Abstract

A wave optical description of the effect of the primary aberrations on the temporal and spatial shape of an ultrashort pulse is presented. The calculations are based on the diffraction theory of aberrations investigated by Nijboer and Zernike, leading to an effective numerical treatment of Seidel aberrations. The explicit form of the recurrence relations for the coefficients of the circular polynomial expansion are published, as far as we know, for the first time. Comparisons between the results of wave optical and geometrical optical formulas are shown. The appearance of a boundary diffraction wave pulse, known from the aberration-free case, is also demonstrated.

© 2013 Optical Society of America

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2011 (3)

2010 (4)

2009 (4)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

2008 (2)

Y. T. Alvin, H. Gibbs, J. J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B 14, 119–131 (2008).
[CrossRef]

P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16, 13663–13675 (2008).
[CrossRef]

2007 (2)

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15, 10219–10230 (2007).
[CrossRef]

2006 (4)

W. Amir, T. A. Planchon, C. G. Durfee, J. A. Squier, P. Gabolde, R. Trebino, and M. Müller, “Simultaneous visualization of spatial and chromatic aberrations by two-dimensional Fourier transform spectral interferometry,” Opt. Lett. 31, 2927–2929 (2006).
[CrossRef]

H.-M. Heuck, P. Neumayer, T. Kuehl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef]

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

2004 (1)

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

2003 (4)

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

B. Schenkel, J. Biegert, U. Keller, C. Vozzi, M. Nisoli, G. Sansone, S. Stagira, S. D. Silvestri, and O. Svelto, “Generation of 3.8 fs pulses from adaptive compression of a cascaded hollow fiber supercontinuum,” Opt. Lett. 28, 1987–1989 (2003).
[CrossRef]

D. Zalvidea, “Phase mask for spatial and temporal control of ultrashort light pulses focused by lenses,” J. Opt. Soc. Am. A 20, 1981–1986 (2003).
[CrossRef]

D. Zalvidea and E. E. Sicre, “Ultrashort light pulse propagation in aberrant optical systems: spatial–temporal analysis,” J. Opt. A 5S310 (2003).
[CrossRef]

2001 (1)

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

2000 (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

1999 (1)

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

1996 (2)

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Zs. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

1994 (1)

Z. L. Horvath and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

1993 (3)

Z. L. Horvath and Zs. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

M. Kempe and W. Rudolph, “Impact of chromatic and spherical-aberration on the focusing of ultrashort light-pulses by lenses,” Opt. Lett. 18, 137–139 (1993).
[CrossRef]

M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
[CrossRef]

1992 (4)

Zs. Bor and Z. L. Horvath, “Distortion of femtosecond pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

A. Federico and O. Martinez, “Distortion of femtosecond pulses due to chromatic aberration in lenses,” Opt. Commun. 91, 104–110 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser-pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

J. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

1989 (3)

1988 (1)

Zs. Bor, “Distortion of femtosecond laser-pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907 (1988).
[CrossRef]

Alvin, Y. T.

Y. T. Alvin, H. Gibbs, J. J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B 14, 119–131 (2008).
[CrossRef]

Amir, W.

Andegeko, Y.

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

Apolonski, A.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Baltuška, A.

Benedetti, C.

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Biegert, J.

Blinder, P.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

Bor, Zs.

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Zs. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

Z. L. Horvath and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Z. L. Horvath and Zs. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Zs. Bor and Z. L. Horvath, “Distortion of femtosecond pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Zs. Bor, “Distortion of femtosecond pulse in lenses,” Opt. Lett. 14, 119–121 (1989).
[CrossRef]

Zs. Bor, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. 14, 862–864 (1989).
[CrossRef]

Zs. Bor, “Distortion of femtosecond laser-pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907 (1988).
[CrossRef]

Z. L. Horvath, A. P. Kovacs, and Zs. Bor, “Distortion of ultrashort pulses caused by aberrations,” in 15th International Conference on Ultrafast Phenomena, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper ThD16.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), Chap. 9.

