Abstract

Many applications of ultrashort laser pulses require manipulation and control of the pulse parameters by propagating them through different optical components before the target. This requires methods of simulating the pulse propagation taking into account all effects of dispersion, diffraction, and system aberrations. In this paper, we propose a method of propagating ultrashort pulses through a real optical system by using the Gaussian pulsed beam decomposition. An input pulse with arbitrary spatial and temporal (spectral) profiles is decomposed into a set of elementary Gaussian pulsed beams in the spatiospectral domain. The final scalar electric field of the ultrashort pulse after propagation is then obtained by performing the phase correct superposition of the electric fields all-Gaussian pulsed beams, which are propagated independently through the optical system. We demonstrate the application of the method by propagating an ultrashort pulse through a focusing aspherical lens with large chromatic aberration and a Bessel-X pulse generating axicon lens.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  32. Thorlabs, Ax252-b 2.0°, 650–1050  nm AR coated UVFS, Ø1″ (Ø25.4  mm) Axicon (2009).
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    [Crossref]
  34. P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
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    [Crossref]

2019 (1)

2018 (1)

2017 (2)

N. Worku and H. Gross, “Vectorial field propagation through high NA objectives using polarized Gaussian beam decomposition,” Proc. SPIE 10347, 103470W (2017).
[Crossref]

M. Cywiak, D. Cywiak, and E. Yáñez, “Finite Gaussian wavelet superposition and Fresnel diffraction integral for calculating the propagation of truncated, non-diffracting and accelerating beams,” Opt. Commun. 405, 132–142 (2017).
[Crossref]

2016 (1)

2015 (1)

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

2014 (1)

2010 (2)

F. Wyrowski, C. Hellmann, R. Krieg, and H. Schweitzer, “Modeling the propagation of ultrashort pulses through optical systems,” Proc. SPIE 7589, 75890O (2010).
[Crossref]

M. Cywiak, A. Morales, M. Servn, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18, 19141–19155 (2010).
[Crossref]

2009 (1)

2008 (1)

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. A 23, 2227–2236 (2008).
[Crossref]

2005 (2)

2004 (1)

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
[Crossref]

2002 (1)

Y. Cai and Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[Crossref]

2001 (1)

M. J. Bastiaans, “Gabor’s signal expansion based on a nonorthogonal sampling geometry,” Proc. SPIE 4392, 46–60 (2001).
[Crossref]

2000 (1)

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[Crossref]

1999 (1)

C. Caron and R. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46, 1881–1891 (1999).
[Crossref]

1998 (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086 (1998).
[Crossref]

1997 (2)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

1996 (1)

1995 (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 785–798 (1995).
[Crossref]

1993 (1)

1990 (2)

A. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[Crossref]

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[Crossref]

1989 (1)

1986 (1)

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–52 (1986).
[Crossref]

1970 (1)

1946 (1)

D. Gabor, “Theory of communication. Part 1: the analysis of information,” J. Inst. Electr. Eng. Part 3 93, 429–441 (1946).
[Crossref]

Akturk, S.

Alda, J.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 785–798 (1995).
[Crossref]

April, A.

A. April, “Ultrashort, strongly focused laser pulses in free space,” in Coherence and Ultrashort Pulse Laser Emission (IntechOpen, 2010), pp. 355–382.

Bastiaans, M. J.

M. J. Bastiaans, “Gabor’s signal expansion based on a nonorthogonal sampling geometry,” Proc. SPIE 4392, 46–60 (2001).
[Crossref]

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 785–798 (1995).
[Crossref]

Bor, Z.

Bowlan, P.

Burghoff, J.

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
[Crossref]

Cai, Y.

Y. Cai and Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[Crossref]

Caron, C.

C. Caron and R. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46, 1881–1891 (1999).
[Crossref]

Collins, S. A.

Cywiak, D.

M. Cywiak, D. Cywiak, and E. Yáñez, “Finite Gaussian wavelet superposition and Fresnel diffraction integral for calculating the propagation of truncated, non-diffracting and accelerating beams,” Opt. Commun. 405, 132–142 (2017).
[Crossref]

Cywiak, M.

M. Cywiak, D. Cywiak, and E. Yáñez, “Finite Gaussian wavelet superposition and Fresnel diffraction integral for calculating the propagation of truncated, non-diffracting and accelerating beams,” Opt. Commun. 405, 132–142 (2017).
[Crossref]

M. Cywiak, A. Morales, M. Servn, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18, 19141–19155 (2010).
[Crossref]

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[Crossref]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[Crossref]

Fuchs, U.

