Abstract

Recently, using parageometrical optics concepts, a hybrid, diffractive-refractive, lens triplet has been suggested to significantly improve the spatiotemporal resolution of light spots in multifocal processing with femtosecond laser pulses. Here, we carry out a rigorous wave-optics analysis, including the spatiotemporal nature of the wave equation, to elucidate both the spatial extent of the diffractive spots and the temporal duration of the pulse at the output plane. Specifically, we show nearly transform-limited behavior of diffraction maxima. Moreover, the temporal broadening of the pulse is related to the group velocity dispersion, which can be pre-compensated for in practical applications. Finally, some numerical simulations of the spatiotemporal wave field at the output plane in a realistic case are provided.

© 2007 Optical Society of America

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References

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  1. Z. Bor, "Distorsion of femtosecond laser pulses in lenses and lens systems," J. Mod. Opt. 35, 1907-1918 (1988).
    [CrossRef]
  2. Z. Bor, "Distorsion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
    [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. M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
    [CrossRef]
  5. T. E. Sharp and P. J. Wisoff, "Analysis of lens and zone plate combinations for achromatic focusing of ultrashort laser pulses," Appl. Opt. 31, 2765-2769 (1992).
    [CrossRef] [PubMed]
  6. E. Ibragimov, "Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements," Appl. Opt. 34, 7280-7285 (1995).
    [CrossRef] [PubMed]
  7. R. Piestun and D. A. B. Miller, "Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements," Opt. Lett. 26, 1373-1375 (2001).
    [CrossRef]
  8. U. Fuchs, U. D. Zeitner, and A. Tünnermann,"Hybrid optics for focusing ultrashort laser pulses," Opt. Lett. 31, 1516-1518 (2006).
    [CrossRef] [PubMed]
  9. M. Kempe and W. Rudolph, "Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses," Opt. Lett. 18, 137-139 (1993).
    [CrossRef] [PubMed]
  10. M. Kempe and W. Rudolph, "Femtosecond pulses in the focal region of lenses," Phys. Rev. A 48, 4721-4729 (1993).
    [CrossRef] [PubMed]
  11. 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]
  12. U. Fuchs, U. D. Zeitner, and A. Tünnermann, "Ultra-short pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005).
    [CrossRef] [PubMed]
  13. S. Nolte, "Micromachining" in Ultrafast Optics: Technology and Application, M. E. Fermann, A. Galva-nauskas and G. Sucha, eds., (Marcel Dekker, New York, 2003).
  14. C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
    [CrossRef]
  15. Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
    [CrossRef]
  16. L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, and F. S. Pavone, "Multiphoton multifocal microscopy exploiting a diffractive optical element," Opt. Lett. 28, 1918-1920 (2003).
    [CrossRef] [PubMed]
  17. S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
    [CrossRef]
  18. Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
    [CrossRef]
  19. S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
    [CrossRef] [PubMed]
  20. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, 2003).
    [CrossRef]
  21. J. Amako, K. Nagasaka, and N. Kazuhiro, "Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements," Opt. Lett. 27, 969-971 (2002).
    [CrossRef]
  22. Y. Kuroiwa, N. Takeshima, Y. Narita, S. Tanaka, and K. Hirao, "Arbitrary micropatterning method in femtosecond laser microprocessing using diffractive optical elements," Opt. Express 12, 1908-1915 (2004).
    [CrossRef] [PubMed]
  23. G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
    [CrossRef]
  24. G. Mínguez-Vega, J. Lancis, J. Caraquitena, V. Torres-Company, and P. Andrés, "High spatiotemporal resolution in multifocal processing with femtosecond laser pulses," Opt. Lett. 31, 2631-2633 (2006).
    [CrossRef] [PubMed]
  25. O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
    [CrossRef]

2006 (3)

2005 (4)

U. Fuchs, U. D. Zeitner, and A. Tünnermann, "Ultra-short pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005).
[CrossRef] [PubMed]

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

J. Amako, K. Nagasaka, and N. Kazuhiro, "Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements," Opt. Lett. 27, 969-971 (2002).
[CrossRef]

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

2001 (1)

1999 (1)

1998 (1)

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

1995 (1)

1993 (2)

1992 (3)

1989 (1)

1988 (2)

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

O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
[CrossRef]

Amako, J.

Andrés, P.

Antolini, R.

Bor, Z.

Z. Bor, "Distorsion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
[CrossRef] [PubMed]

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

Caraquitena, J.

Choudhury, A.

Dai, E.