Bourassin-Bouchet, C.

Bowlan, P.

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Bruce, N. C.

Bulanov, S. V.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Burgdorfer, J.

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Cavalieri, A. L.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Chavel, P.

Dantus, M.

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

de Rossi, S.

Delmotte, F.

Dombi, P.

P. Dombi, S. E. Irvine, P. Rácz, M. Lenner, N. Kroó, G. Farkas, A. Mitrofanov, A. Baltuška, T. Fuji, F. Krausz, and A. Y. Elezzabi, “Observation of few-cycle, strong-field phenomena in surface plasmon fields,” Opt. Express 18, 24206–24212 (2010).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef]

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Durfee, C. G.

Elezzabi, A. Y.

P. Dombi, S. E. Irvine, P. Rácz, M. Lenner, N. Kroó, G. Farkas, A. Mitrofanov, A. Baltuška, T. Fuji, F. Krausz, and A. Y. Elezzabi, “Observation of few-cycle, strong-field phenomena in surface plasmon fields,” Opt. Express 18, 24206–24212 (2010).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef]

Fagerholm, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Farkas, G.

P. Dombi, S. E. Irvine, P. Rácz, M. Lenner, N. Kroó, G. Farkas, A. Mitrofanov, A. Baltuška, T. Fuji, F. Krausz, and A. Y. Elezzabi, “Observation of few-cycle, strong-field phenomena in surface plasmon fields,” Opt. Express 18, 24206–24212 (2010).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef]

Federico, A.

A. Federico and O. Martinez, “Distortion of femtosecond pulses due to chromatic aberration in lenses,” Opt. Commun. 91, 104–110 (1992).
[CrossRef]

Fieß, M.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Friberg, A. T.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Fuchs, U.

Fuji, T.

Gabolde, P.

Garduno-Mejia, J.

Gibbs, H.

Y. T. Alvin, H. Gibbs, J. J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B 14, 119–131 (2008).
[CrossRef]

Gil, M. A.

Gong, Q.

S. Wang and Q. Gong, “Progress in femtochemistry and femtobiology,” Sci. China Phys. Mech. Astron. 54, 2103–2108 (2011).
[CrossRef]

Gonzalez-Galicia, M. A.

Goulielmakis, E.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Greenleaf, J. F.

J. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

Helml, W.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Heuck, H.-M.

H.-M. Heuck, P. Neumayer, T. Kuehl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[CrossRef]

Horvath, B.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Horvath, Z. L.

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Zs. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

Z. L. Horvath and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Z. L. Horvath and Zs. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Zs. Bor and Z. L. Horvath, “Distortion of femtosecond pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Z. L. Horvath, A. P. Kovacs, and Zs. Bor, “Distortion of ultrashort pulses caused by aberrations,” in 15th International Conference on Ultrafast Phenomena, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper ThD16.

Horváth, Z. L.

K. Mecseki, A. P. Kovács, and Z. L. Horváth, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” AIP Conf. Proc. 1228, 190–196 (2010).
[CrossRef]

Hu, J. J.

Y. T. Alvin, H. Gibbs, J. J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B 14, 119–131 (2008).
[CrossRef]

Huttunen, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Irvine, S. E.

P. Dombi, S. E. Irvine, P. Rácz, M. Lenner, N. Kroó, G. Farkas, A. Mitrofanov, A. Baltuška, T. Fuji, F. Krausz, and A. Y. Elezzabi, “Observation of few-cycle, strong-field phenomena in surface plasmon fields,” Opt. Express 18, 24206–24212 (2010).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef]

Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

Jin, Y. S.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

Keller, U.

Kempe, M.

Kienberger, R.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Klebniczki, J.

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Kleinfeld, D.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

Kovacs, A. P.

Z. L. Horvath, A. P. Kovacs, and Zs. Bor, “Distortion of ultrashort pulses caused by aberrations,” in 15th International Conference on Ultrafast Phenomena, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper ThD16.

