Abstract

Experimental studies of ultrafast beam shaping have come about from the need to compensate diffraction-induced dispersive effects in femtosecond laser beams. From a theoretical point of view, chromatic matching of diffracted spherical waves in the vicinity of the geometrical focus is attained by applying conveniently dispersive boundary conditions in the far-field zone, a subject thoroughly analyzed in the paraxial regime. For applications demanding high spatial resolution, however, high-numerical-aperture microscope objectives may be employed instead and would lead to nonparaxiality of the focal wavefields. These circumstances have motivated our investigation. Concretely we report on prerequisites for spectral invariance extended to wide-angle geometries, which provides stabilization of the spatiotemporal response in the Fourier plane. In this context, general boundary conditions are given in the frame of the Debye representation of wavefields. Features of this sort of dynamic apodization (spatial filtering) leading to perfect achromatization are described in detail.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

G. Nyitray, V. Mathew, and S. V. Kukhlevsky, “Generation and interference collapse of distortion-less fs pulses in free space by Fresnel sources,” Opt. Commun. 281, 1082-1086 (2008).
[CrossRef]

2007 (5)

2006 (5)

2005 (1)

2004 (1)

2003 (3)

2002 (5)

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, “Correlation between spatial and temporal uncertainties of a wave packet,” Opt. Commun. 209, 377-382 (2002).
[CrossRef]

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218-2222 (2002).
[CrossRef]

J. Amako, K. Nagasaka, and N. Kazuhiro, “Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements,” Opt. Lett. 27, 969-971 (2002).
[CrossRef]

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]

2001 (3)

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

B. K. A. Ngoi, K. Venkatakrishnan, B. Tan, P. Stanley, and L. E. N. Lim, “Angular dispersion compensation for acousto-optic devices used for ultrashort-pulsed laser micromachining,” Opt. Express 9, 200-206 (2001).
[CrossRef] [PubMed]

S. V. Kukhlevsky, G. Nyitray, and V. L. Kantsyrev, “Fields of optical waveguides as waves in free space,” Phys. Rev. E 64, 026603 (2001).
[CrossRef]

2000 (1)

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862-873 (2000).
[CrossRef]

1998 (2)

1997 (2)

1996 (2)

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. A: Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771-778 (1996).
[CrossRef]

1994 (1)

E. Heyman and T. Melamed, “Certain consideration in aperture synthesis for ultra wideband/short-pulsed fields,” IEEE Trans. Antennas Propag. AP-42, 518-525 (1994).
[CrossRef]

1993 (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mol. Spectrosc. 40, 1631-1651 (1993).

1981 (1)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-210 (1981).
[CrossRef]

Amako, J.

Andrés, P.

Antolini, R.

Besieris, I. M.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. A: Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Bor, Z.

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

Caballero, M. T.

C. J. Zapata-Rodríguez and M. T. Caballero, “Isotropic compensation of diffraction-driven angular dispersion,” Opt. Lett. 32, 2472-2474 (2007).
[CrossRef] [PubMed]

C. J. Zapata-Rodríguez and M. T. Caballero, “Ultrafast beam shaping with high-numerical-aperture microscope objectives,” Opt. Express 15, 308-313 (2007).
[CrossRef]

Caraquitena, J.

Chen, W. R.

Choudhury, A.

Dai, E.

Eberhardt, W.

Feng, S.

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862-873 (2000).
[CrossRef]

Froner, E.

Gan, X. S.

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]

Gu, M.

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771-778 (1996).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mol. Spectrosc. 40, 1631-1651 (1993).

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

Heyman, E.

E. Heyman and T. Melamed, “Certain consideration in aperture synthesis for ultra wideband/short-pulsed fields,” IEEE Trans. Antennas Propag. AP-42, 518-525 (1994).
[CrossRef]

Horváth, Z. L.

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

Jacquemin, R.

Jacques, S. L.

Jiang, Z.

Kantsyrev, V. L.

S. V. Kukhlevsky, G. Nyitray, and V. L. Kantsyrev, “Fields of optical waveguides as waves in free space,” Phys. Rev. E 64, 026603 (2001).
[CrossRef]

Kaplan, A. E.

Kazuhiro, N.

Kukhlevsky, S. V.

G. Nyitray, V. Mathew, and S. V. Kukhlevsky, “Generation and interference collapse of distortion-less fs pulses in free space by Fresnel sources,” Opt. Commun. 281, 1082-1086 (2008).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, “Correlation between spatial and temporal uncertainties of a wave packet,” Opt. Commun. 209, 377-382 (2002).
[CrossRef]

S. V. Kukhlevsky, G. Nyitray, and V. L. Kantsyrev, “Fields of optical waveguides as waves in free space,” Phys. Rev. E 64, 026603 (2001).
[CrossRef]

G. Nyitray and S. V. Kukhlevsky, “Distortion-free tight confinement and step-like decay of fs pulses in free space,” http://arxiv.org/abs/physics/0310057v1.