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

Froner, E.

Fuchs, U.

Gil, M. A.

Hasegawa, S.

Hayasaki, Y.

S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
[CrossRef] [PubMed]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Hirao, K.

Ibragimov, E.

Juodkazis, S.

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Kamlage, G.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Kazuhiro, N.

Kempe, M.

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

M. Kempe and W. Rudolph, "Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses," Opt. Lett. 18, 137-139 (1993).
[CrossRef] [PubMed]

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

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]

Kuroiwa, Y.

Lancis, J.

Li, G.

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

Maeda, M.

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

Martínez, O. E.

O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
[CrossRef]

Matsuo, S.

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Mattei, G. O.

Miller, D. A. B.

Mínguez-Vega, G.

Misawa, H.

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Momma, C.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Nagasaka, K.

Nakata, Y.

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

Narita, Y.

Nishida, N.

S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
[CrossRef] [PubMed]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Nolte, S.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Okada, T.

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

Pavone, F. S.

Piestun, R.

Rudolph, W.

Sacconi, L.

Sharp, T. E.

Stamm, U.

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

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]

Sugimoto, T.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Taghizadeh, M. R.

Takeshima, N.

Takita, A.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Tanaka, S.

Torres-Company, V.

Tünnermann, A.

von Alvensleben, G.

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

Wilhelmi, B.

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]

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

Wisoff, P. J.

Zeitner, U. D.

Zhou, C.

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. A (2)

C. Momma, S. Nolte, G. Kamlage, G. von Alvensleben, and A. Tünnermann, "Beam delivery of femtosecond laser radiation by diffractive optical elements," Appl. Phys. A 67, 517-520 (1998).
[CrossRef]

S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser microfabrication of periodic structures using a microlens array," Appl. Phys. A 80, 683-685 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, "Variable holographic femtosecond laser processing by use of spatial light modulator," Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Y. Nakata, T. Okada, and M. Maeda, "Fabrication of dot matrix, comb, and nanowire structures using laser ablation by interfered femtosecond laser beams," Appl. Phys. Lett. 81, 4239-4241 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

O. E. Martínez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530-2536, (1988).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (1)

G. Li, C. Zhou, and E. Dai, "Splitting of femtosecond laser pulses by using a Damman grating and compensation gratings," J. Opt. Soc. Am. A 4, 767-772 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

M. Kempe, U. Stamm, and B. Wilhelmi, "Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture," Opt. Commun. 89, 119-125 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

U. Fuchs, U. D. Zeitner, and A. Tünnermann,"Hybrid optics for focusing ultrashort laser pulses," Opt. Lett. 31, 1516-1518 (2006).
[CrossRef] [PubMed]

S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006).
[CrossRef] [PubMed]

G. Mínguez-Vega, J. Lancis, J. Caraquitena, V. Torres-Company, and P. Andrés, "High spatiotemporal resolution in multifocal processing with femtosecond laser pulses," Opt. Lett. 31, 2631-2633 (2006).
[CrossRef] [PubMed]

Z. Bor, "Distorsion of femtosecond laser pulses in lenses," Opt. Lett. 14, 119-121 (1989).
[CrossRef] [PubMed]

M. Kempe and W. Rudolph, "Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses and lenses," Opt. Lett. 18, 137-139 (1993).
[CrossRef] [PubMed]

R. Piestun and D. A. B. Miller, "Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements," Opt. Lett. 26, 1373-1375 (2001).
[CrossRef]

J. Amako, K. Nagasaka, and N. Kazuhiro, "Chromatic-distorsion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements," Opt. Lett. 27, 969-971 (2002).
[CrossRef]

L. Sacconi, E. Froner, R. Antolini, M. R. Taghizadeh, A. Choudhury, and F. S. Pavone, "Multiphoton multifocal microscopy exploiting a diffractive optical element," Opt. Lett. 28, 1918-1920 (2003).
[CrossRef] [PubMed]

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]

Other (2)

S. Nolte, "Micromachining" in Ultrafast Optics: Technology and Application, M. E. Fermann, A. Galva-nauskas and G. Sucha, eds., (Marcel Dekker, New York, 2003).

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, 2003).
[CrossRef]

Supplementary Material (1)

» Media 1: AVI (752 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Schematic diagram of a Fourier beam splitter. The system, form the input grating to the output plane, is fullydescribed through the ABCDEF ray transfer matrix.

Fig. 2.
Fig. 2.

Sketch of the conventional grating based multifocal device.

