Abstract

The field amplitude associated with ultrashort light pulses was analyzed by using the phase-space formalism of the Wigner distribution function (WDF). The diffraction integral was properly modified to take into account the dispersion effects (up to second order). A two-dimensional WDF associated with a reduced pupil function was derived, from which the on-axis irradiance was obtained for varying times. A two-dimensional and rotationally symmetric quartic-phase mask to control the temporal stretching of femtosecond light pulses passing through optical systems was proposed and analyzed. A Gaussian spatial and temporal pulse passing through a single lens with and without the phase mask was investigated.

© 2003 Optical Society of America

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References

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  1. G. D. Reid, Klaas Wynne, “Ultrafast laser technology and spectroscopy,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, UK, 2000), pp. 13644–13670.
  2. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).
  3. Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
    [CrossRef]
  4. Z. L. Horváth, Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601, 1–11 (2001).
    [CrossRef]
  5. Z. L. Horváth, Zs. Bor, “Dispersed femtosecond pulses in the vicinity of focus,” Opt. Commun. 111, 478–482 (1994).
    [CrossRef]
  6. Zs. Bor, Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
    [CrossRef]
  7. M. Kempe, W. Rudolph, “Impact of chromatic aberration and spherical aberration on the focusing of ultrashort light pulses by lenses,” Opt. Lett. 18, 137–139 (1993).
    [CrossRef] [PubMed]
  8. M. Kempe, W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
    [CrossRef] [PubMed]
  9. M. Kempe, U. Stamm, B. Wilhelmi, 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]
  10. Zs. Bor, Z. Gogolak, G. Szabo, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. 14, 862–864 (1989).
    [CrossRef] [PubMed]
  11. J. Ojeda-Castañeda, P. Andrés, E. Montes, “Phase-space representation of the Strehl ratio: ambiguity function,” J. Opt. Soc. Am. A 4, 313–317 (1987).
    [CrossRef]
  12. H. O. Bartelt, J. Ojeda-Castañeda, E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
    [CrossRef] [PubMed]
  13. K. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  14. D. Zalvidea, C. Colautti, E. E. Sicre, “Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function,” J. Opt. Soc. Am. A 17, 867–873 (2000).
    [CrossRef]
  15. D. Zalvidea, E. E. Sicre, “Phase pupil function for focal depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
    [CrossRef]
  16. D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
    [CrossRef]
  17. M. Born, E. Wolf, eds., Principles of Optics (Pergamon, Oxford, UK, 1983).
  18. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, Vol. XXXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), pp. 1–56.
  19. Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
    [CrossRef]
  20. J. Gaskill, ed., Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 253–257.
  21. A. Efimov, C. Schaffer, D. H. Reitze, “Programmable shaping of ultrabroad-bandwidth pulses from a Ti:sapphire laser,” J. Opt. Soc. Am. B 12, 1968–1980 (1995).
    [CrossRef]
  22. D. Zalvidea, E. E. Sicre, “Space-temporal analysis of ultra-short light pulse propagation in aberrated optical systems,” in 19th Congress of the International Comission for Optics, A. Consortini, G. C. Rihini, eds., Proc. SPIE4829, 349–350 (2002).

2002 (1)

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

2001 (1)

Z. L. Horváth, Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601, 1–11 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (2)

1995 (1)

1994 (1)

Z. L. Horváth, Zs. Bor, “Dispersed femtosecond pulses in the vicinity of focus,” Opt. Commun. 111, 478–482 (1994).
[CrossRef]

1993 (2)

1992 (2)

M. Kempe, U. Stamm, B. Wilhelmi, 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]

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

1989 (1)

1988 (1)

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

1987 (1)

1984 (1)

Alonso, M. A.

Andrés, P.

Bartelt, H. O.

Bor, Z.

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

Bor, Zs.

Z. L. Horváth, Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601, 1–11 (2001).
[CrossRef]

Z. L. Horváth, Zs. Bor, “Dispersed femtosecond pulses in the vicinity of focus,” Opt. Commun. 111, 478–482 (1994).
[CrossRef]

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

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

Colautti, C.

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, Vol. XXXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), pp. 1–56.

Efimov, A.

Forbes, G. W.

Gogolak, Z.

Granieri, S.

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

Horváth, Z. L.

Z. L. Horváth, Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601, 1–11 (2001).
[CrossRef]

Z. L. Horváth, Zs. Bor, “Dispersed femtosecond pulses in the vicinity of focus,” Opt. Commun. 111, 478–482 (1994).
[CrossRef]

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

Jaroszewicz, Z.

Kempe, M.

