Abstract

There are three main effects that affect the femtosecond pulse focusing process near the focal plane of a refractive lens: the group velocity dispersion (GVD), the propagation time difference (PTD), and the aberrations of the lens. In this paper we study in detail these effects generated by nonideal achromatic doublets based on a Fourier-optical analysis and Seidel aberration theory considering lens material, wavelength range, lens surface design, and temporally and spatially uniform and Gaussian intensity distributions. We show that the residual chromatic aberration in achromatic lenses, which has been neglected so far, has a considerable effect on the focusing of pulses shorter than 20fs in the spectral range between the UV and IR, 300 to 1100nm, and is particularly important in the blue and UV spectral range. We present a general fitted function for an estimation of the pulse stretching parameter, which depends only on the numerical aperture and focal length of the doublet as well as the wavelength of the carrier of the pulse.

© 2009 Optical Society of America

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References

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  1. M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).
  2. M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721-4729 (1993).
    [CrossRef] [PubMed]
  3. 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] [PubMed]
  4. G. O. Mattei and M. A. Gil, “Spherical aberration in spatial and temporal transforming lenses of femtosecond laser pulses,” Appl. Opt. 38, 1058-1064 (1999).
    [CrossRef]
  5. 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]
  6. Z. Bor, Z. Gogolak, and G. Szabo, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. 14, 862-864 (1989).
    [CrossRef] [PubMed]
  7. Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907-1918 (1988).
    [CrossRef]
  8. M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
    [CrossRef]
  9. M. J. Kidger, Fundamental Optical Design (SPIE, 2002).
  10. Zs. Bor and Z. L. Horvarth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249-258 (1992).
    [CrossRef]
  11. Z. L. Horváth and Zs. Bor “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 1-11(2001).
    [CrossRef]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  13. D. T. Reid, W. Sibbett, J. M. Dudley, L. P. Barry, B. Thomsen, and J. D. Harvey “Commercial semiconductor devices for two photon absorption autocorrelation of ultrashort light pulses,” Appl. Opt. 37, 8142-8144 (1998).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics (Pergamon, 1989).
  15. M. Kempe and W. Rudolph, “Microscopy with ultrashort light pulses,” Molec. Cryst. Liq. Cryst. Sect. B Nonlinear Opt. 7, 129-151 (1994).
  16. J. Jasapara and W. Rudolph, “Characterization of sub-10-fs pulse focusing with high-numerical-aperture microscope objectives,” Opt. Lett. 24, 777-779 (1999).
    [CrossRef]
  17. C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
    [CrossRef]
  18. A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
    [CrossRef] [PubMed]
  19. C. Bauer, J.-P. Abid, and H. H. Girault, “Role of adsorbates on dynamics of hot-electron (type I and II) thermalization within gold nanoparticles,” C. R. Chimie 9, 261-267(2006).
    [CrossRef]
  20. P. K. Jain, W. Qian, and M. A. El-Sayed, “Ultrafast electron relaxation dynamics in coupled metal nanoparticles in aggregates,” J. Phys. Chem. B 110, 136-142 (2006).
    [CrossRef] [PubMed]
  21. C. Bauer, J. P. Abid, and H. H. Girault, “Size dependence investigations of hot electron cooling dynamics in metal/adsorbates nanoparticles,” Chem. Phys. 319, 409-421(2005).
    [CrossRef]
  22. J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
    [CrossRef]

2008

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
[CrossRef]

2006

C. Bauer, J.-P. Abid, and H. H. Girault, “Role of adsorbates on dynamics of hot-electron (type I and II) thermalization within gold nanoparticles,” C. R. Chimie 9, 261-267(2006).
[CrossRef]

P. K. Jain, W. Qian, and M. A. El-Sayed, “Ultrafast electron relaxation dynamics in coupled metal nanoparticles in aggregates,” J. Phys. Chem. B 110, 136-142 (2006).
[CrossRef] [PubMed]

2005

C. Bauer, J. P. Abid, and H. H. Girault, “Size dependence investigations of hot electron cooling dynamics in metal/adsorbates nanoparticles,” Chem. Phys. 319, 409-421(2005).
[CrossRef]

2003

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

2001

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

2000

J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
[CrossRef]

1999

1998

1994

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

M. Kempe and W. Rudolph, “Microscopy with ultrashort light pulses,” Molec. Cryst. Liq. Cryst. Sect. B Nonlinear Opt. 7, 129-151 (1994).

