## High localization, focal depth and contrast by means of nonlinear Bessel Beams

Optics Express, Vol. 13, Issue 16, pp. 6160-6167 (2005)

http://dx.doi.org/10.1364/OPEX.13.006160

Acrobat PDF (687 KB)

### Abstract

We show an experimental and computational comparison between the resolution power, the contrast and the focal depth of a nonlin-early propagated diffraction-free beam and of other beams (a linear and a nonlinearly propagated Gaussian pulse): launching a nondiffractive Bessel pulse in a solution of Coumarine 120 in methanol creates a high contrast, 40 mm long, 10 *μ*m width fluorescence channel excited by 3-photon absorption process. This fluorescence channel exhibits the same contrast and resolution of a tightly focused Gaussian pulse, but reaches a focal depth that outclasses by orders of magnitude that reached by an equivalent Gaussian pulse.

© 2005 Optical Society of America

## 1. Introduction

1. J. Durnin, J.J. Micheli, and J.H. Heberly: “Diffraction-Free Beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987) [CrossRef] [PubMed]

2. H. Sõnajalg, M. Rätsep, and P. Saari: “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in dispersive medium,” Opt. Lett. **22**, 310–312 (1996) [CrossRef]

3. D. McGloin and K. Dholakia: “Bessel beams: diffraction in a new light,” Contemporary Physics **46**, 15–28 (2005) [CrossRef]

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

5. 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 **67**, 063820 (2003). [CrossRef]

6. T. Wulle and S. Herminghaus: “Nonlinear Optics of Bessel Beams,” Phys. Rev. Lett. **70**, 1401–1403 (1993) [CrossRef] [PubMed]

7. R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, and A. Stabinis: “Traveling wave optical parametric generator pumped by a conical beam,” Opt. Commun. **146**, 253–256 (1998) [CrossRef]

8. M.A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani: “Nonlinear Unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. **93**, 153902 (2004) [CrossRef] [PubMed]

## 2. Nonlinear loss localization: experimental evidence

*μ*m FWHM central spike (as also experimentally verified), whose energy (within the first-zero circle) is of 15

*μ*J. A schematic description of the experimental apparatus is presented in Fig. 1.

*via*three-photon absorption (TPA) (the Coumarone absorption peak is located at 350 nm). The detection has been performed by side imaging via a commercial CCD photo camera (Canon-EOS D30). Fig. 2(b) illustrates for comparison the fluorescence channel generated when a 1 ps, 70

*μ*J, 1055 nm, 20

*μ*m FWHM pulsed Gaussian beam was launched with the beam waist located inside the cuvette. As expected, a high contrast, high focal depth, ultra thin fluorescence channel is excited in the case of Bessel-beam illumination.

*μ*m FWHM central spike. Owing to the very low power, only linear interaction (

*e*.

*g*. one-photon scattering process) is expected to occur.

## 3. Nonlinear loss localization: model and simulations

*μ*m ,energy 1 mJ) and of pulsed Gaussian beam (wavelength 1055 nm, pulse duration 1 ps, FWHM 20

*μ*m , energy 15

*μ*J) in a Coumarine 120 methanol solution. The energy and the dimensions of the Gaussian beam have been chosen to fit the characteristics of the Bessel beam’s central spot. Note that if we focus the entire 1 mJ energy in the same 20

*μ*m diameter we induce material breakdown. Moreover, the same amount of energy focused in much broader areas leads to multiple filamentation [9

9. H. Schroeder and S.L. Chin: “Visualization of the evolution of multiple filaments in methanol,” Opt. Commun. **234**399–406 (2004) [CrossRef]

*n*

_{2}= 4.7000e-16

*cm*

^{2}/

*W*) and 3-photon absorption. The propagation equation solved by the code is an extended nonlinear Schrödinger equation [10

10. T. Brabec and F. Krausz: “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. **78**, 3282–3285 (1997) [CrossRef]

*z*axis of the envelope of the laser field with central frequency

*ω*

_{0}:

