0exp(4kc0zR)+dfxexp(ifxx)×exp(iw04fx4)1/ζexp[iζ(z+4ic0zR)].
(t2+2/z2+k2)E(x,y,z)=Scs(2)×exp[D(x,y)]δ(x)δ(y)δ(zzcs),
E(x,y,0)=Pe(xw0,c0)Pe(yw0,c0).
(t2+2/z2+k2)G(x,y,z)=δ(x)δ(y)δ(zzcs).
G(x,y,z)=exp(ikR)/(4πR),
E(x,y,z)=2iπkw02exp(4kc0zR)×exp[D(x,y)][exp(ikR)R].
E(x,z)=w0exp(4kc0zR)+dfxexp(ifxx)×exp[ifx2z/(2k)+iW0fx4]×[1+fx2/(2k2)+3fx4/(8k4)+(5/k+iz)fx6/(16k5)],
E(x,z)=w0/W01/4exp(4kc0zR)×[11/(2k2)2/x2+3/(8k4)4/x4(5/k+iz)/(16k5)6/x6]×Pe(x/W01/4,z/(2kW0)).
E(x,y,z)=w02/W0exp(4kc0zR)[11/(2k2)t2+3/(8k4)t4(5/k+iz)/(16k5)t6iz/(4k3)(11/(2k2)t2)4/(x2y2)z2/(32k6)8/(x4y4)]×Pe(x/W01/4,z/(2kW0))×Pe(y/W01/4,z/(2kW0)).

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