Abstract

We analyze the spatio-temporal intensity of sub-20 femtosecond pulses with a carrier wavelength of 810 nm along the optical axis of low numerical aperture achromatic and apochromatic doublets designed in the IR region by using the scalar diffraction theory. The diffraction integral is solved by expanding the wave number around the carrier frequency of the pulse in a Taylor series up to third order, and then the integral over the frequencies is solved by using the Gauss-Legendre quadrature method. The numerical errors in this method are negligible by taking 96 nodes and the computational time is reduced by 95% compared to the integration method by rectangles. We will show that the third-order group velocity dispersion (GVD) is not negligible for 10 fs pulses at 810 nm propagating through the low numerical aperture doublets, and its effect is more important than the propagation time difference (PTD). This last effect, however, is also significant. For sub-20 femtosecond pulses, these two effects make the use of a pulse shaper necessary to correct for second and higher-order GVD terms and also the use of apochromatic optics to correct the PTD effect. The design of an apochromatic doublet is presented in this paper and the spatio-temporal intensity of the pulse at the focal region of this doublet is compared to that given by the achromatic doublet.

© 2012 Optical Society of America

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  1. M. Kempe 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]
  2. M. Rosete-Aguilar, “Analytical method for calculating the electric field envelope of ultrashort pulses by approximating the wavenumber up to third order,” Appl. Opt. 49, 2463–2468 (2010).
    [CrossRef]
  3. F. C. Estrada-Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third-order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825–834 (2011).
    [CrossRef]
  4. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I: Tolerance conditions for secondary spectrum, spherochromatism and fifth-order spherical aberration,” Appl. Opt. 40, 5663–5676 (2001).
    [CrossRef]
  5. M. Rosete-Aguilar and J. L. Rayces, “Selection of glasses for achromatic doublets with reduced secondary spectrum. II. Application of the method for selecting pairs of glasses with reduced secondary spectrum,” Appl. Opt. 40, 5677–5692 (2001).
    [CrossRef]
  6. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).
  7. John H. Mathews and Kurtis D. Fink, “Integración numérica,” in Métodos Numéricos con MATLAB (Prentice Hall, 1999), pp. 371–432.
  8. J. C. Diels and W. Rudolph, “Femtosecond optics,” in Ultrashort Pulse Phenomena (Elsevier, 2006), pp. 61–142.
  9. Zs. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical PhysicsJ. L. Martin, A. Migus, G. A. Mourous, and A. H. Zewail, eds. (Springer-Verlag, 1993).
  10. V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
    [CrossRef]
  11. J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).
  12. Z. L. Horváth and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
    [CrossRef]
  13. Z. L. Horváth and Zs. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
    [CrossRef]
  14. Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
    [CrossRef]
  15. J. L. Rayces and M. Rosete-Aguilar, “Differential equation for normal glass dispersion and evaluation of the secondary spectrum,” Appl. Opt. 38, 2028–2039 (1999).
    [CrossRef]
  16. J. Garduño-Mejía, A. Greenaway, and D. T. Reid, “Designer femtosecond pulses using adaptive optics,” Opt. Express 11, 2030–2040 (2003).
    [CrossRef]

2011

F. C. Estrada-Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third-order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825–834 (2011).
[CrossRef]

2010

2008

V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
[CrossRef]

2006

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

2004

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

2003

2001

1999

1994

Z. L. Horváth and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

1992

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).

Andegeko, Y.

V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
[CrossRef]

Bor, Zs.

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

Z. L. Horváth and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Zs. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical PhysicsJ. L. Martin, A. Migus, G. A. Mourous, and A. H. Zewail, eds. (Springer-Verlag, 1993).

Dantus, M.

V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
[CrossRef]

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

De la Cruz, B. W. J. M.

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

Diels, J. C.

J. C. Diels and W. Rudolph, “Femtosecond optics,” in Ultrashort Pulse Phenomena (Elsevier, 2006), pp. 61–142.

Estrada-Silva, F. C.

F. C. Estrada-Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third-order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825–834 (2011).
[CrossRef]

Fink, Kurtis D.

