Abstract

Beams which follow curved trajectories are useful in a variety of applications, but up to now have been realized mainly in linear media. We demonstrate theoretically and experimentally the generation of second-harmonic (SH) beams which follow arbitrary convex caustic trajectories. These beams are created in a nonlinear photonic crystal with a second-order susceptibility having a tailored pattern; hence, the SH beam follows the desired trajectory after exiting the crystal. The same crystal can incorporate more than one trajectory, enabling the nonlinear creation of bottle beams as well as beams with switchable caustic trajectories that can be controlled by the phase-matching conditions.

© 2017 Optical Society of America

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References

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2016 (1)

2015 (4)

A. Libster-Hershko, S. Trajtenberg-Mills, and A. Arie, Opt. Lett. 40, 1944 (2015).
[Crossref]

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

U. Bar-Ziv, A. Postan, and M. Segev, Phys. Rev. B 92, 100301 (2015).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, Laser Photon. Rev. 9, L40 (2015).
[Crossref]

2014 (3)

I. Epstein and A. Arie, Phys. Rev. Lett. 112, 023903 (2014).
[Crossref]

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

2013 (4)

A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. M. Dudley, and F. Courvoisier, J. Eur. Opt. Soc. Rapid Publ. 8, 13019 (2013).
[Crossref]

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013).
[Crossref]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

2012 (1)

2011 (3)

L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, Opt. Express 19, 16455 (2011).
[Crossref]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

2010 (2)

2009 (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[Crossref]

2008 (1)

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

2007 (1)

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

2006 (1)

2003 (1)

D. G. Grier, Nature 424, 810 (2003).
[Crossref]

2000 (1)

1979 (2)

W. H. Lee, Appl. Opt. 18, 3661 (1979).
[Crossref]

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

Arie, A.

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, Laser Photon. Rev. 9, L40 (2015).
[Crossref]

A. Libster-Hershko, S. Trajtenberg-Mills, and A. Arie, Opt. Lett. 40, 1944 (2015).
[Crossref]

I. Epstein and A. Arie, Phys. Rev. Lett. 112, 023903 (2014).
[Crossref]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013).
[Crossref]

I. Dolev, T. Ellenbogen, and A. Arie, Opt. Lett. 35, 1581 (2010).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[Crossref]

D. Kasimov, A. Arie, E. Winebrand, G. Rosenman, A. Bruner, P. Shaier, and D. Eger, Opt. Express 14, 9371 (2006).
[Crossref]

Arlt, J.

Balazs, N. L.

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

Bandres, M. A.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

Bar-Ziv, U.

U. Bar-Ziv, A. Postan, and M. Segev, Phys. Rev. B 92, 100301 (2015).
[Crossref]

Baumgartl, J.

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

Berry, M. V.

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

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Broky, J.

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

Bruner, A.

Burnett, I.

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Cheng, J.

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Christodoulides, D. N.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

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

Cižmár, T.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Coll-Lladó, C.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Courvoisier, F.

Dalgarno, H. I. C.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Dholakia, K.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

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

Dogariu, A.

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

Dolev, I.

Dudley, J. M.

Eger, D.

Ellenbogen, T.

I. Dolev, T. Ellenbogen, and A. Arie, Opt. Lett. 35, 1581 (2010).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[Crossref]

Epstein, I.

I. Epstein and A. Arie, Phys. Rev. Lett. 112, 023903 (2014).
[Crossref]

Fejer, M. M.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, in Summaries of Papers Presented at the Lasers and Electro-Optics (CLEO ’02), OSA Technical Digest (Optical Society of America, 2002), pp. 377–378.

Ferrier, D. E. K.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Froehly, L.

Fu, S.

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

Furfaro, L.

Gallmann, L.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[Crossref]

Giust, R.

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

Greenfield, E.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Grier, D. G.

D. G. Grier, Nature 424, 810 (2003).
[Crossref]

Gunn-Moore, F. J.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Heyman, E.

Hu, Y.

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Hum, D. S.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, in Summaries of Papers Presented at the Lasers and Electro-Optics (CLEO ’02), OSA Technical Digest (Optical Society of America, 2002), pp. 377–378.

Jacquot, M.

Juwiler, I.

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, Laser Photon. Rev. 9, L40 (2015).
[Crossref]

Kaganovsky, Y.

Kaminer, I.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

Kasimov, D.

Keller, U.

Kurz, J. R.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, in Summaries of Papers Presented at the Lasers and Electro-Optics (CLEO ’02), OSA Technical Digest (Optical Society of America, 2002), pp. 377–378.

Lacourt, P. A.

Lee, W. H.

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

Li, L.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

Li, T.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

Libster, A.

A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013).
[Crossref]

Libster-Hershko, A.

Lilach, Y.

A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013).
[Crossref]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

Lu, J.

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Mathis, A.

Mayer, B. W.

Mazilu, M.

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

Mills, M.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

Nylk, J.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Padgett, M. J.

Phillips, C. R.

Postan, A.

U. Bar-Ziv, A. Postan, and M. Segev, Phys. Rev. B 92, 100301 (2015).
[Crossref]

Qiu, X.