Kovács, A. P.

K. Mecseki, A. P. Kovács, and Z. L. Horváth, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” AIP Conf. Proc. 1228, 190–196 (2010).
[CrossRef]

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Krausz, F.

P. Dombi, S. E. Irvine, P. Rácz, M. Lenner, N. Kroó, G. Farkas, A. Mitrofanov, A. Baltuška, T. Fuji, F. Krausz, and A. Y. Elezzabi, “Observation of few-cycle, strong-field phenomena in surface plasmon fields,” Opt. Express 18, 24206–24212 (2010).
[CrossRef]

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Kroó, N.

Kuehl, T.

H.-M. Heuck, P. Neumayer, T. Kuehl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[CrossRef]

Kurdi, G.

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Larson, A. M.

Y. T. Alvin, H. Gibbs, J. J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B 14, 119–131 (2008).
[CrossRef]

Lemell, C.

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Lenner, M.

Liseykina, T. V.

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Lõhmus, M.

Londrillo, P.

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Lovozoy, V. V.

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

Lu, J.

J. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

Macchi, A.

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Martinez, O.

A. Federico and O. Martinez, “Distortion of femtosecond pulses due to chromatic aberration in lenses,” Opt. Commun. 91, 104–110 (1992).
[CrossRef]

Mattei, G. O.

Mecseki, K.

K. Mecseki, A. P. Kovács, and Z. L. Horváth, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” AIP Conf. Proc. 1228, 190–196 (2010).
[CrossRef]

Migliori, B. J.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

Mitrofanov, A.

Morgan, D. P.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Müller, M.

Neev, J.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

Neumayer, P.

H.-M. Heuck, P. Neumayer, T. Kuehl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[CrossRef]

Nisoli, M.

Ortega-Martinez, R.

Peng, X.

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

Pervak, V.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Pestov, D.

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

Piksarv, P.

Planchon, T. A.

Prata, A.

Rácz, P.

Reivelt, K.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Rosete-Aguilar, M.

Rudolph, W.

Rusch, W. V. T.

Saari, P.

P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, “Basic diffraction phenomena in time domain,” Opt. Express 18, 11083–11088 (2010).
[CrossRef]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Salomaa, M. M.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Sansone, G.

Schenkel, B.

Schultze, M.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Sgattoni, A.

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Sicre, E. E.

D. Zalvidea and E. E. Sicre, “Ultrashort light pulse propagation in aberrant optical systems: spatial–temporal analysis,” J. Opt. A 5S310 (2003).
[CrossRef]

Silvestri, S. D.

Squier, J. A.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

W. Amir, T. A. Planchon, C. G. Durfee, J. A. Squier, P. Gabolde, R. Trebino, and M. Müller, “Simultaneous visualization of spatial and chromatic aberrations by two-dimensional Fourier transform spectral interferometry,” Opt. Lett. 31, 2927–2929 (2006).
[CrossRef]

Stagira, S.

Stamm, U.

Svelto, O.

Tajima, T.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Tempea, G.

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Trebino, R.

Tsai, P. S.

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

Turchetti, G.

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Udem, T.

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Uiberacker, M.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Valtna-Lukner, H.

Veisz, L.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Vozzi, C.

Wang, S.

S. Wang and Q. Gong, “Progress in femtochemistry and femtobiology,” Sci. China Phys. Mech. Astron. 54, 2103–2108 (2011).
[CrossRef]

Wilhelmi, B.

Wittrock, U.

H.-M. Heuck, P. Neumayer, T. Kuehl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), Chap. 9.

Yakovlev, V. S.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Zalvidea, D.

D. Zalvidea and E. E. Sicre, “Ultrashort light pulse propagation in aberrant optical systems: spatial–temporal analysis,” J. Opt. A 5S310 (2003).
[CrossRef]

D. Zalvidea, “Phase mask for spatial and temporal control of ultrashort light pulses focused by lenses,” J. Opt. Soc. Am. A 20, 1981–1986 (2003).
[CrossRef]

Zeitner, U. D.

