Abstract

We carry out a complete spatio-temporal characterization of the electric field of an ultrashort laser pulse after passing through a diffractive optical element composed of several binary amplitude concentric rings. Analytical expressions for the total diffraction field in the time and spectral domain are provided, using the Rayleigh-Sommerfeld formulation of the diffraction. These expressions are experimentally validated. The spatio-temporal amplitude and phase structure of the pulse are measured at different planes beyond the diffractive optical element using spatially-resolved spectral interferometry assisted by an optical fiber coupler (STARFISH). Our results allow corroborating theoretical predictions on the presence of multiple pulses or complex spectral distributions due to the diffraction-induced effects by the hard-edge ring apertures.

© 2010 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Lefrançois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11(10), 1114–1122 (2003).
    [CrossRef] [PubMed]
  2. Z. Jiang, R. Jacquemin, and W. Eberhardt, “Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture,” Appl. Opt. 36(19), 4358–4361 (1997).
    [CrossRef] [PubMed]
  3. J. Li, H. Zhang, D. R. Alexander, D. W. Doerr, and N. R. Tadepalli, “Diffraction characteristics of 10fs laser pulses passing through an aperture,” J. Opt. Soc. Am. A 22(7), 1304–1310 (2005).
    [CrossRef]
  4. H. E. Hwang, G. H. Yang, and P. Han, “Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture,” Opt. Eng. 42(3), 686–695 (2003).
    [CrossRef]
  5. O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, M. Fernández-Alonso, and V. Climent, “On-axis diffraction of an ultrashort light pulse by circularly symmetric hard apertures,” Opt. Express 15(8), 4546–4556 (2007).
    [CrossRef] [PubMed]
  6. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
    [CrossRef] [PubMed]
  7. Z. Liu and B. Lü, “Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture,” Optik (Stuttg.) 115(10), 447–454 (2004).
  8. M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13(4), 771–778 (1996).
    [CrossRef]
  9. R. Netz and T. Feurer, “Diffraction of ultrashort laser pulses and applications for measuring pulse front distortion and pulse width,” Appl. Phys. B 70, 813–819 (2000).
  10. S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
    [CrossRef]
  11. H. Zhang, J. Li, D. W. Doerr, and D. R. Alexander, “Diffraction characteristics of a Fresnel zone plate illuminated by 10 fs laser pulses,” Appl. Opt. 45(33), 8541–8546 (2006).
    [CrossRef] [PubMed]
  12. C. J. Zapata-Rodríguez, “Temporal effects in ultrashort pulsed beams focused by planar diffracting elements,” J. Opt. Soc. Am. A 23(9), 2335–2341 (2006).
    [CrossRef]
  13. R. Ashman and M. Gu, “Effect of ultrashort pulsed illumination on foci caused by a Fresnel zone plate,” Appl. Opt. 42(10), 1852–1855 (2003).
    [CrossRef] [PubMed]
  14. J. Pearce and D. Mittleman, “Defining the Fresnel zone for broadband radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056602 (2002).
    [CrossRef]
  15. O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, and V. Climent, “Focusing and spectral characteristics of periodic diffractive optical elements with circular symmetry under femtosecond pulsed illumination,” J. Opt. Soc. Am. A 24(11), 3600–3605 (2007).
    [CrossRef]
  16. O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, E. Tajahuerce, and V. Climent, “Spectral analysis of femtosecond pulse diffraction through binary diffractive optical elements: theory and experiment,” Opt. Express 16(4), 2541–2546 (2008).
    [CrossRef] [PubMed]
  17. M. Born, and E. Wolf, Principles of Optics, 7th ed. (Cambridge University press, 2005).
  18. G. Mínguez-Vega, O. Mendoza-Yero, J. Lancis, and V. Climent, “Proposal for the generation of THz bursts and codes of shaped femtosecond pulses using binary mask,” IEEE Photon. Technol. Lett. 19(21), 1732–1734 (2007).
    [CrossRef]
  19. B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
    [CrossRef]
  20. S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27(14), 1211–1213 (2002).
    [CrossRef]
  21. X. Hu and J. Pu, “Spectral anomalies of polychromatic, spatially coherent light diffracted by an annular apertures in the far field,” Chin. Opt. Lett. 3, 418–421 (2005).
  22. J. Pu, C. Cai, and S. Nemoto, “Spectral anomalies in Young’s double-slit interference experiment,” Opt. Express 12(21), 5131–5139 (2004).
    [CrossRef] [PubMed]
  23. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Direct space time-characterization of the electric fields of ultrashort optical pulses,” Opt. Lett. 27(7), 548–550 (2002).
    [CrossRef]
  24. D. J. Kane and R. Trebino, “Characterization of Arbitrary Femtosecond Pulses Using Frequency-Resolved Optical Gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
    [CrossRef]
  25. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23(10), 792–794 (1998).
    [CrossRef]
  26. P. Gabolde and R. Trebino, “Single-shot measurement of the full spatio-temporal field of ultrashort pulses with multi-spectral digital holography,” Opt. Express 14(23), 11460–11467 (2006).
    [CrossRef] [PubMed]
  27. P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15(16), 10219–10230 (2007).
    [CrossRef] [PubMed]
  28. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16(18), 13663–13675 (2008).
    [CrossRef] [PubMed]
  29. F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express 17(12), 9804–9809 (2009).
    [CrossRef] [PubMed]
  30. P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, “Basic diffraction phenomena in time domain,” Opt. Express 18(11), 11083–11088 (2010).
    [CrossRef] [PubMed]
  31. B. Alonso, Í. J. Sola, Ó. Varela, J. Hernández-Toro, C. Méndez, J. San Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by Fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B 27(5), 933–940 (2010).
    [CrossRef]
  32. L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12), 2467–2474 (1995).
    [CrossRef]

