## Propagation characteristics of Airy-Bessel wave packets in free space |

Optics Express, Vol. 21, Issue 4, pp. 4481-4492 (2013)

http://dx.doi.org/10.1364/OE.21.004481

Acrobat PDF (933 KB)

### Abstract

Airy-Bessel configuration wave packets are believed to be exotic localized linear light bullets (LLBs) without spatiotemporal spread during propagation in free space. By carefully studying the propagation of ideal Airy-Bessel wave packets (ABWs) in free space, several new results were obtained. Cubic spatially induced dispersion (SID) slightly broadens Airy pulses while quadratic SID cannot temporally change ABWs transmission modes. Hence, ABWs, although remaining as superior localized linear wave packets, cannot be regarded as absolute LLBs. Moreover, cubic SID also decreases the longitudinal acceleration of the Airy pulse peak during propagation.

© 2013 OSA

## 1. Introduction

1. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

3. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express **17**(17), 14948–14955 (2009). [CrossRef] [PubMed]

4. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt. **7**(5), R53–R72 (2005). [CrossRef]

1. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

1. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

2. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

6. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

7. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

6. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

8. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

9. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

12. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A **17**(2), 294–303 (2000). [CrossRef] [PubMed]

6. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

8. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

13. P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A **77**(4), 043814 (2008). [CrossRef]

17. S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. B: Quantum Semiclass. **6**(11), 477–481 (2004). [CrossRef]

18. G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. **167**(1-6), 15–22 (1999). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

**58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

19. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. **26**(17), 1364–1366 (2001). [CrossRef] [PubMed]

20. W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A **19**(1), 49–53 (2002). [CrossRef] [PubMed]

2. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

7. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

27. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum Generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. **107**(24), 243901 (2011). [CrossRef] [PubMed]

**4**(2), 103–106 (2010). [CrossRef]

27. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum Generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. **107**(24), 243901 (2011). [CrossRef] [PubMed]

29. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A **18**(10), 2594–2600 (2001). [CrossRef] [PubMed]

## 2. Theory

18. G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. **167**(1-6), 15–22 (1999). [CrossRef]

24. Z. J. Ren, H. Z. Jin, Y. L. Shi, J. C. Xu, W. D. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading Airy-Bessel wave packets,” J. Opt. Soc. Am. A **29**(6), 848–853 (2012). [CrossRef] [PubMed]

30. B. D. Lü and Z. J. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A **20**(3), 582–587 (2003). [CrossRef] [PubMed]

*U*is the Fourier component of the pulse at the frequency

*ω*. In free space, the wave number is

*c*being light speed in free space.

**58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

*U*(

*r*,

*z*;

*ω*) in the frequency domain can be expressed as [3

3. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express **17**(17), 14948–14955 (2009). [CrossRef] [PubMed]

20. W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A **19**(1), 49–53 (2002). [CrossRef] [PubMed]

30. B. D. Lü and Z. J. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A **20**(3), 582–587 (2003). [CrossRef] [PubMed]

*J*

_{0}(·) represents the zero-order Bessel function of first kind, and

*r*≡(

*x*+

*y*)

^{1/2}is the transverse coordinate in a cylindrical system and

*f*(0,

*ω*–

*ω*

_{0}) is the temporal spectrum distribution of pulsed Bessel beams having a central frequency

*ω*

_{0}at the input plane

*z*= 0.

*r*

_{0}and beams with constant waist

*r*

_{0}[18

18. G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. **167**(1-6), 15–22 (1999). [CrossRef]

*r*

_{0}is proportional to the frequency

*ω*, the pulsed Bessel beam is called a Bessel-X wave, featuring a profile with a long tail that accompanies the main hump. Bessel-X waves can resist the effects of diffraction and dispersion, and their shape in a plane along the wave axis is X-like [21

21. M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A **21**(12), 2455–2463 (2004). [CrossRef] [PubMed]

31. F. W. Wise, “Generation of light bullets,” Physics **3**, 107 (2010). [CrossRef]

*r*

_{0}are more useful in some scientific researches because their beam structures are spatially steady [1

**4**(2), 103–106 (2010). [CrossRef]

19. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. **26**(17), 1364–1366 (2001). [CrossRef] [PubMed]

30. B. D. Lü and Z. J. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A **20**(3), 582–587 (2003). [CrossRef] [PubMed]

*r*

_{0}is constant and independent of the frequency

*ω*. In Eq. (3), the longitudinal wave vector

*β*(

*ω*) is given by [18

**167**(1-6), 15–22 (1999). [CrossRef]

20. W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A **19**(1), 49–53 (2002). [CrossRef] [PubMed]

24. Z. J. Ren, H. Z. Jin, Y. L. Shi, J. C. Xu, W. D. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading Airy-Bessel wave packets,” J. Opt. Soc. Am. A **29**(6), 848–853 (2012). [CrossRef] [PubMed]