Montes, E.

Morales, J.

Ojeda-Castañeda, J.

Reid, G. D.

G. D. Reid, Klaas Wynne, “Ultrafast laser technology and spectroscopy,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, UK, 2000), pp. 13644–13670.

Reitze, D. H.

Rudolph, W.

Schaffer, C.

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).

Sicre, E. E.

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

D. Zalvidea, C. Colautti, E. E. Sicre, “Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function,” J. Opt. Soc. Am. A 17, 867–873 (2000).
[CrossRef]

D. Zalvidea, E. E. Sicre, “Phase pupil function for focal depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
[CrossRef]

H. O. Bartelt, J. Ojeda-Castañeda, E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
[CrossRef] [PubMed]

D. Zalvidea, E. E. Sicre, “Space-temporal analysis of ultra-short light pulse propagation in aberrated optical systems,” in 19th Congress of the International Comission for Optics, A. Consortini, G. C. Rihini, eds., Proc. SPIE4829, 349–350 (2002).

Stamm, U.

Szabo, G.

Wilhelmi, B.

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).

Wolf, K.

Wynne, Klaas

G. D. Reid, Klaas Wynne, “Ultrafast laser technology and spectroscopy,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, UK, 2000), pp. 13644–13670.

Zalvidea, D.

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

D. Zalvidea, C. Colautti, E. E. Sicre, “Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function,” J. Opt. Soc. Am. A 17, 867–873 (2000).
[CrossRef]

D. Zalvidea, E. E. Sicre, “Phase pupil function for focal depth enhancement derived from a Wigner distribution function,” Appl. Opt. 37, 3623–3627 (1998).
[CrossRef]

D. Zalvidea, E. E. Sicre, “Space-temporal analysis of ultra-short light pulse propagation in aberrated optical systems,” in 19th Congress of the International Comission for Optics, A. Consortini, G. C. Rihini, eds., Proc. SPIE4829, 349–350 (2002).

Appl. Opt. (2)

J. Mod. Opt. (1)

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

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

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

Opt. Commun. (3)

D. Zalvidea, S. Granieri, E. E. Sicre, “Space and spectral behaviour of optical systems under broadband illumination by using a Wigner distribution function approach,” Opt. Commun. 204, 99–106 (2002).
[CrossRef]

Z. L. Horváth, Zs. Bor, “Dispersed femtosecond pulses in the vicinity of focus,” Opt. Commun. 111, 478–482 (1994).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. A (1)

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

Phys. Rev. E (1)

Z. L. Horváth, Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601, 1–11 (2001).
[CrossRef]

Other (6)

G. D. Reid, Klaas Wynne, “Ultrafast laser technology and spectroscopy,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, UK, 2000), pp. 13644–13670.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).

M. Born, E. Wolf, eds., Principles of Optics (Pergamon, Oxford, UK, 1983).

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, Vol. XXXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), pp. 1–56.

D. Zalvidea, E. E. Sicre, “Space-temporal analysis of ultra-short light pulse propagation in aberrated optical systems,” in 19th Congress of the International Comission for Optics, A. Consortini, G. C. Rihini, eds., Proc. SPIE4829, 349–350 (2002).

J. Gaskill, ed., Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 253–257.

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Figures (5)

Fig. 1
Fig. 1

Scheme of the optical setup.

Fig. 2
Fig. 2

Relationship between the dispersion coefficients in Eq. (6). Dashed line, ratio of fourth- and second-order terms, solid line ratio of third- and second-order terms. The spectral width of the input pulse is 2.5132741×1014 Hz.

Fig. 3
Fig. 3

Temporal distribution irradiance for three positions along the optical axis for a Gaussian spatial and spectral pulse passing through a BK7 lens: (a) without amplitude and phase mask, (b) with phase mask, α=100.

Fig. 4
Fig. 4

Three-dimensional display of the temporal distribution irradiance along the optical axis for a Gaussian spatial and spectral pulse passing through a BK7 lens with the quartic-phase mask, α=100.

Fig. 5
Fig. 5

Temporal distribution irradiance at the focal point for the quartic-phase mask; as in Fig. 4, with various α.

