Abstract

We show that spatial phase dislocations associated with optical vortices can be embedded in femtosecond laser beams by computer-generated holograms, provided that they are built in a setup compensating for the introduced spatial dispersion of the broad spectrum. We present analytical results describing two possible arrangements: a dispersionless 4f setup and a double-pass grating compressor. Experimental results on the generation of optical vortices in the output beam of a 20 fs Ti:sapphire laser and the proof-of-principle measurements with a broadband-tunable cw Ti:sapphire laser confirm our theoretical predictions.

© 2006 Optical Society of America

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2005 (1)

2004 (6)

2003 (2)

2002 (10)

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

M. V. Berry, "Coloured phase singularities," New J. Phys. 4, 66.1-66.14 (2002).
[CrossRef]

L. Torner and A. P. Sukhorukov, "Quadratic solitons," Opt. Photonics News 13, 42-47 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

A. G. Peele, P. J. McMahon, D. Paterson, Ch. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, "Observation of an x-ray vortex," Opt. Lett. 27, 1752-1754 (2002).
[CrossRef]

D. Ganic, X. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, "Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100%," Opt. Lett. 27, 1351-1353 (2002).
[CrossRef]

M. J. Padgett and L. Allen, "Orbital angular momentum exchange in cylindrical-lens mode converters," J. Opt. B: Quantum Semiclassical Opt. 4, S17-S19 (2002).
[CrossRef]

A. Dreischuh, D. Neshev, G. G. Paulus, F. Grasbon, and H. Walther, "Ring dark solitary waves: experiment versus theory," Phys. Rev. E 66, 066611 (2002).
[CrossRef]

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002).
[CrossRef]

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

2001 (2)

S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, "Soliton algebra by vortex-beam splitting," Opt. Lett. 26, 1004-1006 (2001).
[CrossRef]

B. A. Malomed, G. D. Peng, and P. L. Chu, "Helical versus fundamental solitons in optical fibers," Phys. Scr. 63, 386-390 (2001).
[CrossRef]

2000 (5)

1999 (2)

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

A. Dreischuh, G. G. Paulus, and F. Zacher, "Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations," Appl. Phys. B 69, 107-111 (1999).
[CrossRef]

1998 (3)

1997 (3)

G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, "Optical vortices produced with a nonspiral phase plate," Appl. Opt. 36, 8614-8621 (1997).
[CrossRef]

E. Abramochkin, N. Losevsky, and V. Volostnikov, "Generation of spiral-type laser beams," Opt. Commun. 141, 59-64 (1997).
[CrossRef]

D. V. Petrov, F. Canal, and L. Torner, "A simple method to generate optical beams with a screw phase dislocation," Opt. Commun. 143, 265-267 (1997).
[CrossRef]

1995 (1)

A. M. Weiner, "Femtosecond optical pulse shaping and processing," Prog. Quantum Electron. 19, 161-237 (1995).
[CrossRef]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

1992 (2)

G. A. Swartzlander, Jr., and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

1991 (1)

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

1989 (1)

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725-731 (1989).
[CrossRef]

1988 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 190-191 (1974).

1969 (1)

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

1968 (1)

E. B. Treacy, "Compression of picosecond light pulses," Phys. Lett. 2834-35 (1968).
[CrossRef]

Abramochkin, E.

E. Abramochkin, N. Losevsky, and V. Volostnikov, "Generation of spiral-type laser beams," Opt. Commun. 141, 59-64 (1997).
[CrossRef]

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

Allen, L.

M. J. Padgett and L. Allen, "Orbital angular momentum exchange in cylindrical-lens mode converters," J. Opt. B: Quantum Semiclassical Opt. 4, S17-S19 (2002).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padget, Optical Angular Momentum (Institute of Physics, 2004).

Andersen, D. R.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Armstrong, D. J.

Bang, O.

Barnett, S. M.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Belenkiy, A.

Berry, M. V.

M. V. Berry, "Coloured phase singularities," New J. Phys. 4, 66.1-66.14 (2002).
[CrossRef]

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 190-191 (1974).

Bezuhanov, K.

Briedis, D.

Buryak, A. V.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Canal, F.

