Abstract

We apply well known nonlinear diffraction theory governing focusing of a powerful light beam of arbitrary shape in medium with Kerr nonlinearity to the analysis of femtosecond (fs) laser processing of dielectric in sub-critical (input power less than the critical power of self-focusing) regime. Simple analytical expressions are derived for the input beam power and spatial focusing parameter (numerical aperture) that are required for achieving an inscription threshold. Application of non-Gaussian laser beams for better controlled fs inscription at higher powers is also discussed.

© 2007 Optical Society of America

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  1. K. M. Davis, K. Miura, N. Sugimoto and K. Hirao, "Writing waveguides in glass with a femtosecond. Laser," Opt. Lett. 21,1729 (1996)
    [CrossRef] [PubMed]
  2. E. N. Glezer and E. Mazur, "Ultrafast-laser driven micro-explosions in transparent materials," Appl. Phys. Lett. 71, 882 (1997)
    [CrossRef]
  3. D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, "Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses," Opt. Lett. 24, 1311-1313 (1999)
    [CrossRef]
  4. Y. Kondo, K. Nouchi, T. Mitsuyu, M. Watanabe, P. G. Kazansky, and K. Hirao, "Fabrication of long-period fiber gratings by focused irradiation of infrared femtosecond laser pulses," Opt. Lett. 24, 646-648 (1999)
    [CrossRef]
  5. A.M. Streltsov and N.M. Borrelli, "Study of femtosecond-laser-written waveguides in glasses," J. Opt. Soc. Am. B 19, 2496-2504 (2002).
    [CrossRef]
  6. M. Will, S. Nolte, B. N. Chichkov, and A. Tünnermann, "Optical Properties of Waveguides Fabricated in Fused Silica by Femtosecond Laser Pulses," Appl. Opt. 41, 4360-4364 (2002)
    [CrossRef] [PubMed]
  7. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri and G. Cerlullo, "Femtosecond writing of active optical waveguides with astigmatically shaped beams," J. Opt. Soc. Am. B, 1559 (2003)Q1
    [CrossRef]
  8. C. Florea and K. A. Winick, "Fabrication and Characterization of PhotonicDevices Directly Written in Glass UsingFemtosecond Laser Pulses," IEEE J. Lightwave Technol. 21, 246 (2003)Q2
    [CrossRef]
  9. B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, " Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses," Phys. Rev. Lett. 74, 2248 (1995)
    [CrossRef] [PubMed]
  10. B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Optical ablation by high-power. short-pulse. Lasers," J. Opt. Soc. Am. B 13, 459 (1996)
    [CrossRef]
  11. C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, "Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy," Opt. Lett. 26, 93-95 (2001)
    [CrossRef]
  12. S. Tzortzakis, L. Sudrie, M. Franko, B. Prade, A. Mysrowicz, A. Couairon, and L. Berge, "Self-focusing of few-cycle light pulses in dielectric media," Phys. Rev. Lett. 87, 213902 (2001)
    [CrossRef] [PubMed]
  13. 3D Laser Microfabrication: Principles and Applications, Eds. Hiroaki Misawa and Saulius Juodkazis, Wiley-VCH, 2006
  14. Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, C.R. Thompson, "Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses," IEEE-J. Quantum Electron. 33, 127-137 (1997)
    [CrossRef]
  15. L.V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov.Phys.JETP 20,1307 (1965)Q3
  16. M.V. Ammosov,N.V. Delone, and V.P. Krainov "Tunneling ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov.Phys.JETP 64, 1191 (1986)Q4
  17. V. I. Talanov, "Focusing of light in cubic media," JETP Lett. 11, 199 (1970)
  18. S.N. Vlasov, V.A. Petrishev, and V.I. Talanov, "Average description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971) [Radiophys. and Quantum Electron. 14, 1062 (1974)]Q5
  19. R. Y.  Chiao, E.  Garmire, and C. H.  Townes, "Self-trapping of optical beam," Phys. Rev. Lett.  13, 479 (1964)
    [CrossRef]
  20. L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G.W. ’t Hooft, E. R. Eliel, and G. Fibich, "Collapse of Optical Vortices," Phys. Rev. Lett.,  96, 133901 (2006)
    [CrossRef] [PubMed]