Equations (23)

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2+k2(ω)U(r; ω)=0,
U(x, y, z; ω)--dxdyP0(x, y)exp[iΦ(x, y)]A(ω)×exp[ikld]exp-i(kl-ka) x2+y22 1R1-1R2×expika2z [(x-x)2+(y-y)2],
kl=ωc n(ω)k0n0[1+a1Δω+a2(Δω)2],
ka=ωc=k01+Δωω0,
a1=1ω0+1n0dn(ω)dωω0,
a2=1n0ω0dn(ω)dωω0+12n0d2n(ω)dω2ω0.
1/f0=(n0-1)(1/R1-1/R2).
U(r, θ, z; Δω)=KA(Δω)exp[ik0n0d(a1Δω+a2Δω2)]×expi k02(f0+z) 1+Δωω0r20a02πP0(ρ, ϕ)×exp[iΦ(ρ, ϕ)]exp-ik02f0 1-f0f0+zρ2×exp-ik02f0 a1Δω+a2Δω2-f0Δωω0(f0+z)ρ2×exp-ik0(f0+z) 1+Δωω0rρ cos(ϕ-θ)dϕρdρ,
U(r, θ, z; t)=K-d(Δω)A(Δω)exp[ik0n0dΔω(a1+a2Δω)]×expi k02(f0+z) 1+Δωω0r20a02πρdρP0(ρ)×exp[iΦ(ρ, ϕ)]exp-ik02f0 1-f0f0+zρ2×exp-iΔωt+k02f0 a1+a2Δω-f0ω0(f0+z)ρ2.
I(0, z; t)=|K|2|U(0, z; t)|2.
I(0, z; t)--d(Δω1)d(Δω2)A(Δω1)A(Δω2)×expi(Δω12-Δω22)n0k0da2-Nb22×exp-i(Δω1-Δω2)t-k0n0da1+12Nb1-N f0(f0+z)ω0×G(0, z, Δω1, Δω2),
G(0, z; Δω1, Δω2)=-dζ exp-iζNb2(Δω12-Δω22)+Nb1-N f0(f0+z)ω0(Δω1-Δω2)×-dΔζQζ+Δζ2; ΔωQ*ζ-Δζ2; Δω×exp-iΔζN zz+f0+Nb22 (Δω12+Δω22)×12 Nb1-N f0(f0+z)ω0(Δω1+Δω2),
Wg(x, v)=-gx+x2g*x-x2exp[-i2πvx]dx=-F{g}v+v2F{g}*v-v2×exp[i2πvx]dv.
G(0, z; Δω1, Δω2)=-dζ exp-iζNb2(Δω12-Δω22)+Nb1-N f0(f0+z)ω0(Δω1-Δω2)×WQN zz+f0+Nb22 (Δω12+Δω22)×12Nb1-N f0(f0+z)ω0(Δω1+Δω2).
G(0, z; Δω1, Δω2)=-dζ exp[-iζ(κ1+κ2u)]Wf(ζ; κ3+κ4u)=-duF{Q}κ3+κ4u+u2F{Q}*×κ3+κ4u+u2-dζ×exp[-iζ(κ1+κ2u-u)]=F{Q}κ3+κ4u+κ1+κ2u2×F{Q}*κ3+κ4u-κ1+κ2u2,
u=a2k02f0zz+f0, κ1=Nb2(Δω12-Δω22)+Nn0-1dn(ω)dωω=ω0(Δω1-Δω2), κ2=Δω1-Δω22ω0, κ3=Nb22 (Δω12+Δω22)+N2(n0-1)dn(ω)dωω=ω0(Δω1+Δω2), κ4=12+Δω1+Δω24ω0.
I(0, z; t)-d(Δω1)d(Δω2)A(Δω1)A*(Δω2)×exp-iN2 b2+k0n0da2(Δω12-Δω22)×exp-i(Δω1-Δω2)t-k0n0da1+Nb12-N f02(f0+z)ω0×F{Q}Nb2Δω12+Nn0-1dn(ω)dωω0+u2ω0Δω1×F{Q}*Nb2Δω22+Nn0-1dn(ω)dωω0+u2ω0Δω2.
I(0, z; t)-dϖWhϖ; 2ϖ(n0k0da2-Nb2)-t-k0n0da1+Nb1-a2k0z2f0ω0(f0+z),
h(ϖ; z)=A(ϖ)F{Q}Nb2ϖ2+τ(z)ϖ+a2k0z2f0(f0+z).
τ(z)=Nb1-Nf0(z+f0)ω0.
h(ϖ; z)=A(ϖ)exp-iπαNb2ϖ2+τ(z)ϖ+a2k0z2f0(f0+z)2×Q0(ζ)+n>11(n-1)!-2πiαn-1×F{ζ2nF{Q0}}Nb2ϖ2+τ(z)ϖ+a2k0z2f0(f0+z)1α.
|nth term|max=1(n-1)! 2παn, |α|2π,
|nth term|1(n-1)!.

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