1993

1992

1989

1988

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

Abid, J. P.

C. Bauer, J. P. Abid, and H. H. Girault, “Size dependence investigations of hot electron cooling dynamics in metal/adsorbates nanoparticles,” Chem. Phys. 319, 409-421(2005).
[CrossRef]

Abid, J.-P.

C. Bauer, J.-P. Abid, and H. H. Girault, “Role of adsorbates on dynamics of hot-electron (type I and II) thermalization within gold nanoparticles,” C. R. Chimie 9, 261-267(2006).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Acioli, L. H.

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

Arbouet, A.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Barry, L. P.

Bauer, C.

C. Bauer, J.-P. Abid, and H. H. Girault, “Role of adsorbates on dynamics of hot-electron (type I and II) thermalization within gold nanoparticles,” C. R. Chimie 9, 261-267(2006).
[CrossRef]

C. Bauer, J. P. Abid, and H. H. Girault, “Size dependence investigations of hot electron cooling dynamics in metal/adsorbates nanoparticles,” Chem. Phys. 319, 409-421(2005).
[CrossRef]

Bigot, J. Y.

J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
[CrossRef]

Bor, Z.

Bor, Zs.

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

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1989).

Broker, M.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Celep, G.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Christofilos, D.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Cottancin, E.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Daunois, A.

J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
[CrossRef]

Del Fatti, N.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Dudley, J. M.

El-Sayed, M. A.

P. K. Jain, W. Qian, and M. A. El-Sayed, “Ultrafast electron relaxation dynamics in coupled metal nanoparticles in aggregates,” J. Phys. Chem. B 110, 136-142 (2006).
[CrossRef] [PubMed]

Estrada-Silva, F. C.

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
[CrossRef]

Fermann, M. E.

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

Fujimoto, J. G.

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

Galvanauskas, A.

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

Gaudry, M.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Gil, M. A.

Girault, H. H.

C. Bauer, J.-P. Abid, and H. H. Girault, “Role of adsorbates on dynamics of hot-electron (type I and II) thermalization within gold nanoparticles,” C. R. Chimie 9, 261-267(2006).
[CrossRef]

C. Bauer, J. P. Abid, and H. H. Girault, “Size dependence investigations of hot electron cooling dynamics in metal/adsorbates nanoparticles,” Chem. Phys. 319, 409-421(2005).
[CrossRef]

Gogolak, Z.

Halté, V.

J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
[CrossRef]

Harvey, J. D.

Horvarth, Z. L.

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

Horváth, Z. L.

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

Ippen, E. P.

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

Jain, P. K.

P. K. Jain, W. Qian, and M. A. El-Sayed, “Ultrafast electron relaxation dynamics in coupled metal nanoparticles in aggregates,” J. Phys. Chem. B 110, 136-142 (2006).
[CrossRef] [PubMed]

Jasapara, J.

Kempe, M.

Kidger, M. J.

M. J. Kidger, Fundamental Optical Design (SPIE, 2002).

Langot, P.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Lermé, J.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Maillard, M.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Mattei, G. O.

Merle, J. C.

J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
[CrossRef]

Ortega-Martínez, R.

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
[CrossRef]

Pellarin, M.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Pilen, M. P.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Qian, W.

P. K. Jain, W. Qian, and M. A. El-Sayed, “Ultrafast electron relaxation dynamics in coupled metal nanoparticles in aggregates,” J. Phys. Chem. B 110, 136-142 (2006).
[CrossRef] [PubMed]

Reid, D. T.

Román-Moreno, C. J.