2010 (2)

2009 (1)

2008 (2)

2007 (4)

2006 (5)

P. Gabolde and R. Trebino, “Single-shot measurement of the full spatio-temporal field of ultrashort pulses with multi-spectral digital holography,” Opt. Express 14(23), 11460–11467 (2006).
[CrossRef] [PubMed]

B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
[CrossRef]

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
[CrossRef]

H. Zhang, J. Li, D. W. Doerr, and D. R. Alexander, “Diffraction characteristics of a Fresnel zone plate illuminated by 10 fs laser pulses,” Appl. Opt. 45(33), 8541–8546 (2006).
[CrossRef] [PubMed]

C. J. Zapata-Rodríguez, “Temporal effects in ultrashort pulsed beams focused by planar diffracting elements,” J. Opt. Soc. Am. A 23(9), 2335–2341 (2006).
[CrossRef]

2005 (2)

2004 (2)

J. Pu, C. Cai, and S. Nemoto, “Spectral anomalies in Young’s double-slit interference experiment,” Opt. Express 12(21), 5131–5139 (2004).
[CrossRef] [PubMed]

Z. Liu and B. Lü, “Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture,” Optik (Stuttg.) 115(10), 447–454 (2004).

2003 (3)

2002 (3)

2001 (1)

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

2000 (1)

R. Netz and T. Feurer, “Diffraction of ultrashort laser pulses and applications for measuring pulse front distortion and pulse width,” Appl. Phys. B 70, 813–819 (2000).

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1993 (1)

D. J. Kane and R. Trebino, “Characterization of Arbitrary Femtosecond Pulses Using Frequency-Resolved Optical Gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[CrossRef]

Alexander, D. R.

Alonso, B.

Ashman, R.

Biegert, J.

Bonaretti, F.

Bor, Z.

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

Bowlan, P.

Cai, C.

Callender, C. L.

B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
[CrossRef]

Chen, L. R.

B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
[CrossRef]

Cheriaux, G.

Clerici, M.

Climent, V.

Di Trapani, P.

Doerr, D. W.

Dorrer, C.

Dumais, P.

B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
[CrossRef]

Eberhardt, W.

Faccio, D.

Fernández-Alonso, M.

Feurer, T.

R. Netz and T. Feurer, “Diffraction of ultrashort laser pulses and applications for measuring pulse front distortion and pulse width,” Appl. Phys. B 70, 813–819 (2000).

Fuchs, U.

Gabolde, P.

Gan, X. S.

Gu, M.

Han, P.

H. E. Hwang, G. H. Yang, and P. Han, “Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture,” Opt. Eng. 42(3), 686–695 (2003).
[CrossRef]

Hernández-Toro, J.