**20**(3), 582–587 (2003). [CrossRef] [PubMed]

35. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(1), 1086–1093 (1998). [CrossRef]

*β*is expanded in a Taylor series around the carrier frequency

*ω*

_{0}, similar to classical theory of pulsed beam propagation in dispersive media [18

**167**(1-6), 15–22 (1999). [CrossRef]

19. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. **26**(17), 1364–1366 (2001). [CrossRef] [PubMed]

29. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A **18**(10), 2594–2600 (2001). [CrossRef] [PubMed]

**20**(3), 582–587 (2003). [CrossRef] [PubMed]

*β*

_{0}is the value

*β*at a carrier frequency

*ω*

_{0}. The parameter

*β*

_{m}denotes the

*m*th-order dispersion coefficient; specifically, we obtain where,

*k*

_{0}=

*ω*

_{0}/c = 2

*π*/

*λ*

_{0}. For a given ABW, the central wavelength

*λ*

_{0}is provided. We set the central wavelength

*λ*

_{0}to 800nm in this paper.

*r*

_{0}of pulsed Bessel beams is the only definitive factor of influencing the dispersion coefficient when the central wavelength of pulsed Bessel beams is given. Hence, such dispersion is referred to as SID [18

**167**(1-6), 15–22 (1999). [CrossRef]

**19**(1), 49–53 (2002). [CrossRef] [PubMed]

24. Z. J. Ren, H. Z. Jin, Y. L. Shi, J. C. Xu, W. D. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading Airy-Bessel wave packets,” J. Opt. Soc. Am. A **29**(6), 848–853 (2012). [CrossRef] [PubMed]

35. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(1), 1086–1093 (1998). [CrossRef]

*r*

_{0}and its influence on the pulse shape can be ignored. For focused beams with small beam width

*r*

_{0}, the

*n*-order dispersion value

*β*

_{n}

*z*is still negligible although the coefficient of SID is large. This is primarily because the dispersion is an accumulated process [36], and the effective propagation distance of the focus beams determined by the diffraction length

*L*

_{diff}=

*γr*

_{0}

^{2}(see Ref. 18

**167**(1-6), 15–22 (1999). [CrossRef]

*r*

_{0}, which generally equals to several wavelengths of the carrier, and a very long diffraction length

*z*, theoretically being infinite [6

**58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

**19**(1), 49–53 (2002). [CrossRef] [PubMed]

29. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A **18**(10), 2594–2600 (2001). [CrossRef] [PubMed]

*f*(0,

*ω*–

*ω*

_{0}), i.e., new spectral components are not generated during the propagation of pulsed Bessel beams in free space, which is very consistent with the past research results [18

**167**(1-6), 15–22 (1999). [CrossRef]

22. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E **65**(2), 026606 (2002). [CrossRef] [PubMed]

35. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(1), 1086–1093 (1998). [CrossRef]

*β*

_{n}and the propagation distances

*z*. The change in spectral phase results in a temporal distribution change in pulsed beams. By performing the inverse Fourier transform, the time–space distribution of pulsed Bessel beams is given asEquation (8) shows that the spatial distribution of a pulsed Bessel beam propagates as a shape-preserving Bessel beam structure during propagation in free space. Similarly to the past researches [1

**4**(2), 103–106 (2010). [CrossRef]

**26**(17), 1364–1366 (2001). [CrossRef] [PubMed]

*E*(

*r*,

*z*;

*t*) can be written in the following formApparently, Φ(

*z*,

*t*) represents the temporal profile of pulsed Bessel beams at different propagation distances, as shown by the following equationIn Eq. (11), using methods similar to those being proposed in references [18

**167**(1-6), 15–22 (1999). [CrossRef]

*v*

_{g}, that is

*T*=

*t*–z/

*v*

_{g}=

*t*–

*β*

_{1}

*z*because

*β*

_{1}does not influence the temporal profile. Moreover, similar to material dispersion, the high-order dispersion terms is negligible enough and can be ignored [18

**167**(1-6), 15–22 (1999). [CrossRef]

*T*

_{0}is the full width at half maximum of the Gaussian pulse. By performing Fourier transformation to Eq. (13), then substituting this result into Eq. (12), the temporal distribution of the Gaussian-Bessel beam after propagation at a distance