D. V. Petrov, F. Canal, and L. Torner, "A simple method to generate optical beams with a screw phase dislocation," Opt. Commun. 143, 265-267 (1997).
[CrossRef]

Chang, J.-S.

Chervenkov, S.

Christov, I. P.

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725-731 (1989).
[CrossRef]

Chu, P. L.

B. A. Malomed, G. D. Peng, and P. L. Chu, "Helical versus fundamental solitons in optical fibers," Phys. Scr. 63, 386-390 (2001).
[CrossRef]

Coerwinkel, R. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Cojocaru, C.

Courtial, J.

Crasovan, L.-C.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Danailov, M. B.

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725-731 (1989).
[CrossRef]

Davidson, N.

R. Oron, N. Davidson, A. Friesem, and E. Hasman, "Efficient formation of pure helical laser beams," Opt. Commun. 182, 205-208 (2000).
[CrossRef]

Davis, P.

Denz, C.

A. Desyatnikov, C. Denz, and Yu. S. Kivshar, "Nonlinear optical beams carrying phase dislocations," J. Opt. A 6, S209-S212 (2004).
[CrossRef]

Desyatnikov, A.

A. Desyatnikov, C. Denz, and Yu. S. Kivshar, "Nonlinear optical beams carrying phase dislocations," J. Opt. A 6, S209-S212 (2004).
[CrossRef]

Di Trapani, P.

Dreischuh, A.

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, "Vortices in femtosecond laser fields," Opt. Lett. 29, 1942-1944 (2004).
[CrossRef] [PubMed]

A. Dreischuh, D. Neshev, G. G. Paulus, F. Grasbon, and H. Walther, "Ring dark solitary waves: experiment versus theory," Phys. Rev. E 66, 066611 (2002).
[CrossRef]

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002).
[CrossRef]

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, "Experimental generation of steering odd dark beams of finite length," J. Opt. Soc. Am. B 17, 2011-2017 (2000).
[CrossRef]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

A. Dreischuh, G. G. Paulus, and F. Zacher, "Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations," Appl. Phys. B 69, 107-111 (1999).
[CrossRef]

F. Grasbon, A. Dreischuh, F. Zacher, G. G. Paulus, and H. Walther, "Femtosecond interferometric autocorrelations in the presence of pulse front distortions," in Tenth International School on Quantum Electronics: Laser Physics and Applications, P.A.Atanasov and D.V.Stoyanov, eds., Proc. SPIE 3571, 164-168 (1999).

Edmundson, D.

Firth, W. J.

D. V. Skryabin and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

Franke-Arnold, S.

Freund, I.

Friesem, A.

R. Oron, N. Davidson, A. Friesem, and E. Hasman, "Efficient formation of pure helical laser beams," Opt. Commun. 182, 205-208 (2000).
[CrossRef]

Gan, X.

Ganic, D.

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Gibson, G.

Grasbon, F.

A. Dreischuh, D. Neshev, G. G. Paulus, F. Grasbon, and H. Walther, "Ring dark solitary waves: experiment versus theory," Phys. Rev. E 66, 066611 (2002).
[CrossRef]

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

F. Grasbon, A. Dreischuh, F. Zacher, G. G. Paulus, and H. Walther, "Femtosecond interferometric autocorrelations in the presence of pulse front distortions," in Tenth International School on Quantum Electronics: Laser Physics and Applications, P.A.Atanasov and D.V.Stoyanov, eds., Proc. SPIE 3571, 164-168 (1999).

Gu, M.

Hain, M.

Harvey, E.

Hasman, E.

R. Oron, N. Davidson, A. Friesem, and E. Hasman, "Efficient formation of pure helical laser beams," Opt. Commun. 182, 205-208 (2000).
[CrossRef]

Hayes, J. P.

Heckenberg, N. R.

Heritage, J. P.

Jeng, C.-C.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Jeon, J.-H.

Kaplan, A. E.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Kim, G.-H.

Kirschner, E. M.

Kivshar, Yu. S.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

A. Desyatnikov, C. Denz, and Yu. S. Kivshar, "Nonlinear optical beams carrying phase dislocations," J. Opt. A 6, S209-S212 (2004).
[CrossRef]

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," http://lanl.arxiv.org/abs/nlin.PS/0411024.