Horváth, Z. L.

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

Hu, X.

Hwang, H. E.

H. E. Hwang, G. H. Yang, and P. Han, “Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture,” Opt. Eng. 42(3), 686–695 (2003).
[CrossRef]

Iaconis, C.

Jacquemin, R.

Jiang, Z.

Joffre, M.

Kane, D. J.

D. J. Kane and R. Trebino, “Characterization of Arbitrary Femtosecond Pulses Using Frequency-Resolved Optical Gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[CrossRef]

Kosik, E. M.

Lancis, J.

Lefrançois, M.

Lepetit, L.

Li, J.

Liu, Z.

Z. Liu and B. Lü, “Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture,” Optik (Stuttg.) 115(10), 447–454 (2004).

Lõhmus, M.

Lü, B.

Z. Liu and B. Lü, “Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture,” Optik (Stuttg.) 115(10), 447–454 (2004).

Méndez, C.

Mendoza-Yero, O.

Mínguez-Vega, G.

Mittleman, D.

J. Pearce and D. Mittleman, “Defining the Fresnel zone for broadband radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056602 (2002).
[CrossRef]

Nemoto, S.

Netz, R.

R. Netz and T. Feurer, “Diffraction of ultrashort laser pulses and applications for measuring pulse front distortion and pulse width,” Appl. Phys. B 70, 813–819 (2000).

Pearce, J.

J. Pearce and D. Mittleman, “Defining the Fresnel zone for broadband radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056602 (2002).
[CrossRef]

Pereira, S. F.

Piksarv, P.

Ponomarenko, S. A.

Pu, J.

Roso, L.

Saari, P.

San Román, J.

Sola, Í. J.

Tadepalli, N. R.

Tajahuerce, E.

Trebino, R.

Valtna-Lukner, H.

Varela, Ó.

Veetil, S. P.

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
[CrossRef]

Vijayan, C.

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
[CrossRef]

Viswanathan, N. K.

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
[CrossRef]

Walmsley, I. A.

Wolf, E.

Wyrowski, F.

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
[CrossRef]

Xia, B.

B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
[CrossRef]

Yang, G. H.

H. E. Hwang, G. H. Yang, and P. Han, “Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture,” Opt. Eng. 42(3), 686–695 (2003).
[CrossRef]

Zaïr, A.

Zapata-Rodríguez, C. J.

Zeitner, U. D.

Zhang, H.

Appl. Opt. (3)

Appl. Phys. B (1)

R. Netz and T. Feurer, “Diffraction of ultrashort laser pulses and applications for measuring pulse front distortion and pulse width,” Appl. Phys. B 70, 813–819 (2000).

Appl. Phys. Lett. (1)

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, “Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities,” Appl. Phys. Lett. 89(4), 041119 (2006).
[CrossRef]

Chin. Opt. Lett. (1)

Electron. Lett. (1)

B. Xia, L. R. Chen, P. Dumais, and C. L. Callender, “Ultrafast pulse train generation with binary code patterns using planar lightwave circuits,” Electron. Lett. 42(19), 1119–1120 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. J. Kane and R. Trebino, “Characterization of Arbitrary Femtosecond Pulses Using Frequency-Resolved Optical Gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. Mínguez-Vega, O. Mendoza-Yero, J. Lancis, and V. Climent, “Proposal for the generation of THz bursts and codes of shaped femtosecond pulses using binary mask,” IEEE Photon. Technol. Lett. 19(21), 1732–1734 (2007).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (2)

Opt. Eng. (1)

H. E. Hwang, G. H. Yang, and P. Han, “Near-field diffraction characteristics of a time-dependence Gaussian-shape pulsed beam from a circular aperture,” Opt. Eng. 42(3), 686–695 (2003).
[CrossRef]

Opt. Express (9)

M. Lefrançois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11(10), 1114–1122 (2003).
[CrossRef] [PubMed]

J. Pu, C. Cai, and S. Nemoto, “Spectral anomalies in Young’s double-slit interference experiment,” Opt. Express 12(21), 5131–5139 (2004).
[CrossRef] [PubMed]

P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, “Basic diffraction phenomena in time domain,” Opt. Express 18(11), 11083–11088 (2010).
[CrossRef] [PubMed]