AIP Conf. Proc. (1)

K. Mecseki, A. P. Kovács, and Z. L. Horváth, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” AIP Conf. Proc. 1228, 190–196 (2010).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (2)

H.-M. Heuck, P. Neumayer, T. Kuehl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[CrossRef]

V. S. Yakovlev, P. Dombi, G. Tempea, C. Lemell, J. Burgdorfer, T. Udem, and A. Apolonski, “Phase-stabilized 4 fs pulses at the full oscillator repetition rate for a photoemission experiment,” Appl. Phys. B 76, 329–332 (2003).
[CrossRef]

Braz. J. Phys. (1)

Zs. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

Curr. Opin. Biotechnol. (1)

P. S. Tsai, P. Blinder, B. J. Migliori, J. Neev, Y. S. Jin, J. A. Squier, and D. Kleinfeld, “Plasma-mediated ablation: an optical tool for submicrometer surgery on neuronal and vascular systems,” Curr. Opin. Biotechnol. 20, 90–99 (2009).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

J. Biomed. Opt. (1)

X. Peng, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef]

J. Mod. Opt. (1)

Zs. Bor, “Distortion of femtosecond laser-pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907 (1988).
[CrossRef]

J. Opt. A (1)

D. Zalvidea and E. E. Sicre, “Ultrashort light pulse propagation in aberrant optical systems: spatial–temporal analysis,” J. Opt. A 5S310 (2003).
[CrossRef]

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

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

New J. Phys. (1)

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

C. Benedetti, P. Londrillo, T. V. Liseykina, A. Macchi, A. Sgattoni, and G. Turchetti, “Ion acceleration by petawatt class laser pulses and pellet compression in a fast ignition scenario,” Nucl. Instrum. Methods Phys. Res. A 606, 89–93 (2009).
[CrossRef]

Opt. Commun. (5)

Zs. Bor and Z. L. Horvath, “Distortion of femtosecond pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Z. L. Horvath and Zs. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Z. L. Horvath and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

A. Federico and O. Martinez, “Distortion of femtosecond pulses due to chromatic aberration in lenses,” Opt. Commun. 91, 104–110 (1992).
[CrossRef]

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (1)

M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
[CrossRef]

Phys. Rev. E (2)

Z. L. Horvath and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef]

Rev. Mod. Phys. (3)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Sci. China Phys. Mech. Astron. (1)

S. Wang and Q. Gong, “Progress in femtochemistry and femtobiology,” Sci. China Phys. Mech. Astron. 54, 2103–2108 (2011).
[CrossRef]

Tissue Eng. Part B (1)

Y. T. Alvin, H. Gibbs, J. J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B 14, 119–131 (2008).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), Chap. 9.

Z. L. Horvath, A. P. Kovacs, and Zs. Bor, “Distortion of ultrashort pulses caused by aberrations,” in 15th International Conference on Ultrafast Phenomena, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper ThD16.

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Figures (5)

Fig. 1.
Fig. 1.

Interpretation of the aberration function.

Fig. 2.
Fig. 2.

Choice of the reference frame and notation related to the calculations.

Fig. 3.
Fig. 3.

Intensity distribution of a pulse with temporal duration τ=2T0 at the time t=7000T0,2394T0,1800T0, and 500T0 in the presence of primary spherical aberration characterized by A040=6λ0. The dashed curve shows the pulse front predicted by geometrical optics. The continuous lines indicate the geometrical caustic. The pulse is in front of the focal region in case (a) and behind the focal region in case (d). In case (b) the time was chosen so that the light pulse propagating along the marginal rays reaches the optical axis. Inset (c) shows the case when the pulse is going along rays that are focused into a point between the marginal and the paraxial focus.

Fig. 4.
Fig. 4.

