## Evolution of subcycle pulses in nonparaxial Gaussian beams

Optics Express, Vol. 8, Issue 11, pp. 590-598 (2001)

http://dx.doi.org/10.1364/OE.8.000590

Acrobat PDF (271 KB)

### Abstract

A simple but exact treatment of spatiotemporal behavior of ultrawideband pulses under an arbitrarily tight focusing is developed. The model makes use of the oblate spheroidal coordinate system to represent free scalar field as if generated by a point-like source-and-sink pair placed at a complex location. The results, illustrated by animated 3D plots, demonstrate characteristic temporal reshaping of the pulses in the course of propagation through the focus, which is a spectacular manifestation of the Gouy phase shift. It is shown that the salient features of the reshaping, which were recently established for the paraxial limit, remain valid beyond it. The treatment is particularly applicable to an ultrawideband isodiffracting ultrashort terahertz-domain or light pulses in high-aperture resonators, such as microcavities, and it is usable in femto- and attosecond optics in general.

© Optical Society of America

## 1. Introduction

^{-½}(i.e.

*f*/0.5), which is far from being a limit for various devices nowadays. That is why there has been a considerable and growing interest in the extending of the model of the Gaussian beam beyond the paraxial regime and a variety of infinite-series and more or less approximate solutions have been proposed (see [1

1. C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A **4**, 1354–1360 (1987). [CrossRef]

3. Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. **25**, 1792–1794 (2000). [CrossRef]

4. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. **7**, 684–685 (1971). [CrossRef]

5. R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,”, Phys. Rev. E **54**, 889–896 (1996). [CrossRef]

10. S. M. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E **61**, 862–873 (2000). [CrossRef]

5. R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,”, Phys. Rev. E **54**, 889–896 (1996). [CrossRef]

8. S. M. Feng, H. G. Winful, and R. W. Hellwarth “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E **59**, 4630–4649 (1999). [CrossRef]

11. B. T. Landesman and H. H. Barret, “Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation,” J. Opt. Soc. Am. A **5**, 1610–1619 (1988). [CrossRef]

2. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A **57**, 2971–2979 (1998). [CrossRef]

3. Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. **25**, 1792–1794 (2000). [CrossRef]

12. P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in *Proc. Intern. Conf*. “Time’s Arrows, Quantum Measurements and Superluminal Behaviour,” *Naples, Oct.2–5, 2000* (to be published by the Italian NCR), Los Alamos preprint archive, http://xxx.lanl.gov/abs/physics/0103054

13. E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A **4**, 2081–2091 (1987). [CrossRef]

13. E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A **4**, 2081–2091 (1987). [CrossRef]

16. K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A **17**, 1785–1790 (2000). [CrossRef]

^{rd}section.

## 2. The model

*via*the Dirac delta functions ρ(

*t*,

**r**)=δ(

*t*) δ(

**r**). As it is well known from electrodynamics (or –

*mutatis mutandis*– from acoustics), such source emits a δ-pulse-shaped wave, spherically expanding from the origin at the positive times

*t*. This elementary wave – the retarded Green function of the wave equation, which in the Gauss units reads as G

_{+}=(

*c*/4

*πR*) δ(

*R*-

*ct*),

*R*being the distance between the field and the source points – allows one to express any emitted field as the 4D-convolution with the corresponding source distribution ρ(

*t*,

**r**). Due to the temporal symmetry of the HWE, one can in the same way deal with an elementary wave, spherically collapsing onto the origin – with the advanced Green function G

_{-}=(

*c*/4

*πR*) δ(

*R*+

*ct*), in which case the point looks like a sink rather than a source. The difference between these Green functions, which in relativistic field theories is usually denoted by

*D*:

*t*) converging to the origin (the right term) and then (at positive times

*t*) diverging from it.

