## Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes |

JOSA A, Vol. 30, Issue 2, pp. 215-226 (2013)

http://dx.doi.org/10.1364/JOSAA.30.000215

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### Abstract

The Laguerre–Gaussian (LG) beam expansion is described as a numerical and physical model of paraxial ultrashort pulse diffraction in the time domain. An overview of the dynamics of higher-order ultrashort planar LG modes is given through numerical simulations, and the finite width of these beams is shown to induce a dispersive-like axial broadening of the fields, which creates related variations in the on-axis amplitude of such pulses. The propagation of a pulsed plane wave scattered at an aperture is then illustrated as a finite weighted sum of individual planar LG pulses, which allows for intuitive illustration of the convergence of this expansion technique. By applying such an expansion to diffraction at a hard aperture, the planar pulsed LG beams are described as the paraxial analogs of the Bessel boundary waves typically observed in such situations, with both exhibiting superluminal group velocities along the optical axis. Numerical results of pulse diffraction at an aperture highlight the suitability of the LG expansion method for efficient and practical simulation of ultrashort fields in the paraxial regime.

© 2013 Optical Society of America

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.5080) Diffraction and gratings : Phase shift

(260.1960) Physical optics : Diffraction theory

(260.2030) Physical optics : Dispersion

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 17, 2012

Revised Manuscript: December 6, 2012

Manuscript Accepted: December 9, 2012

Published: January 14, 2013

**Citation**

Ronan J. Mahon and J. Anthony Murphy, "Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes," J. Opt. Soc. Am. A **30**, 215-226 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-2-215

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### References

- M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011). [CrossRef]
- M. F. Kling and M. J. J. Vrakking, “Attosecond electron dynamics,” Annu. Rev. Phys. Chem. 59, 463–492 (2008). [CrossRef]
- R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991). [CrossRef]
- S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001). [CrossRef]
- P. M. Goorjian and A. Taflove, “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992). [CrossRef]
- A. M. Nugrowati, S. F. Pereira, and A. S. van de Nes, “Near and intermediate fields of an ultrashort pulse transmitted through Young’s double-slit experiment,” Phys. Rev. A 77, 053810 (2008). [CrossRef]
- A. Gürtler, C. Winnewisser, H. Helm, and P. U. Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74–83 (2000). [CrossRef]
- E. Budiarto, N.-W. Pu, S. Jeong, and J. Bokor, “Near-field propagation of terahertz pulses from a large-aperture antenna,” Opt. Lett. 23, 213–215 (1998). [CrossRef]
- M. Lefrançois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11, 1114–1122 (2003). [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]
- Z. Jiang, R. Jacquemin, and W. Eberhardt, “Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture,” Appl. Opt. 36, 4358–4361 (1997). [CrossRef]
- 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, 771–778 (1996). [CrossRef]
- D. P. Kelly, B. M. Hennelly, A. Grün, and K. Unterrainer, “Numerical sampling rules for paraxial regime pulse diffraction calculations,” J. Opt. Soc. Am. A 25, 2299–2308 (2008). [CrossRef]
- Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001). [CrossRef]
- 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]
- P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, “Basic diffraction phenomena in time domain,” Opt. Express 18, 11083–11088 (2010). [CrossRef]
- M. Lõhmus, P. Bowlan, P. Piksarv, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort optical pulses from circularly symmetric binary phase gratings,” Opt. Lett. 37, 1238–1240 (2012). [CrossRef]
- P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012). [CrossRef]
- 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, 4546–4556 (2007). [CrossRef]
- C. J. Zapata-Rodríguez, “Temporal effects in ultrashort pulsed beams focused by planar diffracting elements,” J. Opt. Soc. Am. A 23, 2335–2341 (2006). [CrossRef]
- P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
- M. A. Porras, R. Borghi, and M. Santarsiero, “Superluminality in Gaussian beams,” Opt. Commun. 203, 183–189 (2002). [CrossRef]
- R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197–203(1996). [CrossRef]
- A. E. Siegman, Lasers (University Science, 1986).
- J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993). [CrossRef]
- N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003). [CrossRef]
- E. Cavanagh and B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980). [CrossRef]
- Y. Liu and B. Lü, “Truncated Hermite–Gauss series expansion and its application,” Optik 117, 437–442 (2006). [CrossRef]
- E. Cagniot, M. Fromager, and K. Ait-Ameur, “Modeling the propagation of apertured high-order Laguerre–Gaussian beams by a user-friendly version of the mode expansion method,” J. Opt. Soc. Am. A 27, 484–491 (2010). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]
- R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983). [CrossRef]
- I. S. Gradstehyn and I. M. Rhyzik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), p. 811.
- S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001). [CrossRef]
- R. J. Mahon and J. A. Murphy, “Gaussian beam mode analysis of optical pulses,” Proc. SPIE 8171, 81710H (2011). [CrossRef]
- Y. D. Liu and C. Gao, “Study on the time-varying and propagating characteristics of ultrashort pulse Laguerre–Gaussian beam,” Opt. Express 18, 12104–12110 (2010). [CrossRef]
- E. Heyman and I. Beracha, “Complex multipole pulsed beams and Hermite pulsed beams: a novel expansion scheme for transient radiation from well-collimated apertures,” J. Opt. Soc. Am. A 9, 1779–1973 (1992). [CrossRef]
- S. Feng and H. G. Winful, “Higher-order transverse modes of ultrashort isodiffracting pulses,” Phys. Rev. E 63, 046602 (2001). [CrossRef]
- R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992). [CrossRef]
- A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single cycle terahertz pulses,,” Phys. Rev. Lett. 83, 3410–3413 (1999). [CrossRef]
- Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999). [CrossRef]
- S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998). [CrossRef]
- Z. Yang, Z. Yang, and S. Zhang, “Carrier-envelope phase of ultrashort pulsed Laguerre–Gaussian beam,” Chin. Opt. Lett. 6, 189–191 (2008). [CrossRef]
- M. V. Vasnetsov and V. A. Pas’ko, “Group velocity of Gaussian beams,” Ukr. J. Phys. 54, 50–52 (2009).
- Q. Zou and B. Lü, “Anomalous spectral behaviour near phase singularities in diffraction of pulsed Laguerre–Gaussian beams,” J. Opt. A 8, 531–536 (2006).
- W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A 19, 49–53 (2002). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
- D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005). [CrossRef]
- Q. Zou and B. Lü, “Temporal and spectral properties of ultrashort pulsed Laguerre–Gaussian beams in dispersive media,” Optik 118, 83–87 (2007). [CrossRef]
- R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2007), pp. 379–380.
- J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988). [CrossRef]
- D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004). [CrossRef]
- M. Zamboni-Rached, E. Recami, and M. Balma, “A simple and effective method for the analytic description of important optical beams, when truncated by finite apertures,” Appl. Opt. 51, 3370–3379 (2012). [CrossRef]
- M. Zamboni-Rached, K. Z. Nóbrega, and C. A. Dartora, “Analytic description of Airy-type beams when truncated by finite apertures,” Opt. Express 20, 19972–19977 (2012). [CrossRef]

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