John H. Mathews and Kurtis D. Fink, “Integración numérica,” in Métodos Numéricos con MATLAB (Prentice Hall, 1999), pp. 371–432.

Garduño-Mejía, J.

F. C. Estrada-Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third-order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825–834 (2011).
[CrossRef]

J. Garduño-Mejía, A. Greenaway, and D. T. Reid, “Designer femtosecond pulses using adaptive optics,” Opt. Express 11, 2030–2040 (2003).
[CrossRef]

Greenaway, A.

Gunn, J. M.

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

Horváth, Z. L.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

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

Z. L. Horváth and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Zs. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical PhysicsJ. L. Martin, A. Migus, G. A. Mourous, and A. H. Zewail, eds. (Springer-Verlag, 1993).

Kempe, M.

Klebniczki, J.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Kovács, A. P.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Kurdi, G.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Lozovoy, V. V.

V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
[CrossRef]

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

Mathews, John H.

John H. Mathews and Kurtis D. Fink, “Integración numérica,” in Métodos Numéricos con MATLAB (Prentice Hall, 1999), pp. 371–432.

Rayces, J. L.

Reid, D. T.

Rosete-Aguilar, M.

Rudolph, W.

M. Kempe 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]

J. C. Diels and W. Rudolph, “Femtosecond optics,” in Ultrashort Pulse Phenomena (Elsevier, 2006), pp. 61–142.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).

Xu, B. W.

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

Zhu, X.

V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
[CrossRef]

Appl. Opt.

Chem. Phys.

V. V. Lozovoy, Y. Andegeko, X. Zhu, and M. Dantus, “Applications of ultrashort shaped pulses in microscopy and for controlling chemical reactions,” Chem. Phys. 350, 118–124 (2008).
[CrossRef]

J. Mod. Opt.

F. C. Estrada-Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third-order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825–834 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horváth and Zs. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Opt. Express

Phys. Rev. E

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

Proc. SPIE

J. M. Gunn, B. W. Xu, B. W. J. M. De la Cruz, V. V. Lozovoy, and M. Dantus, “The MIIPS method for simultaneous phase measurement and compensation of femtosecond laser pulses and its role in two-photon microscopy and imaging,” Proc. SPIE 6108, 61080C (2006).

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).

John H. Mathews and Kurtis D. Fink, “Integración numérica,” in Métodos Numéricos con MATLAB (Prentice Hall, 1999), pp. 371–432.

J. C. Diels and W. Rudolph, “Femtosecond optics,” in Ultrashort Pulse Phenomena (Elsevier, 2006), pp. 61–142.

Zs. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical PhysicsJ. L. Martin, A. Migus, G. A. Mourous, and A. H. Zewail, eds. (Springer-Verlag, 1993).

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Figures (7)

Fig. 1.
Fig. 1.

Pulses for different positions along the optical axis around the paraxial focal point, z = 0 , of an achromatic doublet with a focal length of 40 mm and a diameter of 12 mm. The incident pulses have a temporal duration of 10 fs and a carrier wavelength of 810 nm. We have assumed homogeneous illumination on the lens.

Fig. 2.
Fig. 2.

The quality of the signal 1 / τ p τ v , and the mean square deviations of the time τ p and space τ v intensity distribution as a function of defocus z for 10 fs pulses at 810 nm. The same line symbols used in the quality of the signal plots are used for τ p and τ v plots.

Fig. 3.
Fig. 3.

The quality of the signal 1 / τ p τ v and the mean square deviations of the time τ p and space τ v intensity distribution as a function of defocus z for 4.5 fs pulses at 810 nm. The same line symbols used in the quality of the signal plots are used for τ p and τ v plots.

Fig. 4.
Fig. 4.

Pulses for the position where the quality of the signal is maximum for the 10 fs pulses at 810 nm presented in Fig. 2.

Fig. 5.
Fig. 5.

Quality of the signal 1 / τ p τ v as a function of defocus z for 20 fs, 10 fs and 4.5 fs pulses at 810 nm incident of the achromatic doublet when (a) GVD is zero to all-orders, GVD = 0 and (b) when only third-order GVD is nonzero, GVD = 1 .

