Abstract

For optical pulses shorter than 20fs duration or highly dispersive materials in the visible range of the spectrum, high-order terms in the Taylor expansion for the wave vector, around the carrier frequency, should be considered. By expanding the wave vector near the center of optical frequency ω0 in a Taylor series up to the third-order approximation, we present an analytical method for calculating the electric field envelope of a pulse after it has propagated through a medium that contributes second- and third-order group velocity dispersion. To verify the method we present some examples for both 20 and 15fs pulses propagating through pieces of glass made of low and high dispersive material. Limitations of the method are discussed.

© 2010 Optical Society of America

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References

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  1. M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A  48, 4721–4729 (1993).
    [CrossRef] [PubMed]
  2. 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]
  3. 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]
  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. 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]
  6. Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt.  35, 1907–1918 (1988).
    [CrossRef]
  7. 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]
  8. Z. Bor and Z. L. Horvath, “Distortion of femtosecond pulses in lenses: wave optical description,” Opt. Commun.  94, 249–258 (1992).
    [CrossRef]
  9. Z. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E  63, 026601(2001).
    [CrossRef]
  10. F. C. Estrada-Silva, J. Garduño-Mejía, M. Rosete-Aguilar, J. Román-Moreno, and R. Ortega-Martínez, “Aberration effects on femtosecond pulses generated by nonideal achromatic doublets,” Appl. Opt.  48, 4723–4732 (2009).
    [CrossRef] [PubMed]
  11. A. Yariv and P. Yeh, “Chromatic dispersion and polarization mode dispersion in fibers,” in Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).
  12. J. C. Diels and W. Rudolph, “Fundamentals: characteristics of femtosecond light pulses,” in Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).
  13. Optical Schott Glass, www.us.schott.com/optics.devices(2009).
  14. Edmund optics catalog, www.edmundoptics.com (2010).

2009 (1)

2008 (1)

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]

2001 (1)

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

1999 (1)

1993 (2)

1992 (2)

1989 (1)

1988 (1)

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

Bor, Z.

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

Gogolak, Z.

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]

Szabo, G.

Bor, Z.

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

Bor, Z.

Diels, J. C.

J. C. Diels and W. Rudolph, “Fundamentals: characteristics of femtosecond light pulses,” in Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).

Estrada-Silva, F. C.

F. C. Estrada-Silva, J. Garduño-Mejía, M. Rosete-Aguilar, J. Román-Moreno, and R. Ortega-Martínez, “Aberration effects on femtosecond pulses generated by nonideal achromatic doublets,” Appl. Opt.  48, 4723–4732 (2009).
[CrossRef] [PubMed]

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]

Garduño-Mejía, J.

Gil, M. A.

Horvath, Z.  L.

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

Horvath, Z.

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

Kempe, M.

Mattei, G. O.

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]

Ortega-Martínez, R.

Román-Moreno, J.

Rosete-Aguilar, M.

F. C. Estrada-Silva, J. Garduño-Mejía, M. Rosete-Aguilar, J. Román-Moreno, and R. Ortega-Martínez, “Aberration effects on femtosecond pulses generated by nonideal achromatic doublets,” Appl. Opt.  48, 4723–4732 (2009).
[CrossRef] [PubMed]

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.

Rudolph, W.

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]

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

J. C. Diels and W. Rudolph, “Fundamentals: characteristics of femtosecond light pulses,” in Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).

Stamm, U.

Wilhelmi, B.

Yariv, A.

A. Yariv and P. Yeh, “Chromatic dispersion and polarization mode dispersion in fibers,” in Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).

Yeh, P.

A. Yariv and P. Yeh, “Chromatic dispersion and polarization mode dispersion in fibers,” in Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).

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. B (1)

Laser Phys. (1)

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]

Opt. Commun. (1)

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

Phys. Rev. E (1)

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

Other (4)

A. Yariv and P. Yeh, “Chromatic dispersion and polarization mode dispersion in fibers,” in Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).

J. C. Diels and W. Rudolph, “Fundamentals: characteristics of femtosecond light pulses,” in Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).

Optical Schott Glass, www.us.schott.com/optics.devices(2009).

Edmund optics catalog, www.edmundoptics.com (2010).

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Figures (2)

Fig. 1
Fig. 1

Intensity of pulses after propagating a distance L = 10 mm in a piece of glass. The central wavelength is λ = 0.8 μm , and the input unchirped pulse has an initial duration time of τ = 20 fs . The thicker solid curve shows the solution by using the exact phase. The thinner solid curve shows the solution by expanding the phase up to second order, i.e., the solution given by the finite integral with N = 0 , Eq. (14). The thin curve with circles shows the third-order solution given by the finite integral with N = 23 , Eq. (14).