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
[CrossRef]

Rosete-Aguilar, M.

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
[CrossRef]

Rudolph, W.

Sibbett, W.

Stamm, U.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Sucha, G.

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

Sun, C. K.

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

Szabo, G.

Thomsen, B.

Treguer, M.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Vallée, F.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

Voisin, C.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Wilhelmi, B.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1989).

Appl. Opt.

C. R. Chimie

C. Bauer, J.-P. Abid, and H. H. Girault, “Role of adsorbates on dynamics of hot-electron (type I and II) thermalization within gold nanoparticles,” C. R. Chimie 9, 261-267(2006).
[CrossRef]

Chem. Phys.

C. Bauer, J. P. Abid, and H. H. Girault, “Size dependence investigations of hot electron cooling dynamics in metal/adsorbates nanoparticles,” Chem. Phys. 319, 409-421(2005).
[CrossRef]

J. Y. Bigot, V. Halté, J. C. Merle, and A. Daunois, “Electron dynamics in metallic nanoparticles,” Chem. Phys. 251, 181-203 (2000).
[CrossRef]

J. Mod. Opt.

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

J. Opt. Soc. Am. B

J. Phys. Chem. B

P. K. Jain, W. Qian, and M. A. El-Sayed, “Ultrafast electron relaxation dynamics in coupled metal nanoparticles in aggregates,” J. Phys. Chem. B 110, 136-142 (2006).
[CrossRef] [PubMed]

Laser Phys.

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223-231 (2008).
[CrossRef]

Molec. Cryst. Liq. Cryst. Sect. B Nonlinear Opt.

M. Kempe and W. Rudolph, “Microscopy with ultrashort light pulses,” Molec. Cryst. Liq. Cryst. Sect. B Nonlinear Opt. 7, 129-151 (1994).

Opt. Commun.

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

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. B

C. K. Sun, F. Vallée, L. H. Acioli, E. P. Ippen, and J. G. Fujimoto, “Femtosecond-tunable measurement of electron thermalization in gold,” Phys. Rev. B 50, 15337-15348(1994).
[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

A. Arbouet, C. Voisin, D. Christofilos, P. Langot, N. Del Fatti, F. Vallée, J. Lermé, G. Celep, E. Cottancin, M. Gaudry, M. Pellarin, M. Broker, M. Maillard, M. P. Pilen, and M. Treguer, “Electron-phonon scattering in metal clusters,” Phys. Rev. Lett. 90, 177401 (2003).
[CrossRef] [PubMed]

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1989).

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

M. J. Kidger, Fundamental Optical Design (SPIE, 2002).

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Figures (14)

Fig. 1
Fig. 1

System coordinates.

Fig. 2
Fig. 2

Marginal pulses are focused at a different plane with respect to the paraxial pulses, producing a relative delay.

Fig. 3
Fig. 3

Angle definition for Seidel coefficients calculation.

Fig. 4
Fig. 4

Time versus space distortion at the paraxial focal point for a chirped precompensated 20 fs pulse at 810 nm . (a) A 2 mm and (b) a 12 mm diameter intensity Gaussian distribution input beam has been used in the simulation. (c) A 12 mm diameter beam with uniform illumination has been considered.

Fig. 5
Fig. 5

(a)  t / T versus I ( t ) time pulse intensity distribution (photon flux) at the paraxial focal plane for a chirped on-axis precompensated 20 fs pulse at 810 nm . Inset is the corresponding autocorrelation of (a). (b) v versus I ( v ) spatial distribution. The squares and triangles represent 2 and 12 mm beam diameter Gaussian intensity distribution, respectively, and the circles show a 12 mm diameter beam with uniform illumination.

Fig. 6
Fig. 6

t / T versus I ( t ) time pulse intensity distribution (photon flux) at paraxial focal plane for a chirped on-axis precompensated 20 fs pulse at 405 nm .