Fig. 3.
Fig. 3.

Representation of the dispersion-compensation hybrid device for spot array generation.

Fig. 4.
Fig. 4.

Relative stretching versus the input pulse width for the fifth-order diffraction maximum (n=5) after focusing with an achromatic doublet (dashed line) and the system proposed in Fig. 3 (continuous line) for: (a) the spatial domain and (b) the temporal domain.

Fig. 5.
Fig. 5.

Integrated intensity profiles of the 5th diffraction order of a 100 lines/inch diffractive grating after focusing with an achromatic doublet (dashed line), the system of Fig. 3 (continuous line) and a monochromatic wave of 800 nm (dash/dot line) for an input pulse duration of: (a) σt = 50 fs and (b) σt = 21.24 fs.

Fig. 6.
Fig. 6.

(752 KB) This video shows the spatiotemporal profiles of the 5th diffraction order of a 100 lines/inch diffractive grating for an input pulse width that varies from σt = 200 to 50 fs. The right part of the figure is obtained after focusing with an achromatic lens double and the left part by focusing with the DOE-based system. Note that the time origin is chosen arbitrarily [Media 1]

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

U out x ω = ω 2 πcBi exp [ i ωL c ] exp [ i D B ω 2 c x 2 ] ×
× U in ( x′ ; ω ) exp [ i A B ω 2 c x′ 2 ] exp [ i ω cB x′ ( x E ) ] dx .
E = nB 2 πc ωp .
I out x t = ʃ U out x ω exp [ i ω ˜ t ] d ω ˜ 2 ,
U out x ω = ω 2 πcBi exp [ i ωL c ] exp [ σ t 2 ω ˜ 2 ] ×
exp [ i ω B 2 c ( D x 2 σ x 2 A ( x E ) 2 4 σ′ 2 ) ] exp [ ( x E ) 2 4 σ′ 2 ] ,
ωL c α o + α 1 ω ˜ + α 2 2 ω ˜ 2 and
ω B 2 c ( D x 2 σ x 2 A ( x E ) 2 4 σ′ 2 ) β o ( x ) + β 1 ( x ) ω ˜ + β 2 ( x ) 2 ω ˜ 2 ,
α i = i ω i ( ωL c ) ω = ω o and β i ( x ) = i ω i ( ω B 2 c { D x 2 σ x 2 A ( x E ) 2 4 σ′ 2 } ) ω = ω o .
E 2 πcn p ω o B o [ 1 + ω ˜ ( 1 B o B ω ω = ω o 1 ω o ) ] .
U out x ω = ω o 2 πc B o i exp [ i ( α o + β o ( x ) ) ] exp [ i ( α 1 + β 1 ( x ) ) ω ˜ ] exp [ i ( α 2 + β 2 ( x ) ) ω ˜ 2 2 ] ×
exp [ ( σ t 2 + E 1 2 4 σ′ 2 ( ω o ) ) ω ˜ 2 ] exp [ ( x E o ) 2 4 σ′ 2 ( ω o ) ] exp [ 2 ( x E o ) E 1 ω ˜ 4 σ′ 2 ( ω o ) ] .
A B C D = 1 f o f ( ω ) f o ( 2 f o f ( ω ) ) 1 f ( ω ) 1 f o f ( ω ) .
E = 2 cn π f o p ω o ( 1 ω ˜ ω o ) and σ′ ( ω o ) = 1 2 c f o σ x ω o .
I out x′ t = exp [ ( x E o ) 2 2 σ′ x 2 ] exp [ ( t α 1 ) 2 2 σ′ t 2 ] ,
σ′ t 2 = σ t 2 ( 1 + 4 n 2 π 2 σ x 2 p 2 ω o 2 σ t 2 ) and σ′ x 2 = σ 2 ( ω o ) ( 1 + 4 n 2 π 2 σ x 2 p 2 ω o 2 σ t 2 ) .
A B C D = 1 d 0 1 1 0 ω o Z′ o ω 1 1 d 0 1 1 0 ω o Z o ω 1
1 f o d 0 1 1 0 1 f ( ω ) 1 1 z 0 1 .
E = E o = 2 πnc Z o f o p ω o ( d + 2 Z o ) , and σ′ ( ω o ) = c f o Z o 2 ω o σ x ( d + 2 Z o ) .
I out x t = exp [ ( x E o ) 2 2 σ′ 2 ( ω o ) ] exp [ ( t α 1 ) 2 2 σ′ t 2 ]
I ( t ) = I out x t dt .

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