*z*can be calculated. References [18

**167**(1-6), 15–22 (1999). [CrossRef]

21. M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A **21**(12), 2455–2463 (2004). [CrossRef] [PubMed]

**167**(1-6), 15–22 (1999). [CrossRef]

21. M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A **21**(12), 2455–2463 (2004). [CrossRef] [PubMed]

**4**(2), 103–106 (2010). [CrossRef]

2. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

*T*

_{0}is the temporal scaling parameter that approximately determines the full width at half maximum (FWHM) of the main lobe width of the Airy pulse, since the main lobe of the square of the Airy function has a Gaussian pulse-like intensity distribution [37

37. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express **19**(23), 23132–23139 (2011). [CrossRef] [PubMed]

*f*(0,

*ω*) is the Fourier transform of the incident pulse shape Φ(0,

*T*), we obtainSubstituting Eq. (16) into Eq. (12), the following equation is givenwhere, it is readily known that sgn(

*β*

_{2}) = –1 and sgn(

*β*

_{3}) = 1 from Eqs. (6.2) and (6.3). It is worth noticing that Eq. (17) is also used in examining Airy pulse propagation in linear dispersive media when

*β*

_{2}and

*β*

_{3}are respectively quadratic and cubic dispersion coefficient of media [38

38. I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(4), 046605 (2008). [CrossRef] [PubMed]

*β*

_{2}and

*β*

_{3}representing respectively the quadratic and cubic dispersion coefficients of media.

38. I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(4), 046605 (2008). [CrossRef] [PubMed]

## 3. Analysis and discussion

### 3.1 The influence of quadratic SID

*β*

_{3}= 0, that is

*L*

_{3}= ∞. Equation (20) can then be written aswhere, the phase term

*z*in free space. Substituting Eq. (21) into Eq. (10), we know that ideal ABWs can spatiotemporally propagate undistorted at infinite distances from the source. However, a longitudinal displacement of time also exists, which indicates a temporal longitudinal acceleration in the absence of any external potential [1

**4**(2), 103–106 (2010). [CrossRef]

7. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

39. K. Y. Kim, C. Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express **19**(3), 2286–2293 (2011). [CrossRef] [PubMed]

**105**(25), 253901 (2010). [CrossRef] [PubMed]

**47**(3), 264–267 (1979). [CrossRef]

8. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

*z*

^{2}/4

*L*

_{2}

^{2}. The main lobe of the Airy pulse moves against the oscillating tail of the Airy pulse. All the sidelobes contribute to the main one through interference. The temporal acceleration of Airy pulses is physically different from the spatial acceleration of Airy beams. The spatial acceleration of an Airy beam means bending its trajectory in space, whereas the temporal acceleration of an Airy pulse means changing its actual group velocity [37

37. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express **19**(23), 23132–23139 (2011). [CrossRef] [PubMed]

**4**(2), 103–106 (2010). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

### 3.2 The influence of cubic SID

*β*

_{2}= 0, i.e.,

*L*

_{2}= ∞, Eq. (20) can be written asEquation (22) indicates that the main lobe width of the Airy pulse gradually increases as (1 +

*z*/2

*L*

_{3})

^{1/3}during propagation, and the amplitude is correspondingly reduced which is in accordance with the law of conservation of energy. The value of

*L*

_{3}determines the influence extent of the cubic SID [36]. Therefore, despite being ideal ABWs with infinite energy, the temporal Airy distribution pulse is not an absolutely undistorted under the influence of cubic SID that inevitably exists because of space–time coupling effects [27

27. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum Generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. **107**(24), 243901 (2011). [CrossRef] [PubMed]

**18**(10), 2594–2600 (2001). [CrossRef] [PubMed]

41. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. **12**(9), 093001 (2010). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

*L*

_{2}<<

*L*

_{3}. Even when the propagation distance is one cubic SID length

*L*

_{3}, the pulse width only broadens 1.140 times. In some scientific research fields, ABWs can be regarded as spatiotemporal localized LLBs. However, we cannot always regard ABWs as LLBs in all situations because the influence of cubic SID on ABWs cannot be always ignored. What situations where ideal ABWs can be regarded as LLBs, and what situations where ideal ABWs cannot be regarded as LLBs, are worth investigating and will be expounded on in the following.

*r*

_{0}and

*T*

_{0}is used in such applied fields as long distance laser-induced lighting, laser filamentation, optical manipulation, and signal transmission in vacuum, the influence of cubic SID on the Airy pulse must be considered.