Ko, K.-H.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Królikowski, W.

Lai, B.

Law, C. T.

G. A. Swartzlander, Jr., and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

Leach, J.

J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1 (2003).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Lee, J.-H.

Lee, W.-H.

W.-H. Lee, "Computer generated holograms: techniques and applications" in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 1978), Vol. 16.
[CrossRef]

Lindner, F.

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

Liu, Y.

Losevsky, N.

E. Abramochkin, N. Losevsky, and V. Volostnikov, "Generation of spiral-type laser beams," Opt. Commun. 141, 59-64 (1997).
[CrossRef]

Malomed, B. A.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

B. A. Malomed, G. D. Peng, and P. L. Chu, "Helical versus fundamental solitons in optical fibers," Phys. Scr. 63, 386-390 (2001).
[CrossRef]

Mancuso, A. P.

Martorell, J.

Mazilu, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

McDuff, R.

McMahon, P. J.

McNulty, I.

Mihalache, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Minardi, S.

Molina-Terriza, G.

Moon, H.-J.

Motzek, K.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Neshev, D.

Nugent, K. A.

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 190-191 (1974).

Oron, R.

R. Oron, N. Davidson, A. Friesem, and E. Hasman, "Efficient formation of pure helical laser beams," Opt. Commun. 182, 205-208 (2000).
[CrossRef]

Padget, M. J.

L. Allen, S. M. Barnett, and M. J. Padget, Optical Angular Momentum (Institute of Physics, 2004).

Padgett, M. J.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, A. Pas'ko, S. M. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbotal angular momentum," Opt. Express 12, 5448-5456 (2004).
[CrossRef] [PubMed]

J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1 (2003).
[CrossRef]

M. J. Padgett and L. Allen, "Orbital angular momentum exchange in cylindrical-lens mode converters," J. Opt. B: Quantum Semiclassical Opt. 4, S17-S19 (2002).
[CrossRef]

Pas'ko, A.

Paterson, D.

Paulus, G. G.

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, "Vortices in femtosecond laser fields," Opt. Lett. 29, 1942-1944 (2004).
[CrossRef] [PubMed]

A. Dreischuh, D. Neshev, G. G. Paulus, F. Grasbon, and H. Walther, "Ring dark solitary waves: experiment versus theory," Phys. Rev. E 66, 066611 (2002).
[CrossRef]

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002).
[CrossRef]

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, "Experimental generation of steering odd dark beams of finite length," J. Opt. Soc. Am. B 17, 2011-2017 (2000).
[CrossRef]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

A. Dreischuh, G. G. Paulus, and F. Zacher, "Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations," Appl. Phys. B 69, 107-111 (1999).
[CrossRef]

F. Grasbon, A. Dreischuh, F. Zacher, G. G. Paulus, and H. Walther, "Femtosecond interferometric autocorrelations in the presence of pulse front distortions," in Tenth International School on Quantum Electronics: Laser Physics and Applications, P.A.Atanasov and D.V.Stoyanov, eds., Proc. SPIE 3571, 164-168 (1999).

Peele, A. G.

Peng, G. D.

B. A. Malomed, G. D. Peng, and P. L. Chu, "Helical versus fundamental solitons in optical fibers," Phys. Scr. 63, 386-390 (2001).
[CrossRef]

Petersen, D. E.

Petrov, D. V.

Recolons, J.

Regan, J. J.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Rozas, D.

Sacks, Z. S.

Schätzel, M. G.

Schmit, J.

G. A. Swartzlander, Jr., and J. Schmit, "Temporal correlation vortices and topological dispersion," Phys. Rev. Lett. 93093901 (2004).
[CrossRef] [PubMed]

Shih, M.-F.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Shum, P.

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

Smith, A. V.

Smith, C. P.

Somalingam, S.

Stankovic, S.

Sukhorukov, A. P.

L. Torner and A. P. Sukhorukov, "Quadratic solitons," Opt. Photonics News 13, 42-47 (2002).
[CrossRef]

Sun, X. W.

Swartzlander, G. A.