Fig. 7
Fig. 7

Time versus space distortion at the paraxial focal point for 200 fs pulse at 810 nm . (a) A 2 mm and (b) a 12 mm diameter intensity Gaussian distribution input beam has been used in the simulation. (c) A 12 mm beam diameter with uniform illumination has been considered.

Fig. 8
Fig. 8

(a)  t / T versus I ( t ) time pulse intensity distribution (photon flux) at the focal plane for 200 fs pulse at 810 nm . Inset is the corresponding autocorrelation of (a). (b) v versus I ( v ) spatial distribution. The squares and triangles represent 2 and 12 mm beam diameter Gaussian intensity distribution, respectively, and the circles a 12 mm beam diameter with uniform illumination.

Fig. 9
Fig. 9

Comparison of experimental results against theoretical calculations for 200 fs pulses for two width waist.

Fig. 10
Fig. 10

(a) v versus I ( v ) spatial distribution and t / T versus intensity autocorrelation of I ( t ) at the focal plane for a chirped precompensated 20 fs pulse at 810 nm . (b) Corresponding results for 200 fs pulse at 810 nm . Both cases were estimated for Gaussian illumination input and w 0 = 6 mm . The square symbols represent the case when both PTD and spherical aberration effects have been considered; triangles represent the case when only PTD has been eliminated and the circles represent the case when both PTD and spherical aberration have been eliminated.

Fig. 11
Fig. 11

Spherical aberration for an achromatic doublet as a function of the input beam radius for different wavelengths between 300 and 1100 nm for two different lens designs NIR and VIS: (a) NIR lens and (b) VIS results.

Fig. 12
Fig. 12

τ / T versus wavelength for different input pulse widths between 20 and 250 fs . Solid lines correspond to the NIR lens and dotted lines for the VIS results.

Fig. 13
Fig. 13

Wavelength versus τ for different achromatic doublets’ focal lengths at 810 nm and constant N . A . = 0.2 . Solid lines represent numerical model estimations for the fitted functions for f = 60 mm (circles), f = 30 mm (triangles), and f = 15 mm (squares).

Fig. 14
Fig. 14

Wavelength versus τ for an achromatic doublet with a focal length of 30 mm at ( 800 nm ) and a numerical aperture of N . A . = 0.2 . The solid line represents numerical model estimation [Eq. (25)]: the triangle symbols correspond to the fitted function [Eq. (40)]; the circles correspond to the Bor approximation [Eq. (42)]; the square symbols correspond to the Jasapara approximation [Eq. (43)].

Tables (2)

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Table 1 Achromatic Doublets Design Parameters

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Table 2 Fitting Coefficients

Equations (43)

