## Influence of the higher-orders of diffraction on the pattern evolution for tightly focused beams |

Optics Express, Vol. 19, Issue 12, pp. 11170-11181 (2011)

http://dx.doi.org/10.1364/OE.19.011170

Acrobat PDF (1086 KB)

### Abstract

The mechanism of the nonparaxial propagation for tightly focused beams is investigated in the view of the influence of the higher-orders of diffraction (HOD). The HOD induce novel propagation characteristics which are crucially different from those predicted by the traditional paraxial theory. Based on the management of HOD, we propose an approach on controlling the intensity pattern of the focus to satisfy the application requirements.

© 2011 OSA

## 1. Introduction

1. M. A. Bandres and J. C. Gutiérrez-Vega, “Cartesian beams,” Opt. Lett. **32**, 3459–3461 (2007). [CrossRef] [PubMed]

3. M. A. Bandres and J. C. Gutiérrez-Vega, “Generalized Ince Gaussian beams,” Proc. SPIE **6290**, 62900S (2006). [CrossRef]

4. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. **29**, 1968–1970 (2004). [CrossRef] [PubMed]

5. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. **86**, 5251–5254 (2001). [CrossRef] [PubMed]

6. J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E **66**, 066501 (2002). [CrossRef]

7. P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. **53**, 2146–2148 (1988). [CrossRef]

8. M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A **11**, 1365–1370 (1975). [CrossRef]

11. D. Deng, Q. Guo, S. Lan, and X. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space,” J. Opt. Soc. Am. A **24**, 3317–3325 (2007). [CrossRef]

12. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. **177**, 9–13 (2000). [CrossRef]

13. S. Sepke and D. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. **31**, 1447–1449 (2006). [CrossRef] [PubMed]

14. Y. I. Salamin, “Fields of a focused linearly polarized Gaussian beam: truncated series versus the complex-source-point spherical-wave representation,” Opt. Lett. **34**, 683–685 (2009). [CrossRef] [PubMed]

15. S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A **82**, 063820 (2010). [CrossRef]

*-order diffraction. For the management of the 2*

^{nd}*-order diffraction, as well as its counterpart in temporal domain, i.e., the 2*

^{nd}*-order dispersion, there are various approaches developed. It is the standard technique the use of diffraction gratings (lenses) to compress a broadened pulse (beam). In nonlinear media, the self-focusing balances the 2*

^{nd}*-order diffraction (dispersion) so that during propagation the beam keeps invariant and becomes a soliton [16]. In linear and dispersive medium, specific transverse beam profiles (temporal pulse forms) have been developed to exhibit pseudo-diffraction-free (pseudo-dispersion-free) properties [17*

^{nd}17. K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E **65**, 046622 (2002). [CrossRef]

18. S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. **27**, 2167–2169 (2002) [CrossRef]

20. M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. **206**, 235–241 (2003). [CrossRef]

21. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. **26**, 1364–1366 (2001) [CrossRef]

22. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E **69**, 066606 (2004). [CrossRef]

19. S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. **240**, 1–8 (2004). [CrossRef]

23. R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. **227**, 237–243 (2003). [CrossRef]

*-order diffraction or dispersion indicate that the management of the HOD would induce novel phenomena and might be of interest for real applications.*

^{nd}## 2. Preliminaries

**E**

_{⊥}and the longitudinal component

**E**

_{‖}connect with each other based on the relation [12

12. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. **177**, 9–13 (2000). [CrossRef]

**E**

_{⊥,‖}can be regarded as the superposition of various angular spectrum components, each of which represents a planar wave propagating in a certain direction. Every spectrum

**Ẽ**

_{⊥,‖}experiences a corresponding phase shift when it propagates from

*z*=

*z*

_{1}to

*z*=

*z*

_{2}, i.e., where

**k**=

_{r}*k*

_{x}**e**

*+*

_{x}*k*

_{y}**e**

*,*

_{y}*z*=

*z*

_{2}–

*z*

_{1}. According to Eq.(1), at every plane, the longitudinal spectrum component can be evaluated from the transverse spectrum component with the relation At the plane