G. A. Swartzlander, Jr., and J. Schmit, "Temporal correlation vortices and topological dispersion," Phys. Rev. Lett. 93093901 (2004).
[CrossRef] [PubMed]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., "Holographic formation of optical-vortex filaments," J. Opt. Soc. Am. B 15, 2226-2234 (1998).
[CrossRef]

G. A. Swartzlander, Jr., and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Torner, L.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

L. Torner and A. P. Sukhorukov, "Quadratic solitons," Opt. Photonics News 13, 42-47 (2002).
[CrossRef]

S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, "Soliton algebra by vortex-beam splitting," Opt. Lett. 26, 1004-1006 (2001).
[CrossRef]

G. Molina-Terriza, J. Recolons, and L. Torner, "The curious arithmetic of optical vortices," Opt. Lett. 25, 1135-1137 (2000).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, "Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal," Opt. Lett. 23, 1444-1446 (1998).
[CrossRef]

D. V. Petrov, F. Canal, and L. Torner, "A simple method to generate optical beams with a screw phase dislocation," Opt. Commun. 143, 265-267 (1997).
[CrossRef]

Torres, J. P.

Towers, I.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Tran, Ch. Q.

Treacy, E. B.

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

E. B. Treacy, "Compression of picosecond light pulses," Phys. Lett. 2834-35 (1968).
[CrossRef]

Tschudi, T.

van der Veen, H. E.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Vasnetsov, M.

Vilaseca, R.

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Volostnikov, V.

E. Abramochkin, N. Losevsky, and V. Volostnikov, "Generation of spiral-type laser beams," Opt. Commun. 141, 59-64 (1997).
[CrossRef]

Walther, H.

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, "Vortices in femtosecond laser fields," Opt. Lett. 29, 1942-1944 (2004).
[CrossRef] [PubMed]

A. Dreischuh, D. Neshev, G. G. Paulus, F. Grasbon, and H. Walther, "Ring dark solitary waves: experiment versus theory," Phys. Rev. E 66, 066611 (2002).
[CrossRef]

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002).
[CrossRef]

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, "Experimental generation of steering odd dark beams of finite length," J. Opt. Soc. Am. B 17, 2011-2017 (2000).
[CrossRef]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

F. Grasbon, A. Dreischuh, F. Zacher, G. G. Paulus, and H. Walther, "Femtosecond interferometric autocorrelations in the presence of pulse front distortions," in Tenth International School on Quantum Electronics: Laser Physics and Applications, P.A.Atanasov and D.V.Stoyanov, eds., Proc. SPIE 3571, 164-168 (1999).

Wang, Q.

Weiner, A. M.

A. M. Weiner, "Femtosecond optical pulse shaping and processing," Prog. Quantum Electron. 19, 161-237 (1995).
[CrossRef]

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, "High-resolution femtosecond pulse shaping," J. Opt. Soc. Am. B 5, 1563-1572 (1988).
[CrossRef]

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Yakimenko, A. I.

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," http://lanl.arxiv.org/abs/nlin.PS/0411024.

Yin, H.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

Zacher, F.

A. Dreischuh, G. G. Paulus, and F. Zacher, "Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations," Appl. Phys. B 69, 107-111 (1999).
[CrossRef]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

F. Grasbon, A. Dreischuh, F. Zacher, G. G. Paulus, and H. Walther, "Femtosecond interferometric autocorrelations in the presence of pulse front distortions," in Tenth International School on Quantum Electronics: Laser Physics and Applications, P.A.Atanasov and D.V.Stoyanov, eds., Proc. SPIE 3571, 164-168 (1999).

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," http://lanl.arxiv.org/abs/nlin.PS/0411024.