Intensity distribution in planes given by ψ=0° (meridional plane), ψ=45°, and ψ=90° (sagital plane) for a pulse with temporal duration τ=2T0 at the time t=2000T0,200T0,0,200T0, and 2000T0 in the presence of primary coma characterized by A031=2.5λ0. The dashed curve shows the pulse front predicted by geometrical optics. In cases (a) and (e) the pulse is far from the paraxial focal plane. In inset (a) it is in front of the focus, and in inset (e) it is behind. Cases (b)–(d) show that in the vicinity of the paraxial image point the shape of the pulse front in the meridional plane forms a letter V. One can conclude that in the domain y>0 there are points close to the paraxial image point and the meridional plane at which the pulse passes through twice. In case (c) the pulse is at the circle of least confusion (in the middle between the sagital and the meridional focal lines).

Fig. 5.
Fig. 5.

Intensity distribution in planes given by ψ=0° (meridional plane), ψ=45°, and ψ=90° (sagital plane) for a pulse with temporal duration τ=2T0 at the time t=2000T0,0,150T0,300T0, and 2000T0 in the presence of primary astigmatism characterized by A022=1.5λ0. The dashed curve shows the pulse front predicted by geometrical optics. In cases (a) and (e) the pulse is far from the paraxial focal plane. In inset (a) it is in front of the focus, and in inset (e) it is behind. In case (b) and (d) the pulse is situated at the sagital and meridional (tangential) focal lines, respectively. In case (c) the pulse is at the circle of least confusion (in the middle between the sagital and the meridional focal lines).

Equations (86)