2006 (1)

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G.W. ’t Hooft, E. R. Eliel, and G. Fibich, "Collapse of Optical Vortices," Phys. Rev. Lett.,  96, 133901 (2006)
[CrossRef] [PubMed]

2003 (2)

R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri and G. Cerlullo, "Femtosecond writing of active optical waveguides with astigmatically shaped beams," J. Opt. Soc. Am. B, 1559 (2003)Q1
[CrossRef]

C. Florea and K. A. Winick, "Fabrication and Characterization of PhotonicDevices Directly Written in Glass UsingFemtosecond Laser Pulses," IEEE J. Lightwave Technol. 21, 246 (2003)Q2
[CrossRef]

2002 (2)

2001 (2)

C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, "Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy," Opt. Lett. 26, 93-95 (2001)
[CrossRef]

S. Tzortzakis, L. Sudrie, M. Franko, B. Prade, A. Mysrowicz, A. Couairon, and L. Berge, "Self-focusing of few-cycle light pulses in dielectric media," Phys. Rev. Lett. 87, 213902 (2001)
[CrossRef] [PubMed]

1999 (2)

1997 (2)

E. N. Glezer and E. Mazur, "Ultrafast-laser driven micro-explosions in transparent materials," Appl. Phys. Lett. 71, 882 (1997)
[CrossRef]

Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, C.R. Thompson, "Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses," IEEE-J. Quantum Electron. 33, 127-137 (1997)
[CrossRef]

1996 (2)

1995 (1)

B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, " Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses," Phys. Rev. Lett. 74, 2248 (1995)
[CrossRef] [PubMed]

1986 (1)

M.V. Ammosov,N.V. Delone, and V.P. Krainov "Tunneling ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov.Phys.JETP 64, 1191 (1986)Q4

1971 (1)

S.N. Vlasov, V.A. Petrishev, and V.I. Talanov, "Average description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971) [Radiophys. and Quantum Electron. 14, 1062 (1974)]Q5

1970 (1)

V. I. Talanov, "Focusing of light in cubic media," JETP Lett. 11, 199 (1970)

1965 (1)

L.V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov.Phys.JETP 20,1307 (1965)Q3

1964 (1)

R. Y.  Chiao, E.  Garmire, and C. H.  Townes, "Self-trapping of optical beam," Phys. Rev. Lett.  13, 479 (1964)
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. N. Glezer and E. Mazur, "Ultrafast-laser driven micro-explosions in transparent materials," Appl. Phys. Lett. 71, 882 (1997)
[CrossRef]

IEEE J. Lightwave Technol. (1)

C. Florea and K. A. Winick, "Fabrication and Characterization of PhotonicDevices Directly Written in Glass UsingFemtosecond Laser Pulses," IEEE J. Lightwave Technol. 21, 246 (2003)Q2
[CrossRef]

IEEE-J. Quantum Electron. (1)

Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, C.R. Thompson, "Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses," IEEE-J. Quantum Electron. 33, 127-137 (1997)
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

S.N. Vlasov, V.A. Petrishev, and V.I. Talanov, "Average description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971) [Radiophys. and Quantum Electron. 14, 1062 (1974)]Q5

J. Opt. Soc. Am. B (3)

B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Optical ablation by high-power. short-pulse. Lasers," J. Opt. Soc. Am. B 13, 459 (1996)
[CrossRef]

R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri and G. Cerlullo, "Femtosecond writing of active optical waveguides with astigmatically shaped beams," J. Opt. Soc. Am. B, 1559 (2003)Q1
[CrossRef]

A.M. Streltsov and N.M. Borrelli, "Study of femtosecond-laser-written waveguides in glasses," J. Opt. Soc. Am. B 19, 2496-2504 (2002).
[CrossRef]

JETP Lett. (1)

V. I. Talanov, "Focusing of light in cubic media," JETP Lett. 11, 199 (1970)

Opt. Lett. (4)

Phys. Rev. Lett. (4)