Appl. Opt. (2)

Appl. Phys. B (1)

A. Dreischuh, G. G. Paulus, and F. Zacher, "Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations," Appl. Phys. B 69, 107-111 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

F. Lindner, G. G. Paulus, F. Grasbon, A. Dreischuh, and H. Walther, "Dispersion control in a 100-kHz-repetition-rate 30-fs Ti:sapphire regenerative amplifier system," IEEE J. Quantum Electron. 38, 1465-1470 (2002).
[CrossRef]

J. Mod. Opt. (1)

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725-731 (1989).
[CrossRef]

J. Opt. A (1)

A. Desyatnikov, C. Denz, and Yu. S. Kivshar, "Nonlinear optical beams carrying phase dislocations," J. Opt. A 6, S209-S212 (2004).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

M. J. Padgett and L. Allen, "Orbital angular momentum exchange in cylindrical-lens mode converters," J. Opt. B: Quantum Semiclassical Opt. 4, S17-S19 (2002).
[CrossRef]

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

J. Opt. Soc. Am. B (4)

New J. Phys. (2)

M. V. Berry, "Coloured phase singularities," New J. Phys. 4, 66.1-66.14 (2002).
[CrossRef]

J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1 (2003).
[CrossRef]

Opt. Commun. (5)

D. V. Petrov, F. Canal, and L. Torner, "A simple method to generate optical beams with a screw phase dislocation," Opt. Commun. 143, 265-267 (1997).
[CrossRef]

R. Oron, N. Davidson, A. Friesem, and E. Hasman, "Efficient formation of pure helical laser beams," Opt. Commun. 182, 205-208 (2000).
[CrossRef]

E. Abramochkin, N. Losevsky, and V. Volostnikov, "Generation of spiral-type laser beams," Opt. Commun. 141, 59-64 (1997).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Opt. Express (3)

Opt. Lett. (8)

Opt. Photonics News (1)

L. Torner and A. P. Sukhorukov, "Quadratic solitons," Opt. Photonics News 13, 42-47 (2002).
[CrossRef]

Phys. Lett. (1)

E. B. Treacy, "Compression of picosecond light pulses," Phys. Lett. 2834-35 (1968).
[CrossRef]

Phys. Rev. E (3)

D. V. Skryabin and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999).
[CrossRef]

A. Dreischuh, D. Neshev, G. G. Paulus, F. Grasbon, and H. Walther, "Ring dark solitary waves: experiment versus theory," Phys. Rev. E 66, 066611 (2002).
[CrossRef]

Phys. Rev. Lett. (6)

G. A. Swartzlander, Jr., and J. Schmit, "Temporal correlation vortices and topological dispersion," Phys. Rev. Lett. 93093901 (2004).
[CrossRef] [PubMed]

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Phys. Scr. (1)

B. A. Malomed, G. D. Peng, and P. L. Chu, "Helical versus fundamental solitons in optical fibers," Phys. Scr. 63, 386-390 (2001).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 190-191 (1974).

Prog. Quantum Electron. (1)

A. M. Weiner, "Femtosecond optical pulse shaping and processing," Prog. Quantum Electron. 19, 161-237 (1995).
[CrossRef]

Other (5)

W.-H. Lee, "Computer generated holograms: techniques and applications" in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 1978), Vol. 16.
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padget, Optical Angular Momentum (Institute of Physics, 2004).

F. Grasbon, A. Dreischuh, F. Zacher, G. G. Paulus, and H. Walther, "Femtosecond interferometric autocorrelations in the presence of pulse front distortions," in Tenth International School on Quantum Electronics: Laser Physics and Applications, P.A.Atanasov and D.V.Stoyanov, eds., Proc. SPIE 3571, 164-168 (1999).

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," http://lanl.arxiv.org/abs/nlin.PS/0411024.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

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Figures (15)

Fig. 1
Fig. 1

CGH for generating a 1D ODB in a general (nonperpendicular) orientation of the dislocation axis with respect to the grating stripes.

Fig. 2
Fig. 2

Gray-scale images of the 1D ODB 17 (top) and 35 cm (bottom) behind a single CGH, for cw (left) and femtosecond (fs) laser beams (right). The dashed horizontal line marks the position where the cross section of the light intensity distributions has been taken (see Fig. 4).

Fig. 3
Fig. 3

Vertical cross sections of the images shown in Fig. 2. Solid squares, cw regime; open circles, femtosecond regime. Propagation distances are 17 (top) and 35 cm (bottom).

Fig. 4
Fig. 4

Left graph: cross sections of the bottom images in Fig. 2 taken parallel to the 1D ODB. Right: numerical results for a propagation distance z = 0.6 L D . Solid squares and solid curve, cw regime; open circles and dashed curve, femtosecond regime. The transverse coordinate is in CCD-camera pixels for the experimental profiles and in arbitrary units for the numerical ones.