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Rodríguez-Lara, B. M.

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

Rosenman, G.

Saltzman, A. J.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, in Summaries of Papers Presented at the Lasers and Electro-Optics (CLEO ’02), OSA Technical Digest (Optical Society of America, 2002), pp. 377–378.

Schober, A. M.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, in Summaries of Papers Presented at the Lasers and Electro-Optics (CLEO ’02), OSA Technical Digest (Optical Society of America, 2002), pp. 377–378.

Segev, M.

U. Bar-Ziv, A. Postan, and M. Segev, Phys. Rev. B 92, 100301 (2015).
[Crossref]

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Shaier, P.

Shapira, A.

A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013).
[Crossref]

Shemer, L.

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

Siviloglou, G. A.

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

Trajtenberg-Mills, S.

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, Laser Photon. Rev. 9, L40 (2015).
[Crossref]

A. Libster-Hershko, S. Trajtenberg-Mills, and A. Arie, Opt. Lett. 40, 1944 (2015).
[Crossref]

Tsur, Y.

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

Vettenburg, T.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[Crossref]

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Wang, S. M.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

Winebrand, E.

Zhang, C.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

Zhao, S.

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Zhou, J.

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

Zhu, S. N.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

Am. J. Phys. (1)

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

Appl. Opt. (1)

J. Eur. Opt. Soc. Rapid Publ. (1)

A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. M. Dudley, and F. Courvoisier, J. Eur. Opt. Soc. Rapid Publ. 8, 13019 (2013).
[Crossref]

Laser Photon. Rev. (1)

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, Laser Photon. Rev. 9, L40 (2015).
[Crossref]

Nat. Methods (1)

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, Nat. Methods 11, 541 (2014).
[Crossref]

Nat. Photonics (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[Crossref]

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

Nature (2)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[Crossref]

D. G. Grier, Nature 424, 810 (2003).
[Crossref]

Opt. Commun. (1)

A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Opt. Photon. News (1)

M. A. Bandres, I. Kaminer, M. Mills, B. M. Rodríguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Photon. News 24(6), 30 (2013).
[Crossref]

Phys. Rev. B (1)

U. Bar-Ziv, A. Postan, and M. Segev, Phys. Rev. B 92, 100301 (2015).
[Crossref]

Phys. Rev. Lett. (5)

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
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[Crossref]

I. Epstein and A. Arie, Phys. Rev. Lett. 112, 023903 (2014).
[Crossref]

Sci. Rep. (1)

S. Zhao, Y. Hu, J. Lu, X. Qiu, J. Cheng, and I. Burnett, Sci. Rep. 4, 6628 (2014).
[Crossref]

Other (2)

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, in Summaries of Papers Presented at the Lasers and Electro-Optics (CLEO ’02), OSA Technical Digest (Optical Society of America, 2002), pp. 377–378.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

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Figures (5)

Fig. 1.
Fig. 1. (a) Schematic diagram of our experiment: the input laser beam (red) incident on the curved modulation of the crystal generates the SH beam (green) which follows a curved trajectory. (b) Switchable trajectories: the trajectory of the SH beam can be controlled by changing the temperature for a crystal with two different regions, each phase matching the fundamental beam at different temperatures.
Fig. 2.
Fig. 2. Generation of a SH beam with a polynomial trajectory. (a) Microscopic photo of fabricated crystal. (b1) simulation and (b2) experimental results of SH beam propagation after the crystal, following the polynomial trajectory ax4. The designed trajectory is shown as a white dotted line.
Fig. 3.
Fig. 3. SH generation of bottle beams. (a) Microscopic photo of fabricated crystal showing the two different curved periodic domains spatially separated. (b) Illustration of the crystal and pump beam, drawn to scale with the y axis. (c1) Simulation and (c2) experimental results for two different bottle beams, with parabolic (top) and quartic (bottom) trajectories. The dotted line shows the planned trajectory.
Fig. 4.
Fig. 4. Thermal switching of the caustic trajectory. (a) Microscopic photo of fabricated crystal (top), showing the two different curved periodic domains with different poling periods. Bottom shows the designed pattern, with the dotted square showing the location of the microscopic image in the crystal. (b1) Simulation and (b2) experimental results for 50°C (top) and 150°C (bottom). The dotted line shows the planned cubic trajectory.
Fig. 5.
Fig. 5. Schematic illustration of the limitations described in the text: (a) corresponds to the limitation arising from the minimal resolution achievable by poling-RES and the constant poling period in x, Λ; (b) corresponds to the limitation arising from the input beam waist W0 and the propagation length Δx.

Equations (7)

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dϕ(y)dy=ksinθ=kc(x)1+[c(x)]2,
χ(2)(y,x)=χ(2)sign{cos[2πΛ(Δk)x+ϕ(y)]},
Δk(λ,T)=4πλ[n(λ,T)n(λ2,T)].
φ(y)=k2ωn2y2[a(1n)/y]1/n(2n1)(1n).
θ=cos1(RES·Δkπ),
I=κ[sin((Δk2π/Λ)L/2)(Δk2π/Λ)L/2]2,
Δk1(λ1,T1)Δk2(λ2,T2)=2πL.

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