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U ( x 2 , y 2 , z ; Δ ω ) d x 1 d y 1 U 0 ( x 1 , y 1 ) P ( x 1 , y 1 ) A ( Δ ω ) exp { i Θ ( x 1 , y 1 ; w 0 ) } exp { i Φ ( x 1 , y 1 ) } exp { i k a 2 z [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ] } .
1 f 0 = 1 f 01 + 1 f 02 ,
1 f 01 = ( n 1 1 ) ( 1 R 1 1 R 2 ) ,
1 f 02 = ( n 2 1 ) ( 1 R 2 1 R 3 ) ,
ϕ 0 ( 4 ) ( x 1 , y 1 ; w 0 ) = 1 8 S Tot ( w 0 ) ( x 1 + y 1 ) 4 ,
S ( w 0 ) = ( 1 4 ) ( w 0 4 f 3 ) [ ( n n 1 ) 2 + ( n + 2 n ( n 1 ) 2 ) ( B + ( 2 ( n 2 1 ) C n + 2 ) 2 n C 2 n + 2 ) ] ,
B = ( ς 1 + ς 2 ) ( ς 1 ς 2 ) ,
C = ( μ 1 + μ 2 ) ( μ 1 μ 2 ) ,
B 1 = ( ς 1 + ς 2 ) ( ς 1 ς 2 ) , B 2 = ( ς 2 + ς 3 ) ( ς 2 ς 3 ) ,
μ 1 = μ 1 = tan 1 ( w 0 f 01 ) ,
μ 2 = tan 1 ( w 0 f 02 ) .
C 2 = μ 1 + μ 2 μ 1 μ 2 .
Θ ( x 1 , y 1 ; w 0 ) = k 0 ϕ 0 ( 4 ) ( x 1 , y 1 ; w 0 ) .
P ( x 1 , y 1 ) = { 1 , if   x 1 2 + y 1 2 r 1 2 = ( r ρ ) 2 0 , otherwise .
Φ ( x 1 , y 1 ) = ( k 1 d 1 + k 2 d 2 ) ( x 1 2 + y 1 2 ) 2 [ k 0 R 1 ( n 1 1 ) β 1 k 0 R 2 ( n 1 1 ) β 1 + k 0 R 2 ( n 2 1 ) β 2 k 0 R 3 ( n 2 1 ) β 2 ] ,
β 1 = 1 + b 1 1 Δ ω + b 2 1 ( Δ ω ) 2 ,
β 2 = 1 + b 1 2 Δ ω + b 2 2 ( Δ ω ) 2 ,
k j = ω c n j ( ω ) k 0 n 0 j [ 1 + a 1 j Δ ω + a 2 j ( Δ ω ) 2 ] ,
a 1 j = 1 ω 0 + 1 n j d n j d ω | ω 0 ,
a 2 j = 1 ω 0 n j d n j d ω | ω 0 + 1 2 n j d 2 n j d ω 2 | ω 0 .
U ( x 2 , y 2 , z ; Δ ω ) exp { i k 0 [ n 1 d 1 ( 1 + a 1 1 Δ ω + a 2 1 Δ ω 2 ) + n 2 d 2 ( 1 + a 1 2 Δ ω + a 2 2 Δ ω 2 ) ] } d x 1 d y 1 U 0 ( x 1 , y 1 ) P ( x 1 , y 1 ) A ( Δ ω ) exp { i Θ ( x 1 , y 1 ; w 0 ) } exp [ i k 0 ( x 1 2 + y 1 2 2 ) ( 1 f 0 1 z ) ] × exp { i k 0 ( x 1 2 + y 1 2 ) 2 Δ ω [ ( n 1 1 ) R 1 ( β 1 1 ) Δ ω ( n 1 1 ) R 2 ( β 1 1 ) Δ ω + ( n 2 1 ) R 2 ( β 2 1 ) Δ ω ( n 2 1 ) R 3 ( β 2 1 ) Δ ω 1 z ω ] } exp { i k 0 ( 1 + Δ ω ω ) 2 f 0 ( x 2 2 + y 2 2 ) } exp { i k 0 ( 1 + Δ ω ω ) f 0 ( x 1 x 2 + y 1 y 2 ) } .
U ( u , v , z ; Δ ω ) exp { i k 0 [ n 1 d 1 ( 1 + a 1 1 Δ ω + a 2 1 Δ ω 2 ) + n 2 d 2 ( 1 + a 1 2 Δ ω + a 2 2 Δ ω 2 ) ] } 0 r r d r U 0 ( r ) P ( r ) A ( Δ ω ) exp { i Θ ( r ; w 0 ) } exp { i ( u 2 ) r 2 } × exp { i k 0 r 2 2 Δ ω [ ( n 1 1 ) R 1 ( β 1 1 ) Δ ω ( n 1 1 ) R 2 ( β 1 1 ) Δ ω + ( n 2 1 ) R 2 ( β 2 1 ) Δ ω ( n 2 1 ) R 3 ( β 2 1 ) Δ ω 1 z ω ] } × exp { i v 2 4 N ( 1 + Δ ω ω ) } J 0 [ v r ( 1 + Δ ω ω ) ] ,