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E0(Q,t)=Ah(t+R/c)R,
h(t)=b(t)eiω0t
E(Q,t)=Ah(tΔt+R/c)R=Ah(t+(RΦ)/c)R.
E(Q,t)=F1{U(Q,ω)}=12πU(Q,ω)eiωtdω,
U(Q,ω)=F{E(Q,t)}=E(Q,t)eiωtdt.
U(Q,ω)=B(ωω0)Aeik(RΦ)R,
U(P,ω)=iωa22cR2AB(ωω0)eiωz/cY(u,v,ψ,Φ),
Y(u,v,ψ,Φ)=1π0102πei[kΦ(ρ,θ)vρcos(θψ)(u/2)ρ2]dθρdρ,
u=k(a/R)2z,
v=k(a/R)x2+y2=k(a/R)r,
E(P,t)=F1{U(P,ω)}=12πU(P,ω)eiωtdω
=ia2A4πcR2eiω0(tz/c)(ω0+Δω)B(Δω)Y(u,v,ψ,Φ)eiΔω(tz/c)d(Δω),
(x˜,y˜,z˜,t˜)=(xλ0,yλ0,zλ0,tT0),
b(t)=b˜(γt˜)=b˜(γt/T0).
E(P,t)=iπa2Aλ0γR2ei2π(t˜z˜)(1+Ω)B˜(2πΩ/γ)Y(u,v,ψ,Φ)ei2πΩ(t˜z˜)dΩ,
u=2π(1+Ω)(a/R)2z˜,
v=2π(1+Ω)(a/R)r˜,
b˜(t)=exp(t2),
B˜(ω)=πexp(ω2/4),
γ=2ln2T0τ=2ln2N,
Φlmn(ρ,θ)=Almnρncosmθ,
Φlmn(ρ,θ)=AlmnRnm(ρ)cosmθ,
Φ=Φ+Hρ2+Kρsinθ+Lρcosθ+M,
Y(u,v,ψ,Φ)=eikMY(u,v,ψ,Φ),
u=u+2kH,
vsinψ=vsinψ+kK,
vcosψ=vcosψ+kL.
Y(x,y,z,Φ)=eikMY(x+(R/a)K,y+(R/a)L,z+(R/a)2H,Φ).
Y(u,v,ψ,Φlmn)=p=0Cp(iαlnm)p0qpqp(mod2)(i)qmDpqIpq(n,m)(u,v)cos(qmψ),
Dpq=(p(pq)/2),
Ipq(n,m)(u,v)=01ei(u/2)ρ2[Rnm(ρ)]pJqm(ρv)ρdρ,
Ipq(n,m)(u,v)=eiu/4s=0(i)s(2s+1)js(u/4)jNpwsj(p)mq(1)(wsj(p)mq)/2Aj(p,q)Jwsj(p)+1(v)v,
[Rnm(ρ)]pR2s0(ρ)=jNpwsj(p)mqAj(p,q)Rwsj(p)mq(ρ).
2ρRnm(ρ)=nmn+1Rn1m+1(ρ)+n+m+2n+1Rn+1m+1(ρ),
4(n+1)ρ2Rnm(ρ)=n2m2nRn2m(ρ)+[(n+m)2n+(nm+2)2n+2]Rnm(ρ)+(n+2)2m2n+2Rn+2m(ρ).
Ip(4,0)(u,v)=eiu/4s=0(i)s(2s+1)js(u/4)j=0s2(pj)02pAj(p)(s)J2[s2(pj)]+1(v)v,
A0(0)(s)=1,
A(1)(s)=316[212s132s+143+12s112s+32+32s+1+12s+3],
Ak(p)(s)=l=02Akl(p1)(s)Al(1)[s2(p1)+2(kl)],
Y(u,v,ψ,Φ040)=p=0(iα040)pp!Ip(4,0)(u,v),
Y(u,v,ψ,Φ040)=eikA040/6Y(u2kA040,v,ψ,Φ040),
Ipq(3,1)(u,v)=eiu/4(1)(pq)/2s=0(i)s(2s+1)js(u/4)j=02s3p+2jq3p(1)jAj(p,q)(2s)J2s3p+2j+1(v)v,
A0(0,0)(s)=1,
A(1,1)(s)=116[63s1+9s+12+3s1+1s+121s+13s+36+9s+1+3s+3],
A(2,0)(s)=164[9278(s3)274(s1)1358(s+1)63s19s+17+278(s3)298(s+1)274(s+3)20+3s13s+37+274(s1)+298(s+1)278(s+5)6+9s+1+3s+39+1358(s+1)+274(s+3)+278(s+5)],
A(2,2)(s)=164[9818(s3)454(s1)1898(s+1)6+9s1+3s+17+818(s3)2558(s+1)454(s+3)209s1+9s+37+454(s1)+2558(s+1)818(s+5)63s+19s+39+1898(s+1)+454(s+3)+818(s+5)],
Ak(p,0)(s)=l=06Akl(p2,0)(s)Al(2,0)(s3(p2)+2(kl))ifq=0,