*z*=

*z*

_{2}, the linear superposition of all angular spectrums yields the field in spatial domain, i.e. Equation (4) is tantamount to

**Ẽ**

_{⊥,‖}(

*z*

_{2}) =

*F̂*

^{−1}{

*F̂*{

**Ẽ**

_{⊥,‖}(

*z*

_{1}) } exp (

*ik*Δ

_{z}*z*) }, where operators

*F̂*and

*F̂*

^{−1}represent the forward and the inverse Fourier transform, respectively. Therefore, in the view of Fourier optics, the nonparaxial propagation from

*z*=

*z*

_{1}to

*z*=

*z*

_{2}results from three steps: i) transform

**E**

_{⊥,‖}(

*z*

_{1}) to the angular spectrum domain; ii) add a phase

*k*Δ

_{z}*z*on the angular spectrum

**E**

_{⊥,‖}(

*z*

_{1}); and iii) inversely transform the angular spectrum at

*z*=

*z*

_{2}to the spatial domain.

*k*Δ

_{z}*z*, which results in the evolution of the beam pattern. In fact, if we make a Taylor expansion on

*k*, we have where

_{z}*β*= −

_{n}*n*!|

*n −*3|!!/

*n*!!

*k*

^{n−}^{1}(

*n*≥ 2). As shown in Eq. (5), during propagation the beam experiences different orders of diffraction (in the following we call

*β*the

_{n}*n*order diffraction), which is similar to the different orders of dispersion for a laser pulse in the fiber. However, there are two differences between the spatial diffraction and the temporal dispersion: i) During propagation in free space the beam experiences only even-order diffraction, whereas the pulse in the fibre experiences both even- and odd-order dispersion; ii) For different media and different carrier wavelengths, the temporal dispersion can be positive or negative, and the dispersion with different orders can be with different signals. However for the diffraction of the beam in free space, the signals of all orders of diffraction are negative (

^{th}*β*< 0), which means that the higher spatial frequency experiences a smaller phase-shift, and all orders of spatial diffraction together cause a negative spatial chirp during propagation.

_{n}*β*

_{2}is necessary to be taken into account. Then Eq. (4) reduces to which is the well-known Fresnel integral that governs the evolution of an arbitrary paraxial beam. In fact, under the paraxial condition, because the angular spectrum is very narrow, the longitudinal component is so weak that can be neglected; and only the transverse component is necessary to be taken into account.

*k*is no longer small enough to justify the truncation of the expansion (5) after the

*β*

_{2}term, the HOD should be included [12

12. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. **177**, 9–13 (2000). [CrossRef]

## 3. Influence of the HOD on the pattern evolution for tightly focused beams

*x*direction) of a Gaussian beam can be written as where

*w*

_{z}_{1}is the beam width,

*R*is the radius of spherical cophasal surface, whose signal is negative because of the concave cophasal surface. ϕ

_{z}_{1}is the initial phase. Correspondingly, the spectrum of the transverse and the longitudinal components are where

### 3.1. Paraxial results

*z*

_{2}, the spectrums of the transverse and the longitudinal components become where we have adopt the approximation

*k*

_{⊥}/

*k*≃

_{z}*k*

_{⊥}/

*k*.

*order diffraction, which does not vary the angular spectrum distribution, but varies the parameter and therefore the radius of the spherical phase distribution in the angular spectrum domain during propagation. In the view of Fourier optics, the Hermite-, Laguerre-, and Ince-Gaussian beams with spherical cophasal surface are the eigen functions of the Fourier transform in different coordinate systems, i.e. where Λ(·) represents an arbitrary Hermite-, Laguerre-, or Ince-Gaussian function. Therefore, an arbitrary beam, which is resulted from the linear superposition of these three types of beams with the same waist location and Rayleigh distance, would remain shape-invariant during propagation under the paraxial approximation. The paraxial propagation only varies the beam width and the radius of the cophasal surface (as shown in row 1 in Fig. 1).*