*D*(

*t,R*) allows one to express any solution of the HWE as the 4D convolution with an appropriately chosen “charge” distribution ρ(t,r), whereas the quantity ρ(t,r) expresses strength of both the source and the sink at the same point. The minus sign between the two terms in Eq. (1), which follows from the requirement that a source-free field cannot have a singular point at R=+0, is crucial and it guarantees the vanishing of the function at t=0. It is just the resulting change of the sign of D(t,R), which takes place when the elementary wave goes through the collapsed stage at the focus, that is responsible for the 90-degrees phase factor known from the Huygens-Fresnel-Kirchhoff principle and for the Gouy phase shift peculiar to all focused waves. Generally a given field does not determine uniquely the distribution of the charge generating the field. Consequently, the sinks and sources need not to cover a distant surrounding surface – the picture commonly associated with Huygens principle, but may be spatio-temporally localized in an appropriate way provided they generate the same given field.

12. P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in *Proc. Intern. Conf*. “Time’s Arrows, Quantum Measurements and Superluminal Behaviour,” *Naples, Oct.2–5, 2000* (to be published by the Italian NCR), Los Alamos preprint archive, http://xxx.lanl.gov/abs/physics/0103054

13. E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A **4**, 2081–2091 (1987). [CrossRef]

*q*is the total sink-and-source strength, the parameter

*t*

_{0}will be specified later on, and where the Lorentzian expressing the temporal dependence has been introduced in the form of the complex analytic signal for mathematical convenience. When expressing the source-free field through the 4D-convolution of the functions

*D*(

*t,R*) and ρ(

*t*,

**r**) like it is commonly done for obtaining source-induced retarded potentials, the spatial integration is trivial and one obtains:

*t*|>Δ) pulse which is odd with respect to time and changes its sign at

*t*=0, while the imaginary part represents a bipolar pulse which is temporally even and therefore unipolar at the origin (such tranformation of pulses will be illustrated by animated plots later on).

*viz*., we shift the “charge” from the origin to an imaginary location

*z*

_{0}=

*id*on the axis

*z*. Naturally, this makes the expression of the distance

*R*between a field point and the “charge” location rather complicated. Fortunately, we can avoid the complications by introducing the system of oblate spheroidal coordinates (OSC) instead of the Cartesian

*x,y,z*by the following relations [3

3. Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. **25**, 1792–1794 (2000). [CrossRef]

11. B. T. Landesman and H. H. Barret, “Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation,” J. Opt. Soc. Am. A **5**, 1610–1619 (1988). [CrossRef]

*d*placed in the equatorial plane (see Fig. 2). The coordinate surfaces φ=

*const*are – just as in the case of spherical coordinates – the meridian planes containing the

*z*axis. The surfaces η≡cosθ=

*const*≠±1 are hyperboloids of revolution of one sheet – just matching the shape of the Gaussian beam waist -whose asymptotes pass through the origin inclined at an angle θ with the

*z*axis, while two degenerate surfaces η=±1 constitute the

*z*axis. The surfaces ξ=

*const*≠0 are oblate ellipsoids having an interfocal spacing of 2

*d*, while the surface ξ=0 is the circular disk.

*R*

^{2}=

*d*

^{2}(ξ-

*iη*)

^{2}, while the length

*d*turns out to be the Rayleigh range of the field in the paraxial limit [3

**25**, 1792–1794 (2000). [CrossRef]

*t*

_{0}=

*d/c*and Δ=

*t*

_{0}+

*a/c*. The latter definition introduces an arbitrary contribution

*a*>0 to the pulselength but assures the fulfilling of the condition Δ>

*t*

_{0}required for finiteness of the field at

**r**=t=0. Thus, from Eq. (3) the model scalar field reads

## 3. Results and discussion

*a*=0.05, where the confocal parameter 2

*d*has been taken for the unit of length. The same relative unit is also used on the length scales of the forthcoming figures. In an absolute scale this unit might possess values from a few centimeters or less – to relate our results to typical terahertz pulses – down to, say, ten microns for a frontier femtosecond optical experiment. Consequently, the ratio

*a/d*=0.05/0.5=0.1 is rather large stressing the nonparaxiality. Eq. (5) does not depend on the angle φ, i.e. the field is axisymmetric around the z axis and it suffices to study the behavior of the pulse in the plane

*y*=0, which is depicted in Fig. 3 with coordinate lines of ξ,θ shown in it. Thus, by making use of the third axis of a 3-D plot for depicting the strength of the field and relating the time

*t*to the frame number of animation, we can visualize all spatio-temporal dependences of Eq. (5) by two video clips (Fig. 4 and 5). The field values have been computed on a 60×60 grid over the range (ξ=0…4.8)×(θ=0…π). Thus, both plots consist of 7200 data points with their density increasing towards the center of the ellipse-bounded region in the z-x plane.