Fig. 6.
Fig. 6.

Quality of the signal 1 / τ p τ v for 10 fs and 4.5 fs pulses at 810 nm incident on the achromatic and apochromatic doublets by assuming that GVD is zero to all-orders.

Fig. 7.
Fig. 7.

Pulses for an ideal lens, A = 0 , PTD = 0 and GVD = 0 and for the real lens, A = 1 , PTD = 1 and GVD = 1 for two defocus positions: z = 1250 μm and z = + 1250 μ m from the paraxial focus, z = 0 .

Tables (3)

Tables Icon

Table 1. Comparison Between Different Integration Methods

Tables Icon

Table 2. Lens Data for the Achromatic and Apochromatic Lenses

Tables Icon

Table 3. Maximum Value of the Quality of the Signal for the Cases Presented in Figs. 2 and 3

Equations (34)

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U ( u , v , z ; t ) K d ( Δ ω ) exp { ( Δ ω ) q } exp { ( Δ ω ) 2 p 2 } exp { ( Δ ω ) 3 s } × 0 1 r d r U 0 ( r ) P ( r ) exp { i Θ ( r ) } J 0 [ v r ] exp { i r 2 u 2 } ,
K = exp { i [ k 0 ( n 1 d 1 + n 2 d 2 ) ] } exp { i ( v 2 4 N ) } ,
q = i ( t τ + r 2 τ ( u ) ) ,
p = T 2 4 i ( δ r 2 δ ) ,
s = i ( γ r 2 γ ) ,
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 1 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 ] ,
γ = ρ 2 k 0 2 [ ( n 1 1 ) b 3 1 R 1 ( n 1 1 ) b 3 1 R 2 + ( n 2 1 ) b 3 2 R 2 ( n 2 1 ) b 3 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 ) ,
γ = k 0 ( n 1 d 1 a 3 1 + n 2 d 2 a 3 2 ) ,
u = ρ 2 k 0 ( 1 f 0 1 z ) ,
v = ρ k 0 r 2 f 0 ,
N = ρ 2 k 0 2 f 0 ( Fresnel Number ) ,
T int = 2 T .
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 ,
a 3 j = 1 2 ω 0 n j d 2 n j d ω 2 | ω 0 + 1 6 n j d 3 n j d ω 3 | ω 0 .
P ( x 1 , y 1 ) = { 1 , if x 1 2 + y 1 2 r 1 2 = ( r ρ ) 2 0 , otherwise r [ 0 , 1 ] .
I ( t ) 0 d v v | U ( v , u , z ; t ) | 2 ,
I ( v ) d t | U ( v , u , z ; t ) | 2 .
exp ( p 2 x 2 ± q x ) d x = π p 2 exp ( q 2 4 p 2 ) .
1 1 f ( x ) d x = k = 1 N w N , k f ( x N , k ) + E N ( f ) = G N ( f ) + E N ( f ) .
E N ( f ) = f 2 N ( c ) 2 2 N 1 ( N ! ) 4 ( ( 2 N ) ! ) 3 ( 2 N + 1 ) ! .
G N ( f ) = w N , 1 f ( x N , 1 ) + w N , 2 f ( x N , 2 ) + + w N , N f ( x N , N ) .
t = a + b 2 + b a 2 x so d t = b a 2 d x .
a b f ( t ) d t = 1 1 f ( a + b 2 + b a 2 x ) b a 2 d x
a b f ( t ) d t = b a 2 k = 1 N w N , k f ( a + b 2 + b a 2 x N , k ) .
τ p = Δ t = [ 1 W t 2 I ( t ) d t 1 W 2 ( t I ( t ) d t ) 2 ] 1 2
I ( t ) = 0 d v v | U ( u , v , z , t ) | 2 ,
τ p = τ p T int .
τ v = Δ v = [ 1 W v 2 I ( v ) d v 1 W 2 ( v I ( v ) d v ) 2 ] 1 2 ,
I ( v ) = 0 d t | U ( u , v , z , t ) | 2 .
S 1 τ p τ v .

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