^{nd}**177**, 9–13 (2000). [CrossRef]

_{z}_{1}+ arctan(

*z*/

_{s}*z*) + arctan [(Δ

_{R}*z*–

*z*)/

_{s}*z*],

_{R}*order diffraction is taken into account and the HOD are neglected, the diffraction-induced negative linear chirp can completely compensate the initial positive linear chirp when it propagates a distance Δ*

^{nd}*z*=

*z*, i.e., According to Eq. (4), at that plane the beam width is Fourier-transform-limited and arrives its minimum

_{s}*w*

_{0}, thus that plane becomes the waist of the focused beam; and the intensity distribution is symmetric about the waist (we will call it the

*pseudo-waist*in the following, because it is not really the waist for a tightly focused beam), as shown in Eqs. (14) and (15).

### 3.2. Influence of the HOD

*-order diffraction but also the HOD play critical roles; and the paraxial theory becomes invalid. According to Eq. (7), the vectorial field in the framework of nonparaxial propagation becomes: where represents the interaction between the initial chirp and all orders of diffraction. It is the HOD that differs the paraxial from nonparaxial propagation. Because of the mathematical complexity, the analytical expressions of the integral equations (16) and (17) are too difficult to get. The result of the example is based on the numerical simulation of the integral equations.*

^{nd}- The HOD induce fourth- and higher-order phase factors in the angular spectrum domain. Mathematically, each Hermite-, Laguerre-, or Ince-Gaussian function with such phase factors is no longer the eigen function of Fourier transform, i.e., Therefore any paraxially shape-invariant beam becomes shape-variant under the nonparaxial condition (e.g. the fundamental Gaussian beam shown in row 2 in Fig. 1).
- At the pseudo-waist, the linear initial positive chirp induced by the focusing lens only balances the 2
-order-diffraction-induced negative chirp, and the HOD are not eliminated, i.e., Therefore the HOD induce a nonlinear negative chirp in angular spectrum domain, i.e., which broadens the beam size. Therefore the beam width is larger than the Fourier-transform-limited one predicted by the paraxial theory. The farther the distance between the entrance plane and the pseudo-waist is, the larger the chirp will be, and in turn the larger the beam width will be resulted in (Fig. 2).^{nd} - There exists a plane where the beam width is the smallest (however it is still larger than the Fourier-Transform-limited one). In the following we call it the
*real-waist*. The real-waist becomes farther and father from the pseudo-waist with the increase of the distance between the entrance plane and the pseudo-waist (Fig. 2). - The phase distribution in angular spectrum domain, i.e., Ω
(_{np}*k*, Δ_{r}*z*), is asymmetric about the pseudo-waist, therefore neither the pattern nor the size of the beam is symmetric about the pseudo-waist. Because Ω(_{np}*k*, Δ_{r}*z*) is asymmetric about the pseudo-waist, the chirp in angular spectrum domain, i.e., is also asymmetric about the pseudo-waist. At the planes after the pseudo-waist, Ω(_{np}*k*, Δ_{r}*z*) decreases monotonically with |*k*| and results in a negative [i.e., Δ_{r}*r*(Δ*z*) = −|Δ*r*(Δ*z*)|] and nonlinear chirp. Therefore the size would be larger than that predicted by the paraxial theory; and the shape is approximate to the paraxial result. However, very differently, at some planes before the pseudo-waist, the phase in angular spectrum domain (Ω) varies nonmonotonically with |_{np}*k*| and results in a s-like distribution of the chirp Δ_{r}*r*. Therefore, at such planes there is the same chirp occurring at three values of*k*. These three angular spectrum components interfere constructively or destructively, depending on their relative phase difference, just like the mechanism for the changes in the pulse spectrum induced by the self-phase modulation in nonlinear fiber [16]. This property is validated by the example of the transverse component of the Gaussian beam (rows 2 and 4 in Fig. 1): at the planes after the pseudo-waist the pattern remains bell-like and the size is larger than that predicted by the paraxial theory; but at some planes before the pseudo-waist, the interference among the three angular spectrum components results in a multi-ringed distribution of the beam in the spatial domain._{r}