Fig. 5
Fig. 5

Gray-scale images of a quasi-2D dark beam formed by crossed 1D phase dislocations, for cw and femtosecond laser beams. Top, 17 cm behind a single CGH; bottom, 35 cm behind.

Fig. 6
Fig. 6

(a) Central cross sections of the images shown in Fig. 5 for z = 35 cm . Solid squares and solid curve, cw regime (horizontal and vertical slices, respectively); open circles and dashed curve, femtosecond regime (horizontal and vertical slices, respectively). (b) Numerically obtained vertical cross sections for propagation distance z = 0.6 L D . Solid curve, cw regime; dashed curve, femtosecond regime. (c) The same as in (b) but in the horizontal direction.

Fig. 7
Fig. 7

Gray-scale images of OV beams 17 (top) and 35 cm (bottom) behind a single CGH, for cw and femtosecond laser beams. Graph: corresponding horizontal cross sections of the OV beams recorded at 35 cm (left) and numerical results for propagation distance 0.6 L D (right). The transverse coordinate is in CCD-camera pixels for the experimental profiles and in arbitrary units for the numerical ones.

Fig. 8
Fig. 8

Illustration of the 4 f setup that is analyzed theoretically. G, diffraction grating; CGH, computer-generated hologram with an encoded phase singularity; L, lenses of focal length f ; D, iris diaphragm. The input, Fourier, and output planes are denoted by indices 0, f , and 1, respectively.

Fig. 9
Fig. 9

Illustration of the folded 4 f setup that is used in the experiment. CGH, computer-generated hologram; L, quartz lens; M, silver-coated mirror.

Fig. 10
Fig. 10

Frames: OVs recorded 35 cm after the 4 f setup in the cw and the femtosecond regimes. Graph: corresponding vertical cross sections of OV beams in the cw and femtosecond regimes (solid squares and open circles, respectively).

Fig. 11
Fig. 11

Comparison between the OV cross sections taken parallel to the stripes of the CGH in the femtosecond regime. Open circles, single CGH; solid squares, folded 4 f setup.

Fig. 12
Fig. 12

Illustration of the double-pass grating compressor that is analyzed theoretically. G, diffraction gratings; CGH, computer-generated hologram with an encoded phase singularity; l, compressor length. The planes of the gratings are indexed successively. The CGH is assumed to stand at the exit of the compressor.

Fig. 13
Fig. 13

Setup of the proof-of-principle experiment. PM, phase masks (phase CGHs); D, iris diaphragm; S, slit; M, removable mirrors (dashed boxes) and folding mirror ensuring vertical offset in the reverse pass through the system; F, filter; L, imaging lens ( f = 2 cm ) ; DPSSL, diode-pumped solid-state laser (Verdi V5); CCD, charge-coupled device camera.

Fig. 14
Fig. 14

Position of the OV at the exit plane of the pulse compressor as a function of the wavelength. Squares and triangles (solid and dashed curves) correspond to dispersion compensated and two uncompensated (zero-order and idler) beams, respectively. The dislocation is reproduced by the first PM.

Fig. 15
Fig. 15

Proof-of-principle simulations with OVs encoded in the (a) first and (b) last diffraction from a PM. Graphs: transverse cross sections of the uncompensated idler (solid curve) and desired OV (dotted curve). Insets: spectrally integrated experimental gray-scale images (idler, left; compensated OV, right).

Equations (23)