Fig. 2
Fig. 2

Intensity of pulses after propagating a distance L = 10 mm in a piece of glass. The central wavelength is λ = 0.8 μm , and the input unchirped pulse has an initial duration time of τ = 15 fs . The thicker solid curve shows the solution by using the exact phase. The thinner solid curve shows the solution by expanding the phase up to second order, i.e., the solution given by the finite integral with N = 0 , Eq. (14). The thin curve with circles shows the third-order solution given by the finite integral with N = 23 , Eq. (14).

Tables (2)

Tables Icon

Table 1 Central Wavelength of 0.800 μm and c = 3 × 10 4 mm/fs

Tables Icon

Table 2 Central Wavelength of 0.800 μm and c = 3 × 10 4 mm/fs

Equations (35)

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E ( z = 0 , t ) = exp ( α t 2 + i ω 0 t ) ,
E ( z = 0 , t ) = F ( Ω ) e i ( ω 0 + Ω ) t d Ω ,
F ( Ω ) = 1 4 π α exp ( Ω 2 4 α ) .
exp [ i k z ] = exp [ i Ψ ] ,
E ( z , t ) = F ( Ω ) exp [ i ( ω 0 + Ω ) t i k ( ω 0 + Ω ) z ] d Ω .
k ( ω 0 + Ω ) k ( ω 0 ) + d k d ω | ω 0 Ω + 1 2 d 2 k d ω 2 | ω 0 Ω 2 + 1 6 d 3 k d ω 3 | ω 0 Ω 3 .
E ( z , t ) = 1 4 π α exp [ i ( k ( ω 0 ) z ω 0 t ) ] exp { [ i d k d ω | ω 0 Ω z i Ω t ( 1 4 α i 1 2 d 2 k d ω 2 | ω 0 z ) Ω 2 + i 1 6 d 3 k d ω 3 | ω 0 Ω 3 z ] } d Ω .
E ( z , t ) = 1 4 π α exp [ i ( Ψ ( ω 0 ) ω 0 t ) ] exp { [ i ( d Ψ d ω | ω 0 t ) Ω ( 1 4 α i 1 2 d 2 Ψ d ω 2 | ω 0 ) Ω 2 + i 1 6 d 3 Ψ d ω 3 | ω 0 Ω 3 ] } d Ω = 1 4 π α exp [ i ( Ψ ( ω 0 ) ω 0 t ) ] exp { [ i m Ω q Ω 2 + r Ω 3 ] } d Ω ,
m = ( t d Ψ d ω | ω 0 ) ,
q = ( 1 4 α i 1 2 d 2 Ψ d ω 2 | ω 0 ) ,
r = i 6 d 3 Ψ d ω 3 | ω 0 .
exp ( i m x q x 2 + r x 3 ) d x = γ ( i m , q ; 0 ) + r γ ( i m , q ; 3 ) + r 2 2 γ ( i m , q ; 6 ) + r 3 6 γ ( i m , q ; 9 ) + r 4 24 γ ( i m , q ; 12 ) + . r N N ! γ ( i m , q ; 3 N ) ,
γ ( i m , q ; 0 ) = π q e m 2 4 q , γ ( i m , q ; 1 ) = i m 2 q γ ( i m , q ; 0 ) , q > 0 , γ ( i m , q ; n ) = n 1 2 q γ ( i m , q ; n 2 ) i m 2 q γ ( i m , q ; n 1 ) , q > 0 , n = 2 , 3 , .
x 1 x 2 exp ( i m x q x 2 + r x 3 ) d x = γ f ( i m , q ; 0 ) + r γ f ( i m , q ; 3 ) + r 2 2 γ f ( i m , q ; 6 ) + r 3 6 γ f ( i m , q ; 9 ) + r 4 24 γ f ( i m , q ; 12 ) + . r N N ! γ f ( i m , q ; 3 N ) ,
γ f ( i m , q ; 0 ) = x 1 x 2 e q x 2 i m x d x = i exp ( m 2 4 q ) π ( erf i ( m 2 i q x 1 2 q ) erf i ( m 2 i q x 2 2 q ) ) 2 q , γ f ( i m , q ; 1 ) = exp ( q x 1 2 i m x 1 ) exp ( q x 2 2 i m x 2 ) 2 q i m 2 q γ f ( i m , q ; 0 ) , γ f ( i m , q ; n ) = ( n 1 ) 2 q γ f ( i m , q ; n 2 ) i m 2 q γ f ( i m , q ; n 1 ) ( x 2 n 1 exp ( q x 2 2 i m x 2 ) x 1 n 1 exp ( q x 1 2 i m x 1 ) 2 q ) .