S. Tzortzakis, L. Sudrie, M. Franko, B. Prade, A. Mysrowicz, A. Couairon, and L. Berge, "Self-focusing of few-cycle light pulses in dielectric media," Phys. Rev. Lett. 87, 213902 (2001)
[CrossRef] [PubMed]

R. Y.  Chiao, E.  Garmire, and C. H.  Townes, "Self-trapping of optical beam," Phys. Rev. Lett.  13, 479 (1964)
[CrossRef]

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G.W. ’t Hooft, E. R. Eliel, and G. Fibich, "Collapse of Optical Vortices," Phys. Rev. Lett.,  96, 133901 (2006)
[CrossRef] [PubMed]

B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, " Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses," Phys. Rev. Lett. 74, 2248 (1995)
[CrossRef] [PubMed]

Sov.Phys.JETP (2)

L.V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov.Phys.JETP 20,1307 (1965)Q3

M.V. Ammosov,N.V. Delone, and V.P. Krainov "Tunneling ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov.Phys.JETP 64, 1191 (1986)Q4

Other (1)

3D Laser Microfabrication: Principles and Applications, Eds. Hiroaki Misawa and Saulius Juodkazis, Wiley-VCH, 2006

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Figures (7)

Fig. 1.
Fig. 1.

Evolution of the pulse energy with distance for different input peak powers. Horizontal dashed line corresponds to the critical power for a given initial Gaussian pulse width (FWHM) of 70 fs.

Fig. 2.
Fig. 2.

Relative role of different physical effects (Eqs (1)) in the process of fs laser radiation interaction with silica. Here the input power Pin = 0.5Pcr and the numerical aperture NA=0.1.

Fig. 3.
Fig. 3.

Product of the maximum field intensity and minimal RMS beam width vs input beam power. The dashed line corresponds to assumption I max × R RMS_minPin /π, the markers are results of numerical simulation of 2d NLSE: the circles- Cs =25 , the crosses- Cs = 50 , the squares- Cs = 100. The solid line is an approximation by the method of the least squares (here x = Pin /Pcr ):

Fig. 4.
Fig. 4.

Numerical solutions of the Eq. 4. Top: Input pulse power (measured in critical power) required to produce intensity above the threshold of inscription is shown as a function of the parameter Q = Ith × Ss /Pcr = Ss /Scr – ratio of the beam spot area at the surface to the “critical” focal area Scr = Pcr /Ith , for different pre-focusing parameters Cs . Bottom picture: isolines of the parameter Q in the plane (Cs , Pin /Pcr ) .

Fig. 5.
Fig. 5.

Numerical aperture NA in the nonlinear regime vs numerical aperture of the linear focusing required to produce the same minimal beam size in the focal point for three values of the input power: 0.1, 0.5 and 0.8 (measured in critical power).

Fig. 6.
Fig. 6.

Dependence of the inscription threshold energy required to produce irreversible change of the refractive index on the numerical aperture of a focusing lens using Gaussian pulse with FWHM = 100 fs. Solid line — Eq. 4, dashed line – approximation (5) effectively used in [11]. Inset shows the interval of NA from 0.3 to 0.6.

Fig. 7.
Fig. 7.

Isolines of the normalized intensity I(r, z) / Ith for initial Gaussian beam (left) and ring beam with m=1 (right); both with the input power Pin / Pcr = 0.54 and pre-focusing parameter Cs = 50. The area with I(r, z) ≥ Ith is shown by red.

Equations (22)