*z*axis and the time, while the other pulse is symmetric – the rules that had been earlier established for the on-axis behavior [6

6. A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B **15**, 951–956 (1998). [CrossRef]

8. S. M. Feng, H. G. Winful, and R. W. Hellwarth “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E **59**, 4630–4649 (1999). [CrossRef]

*a/d*≪1, |η|≈1), the quantity 2ξ-τ can be replaced by 2ξ and we obtain

*const*as far as the second one expresses nothing but the change of the pulses’s overall intensity depending on the distance from the focus. The constant 900 phase factor is unimportant and it results from our particular choice of the phase in Eq. (2). If the last term in the exponent were absent, the phase – and hence the profiles of both the real and the imaginary pulse – would also remain invariant along the hyperbolas. The pulse reshaping is introduced by the last term tan

^{-1}(ξ), which is the Gouy phase shift that any finite wave field obeys upon passing through a focus.

*k*or frequency ω=

*kc*) domain. Namely, the calculation of a

*k*-domain superposition of the monochromatic nonparaxial Gaussian beams expressed in OSC [3

**25**, 1792–1794 (2000). [CrossRef]

*S*(

*k*)sin(

*Rk*), where the complex number

*R*=

*d*(ξ-

*i*η) and

*S*(

*k*) is the spectrum without the exponential factor exp(-

*ak*). There is a general formula in the transform tables for choice

*S*(

*k*)=

*k*, which means that the time-derivative pulses of any order

^{p}*p*can be readily obtained. The setting

*p*=0 reproduces Eq. (5), yet many other interesting exact solutions to the HWE in OSC can be found

*via*the Laplace transform tables.

5. R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,”, Phys. Rev. E **54**, 889–896 (1996). [CrossRef]

8. S. M. Feng, H. G. Winful, and R. W. Hellwarth “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E **59**, 4630–4649 (1999). [CrossRef]

17. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A **16**, 1381–1386 (1999). [CrossRef]

_{00}mode in the paraxial limit, we should replace the “charge” with an electric dipole and a magnetic dipole, oriented along the

*x*and the

*y*axis, respectively, and combined with the sinks possessing the same dipole properties [17

17. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A **16**, 1381–1386 (1999). [CrossRef]

*and*

**E***(or*

**H***), is perhaps the most convenient to be carried out by making use of the Hertz vector technique (like it is done in [8*

**B****59**, 4630–4649 (1999). [CrossRef]

**59**, 4630–4649 (1999). [CrossRef]

*q*

_{1}and

*q*

_{2}) solution to the scalar HWE – if one takes the the pulsewidth parameter

*q*

_{1}equal to

*a*of our model and sets

*q*

_{2}=2

*d*+

*a*, which in the paraxial limit

*a*≪

*d*means that the confocal parameters of both model scalar fields concide. Thus, by Eq. (5) we have expressed the so-called modified power spectrum pulse – a particular version of “electromagnetic directed energy pulse trains” (see [14

14. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A **10**, 75–87 (1993). [CrossRef]

*and*

**E***from the scalar generating field has been accomplished in [8*

**H****59**, 4630–4649 (1999). [CrossRef]

*and*

**E***are second-order spatio-temporal derivatives of the Hertz vector, the spectra of their components acquire the*

**H***k*

^{2}-dependence, thus suppressing the nonparaxiality of the initial scalar field. Fortunately, for the role of the generating scalar field we could take the indefinite integral of Eq. (5), in other words – choose the negative power

*p*in the spectrum

*k*

^{p}of the generating field, which means an arctan-type temporal dependence instead of the Lorentzian one introduced into our model by Eq. (2). Moreover, the above-mentioned general formula in the Laplace transform tables proves the existence of the transform of

*k*

^{p}sin(

*Rk*) even for the case

*p*→-2. Hence, it should be possible, by appropriately changing the initial time dependence in Eq. (2), to reach finally the fields

*and*

**E***possessing spectra, where the low-frequency tail is not suppressed and which therefore represent essentially nonparaxial vector fields.*

**H***via*differentiation/integration and/or the Hilbert transform, one has to insert an appropriate temporal dependence into Eq. (2) or, equivalently, to use an expression for the spectrum S(k), other than the simple powers of the wavenumber or the frequency. Yet in a number of such cases closed-form final expressions can be found with the help of the Laplace transform tables.