*ℰ*̂(

*r*,

*ω*,

*z*) = TF{

*ℰ*

*r*,

*t*,

*z*)} (TF stands for time Fourier transform),

*U*̂(

*ω*) = 1 + (

*ω*-

*ω*

_{0})/

*kv*,

_{g}*v*≡

_{g}*∂ω*/

*∂k*|

_{ω0}is the group velocity and TF{

*N*(

*ℰ*)} the time-Fourier transform of the nonlinear terms. Here

*n*(

*ω*) denotes the linear refraction index. The nonlinear terms is

*ρ*is the molar concentration of the multiphoton absorber (10% in this case). Factor (

*β*(K = 3) is computed with Keldysh formula [11] (

_{k}*β*

_{3}= 2.220710

^{-22}cm

^{5}/J

^{2}). Bessel beam is apodized with a Gaussian mask with a FWHM of 4 mm, so that the input pulse in the simulation is not an idealized radially infinite one, but a real world pulsed Bessel beam.

*NLL*[

*ℰ*(

*t*)] in Eq. (4) and the intensity profiles obtained by integration of the propagation equation Eq. (1). We perform for this analysis a small

*ω*-

*ω*

_{0}expansion for the linear refraction index

*n*(

*ω*) and neglect the departure from unity in the operators

*U*̂ in Eq. (1) and

*T*in Eq. (3) and (4) (which is a good approximation for our long pulses). We obtain the evolution equation for the intensity:

*k*″ =

*∂*

^{2}

*k*/

*∂ω*

^{2}and

*I*is the intensity (

*I*∝

*ℰ*·

*ℰ*

^{*}).

*D*accounts for the intensity redistribution due to diffraction and dispersion while

*ℒ*is the density of three photon absorption. Once calculated the intensity profile

*I*(

*r*,

*z*,

*t*) from Eq. (1), we integrate in time this density:

^{3}). Since the only absorption mechanism is three photon absorption, we expect the fluorescence to be proportional to

*L*(

*z*,

*r*).

## 4. Simulation results

*F*(

*r*,

*z*) = ∫

*dtI*(

*r*,

*t*,

*z*) is plotted as a function of the radial coordinate and the propagation distance for the nonlinear Bessel (Fig. 4(a)) and the Gaussian (Fig. 4(b)) pulses. Figure 5 shows energy losses due to three photon absorption (function

*L*(

*r*,

*z*) defined by Eq. (8)) in J/cm

^{3}as a function of the propagation distance and radius both for the Bessel (Fig. 5(a)) and the Gaussian (Fig. 5(b)) pulse. Figure 6 shows the beam width at half maximum as a function of the propagation distance of the Bessel (Fig. 6(a)) and the Gaussian (Fig. 6(b)) pulses. For comparison, the beam width obtained by linear propagation of the same pulses is plotted in dashed curves.

## 5. Discussion of the results

^{2}for the Bessel pulse, just 0.1 J/cm

^{2}for the Gaussian). The robustness of the conical wave against strongly nonlinear propagation is due to the fact that nonlinear losses are compensated thanks to the continuous refilling of the hot core by linearly propagated energy coming from the outer rings. Figure 5(a), which plots function

*L*(

*r*,

*z*,

*t*) computed on the Bessel beam, clearly shows that absorption involves just the central peak, while outer rings propagate quite untouched. So although 42.5% of the total energy is absorbed in 10 cm, the central peak profile remains unchanged. This is not true for the Gaussian beam, whose energy dissipation process shown in Fig. 5(b) involves all the beam. For this reason dissipation reaches 82.5% and completely destroys the beam in just 4 mm (a crude estimation of the loss can be argued from Fig 5b:

*π*· (3

*μ*m)

^{2}· 0.1cm · 0.1J/cm

^{3}= 10

*μ*J corresponding to 66% energy loss). Moreover, the fact that nonlinear absorption involves just the central peak of the Bessel pulse, implies a great enhancement in contrast for the generated fluorescence with respect to the linear slow decaying Bessel pulse, since fluorescence radial profile maintains Gaussian-like fast decaying (see again Fig. 5(a)) over the 10 cm long propagation.