O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, E. Tajahuerce, and V. Climent, “Spectral analysis of femtosecond pulse diffraction through binary diffractive optical elements: theory and experiment,” Opt. Express 16(4), 2541–2546 (2008).
[CrossRef] [PubMed]

P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16(18), 13663–13675 (2008).
[CrossRef] [PubMed]

F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express 17(12), 9804–9809 (2009).
[CrossRef] [PubMed]

P. Gabolde and R. Trebino, “Single-shot measurement of the full spatio-temporal field of ultrashort pulses with multi-spectral digital holography,” Opt. Express 14(23), 11460–11467 (2006).
[CrossRef] [PubMed]

O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, M. Fernández-Alonso, and V. Climent, “On-axis diffraction of an ultrashort light pulse by circularly symmetric hard apertures,” Opt. Express 15(8), 4546–4556 (2007).
[CrossRef] [PubMed]

P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15(16), 10219–10230 (2007).
[CrossRef] [PubMed]

Opt. Lett. (3)

Optik (Stuttg.) (1)

Z. Liu and B. Lü, “Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture,” Optik (Stuttg.) 115(10), 447–454 (2004).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

J. Pearce and D. Mittleman, “Defining the Fresnel zone for broadband radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056602 (2002).
[CrossRef]

Other (1)

M. Born, and E. Wolf, Principles of Optics, 7th ed. (Cambridge University press, 2005).

Supplementary Material (3)

» Media 1: AVI (2022 KB)     
» Media 2: AVI (770 KB)     
» Media 3: AVI (575 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Movies of the spatio-temporal time evolution of the instantaneous intensity at different spatial regions a) (2.02 MB Media 1) Near-field region, b) (0.77 MB Media 2) Fresnel region, and c) (0.57 MB Media 3) Far-field region.

Fig. 2
Fig. 2

Experimental setup: one replica of the laser pulse is used as the reference and the other replica illuminates the DOE (test beam). The pulses are collected by the arms of the fiber coupler, the reference one controls the relative delay, whereas the test one spatially scans the test beam. The spatially resolved SI is measured after the fiber coupler in the spectrometer. The position of the DOE allows exploring different propagation distances.

Fig. 3
Fig. 3

On-axis irradiance pattern of the DOE for the central wavelength of the pulse. The green stars correspond approximately to the positions of the spatio-temporal reconstructed planes at z = 35mm, z = 141mm and z = 203mm shown later in Fig. 4, 5 and 6, respectively.

Fig. 4
Fig. 4

Simulated and experimental spatially resolved spectrum (a, c), and corresponding spatio-temporal intensity (b, d) for the propagation distance z = 35mm.

Fig. 6
Fig. 6

Simulated and experimental spatially resolved spectrum (a, c), corresponding spatio temporal intensity (b, d) for the propagation distance z = 203mm.

Fig. 5
Fig. 5

Simulated and experimental spatially resolved spectrum (a, c), corresponding spatio-temporal intensity (b, d) for the propagation distance z = 141mm.

Equations (8)

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

U m ( z , R , ω ) U i m ( z , R , ω ) U o m ( z , R , ω )
U i m / o m ( z , R , ω ) = z exp ( i ω c z 2 + r i m / o m 2 ) J 0 ( ω c r i m / o m R z 2 + r i m / o m 2 ) z 2 + r i m / o m 2
S ( z , R , ω ) = S 0 ( ω ) | U ( z , R , ω ) | 2
U t i m e ( z , R , t ) = 1 2 π A ( ω ) U ( z , R , ω ) exp ( i ω t ) d ω
A ( ω ) = u ( t ) exp ( i ω t ) d t
U t i m e ( z , R , t ) = m = 1 N F i m ( z , R , t ) F o m ( z , R , t )
F i m / o m ( z , R , t ) = z z 2 + r i m / o m 2 1 2 π 0 2 π u ( t z 2 + r i m / o m 2 c + r i m / o m R cos θ c z 2 + r i m / o m 2 ) d θ
F i m / o m ( z , R , t ) = z exp [ i ω 0 ( t s i m / o m c ) ] s i m / o m exp [ ( t s i m / o m c ) 2 4 σ 2 ] J 0 ( ω 0 r i m / o m R c s i m / o m )

Metrics