S. V. Kukhlevsky, “Diffraction-free subwavelength-beam optics on a nanometer scale,” in Localized Waves, H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds. (Wiley, 2008) pp. 273-297.
[CrossRef]

Lancis, J.

Lefrancois, M.

Li, G.

Li, Y.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-210 (1981).
[CrossRef]

Lim, L. E. N.

Luo, Q.

Lv, X.

Mathew, V.

G. Nyitray, V. Mathew, and S. V. Kukhlevsky, “Generation and interference collapse of distortion-less fs pulses in free space by Fresnel sources,” Opt. Commun. 281, 1082-1086 (2008).
[CrossRef]

Melamed, T.

E. Heyman and T. Melamed, “Certain consideration in aperture synthesis for ultra wideband/short-pulsed fields,” IEEE Trans. Antennas Propag. AP-42, 518-525 (1994).
[CrossRef]

Mínguez-Vega, G.

Monsoriu, J. A.

Nagasaka, K.

Ngoi, B. K. A.

Nyitray, G.

G. Nyitray, V. Mathew, and S. V. Kukhlevsky, “Generation and interference collapse of distortion-less fs pulses in free space by Fresnel sources,” Opt. Commun. 281, 1082-1086 (2008).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, “Correlation between spatial and temporal uncertainties of a wave packet,” Opt. Commun. 209, 377-382 (2002).
[CrossRef]

S. V. Kukhlevsky, G. Nyitray, and V. L. Kantsyrev, “Fields of optical waveguides as waves in free space,” Phys. Rev. E 64, 026603 (2001).
[CrossRef]

G. Nyitray and S. V. Kukhlevsky, “Distortion-free tight confinement and step-like decay of fs pulses in free space,” http://arxiv.org/abs/physics/0310057v1.

Pavone, F. S.

Pereira, S.

Porras, M. A.

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

Sacconi, L.

Schimmel, H.

Sedky, S. M.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. A: Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Shaarawi, A. M.

A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804-1816 (1997).
[CrossRef]

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. A: Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218-2222 (2002).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mol. Spectrosc. 40, 1631-1651 (1993).

Stanley, P.

Taghizadeh, M. R.

Tan, B.

Torres-Company, V.

Trapani, P. D.

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Valiulis, G.

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Veetil, S. P.

Venkatakrishnan, K.

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]

Winful, H. G.

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862-873 (2000).
[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]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-210 (1981).
[CrossRef]

Wyrowski, F.

Xiong, W. H.

Zapata-Rodríguez, C. J.

Zeng, S. Q.

Zhan, C.

Zhou, C.

Ziolkowski, R. W.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. A: Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

E. Heyman and T. Melamed, “Certain consideration in aperture synthesis for ultra wideband/short-pulsed fields,” IEEE Trans. Antennas Propag. AP-42, 518-525 (1994).
[CrossRef]

J. Mol. Spectrosc. (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mol. Spectrosc. 40, 1631-1651 (1993).

J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Gen. (1)

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. A: Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Opt. Commun. (5)

G. Nyitray, V. Mathew, and S. V. Kukhlevsky, “Generation and interference collapse of distortion-less fs pulses in free space by Fresnel sources,” Opt. Commun. 281, 1082-1086 (2008).
[CrossRef]

S. V. Kukhlevsky and G. Nyitray, “Correlation between spatial and temporal uncertainties of a wave packet,” Opt. Commun. 209, 377-382 (2002).
[CrossRef]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-210 (1981).
[CrossRef]

C. J. Zapata-Rodríguez, “Analytical characterization of spectral anomalies in polychromatic apertured beams,” Opt. Commun. 257, 9-15 (2006).
[CrossRef]

C. J. Zapata-Rodríguez, “Spectral anomalies in supercontinuum focused waves,” Opt. Commun. 263, 131-134 (2006).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. E (6)

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

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086-1093 (1998).
[CrossRef]

S. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862-873 (2000).
[CrossRef]

S. V. Kukhlevsky, G. Nyitray, and V. L. Kantsyrev, “Fields of optical waveguides as waves in free space,” Phys. Rev. E 64, 026603 (2001).
[CrossRef]

M. A. Porras, G. Valiulis, and P. D. Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Other (3)

G. Nyitray and S. V. Kukhlevsky, “Distortion-free tight confinement and step-like decay of fs pulses in free space,” http://arxiv.org/abs/physics/0310057v1.

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

S. V. Kukhlevsky, “Diffraction-free subwavelength-beam optics on a nanometer scale,” in Localized Waves, H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds. (Wiley, 2008) pp. 273-297.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Left column: Schematic representation (a) of 2D X waves and (b) wavefields with transverse isodiffracting behavior. (c) Angular dispersion of a 2D TID wave of k t = 6.8 μ m 1 propagating in vacuum.

Fig. 2
Fig. 2

Instantaneous amplitude E ( x , z ) of (a) a 2D X wave and (b) a TID wavefield, where the carrier frequency ω 0 = 2.36 fs 1 , the pulse width is 9.6 fs , and the angular deviation θ ( ω 0 ) = 59.8 deg .