Gabolde, P.

Gabor, D.

D. Gabor, “Theory of communication. Part 1: the analysis of information,” J. Inst. Electr. Eng. Part 3 93, 429–441 (1946).
[Crossref]

Gómez-Medina, R.

Greynolds, A. W.

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–52 (1986).
[Crossref]

A. W. Greynolds, “Fat rays revisited: a synthesis of physical and geometrical optics with gaußlets,” in International Optical Design Conference (Optical Society of America, 2014), paper ITu1A.3.

Gross, H.

Gu, X.

Hambach, R.

Harvey, J. E.

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Hellmann, C.

F. Wyrowski, C. Hellmann, R. Krieg, and H. Schweitzer, “Modeling the propagation of ultrashort pulses through optical systems,” Proc. SPIE 7589, 75890O (2010).
[Crossref]

Hirlimann, C.

Irvin, R. G.

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Kostenbauder, A.

A. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[Crossref]

Krieg, R.

F. Wyrowski, C. Hellmann, R. Krieg, and H. Schweitzer, “Modeling the propagation of ultrashort pulses through optical systems,” Proc. SPIE 7589, 75890O (2010).
[Crossref]

Lin, Q.

Y. Cai and Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[Crossref]

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[Crossref]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 785–798 (1995).
[Crossref]

Lõhmus, M.

Mahon, R. J.

Marcus, G.

May, M.

Morales, A.

Morhange, J. F.

Murphy, J. A.

Nolte, S.

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
[Crossref]

Pfisterer, R. N.

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Piksarv, P.

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086 (1998).
[Crossref]

Potvliege, R.

C. Caron and R. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46, 1881–1891 (1999).
[Crossref]

Saari, P.

Schimmel, H.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. A 23, 2227–2236 (2008).
[Crossref]

Schweitzer, H.

F. Wyrowski, C. Hellmann, R. Krieg, and H. Schweitzer, “Modeling the propagation of ultrashort pulses through optical systems,” Proc. SPIE 7589, 75890O (2010).
[Crossref]

Servn, M.

Sharma, D. K.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. A 23, 2227–2236 (2008).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Linear pulse propagation,” in Lasers (University Science Books, 1986), Vol. 37, pp. 331–361.

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[Crossref]

Sõnajalg, H.

Trebino, R.

Tünnermann, A.

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

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
[Crossref]

Valtna-Lukner, H.

Veetil, S. P.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. A 23, 2227–2236 (2008).
[Crossref]

Vijayan, C.

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. A 23, 2227–2236 (2008).
[Crossref]

Wang, L.

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[Crossref]

Wang, S.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 785–798 (1995).
[Crossref]

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Weiner, A.

A. Weiner, Ultrafast Optics (Wiley, 2011), Vol. 72.

Will, M.

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
[Crossref]

Worku, N.

N. Worku and H. Gross, “Vectorial field propagation through high NA objectives using polarized Gaussian beam decomposition,” Proc. SPIE 10347, 103470W (2017).
[Crossref]

Worku, N. G.

Wyrowski, F.

F. Wyrowski, C. Hellmann, R. Krieg, and H. Schweitzer, “Modeling the propagation of ultrashort pulses through optical systems,” Proc. SPIE 7589, 75890O (2010).
[Crossref]

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Schimmel, and F. Wyrowski, “Sampling rules in frequency domain for numerical propagation of ultrashort pulses through linear dielectrics,” J. Opt. Soc. Am. A 23, 2227–2236 (2008).
[Crossref]

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Yáñez, E.

M. Cywiak, D. Cywiak, and E. Yáñez, “Finite Gaussian wavelet superposition and Fresnel diffraction integral for calculating the propagation of truncated, non-diffracting and accelerating beams,” Opt. Commun. 405, 132–142 (2017).
[Crossref]

Zeitner, U. D.