## 4. An approach on controlling the intensity pattern: managing the HOD

*order diffraction. For instance, during the process of degenerate optical parametric amplification, the nonlinear interaction modifies the signal and idler pulsed beams on the phase distribution in angular spectrum and frequency spectrum domain, and therefore on the spatial and temporal distribution. In suitably designed case, the spatial diffraction and the temporal dispersion can be suppressed and a simultaneous temporal and spatial localization is reached [23*

^{nd}23. R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. **227**, 237–243 (2003). [CrossRef]

23. R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. **227**, 237–243 (2003). [CrossRef]

*order diffraction is based on the 2*

^{nd}*order nonlinear phase modulation; it shows that the controlling of the phase distribution in angular spectrum domain would be a feasible approach to influence the beam profile in spatial domain. Here we propose a method to manage the HOD with linear optical process based on the theory of Fourier optics.*

^{nd}*z*, we can pre-add a chirp on the angular spectrum of the input field at the entrance plane, i.e. where represents all orders of the HOD. Therefore

_{s}**Ẽ′**

_{⊥,‖}(

*z*

_{1}) is connected with

**Ẽ′**

_{⊥,‖}(

*z*

_{1}) becomes

**E′**

^{(}

^{np}^{)}, based on Eqs. (23) and (24). First, at the pseudo-waist plane where Δ

*z*=

*z*, we have

_{S}*z*

_{1}+

*z*), not only the second-order but also the higher-orders of diffraction are balanced by the pre-added initial chirp in

_{s}**Ẽ′**

_{⊥,‖}(

*z*

_{1}), the beam width arrives its minimum which is the Fourier transform limited one, and the pseudo-waist of

*z*=

*z*

_{1}+

*z*. Therefore the field in spatial domain, which is the Fourier transform of the angular spectrum, is also symmetrical about

_{s}*z*=

*z*

_{1}+

*z*. In addition, for the pre-chirped field, the chirp in spectrum domain becomes Critically different from that for the unpre-chirped field [ shown in Eq. (19)], during the process of focusing (de-focusing) of the beam, i.e., at planes where Δ

_{s}*z*<

*z*(Δ

_{s}*z*>

*z*), Δ

_{s}*r′*(Δ

*z*) always decreases (increases) monotonically with

*k*, which means that the same chirp occurs only at one value of

_{r}*k*. Therefore the influence of Δ

_{r}*r′*(Δ

*z*) on the evolution of the beam is a weak broadening of the size, both before and after the waist.

*β*/

_{n}*n*! before every order of diffraction

^{n}^{+2}

*th*order diffraction to the

*order diffraction, i.e., keeps invariant during propagation, which is crucially different from that for the unpre-chirped beam [Eq. (19)]. On the other hand, according to the theory of Fourier transform, the ratio*

^{n}th*n*– 1)/[(

*n*+ 2)

*k*

^{2}

*w*

^{2}]. Even for a beam which is focused to the size of the wavelength

*λ*, Γ ∼ (

*n*– 1)/[(

*n*+ 2)4

*π*

^{2}] is still of the order of few percent. Therefore, all through the propagation the 2

*order phase*

^{nd}*k*only provide a little influence on the evolution; and thus the nonparaxial propagation of

_{r}**E′**

_{⊥,‖}approximates well to the result of the propagation of

**E**

_{⊥,‖}predicted by the paraxial theory.

**E**

_{⊥}predicted by the paraxial theory (and at the waist plane Δ

*z*=

*z*the two width are identical with each other), and the pseudo-waist becomes the real-waist.