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i A z + 1 2 k 2 A k″ 2 A tt + k 0 n 2 A 2 A = 2 ( 1 + τ e ) ρA β ( K ) 2 A 2 K 2 A
ρ t = 1 n b 2 σ bs E g ρ A 2 + β ( K ) K ħ ω A 2 K
C 1 = λ 0 2 4 π 2 n 0 2 A 0 2 d r P n 2 A 0 4 d r P , C 2 = λ 0 π n 0 ( r arg A 0 2 d r ) ) P .
R 2 z t = R min NL 2 ( 1 + ( z z min _ NL ) 2 Z R _ NL 2 ) ,
R min _ NL 2 = a s 2 [ 1 P ( t ) P cr _ G ] 1 + C s 2 P ( t ) P cr _ G , Z R _ NL 2 = k 2 a s 4 [ 1 P ( t ) P cr _ G ] [ 1 + C s 2 P ( t ) P cr _ G ] 2 , z min _ NL = k a s 2 C s [ 1 + C s 2 P ( t ) P cr _ G ] .
I th = P in π R min 2 = P in [ 1 + C s 2 P in P cr ] S s [ 1 P in P cr ] .
P in P cr = 1 2 { 1 + C s 2 + I th S s P cr [ 1 + C s 2 + I th S s P cr ] 2 4 I th S s P cr }
E in τ π = P in I th S s ( 1 + C s 2 ) + I th S s P cr , P in P cr ( 1 + C s 2 )
I th = P cr π R min 2 × f ( P in P cr ) = P cr × [ 1 + C s 2 P in P cr ] S s × [ 1 P in P cr ] × f ( P in P cr ) P cr × [ 1 + C s 2 ] S s × [ 1 P in P cr ] × f ( P in P cr ) ,
f ( x ) = 1.0073 × ( 1.01 x ) + 0.91445 + 8.567 10 2 ( 1.01 x ) 4.273 10 4 ( 1.01 x ) 2
P in P cr = g ˜ ( C s 2 , I th S s P cr ) g ( I th S s P cr ( 1 + C s 2 ) ) .
A z r = 1 L ( z ) U ( 0 ds L 2 ( z ) z , r L ( z ) ) exp [ i r 2 4 FL ( z ) ] , L ( z ) = 1 + z F , F = a s 2 2 C s .
( Δ z 2 ) 2 = k 2 a s 2 [ 1 + C s 2 P in P cr ] 2 [ P in P cr [ r cr 2 ( 1 + C s 2 P in P cr ) + a s 2 ] a s 2 ]
Δ z = 2 k a min _ lin [ P in P cr ( r cr 2 + a min _ lin 2 ) a min _ lin 2 ] 1 2 , a min _ lin 2 = a s 2 1 + C s 2 .
z diss = z min _ NL Δ z 2 = k a s 2 C s [ 1 + C s 2 P in P cr ] k a s [ P in P cr ( r cr 2 + a s 2 1 + C s 2 ) a s 2 1 + C s 2 ] 1 2 .
R min _ G 2 = λ 0 2 2 π 2 × 1 N A 2 N A 2 × 1 P ( t ) P cr 1 P ( t ) [ P cr ( 1 + π 2 ( f d ) 2 λ 0 2 ( NA 2 1 NA 2 ) 2 ) ] ,
z min _ G = n 0 × ( f d ) 1 P ( t ) [ P cr ( 1 + π 2 ( f d ) 2 λ 0 2 ( NA 2 1 NA 2 ) 2 ) ] , Z R _ G 2 = 2 R min _ G 2 × n 0 2 × 1N A 2 N A 2 1 P ( t ) [ P cr ( 1 + π 2 ( f d ) 2 λ 0 2 ( NA 2 1 NA 2 ) 2 ) ] .
P in = E in τ π = P th = P cr I th I th + 2 π P cr N A 2 [ λ 0 2 ( 1 N A 2 ) ]
E th P cr τ π = 1 1 + 2 π N A 2 1 N A 2 S cr λ 0 2 .
R 2 z t = R min _ NL _ m 2 ( 1 + ( z z min _ m ) 2 Z R _ NL _ m 2 ) ,
R min _ NL _ m 2 = a s 2 ( m + 1 ) [ 1 P ( t ) P cr _ m ] 1 + ( m + 1 ) C s 2 P ( t ) P cr _ m , Z R _ NL _ m 2 = k 2 a s 4 ( m + 1 ) [ 1 P ( t ) P cr _ m ] [ 1 + ( m + 1 ) C s 2 P ( t ) P cr _ m ] 2 ,
z min = k a s 2 C s ( m + 1 ) [ 1 + ( m + 1 ) C s 2 P ( t ) P cr _ m ] , P cr _ m = λ 0 2 2 π n 2 n 0 × 2 2 m ( m ! ) 2 ( 2 m ) ! .

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