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (3)

A. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[Crossref]

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[Crossref]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

J. Inst. Electr. Eng. Part 3 (1)

D. Gabor, “Theory of communication. Part 1: the analysis of information,” J. Inst. Electr. Eng. Part 3 93, 429–441 (1946).
[Crossref]

J. Mod. Opt. (2)

C. Caron and R. Potvliege, “Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46, 1881–1891 (1999).
[Crossref]

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

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

Laser Phys. (1)

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

Opt. Commun. (3)

M. Cywiak, D. Cywiak, and E. Yáñez, “Finite Gaussian wavelet superposition and Fresnel diffraction integral for calculating the propagation of truncated, non-diffracting and accelerating beams,” Opt. Commun. 405, 132–142 (2017).
[Crossref]

Y. Cai and Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[Crossref]

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263–269 (2000).
[Crossref]

Opt. Eng. (1)

J. E. Harvey, R. G. Irvin, and R. N. Pfisterer, “Modeling physical optics phenomena by complex ray tracing,” Opt. Eng. 54, 035105 (2015).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Opt. Quantum Electron. (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27, 785–798 (1995).
[Crossref]

Phys. Rev. E (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086 (1998).
[Crossref]

Proc. SPIE (4)

M. J. Bastiaans, “Gabor’s signal expansion based on a nonorthogonal sampling geometry,” Proc. SPIE 4392, 46–60 (2001).
[Crossref]

N. Worku and H. Gross, “Vectorial field propagation through high NA objectives using polarized Gaussian beam decomposition,” Proc. SPIE 10347, 103470W (2017).
[Crossref]

F. Wyrowski, C. Hellmann, R. Krieg, and H. Schweitzer, “Modeling the propagation of ultrashort pulses through optical systems,” Proc. SPIE 7589, 75890O (2010).
[Crossref]

A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–52 (1986).
[Crossref]

Other (5)

A. April, “Ultrashort, strongly focused laser pulses in free space,” in Coherence and Ultrashort Pulse Laser Emission (IntechOpen, 2010), pp. 355–382.

A. W. Greynolds, “Fat rays revisited: a synthesis of physical and geometrical optics with gaußlets,” in International Optical Design Conference (Optical Society of America, 2014), paper ITu1A.3.

A. E. Siegman, “Linear pulse propagation,” in Lasers (University Science Books, 1986), Vol. 37, pp. 331–361.

Thorlabs, Ax252-b 2.0°, 650–1050  nm AR coated UVFS, Ø1″ (Ø25.4  mm) Axicon (2009).

A. Weiner, Ultrafast Optics (Wiley, 2011), Vol. 72.

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Figures (19)

Fig. 1.
Fig. 1. Schematic diagram showing the ray pulse parameters of a given optical system.
Fig. 2.
Fig. 2. Schematic diagram showing the curved pulse front due to the spatiotemporal coupling in free space.
Fig. 3.
Fig. 3. Simple 2D geometric construction for computation of the complex amplitude of the Gaussian pulsed beam on a given target pixel $ P $ at a given flight time.
Fig. 4.
Fig. 4. (a) Overlapping Gaussian beam foot prints projected on the transversal X–Y plane. (b) The directions (shown in blue arrows) of all Gaussian beams used for decomposition of a given curved wavefront. (c) The local surface patch of a single Gaussian beam.
Fig. 5.
Fig. 5. (a) Parabolic approximations of a spherical wavefront, with radius of 4 mm, by using the curvature at the center (in approx. 1) and by forcing the parabolic wavefront to coincide with the given one at a finite width $ x = 3\;{\rm mm} $ (in approx. 2). (b) The residual error of the two approximations that are shown in (a).
Fig. 6.
Fig. 6. Schematic diagram showing the spectral decomposition of an ultrashort pulse with an arbitrary spectral amplitude and phase profile into set of Gaussian pulses.
Fig. 7.
Fig. 7. (a) Spectral phase of the given pulse together with the quadratic spectral phases (shown for only two sampling points for clarity). (b) The group delays and the group delay dispersion of the Gaussian pulses used for the decomposition.
Fig. 8.
Fig. 8. Spectral (a) amplitude and (b) phase profiles of the given pulse together with that obtained by superposition of the elementary Gaussian pulses each with different delay and chirp parameters. The residual error of the spectral (c) amplitude and (d) phase after Gaussian pulse decomposition.
Fig. 9.
Fig. 9. Temporal amplitude profile of the given pulse with large TOD computed by superposition of the elementary Gaussian pulses.
Fig. 10.
Fig. 10. Same as in Fig. 8 but now computed using zero initial delay and chirp parameters of the individual Gaussian pulses.
Fig. 11.
Fig. 11. Spatiotemporal amplitude profiles of an ultrashort pulse propagating different distances in free space.
Fig. 12.
Fig. 12. (a) Same as in Fig. 11(c) but now computed using the conventional Fourier-transform-based method. (b) The numerical difference between the amplitude profiles in (a) and in Fig. 11(c).
Fig. 13.
Fig. 13. 2D layout of the Geltech Aspheric Lens used for focusing ultrashort pulse.
Fig. 14.
Fig. 14. $ y - z $ cross-sectional view of the spatial intensity profile of the ultrashort pulse near the focal plane of the aspheric lens for different times. The time is chosen in such a way that $ t = 0 $ refers to the pulse intensity maximum reaches at the focus.
Fig. 15.
Fig. 15. (a) $ y $ cross section of the pulse intensity profile at the focal plane computed using Gaussian pulsed beam decomposition (GPBD)- and the Fourier transform (FFT)-based methods. (b) The difference in the pulse intensities obtained using the two methods.
Fig. 16.
Fig. 16. 2D layout of an axicon lens used to generate a Bessel-X pulse (different scales are used for $ z $ and $ y $ axis to make the rays more visible). The dotted lines indicate the locations of the transversal planes where the pulse intensities are computed.
Fig. 17.
Fig. 17. $ y - t $ cross section of the amplitude profile of the Bessel-X pulse on the transversal planes, which are located at (a) 55 mm, (b) 95 mm, and (c) 135 mm from the axicon lens. The white dotted lines indicate the location of $ t = 0 $, and the red dotted lines indicate the location of the pulse peak.
Fig. 18.
Fig. 18. Temporal cross sections of the amplitude profiles shown in Fig. 17 for the on-axis point.
Fig. 19.
Fig. 19. (a) 1D cross section of the Bessel-X pulse on the transversal plane located at $ z = 95\;{\rm mm} $ from the axicon lens computed by using the GPBD method together with that obtained using the analytical propagation formula. (b) The numerical difference of the magnitudes of the amplitude profiles shown in (a).