### 3.3 The influence of quadratic and cubic SID

*a*=

*z*

^{2}/4

*L*

_{2}

^{2}toTherefore, the real transmission trajectory of the Airy pulse peak is different from the traditional research results, in which the influence of cubic SID is ignored [1

**4**(2), 103–106 (2010). [CrossRef]

3. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express **17**(17), 14948–14955 (2009). [CrossRef] [PubMed]

*L*

_{3}is, the larger the degree of difference will become between the trajectory when cubic SID is considered and the trajectory when cubic SID is ignored. According to Eq. (18), the relationship between

*L*

_{2}and

*L*

_{3}can be illustrated asGiven that

*r*

_{0}

^{2}

*k*

_{0}

^{2}>>1, Eq. (29) is transformed intoEquation (30) indicates that the smaller the

*T*

_{0}of an ABW for a fixed

*L*

_{2}is, the larger

*L*

_{3}and the larger influence on the trajectory of the Airy pulse peak will become. For example, the same quadratic SID length

*L*

_{2}and different cubic SID length

*L*

_{3}were used among an ABW with

*r*

_{0}= 5

*μm*and

*T*

_{0}= 80

*fs*, another ABW with

*r*

_{0}= 20

*μm*and

*T*

_{0}= 20

*fs*, and finally one ABW with

*r*

_{0}= 40

*μm*and

*T*

_{0}= 10

*fs*. Figure 1 shows the trajectory of the Airy pulse peak for these three cases with cubic SID being ignored. These propagation distances correspond to different quadratic SID lengths (

*L*

_{2}), and time corresponds to pulse width of ABWs (

*T*

_{0}). Figure 1 further illustrates that the influence of cubic SID on the trajectory of the Airy pulse must be considered, especially for ultra-short pulsed ABWs. Otherwise, the actual temporal acceleration of ABWs must deviate from the theoretical prediction given in the past research results because cubic SID is usually ignored [1

**4**(2), 103–106 (2010). [CrossRef]

**47**(3), 264–267 (1979). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

**4**(2), 103–106 (2010). [CrossRef]

*r*

_{0}is 5

*μm*, and the temporal parameter

*T*

_{0}is 20

*fs*. For these values, one has respectively

*L*

_{2}= 0.435

*m*, and

*L*

_{3}= 6.840

*m*from Eq. (18). Based on Eqs. (14) and (23), the temporal broadening factor of Gaussian-Bessel wave packets and ABWs is given in Fig. 2. For the sake of comparison, the FWHM of the initial Gaussian pulse has been chosen to be the same with the FWHM of the main lobe of Airy pulse at

*z*= 0. The Airy pulse broadens by 1.046 times instead of the Gaussian pulse broadening by 4.702 times than its original width for the same propagation distance of 2

*m*. It can be shown that, compared to the Gaussian distribution pulse, the Airy distribution pulse is quite stable during propagation. A deeply comparison of the propagation of initially Gaussian and finite-energy Airy pulse under the influence of both quadratic and cubic dispersion has also been made by Besieris and Shaarawi (see Ref. 38

38. I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(4), 046605 (2008). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgments

## References and links

1. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

2. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

3. | H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express |

4. | B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt. |

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6. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

7. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

8. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

9. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

10. | V. Pasiskevicius, “Engineering Airy beams,” Nat. Photonics |

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12. | R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A |

13. | P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A |

14. | P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express |

15. | M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Opt. Express |

16. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

17. | S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. B: Quantum Semiclass. |

18. | G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. |

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

20. | W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A |

21. | M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A |

22. | M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E |

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

24. | Z. J. Ren, H. Z. Jin, Y. L. Shi, J. C. Xu, W. D. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading Airy-Bessel wave packets,” J. Opt. Soc. Am. A |

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

26. | R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A |

27. | C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum Generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. |

28. | H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, |

29. | C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A |

30. | B. D. Lü and Z. J. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A |

31. | F. W. Wise, “Generation of light bullets,” Physics |

32. | P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. |

33. | J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control |

34. | P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. |

35. | M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

36. | G. P. Agrawal, |

37. | I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express |

38. | I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

39. | K. Y. Kim, C. Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express |

40. | A. Nerukh, D. Zolotariov, and D. Nerukh, “Time-varying Airy pulses,” IEEE 978–1-4577–0882–4/11 (2011) |

41. | S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(320.5540) Ultrafast optics : Pulse shaping

(070.3185) Fourier optics and signal processing : Invariant optical fields

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 13, 2012

Revised Manuscript: February 4, 2013

Manuscript Accepted: February 6, 2013

Published: February 13, 2013

**Citation**

Zhijun Ren, Qiong Wu, Hefa Mao, Yile Shi, and Changjiang Fan, "Propagation characteristics of Airy-Bessel wave packets in free space," Opt. Express **21**, 4481-4492 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4481

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### References

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