Fig. 3
Fig. 3

Schematic depiction of an off-axis focused wave. Induced by diffraction, plane-wave constituents are phase-matched at P off , a point placed at the transverse r = ( x , y ) plane. Considering the function F ( q ) of Eq. (21), a point P off would be found at a distance Δ k 0 q m from the origin, together with another one symmetrically arranged along the x axis.

Fig. 4
Fig. 4

Temporal dynamics of E in the focal plane of IAS focal fields with off-axis parameter: (a) μ = 0 , (b) μ = 5 Δ , (c) μ = 10 Δ , and (d) μ = 20 Δ . Chromatic phase mismatching inducing spatio-temporal stretching (and subsequent attenuation) of the field grows as focalization is produced at points increasingly far from the optical axis.

Fig. 5
Fig. 5

Time evolution of the IPID field E along the grating pitch direction ( y = 0 ) of the focal plane.

Fig. 6
Fig. 6

Axial response of the instantaneous field E at the local time t = 0 for (a) an IPID wave and (b) an IAS focused field, both at μ = 20 Δ . The IAS wave undergoes a considerable increment of the depth of field.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

k ( ω ) = ω n ( ω ) c ,
exp ( i k z R ) cos ( k t R ) ,
k z = k 1 + k 2 2
k t = k 1 k 2 2
θ ( ω ) = arcsin [ k 0 k ( ω ) sin θ 0 ] ,
k z = k 2 ( ω ) k t 2
cos ( k t r ) S ( ω ) cos [ k z ( ω ) z ω t + φ ( ω ) ] d ω ,
S ( ω ) = ( ω τ 0 ) s exp ( ω τ 0 ) ,
a ( z ) = a 2 ( 0 ) + 4 c 2 z 2 a 2 ( 0 ) ω 2
E ̃ ( R , ω ) = i k ( ω ) 2 π E ̃ 0 ( Ω , ω ) exp [ i k ( ω ) R ] d Ω .
d Ω = sin θ d θ d φ ,
E ̃ 0 ( Ω , ω ) = S ( ω ) P ( Ω )
E ̃ ( R , ω ) k R 1 S ( ω ) P ( Ω ) exp ( i k R ) R ( z < 0 ) ,
P ( Ω ) = i F ( k t k ) cos θ ,
cos θ = k z k
k t = ( k x , k y ) = k sin θ ( cos φ , sin φ )
d Ω = d k t k k z ,
E ̃ ( R , ω ) = S ( ω ) 2 π k ( ω ) F ( k t k ) cos ( k t r ) exp ( i k z z ) d k t ,
E ̃ ( r , ω ) = k ( ω ) S ( ω ) G [ k ( ω ) r ] ,
G ( ρ ) = ( 2 π ) 1 F ( q ) exp ( i q ρ ) d q
k ( ω ) S ( ω ) exp ( i ω t ) d ω ,
F ( q ) = cos ( Δ q x q m ) circ ( q q m ) ,
q = q x 2 + q y 2 ,
G ( ρ ) = G D ( ρ + ρ 0 ) + G D ( ρ ρ 0 ) 2 ,
G D ( ρ ) = q m J 1 ( q m ρ ) ρ .
F off ( q ) = exp ( i μ q x q m ) F ( q ) ,
G off ( ρ ) = G ( ρ ρ off ) ,
E ̃ 0 ( Ω , ω ) = i S ( ω ) F ( k t k 0 ) k z ( ω ) k 0 .
E ̃ ( r , ω ) = k 0 S ( ω ) G ( k 0 r ) ,
x x + μ k 0 q m .
E ̃ in ( r i , ω ) = S in ( ω ) T ( ω , r i ) ,
E ̃ out ( R , ω ) = E ̃ 0 ( Ω , ω ) exp ( i k R ) R ,
S in ( ω ) 2 T ( ω , r i ) 2 d r i = E ̃ 0 ( Ω , ω ) 2 d Ω .
T ( ω , r i ) = i S ( ω ) S in ( ω ) J k z k k 0 2 F ( k t k 0 ) ,
( k t ) ( r i ) = [ k x x i k x y i k y x i k y y i ] .
J = ( k t 2 ) ( r i 2 ) .
T ( ω , r i ) = i S ( ω ) k ( ω ) S in ( ω ) k 0 J 0 cos θ k 0 2 F ( k t k 0 ) ,
S ( ω ) = i γ k 0 k ( ω ) S in ( ω )
r i f = k t k ,
k t = k f r i .
T ( ω , r i ) = ( 1 r i 2 f 2 ) 1 4 F ( r i f M ) ,
J k z = k 3 f 2 .
( k t 2 ) ( r i 2 ) k 2 k t 2 = k 3 f 2 ,
r i f = 2 3 1 [ 1 ( k t k ) 2 ] 3 2 = 2 3 1 cos 3 θ .
T ( ω , r i ) = F ( r i f M ) .
d ( ω M ) d ω = 0 ,

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