Equations (20)

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P = ( x y c t ¯ n θ x n θ y f ¯ / f 0 ) T ,
( r ~ 2 ν ~ 2 ) = ( A ~ B ~ C ~ D ~ ) M ~ ( r ~ 1 ν ~ 1 ) ,
A ~ = r ~ 2 r ~ 1 , B ~ = r ~ 2 ν ~ 1 , C ~ = ν ~ 2 r ~ 1 , a n d D ~ = ν ~ 2 ν ~ 1 .
E ( x , y , z , t ) = e ( U ( r ~ ; z ) exp i ( ω 0 t n k 0 z ) ) ,
U ( r ~ ; z ) = U 0 exp [ i k 0 2 r ~ T Q ~ 1 r ~ ] ,
Q ~ 1 = ( q xx 1 q xy 1 q x τ 1 q yx 1 q yy 1 q y τ 1 q τ x 1 q τ y 1 q τ τ 1 ) .
U 2 ( r ~ 2 ) = U 0 d e t ( A ~ + B ~ Q ~ 1 1 ) exp ( i k 0 2 r ~ 2 T Q ~ 2 1 r ~ 2 ) ,
t p h a s e = n Δ z c ,
t g r o u p = n g Δ z c .
Δ G P L = n g Δ z .
Δ t = r 2 2 cR ( z ) ,
Δ t = n g z ( r ~ 0 ) c ,
z q ( x , y ) = c xx x 2 + c xy x y + c yy y 2 .
c xx = z ϕ ( w x , 0 ) w x 2 , c yy = z ϕ ( 0 , w y ) w y 2 , c xy = z ϕ ( w x , w y ) ( c xx w x 2 + c yy w y 2 ) w x 2 w y 2 ,
q xx 1 = 2 c xx 1 i λ 0 π w x 2 , q yy 1 = 2 c yy 1 i λ 0 π w y 2 , q xy 1 = q yx 1 = c xy .
σ ω = 1 2 ( Ω m a x Ω m i n ) N ω ( ε O L ) ,
ϕ ( ω ) | ω 0 ϕ 0 + ϕ 1 ( ω ω 0 ) + ϕ 2 ( ω ω 0 ) 2 ,
E ( r = 0 , ω ¯ ) exp ( i 1 2 k 0 c 2 q τ τ 1 ω ¯ 2 ) ,
i 1 2 k 0 c 2 q τ τ 1 ω ¯ 2 = ω ¯ 2 σ ω 2 + i ϕ 2 ω ¯ 2 .
q τ τ 1 = i [ 2 k 0 c 2 ( 1 σ ω 2 + i ϕ 2 ) ] 1 .

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