U ( u , v , z ; t ) d ( Δ ω ) exp { i ( Δ ω ) t } U ( u , v , z ; Δ ω ) .
U ( u , v , z ; Δ ω ) exp { i [ k 0 ( n 1 d 1 + n 2 d 2 ) ] } d ( Δ ω ) A ( Δ ω ) 0 r r d r U 0 ( r ) P ( r ) exp { i Θ ( r ; w 0 ) } J 0 [ v r ( 1 + Δ ω ω ) ] exp { i ( v 2 4 N ) ( 1 + Δ ω ω ) } × exp { i Δ ω ( t τ + r 2 τ ( u ) ) } exp { i Δ ω 2 ( δ r 2 δ ) } ,
τ = k 0 ρ 2 2 [ ( n 1 1 ) b 1 1 R 1 ( n 1 1 ) b 1 1 R 2 + ( n 2 1 ) b 1 2 R 2 ( n 2 1 ) b 1 2 R 3 ] [ k 0 ρ 2 2 f 0 ω 0 u 2 ω 0 ] ,
δ = ρ 2 k 0 2 [ ( n 1 1 ) b 2 1 R 1 ( n 1 1 ) b 2 1 R 2 + ( n 2 1 ) b 2 2 R 2 ( n 2 1 ) b 2 2 R 3 ] ,
τ = k 0 ( n 1 d 1 a 1 1 + n 2 d 2 a 1 2 ) ,
δ = k 0 ( n 1 d 1 a 2 1 + n 2 d 2 a 2 2 ) .
U ( u , v , z ; Δ ω ) K d ( Δ ω ) exp { ( Δ ω ) 2 p 2 } exp { ( Δ ω ) q } 0 r r d r U 0 ( r ) P ( r ) exp { i Θ ( r ; w 0 ) } J 0 [ v r ] ,
K = exp { i [ k 0 ( n 1 d 1 + n 2 d 2 ) ] } exp { i ( v 2 4 N ) } ,
p = T 2 4 i ( δ r 2 δ ) ,
q = i ( t τ + r 2 τ ( u ) ) .
- d ( Δ ω ) exp ( p 2 ( Δ ω ) 2 ± q ( Δ ω ) ) = π p exp ( q 2 4 p 2 ) = [ 4 π ( 1 + i ξ ( r ; T ) ) T 2 ( 1 + ξ 2 ( r ; T ) ) ] 1 / 2 exp [ ( t τ + r 2 τ ( u ) ) 2 T 2 ( 1 + ξ 2 ( r ; T ) ) ( 1 + i ξ ( r ; T ) ) ] ,
ξ ( r ; T ) = 4 i ( δ r 2 δ ) T 2 ,
U ( v , u ; t ) K 0 r r d r J 0 [ v r ] exp ( i u 2 r 2 ) [ 1 + i ξ ( r ; T ) 1 + ξ 2 ( r ; T ) ] 1 / 2 exp [ ( t τ + r 2 τ ( u ) ) 2 T 2 ( 1 + ξ 2 ( r ; T ) ) ( 1 + i ξ ( r ; T ) ) ] × exp ( i Θ ( r ; w 0 ) ) U 0 ( r ) P ( r ) .
ξ ( r ; T ) = 4 i ( δ r 2 δ ) T 2 δ ,
A ( t ) = A 0 exp { ( t T [ 1 + ( δ ) 2 ] ) 2 ( 1 + i δ ) } .
I ( t ) 0 d v v | U ( v , u ; t ) | 2 ,
I ( v ) d t | U ( v , u ; t ) | 2 .
τ ( N . A . , f ; λ ) = [ N . A . ( λ ) ] 2 [ c 1 ( λ ) · f ( λ ) + c 2 ( λ ) ] .
c i ( λ ) = p 1 , i λ 5 + p 2 , i λ 4 + p 3 , i λ 3 + p 4 , i λ 2 + p 5 , i λ + p 6 , i .
τ ( N . A . , f ; λ ) = λ 2 c [ N . A . ( λ ) ] 2 d f d λ ,
τ ( N . A . , f ; λ ) = λ 2 c ( [ N . A . ( λ ) ] 2 1 [ N . A . ( λ ) ] 2 ) d f d λ

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