Ak(p,q)(s)=l=03Akl(p1,q1)(s)Gl(s3(p1)+2(kl),q1)ifq0,
G(n,m)=38[1+(m1)(m+1)22(n1)m2(m+2)n+(m1)(m+1)(m+3)2(n+1)13(m1)(m+1)22(n1)+(3m+1)2(m+1)6(n+1)m2(m+2)n+213+m2(m+2)n(3m+1)2(m+1)6(n+1)(m1)(m+1)22(n+3)1(m1)(m+1)(m+3)2(n+1)+m2(m+2)n+2(m1)(m+1)22(n+3)].
Y(u,v,ψ,Φ031)=p=0Cp(iα031)p0qpqp(mod2)(i)qDpqIpq(3,1)(u,v)cos(qψ),
Y(x,y,z,Φ031)=Y(x,y2(R/a)A031/3,z,Φ031),
Ipq(2,2)(u,v)=eiu/4s=0(i)s(2s+1)js(u/4)j=0sp+jq2p(1)jAj(p,q)(2s)J2(sp+j)+1(v)v,
A0(0,0)(s)=1,
A(1,1)(s)=14[13s+121+3s+1],
A(2,0)(s)=132[21s13s+188s+112+1s11s+38+8s+12+3s+1+1s+3],
A(2,2)(s)=132[215s135s+1840s+112+15s115s+38+40s+12+35s+1+15s+3],
Ak(p,0)(s)=l=04Akl(p2,0)(s)Al(2,0)(s2(p2)+2(kl))ifq=0,
Ak(p,q)(s)=l=02Akl(p1,q1)(s)Dl(s2(p1)+2(kl),2(q1))ifq0,
D(n,m)=14[1+m(m+2)n(m+1)(m+3)n+12m(m+2)n+m(m+2)n+21+(m+1)(m+3)n+1m(m+2)n+2].
Y(u,v,ψ,Φ022)=p=0Cp(iα022)p0qpqp(mod2)(1)qDpqIpq(2,2)(u,v)cos(2qψ),
Y(u,v,ψ,Φ022)=Y(ukA022,v,ψ,Φ022),
Y(u,v,ψ,Φ120)=Y(u2kA120,v,ψ,Φ=0).
E(x,y,z,t)=E0(x,y,zΔz120,tΔz120/c),
Δz120=2(R/a)2A120.
Y(x,y,z,Φ111)=Y(x,y(R/a)A111,z,Φ=0).
E(x,y,z,t)=E0(x,yΔy111,z,t),
Δy111=(R/a)A111.
S(R¯,α,θ)=R,
S(R¯,α,θ)=R¯Φ(α,θ),
r⃗(t,α,θ)=r⃗Q¯+c(tR/c)q⃗/|q⃗|,
r⃗Q¯(α,θ)=[R+Φ(α,θ)](sinθsinαe⃗x+cosθsinαe⃗ycosαe⃗z)
q⃗=gradS=(sinθsinα+sinθcosαR+ΦΦα+cosθ(R+Φ)sinαΦθ)e⃗x+(cosθsinα+cosθcosαR+ΦΦαsinθ(R+Φ)sinαΦθ)e⃗y+(cosα+sinαR+ΦΦα)e⃗z
ct0=|q⃗|Φ+(|q⃗|1)R,
r⃗(t,α,θ)=r⃗0+c(tt0)q⃗/|q⃗|,
r⃗0=(sinθcosαΦα+cosθsinαΦθ)e⃗x+(cosθcosαΦαsinθsinαΦθ)e⃗y+sinαΦαe⃗z.
Φlnm(α,θ)=Klnmsinnαcosmθ,
q⃗=(1+4ΦR+Φ)[cosαe⃗zsinα(14Ksin2αR+5Φ)e⃗y],
r⃗0=4Φcosαe⃗z4(ΦKsinα)sinαe⃗y,
ct0=|q⃗|Φ+(|q⃗|1)R=Φ(18ΦKsin2αR8ΦRKsin2αR),
q⃗=(1+3ΦR+Φ){sinθsinα(12KsinαcosθR+4Φ)e⃗x+[cosθsinα(12KsinαcosθR+4Φ)+Ksin2αR+4Φ]e⃗y+cosαe⃗z},
r⃗0=3Φcosαe⃗z+(2Ksinαcosθ3Φ)sinαsinθe⃗x+[(2Ksinαcosθ3Φ)sinαcosθ+Ksin2α]e⃗y,
ct0=|q⃗|Φ+(|q⃗|1)R=Φ+K2sin4α+2ΦKsinαcosθ9Φ22R+9Φ38Φ2KsinαcosθΦK2sin4α2R2+,
q⃗=(1+2ΦR+Φ){sinθsinαe⃗xcosθsinα(12KR+3Φ)e⃗y+cosαe⃗z},
r⃗0=2Φcosαe⃗z2Φsinαsinθe⃗x+2(KΦ)sinαcosθe⃗y
ct0=|q⃗|Φ+(|q⃗|1)R=Φ(12ΦKR2ΦRKR),
zbλ0=(tΔt)/T0112(a/R)2.

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