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E ( x , y , z = s ) = exp ( i k s ) i λ s E ( x 0 , y 0 , 0 ) exp ( i π r 2 λ s ) d x 0 d y 0 .
E ( x 0 , y 0 , 0 ) = T ( x 0 , y 0 ) exp [ ( x 0 2 + y 0 2 ) σ 0 2 ] .
T ( x 0 , y 0 ) = n = C n exp [ i n 2 π ( x 0 d ) ] exp [ i n φ ( x 0 , y 0 ) ] .
T ± ( x 0 , y 0 ) = C 1 exp ( i 2 π x 0 d ) exp { i [ φ 0 + sgn ( y 0 x 0 tan α ) π 2 ] } ,
E ± ( x 0 , y 0 ) = T ± ( x 0 , y 0 ) exp [ ( x 0 2 + y 0 2 ) σ 0 2 ] .
E y 1 > 0 ( x 1 , y 1 ) = exp [ ( x 1 2 + y 1 2 ) σ 0 2 ] T y 1 > 0 ( x 1 , y 1 ) ,
E y 1 < 0 ( x 1 , y 1 ) = exp [ ( x 1 2 + y 1 2 ) σ 0 2 ] T y 1 < 0 ( x 1 , y 1 ) .
E ( x 2 , y 2 ) = exp ( i k s ) i λ s [ 0 E y 1 > 0 ( x 1 , y 1 ) exp ( i π r 21 2 λ s ) + 0 E y 1 < 0 ( x 1 , y 1 ) exp ( i π r 21 2 λ s ) ] d x 1 d y 1 ,
E ( x 2 , y 2 ) = 2 C 1 i λ s exp ( i φ 0 ) exp ( i k s ) exp ( i π r 2 2 λ s ) 0 exp ( r 1 2 σ 0 2 ) exp ( i π r 1 2 λ s ) exp [ i k x 1 ( x 2 λ s d cos α ) ] sin [ k s y 1 ( y 2 + λ s d sin α ) ] d x 1 d y 1 .
y = ( x λ s d ) tan α .
E ( x f , y f ) = 1 λ f E 0 exp [ i k f ( x 0 x f + y 0 y f ) ] d x 0 d y 0 ,
t ( x , y ) = exp ( i k n d 0 ) exp [ i k 2 f ( x 2 + y 2 ) ]
E ( x 0 , y 0 ) = C 1 exp ( x 0 2 + y 0 2 σ 0 2 ) exp ( i 2 π d x 0 ) ,
E ( x f , y f ) = σ 0 2 π λ f C 1 exp [ ( x f λ f d ) 2 + y f 2 ( λ f π σ 0 ) 2 ] .
E ( x , y ) = C 1 λ 2 f 2 exp [ x 2 + y 2 ( β σ 0 ) 2 ] exp ( i 2 π β d x ) ,
E ( x , y ) = C 1 A 1 ( π λ f ) 2 exp [ x 2 + y 2 ( β σ 0 ) 2 ] exp [ i φ ( x , y ) ] exp [ i 2 π d ( 1 + 1 β ) x ] .
T ( x 1 , y 1 ) = C 1 exp [ i ( 2 π d ) x 1 ] ,
T ( x j , y j ) = C 1 exp [ i ( 2 π d ) x j ] , j = 2 , 3 ,
T ( x , y ) = C 1 exp [ i ( 2 π d ) x ] exp [ φ ( x , y ) ] .
E ( x 2 , y 2 ) = C 1 λ 2 s 0 l exp ( i 2 π x 2 d ) exp [ i k ( s 0 + l ) ] × [ C 1 exp ( i 2 π x 1 d ) E ( x 0 , y 0 ) exp ( i π r 10 2 λ s 0 ) d x 0 d y 0 ] exp ( i π r 21 2 λ l ) d x 1 d y 1 ,
E = ( x 2 , y 2 ) = C 1 2 i λ ( s 0 + l ) exp [ i k ( s 0 + l ) ] exp [ i π s 0 l λ ( s 0 + l ) d 2 ] × E ( x 0 , y 0 ) exp [ i 2 π l ( x 2 x 0 ) ( s 0 + l ) d ] exp [ i π r 20 2 λ ( s 0 + l ) ] d x 0 d y 0 .
E = ( x , y ) = C 1 2 i λ ( l 1 + l ) exp [ i k ( l 1 + l ) ] exp [ φ ( x , y ) ] exp [ i π l 1 l λ ( l 1 + l ) d 2 ] E ( x 2 , y 2 ) exp [ i 2 π l ( x x 2 ) d ( l 1 + l ) ] exp [ i π r 2 λ ( l 1 + l ) ] d x 2 d y 2 ,
E ( x , y ) = C 1 4 E diff exp { i k [ s l ( λ d ) 2 ] } exp [ φ ( x , y ) ] .

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