**59**, 4630–4649 (1999). [CrossRef]

14. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A **10**, 75–87 (1993). [CrossRef]

16. K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A **17**, 1785–1790 (2000). [CrossRef]

*and*

**E***, as shown in [8*

**H****59**, 4630–4649 (1999). [CrossRef]

**25**, 1792–1794 (2000). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and Links

1. | C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A |

2. | C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A |

3. | Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. |

4. | G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. |

5. | R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,”, Phys. Rev. E |

6. | A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B |

7. | M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E |

8. | S. M. Feng, H. G. Winful, and R. W. Hellwarth “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E |

9. | Z. L. Horváth and Zs. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E |

10. | S. M. Feng and H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E |

11. | B. T. Landesman and H. H. Barret, “Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation,” J. Opt. Soc. Am. A |

12. | P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in |

13. | E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A |

14. | R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A |

15. | P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. |

16. | K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A |

17. | C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(140.4780) Lasers and laser optics : Optical resonators

(260.2110) Physical optics : Electromagnetic optics

(320.0320) Ultrafast optics : Ultrafast optics

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 11, 2001

Published: May 21, 2001

**Citation**

Peeter Saari, "Evolution of subcycle pulses in nonparaxial Gaussian beams," Opt. Express **8**, 590-598 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-11-590

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### References

- C. J. R. Sheppard and H. J. Matthews, "Imaging in high-aperture optical systems," J. Opt. Soc. Am. A 4, 1354-1360 (1987). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998). [CrossRef]
- Z. Ulanowski and I. K. Ludlow, "Scalar field of nonparaxial Gaussian beams," Opt. Lett. 25, 1792-1794 (2000). [CrossRef]
- G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971). [CrossRef]
- R. W. Hellwarth and P. Nouchi, "Focused one-cycle electromagnetic pulses," Phys. Rev. E 54, 889-896 (1996). [CrossRef]
- A. E. Kaplan, "Diffraction-induced transformation of near-cycle and subcycle pulses," J. Opt. Soc. Am. B 15, 951-956 (1998). [CrossRef]
- M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998). [CrossRef]
- S. M. Feng, H. G. Winful, and R. W. Hellwarth "Spatiotemporal evolution of focused single-cycle electromagnetic pulses," Phys. Rev. E 59, 4630-4649 (1999). [CrossRef]
- Z. L. Horv�th and Zs. Bor, "Reshaping of femtosecond pulses by the Gouy phase shift," Phys. Rev. E 60, 2337-2345 (1999). [CrossRef]
- S. M. Feng and H. G. Winful, "Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses," Phys. Rev. E 61, 862-873 (2000). [CrossRef]
- B. T. Landesman and H. H. Barret, "Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation," J. Opt. Soc. Am. A 5, 1610-1619 (1988). [CrossRef]
- P. Saari, "Superluminal localized waves of electromagnetic field in vacuo," in Proc. Intern. Conf. "Time's Arrows, Quantum Measurements and Superluminal Behaviour," Naples, Oct.2-5, 2000 (to be published by the Italian NCR), Los Alamos preprint archive, http://xxx.lanl.gov/abs/physics/0103054
- E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987). [CrossRef]
- R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of exact solutions to homogeneous wave equations," J. Opt. Soc. Am. A 10, 75-87 (1993). [CrossRef]
- P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997). [CrossRef]
- K. Reivelt and P. Saari, "Optical generation of focus wave modes,"J. Opt. Soc. Am. A 17, 1785-1790 (2000). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999). [CrossRef]

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