d Ψ d ω | ω 0 = L c ( n λ d n d λ ) ,
d 2 Ψ d ω 2 | ω 0 = L λ 3 2 π c 2 d 2 n d λ 2 ,
d 3 Ψ d ω 3 | ω 0 = L c 3 λ 5 ( 2 π ) 2 d 3 n d λ 3 3 L c 3 λ 4 ( 2 π ) 2 d 2 n d λ 2 ,
exp ( i m x q x 2 + r x 3 ) d x = exp ( i m x q x 2 ) exp ( r x 3 ) d x = exp ( i m x q x 2 ) n = 0 N ( r x 3 ) n n ! d x = exp ( i m x q x 2 ) ( 1 + r x 3 + 1 2 r 2 x 6 + 1 6 r 3 x 9 + . ) d x = exp ( i m x q x 2 ) d x + exp ( i m x q x 2 ) ( r x 3 ) d x + exp ( i m x q x 2 ) ( 1 2 r 2 x 6 ) d x + exp ( i m x q x 2 ) ( 1 6 r 3 x 9 ) d x +
G = e q x 2 i m x d x = π q e q 2 4 m , Re [ m ] > 0.
exp ( i m x q x 2 ) ( r x 3 ) d x = r x 3 exp ( i m x q x 2 ) d x .
d d x ( e q x 2 e i m x ) = 2 q x e q x 2 e i m x i m e q x 2 e i m x .
e q x 2 e i m x | = 2 q x e q x 2 e i m x d x i m e q x 2 e i m x d x .
e q x 2 e i m x | z z = e q z 2 i m z e q z 2 + i m z = e q z 2 ( e i m z e i m z ) = 2 e q z 2 cos ( m z ) .
x e q x 2 e i m x d x = i m 2 q e q x 2 e i m x d x = i m 2 q G .
γ ( i m , q ; n ) x n e q x 2 e i m x d x , n = 1 , 2 , 3.
γ ( i m , q ; n ) = n 1 2 q γ ( i m , q ; n 2 ) i m 2 q γ ( i m , q ; n 1 ) , q > 0 , n = 2 , 3 , ,
γ ( i m , q ; 1 ) = i m 2 q γ ( i m , q ; n 1 ) , q > 0 , n = 1 ,
γ ( i m , q ; 0 ) = π q e m 2 4 q .
x n e q x 2 e i m x d x = x e q x 2 x n 1 e i m x d x = d d x ( e q x 2 2 q ) x n 1 e i m x d x = 1 2 q d d x ( e q x 2 ) x n 1 e i m x d x = 1 2 q { [ e q x 2 x n 1 e i m x ] e q x 2 [ ( n 1 ) x n 2 i m x n 1 ] e i m x d x } .
x n e q x 2 e i m x d x = 1 2 q { e q x 2 [ ( n 1 ) x n 2 i m x n 1 ] e i m x d x } = ( n 1 ) 2 q e q x 2 x n 2 e i m x d x i m 2 q e q x 2 x n 1 e i m x d x = n 1 2 q γ ( i m , q ; n 2 ) i m 2 q γ ( i m , q ; n 1 ) .
x 1 x 2 exp ( i m x q x 2 + r x 3 ) d x = x 1 x 2 exp ( i m x q x 2 ) n = 0 N ( r x 3 ) n n ! d x = x 1 x 2 exp ( i m x q x 2 ) d x + x 1 x 2 exp ( i m x q x 2 ) ( r x 3 ) d x + x 1 x 2 exp ( i m x q x 2 ) ( 1 2 r 2 x 6 ) d x + x 1 x 2 exp ( i m x q x 2 ) ( 1 6 r 3 x 9 ) d x +
γ f ( i m , q ; 0 ) = x 1 x 2 e q x 2 i m x d x = i exp ( m 2 4 q ) π ( erf i ( m 2 i q x 1 2 q ) erf i ( m 2 i q x 2 2 q ) ) 2 q .
γ f ( i m , q ; 1 ) = 1 2 q e q x 2 e i m x | x 1 x 2 i m 2 q x 1 x 2 exp ( q x 2 i m x ) d x = exp ( q x 1 2 i m x 1 ) exp ( q x 2 2 i m x 2 ) 2 q i m 2 q γ f ( i m , q ; 0 ) ,
γ f ( i m , q ; n ) = ( n 1 ) 2 q γ f ( i m , q ; n 2 ) i m 2 q γ f ( i m , q ; n 1 ) x 2 n 1 exp ( q x 2 2 i m x 2 ) x 1 n 1 exp ( q x 1 2 i m x 1 ) 2 q .

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