_{s}*z*=

*z*

_{1}) of the tightly focusing region are paraxial and nonparaxial, respectively. Therefore the propagation from

*z*

_{0}to

*z*

_{1}can be described by the theory of the Fourier optics. The propagation of the beam from

*z*

_{0}to

*δ*is a constant phase shift. According to Eqs. (28) and (29), one has

*E*(

*z*

_{1}) is connected with the field at the plane

*z*

_{0}by the relation where Therefore, according to Eq. (30), when an appropriate phase –Δ

*βz*is induced by the phase plate at

_{s}*z*

_{0}, i.e.,

**E′**

_{⊥,‖}(

*k*,

_{x}*k*,

_{y}*z*

_{0}) =

**E**

_{⊥,‖}(

*k*,

_{x}*k*,

_{y}*z*

_{0}) exp(−

*i*Δ

*βz*), the spectrum of the field at

_{s}*z*

_{1}becomes

**Ẽ′**

_{⊥,‖}(

*k*,

_{x}*k*,

_{y}*z*

_{1}) =

**Ẽ**

_{⊥,‖}(

*k*,

_{x}*k*,

_{y}*z*

_{1}) exp(−

*i*Δ

*βz*) correspondingly, i.e., the chirp is pre-added.

_{s}## 5. Conclusion

**Ẽ′**

_{⊥,‖}becomes in good approximation to the result of the propagation of the unpre-chirped field

**E**

_{⊥,‖}predicted by the paraxial theory; and one can get a focused spot which is expected by applications and identical with that predicted by the paraxial theory.

## Acknowledgments

## References and links

1. | M. A. Bandres and J. C. Gutiérrez-Vega, “Cartesian beams,” Opt. Lett. |

2. | M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. |

3. | M. A. Bandres and J. C. Gutiérrez-Vega, “Generalized Ince Gaussian beams,” Proc. SPIE |

4. | N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. |

5. | L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. |

6. | J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E |

7. | P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. |

8. | M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A |

9. | R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. |

10. | Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B |

11. | D. Deng, Q. Guo, S. Lan, and X. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space,” J. Opt. Soc. Am. A |

12. | A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. |

13. | S. Sepke and D. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. |

14. | Y. I. Salamin, “Fields of a focused linearly polarized Gaussian beam: truncated series versus the complex-source-point spherical-wave representation,” Opt. Lett. |

15. | S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A |

16. | G. P. Agrawal, |

17. | K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E |

18. | S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. |

19. | S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. |

20. | M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. |

21. | M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. |

22. | M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E |

23. | R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. |

24. | J. W. Goodman, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 22, 2011

Revised Manuscript: May 14, 2011

Manuscript Accepted: May 17, 2011

Published: May 24, 2011

**Citation**

Daquan Lu, Zhenjun Yang, and Wei Hu, "Influence of the higher-orders of diffraction on the pattern evolution for tightly focused beams," Opt. Express **19**, 11170-11181 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11170

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### References

- M. A. Bandres and J. C. Gutiérrez-Vega, “Cartesian beams,” Opt. Lett. 32, 3459–3461 (2007). [CrossRef] [PubMed]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008). [CrossRef] [PubMed]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Generalized Ince Gaussian beams,” Proc. SPIE 6290, 62900S (2006). [CrossRef]
- N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004). [CrossRef] [PubMed]
- L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]
- J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002). [CrossRef]
- P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988). [CrossRef]
- M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]
- R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28, 774–776 (2003). [CrossRef] [PubMed]
- Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319–326 (2007). [CrossRef]
- D. Deng, Q. Guo, S. Lan, and X. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space,” J. Opt. Soc. Am. A 24, 3317–3325 (2007). [CrossRef]
- A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000). [CrossRef]
- S. Sepke and D. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31, 1447–1449 (2006). [CrossRef] [PubMed]
- Y. I. Salamin, “Fields of a focused linearly polarized Gaussian beam: truncated series versus the complex-source-point spherical-wave representation,” Opt. Lett. 34, 683–685 (2009). [CrossRef] [PubMed]
- S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A 82, 063820 (2010). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic, 2001).
- K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E 65, 046622 (2002). [CrossRef]
- S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002) [CrossRef]
- S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. 240, 1–8 (2004). [CrossRef]
- M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003). [CrossRef]
- M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001) [CrossRef]
- M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004). [CrossRef]
- R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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