*μ*m but the pulse continues its propagation for only 4 equivalent Rayleigh ranges. Indeed the Rayleigh range of a Gaussian profile with 6.5

*μ*m FWHM is 180

*μ*m, about four time less than the propagation length of the filament generated here by competition between Kerr-nonlinearity, diffraction and nonlinear losses [12

12. A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. Di Trapani: “Light filaments without self channeling,” Phys. Rev. Lett. **92**, 253903 (2004) [CrossRef] [PubMed]

*μ*m respectivelly) will induce material breakdown. Lower energies or larger radii will lead to a filamentation regime of a tightly focused pulse that is only apparently stationary over many Rayleigh ranges and that is in fact characterized by many refocusing cycles with very high local intensity peaks [13

13. W. Liu, S.L. Chin, O. Kosareva, I.S. Golubtsov, and V.P. Kandidov: “Multiple refocusing of a femtosecond laser pulse in a dispersive liquid (methanol),” Opt. Commun. **225**, 193–209 (2003) [CrossRef]

## 6. Conclusions

14. M. Erdelyi, Z. L. Horvath, G. Szabo, Zs. Bor, F.K. Tittel, J.R. Cavallaro, and M.C. Smayling: “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B **15**, 287–292 (1997) [CrossRef]

15. J. Amako, D. Sawaki, and E. Fujii: “Microstructuring transparent materials by use of nondiffracting ultrashort pulse beams generated by diffractive optics,” J. Opt. Soc. Am. B **20**, 2562–2568 (2003) [CrossRef]

## References and links

1. | J. Durnin, J.J. Micheli, and J.H. Heberly: “Diffraction-Free Beams,” Phys. Rev. Lett. |

2. | H. Sõnajalg, M. Rätsep, and P. Saari: “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in dispersive medium,” Opt. Lett. |

3. | D. McGloin and K. Dholakia: “Bessel beams: diffraction in a new light,” Contemporary Physics |

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

5. | 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 |

6. | T. Wulle and S. Herminghaus: “Nonlinear Optics of Bessel Beams,” Phys. Rev. Lett. |

7. | R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, and A. Stabinis: “Traveling wave optical parametric generator pumped by a conical beam,” Opt. Commun. |

8. | M.A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani: “Nonlinear Unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. |

9. | H. Schroeder and S.L. Chin: “Visualization of the evolution of multiple filaments in methanol,” Opt. Commun. |

10. | T. Brabec and F. Krausz: “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. |

11. | A. M. Perelomov, V. S. Popov, and M. V. Terent’ev: “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP |

12. | A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. Di Trapani: “Light filaments without self channeling,” Phys. Rev. Lett. |

13. | W. Liu, S.L. Chin, O. Kosareva, I.S. Golubtsov, and V.P. Kandidov: “Multiple refocusing of a femtosecond laser pulse in a dispersive liquid (methanol),” Opt. Commun. |

14. | M. Erdelyi, Z. L. Horvath, G. Szabo, Zs. Bor, F.K. Tittel, J.R. Cavallaro, and M.C. Smayling: “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B |

15. | J. Amako, D. Sawaki, and E. Fujii: “Microstructuring transparent materials by use of nondiffracting ultrashort pulse beams generated by diffractive optics,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 8, 2005

Revised Manuscript: July 27, 2005

Published: August 8, 2005

**Citation**

Paolo Polesana, Daniele Faccio, Paolo Di Trapani, Audrius Dubietis, Algis Piskarskas, Arnaud Couairon, and Miguel Porras, "High localization, focal depth and contrast by means of nonlinear Bessel beams," Opt. Express **13**, 6160-6167 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6160

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### References

- J. Durnin, J.J. Micheli and J.H. Heberly, �??Diffraction-Free Beams,�?? Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- H. Sõnajalg, M. Rätsep and P. Saari, �??Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in dispersive medium,�?? Opt. Lett. 22, 310-312 (1996). [CrossRef]
- D. McGloin and K. Dholakia, �??Bessel beams: diffraction in a new light,�?? Contemporary Physics 46, 15-28 (2005). [CrossRef]
- S. Orlov, A. Piskarskas and A. Stabinis, �??Localized optical subcycle pulses in dispersive media,�?? Opt. Lett. 27, 2167-2169 (2002). [CrossRef]
- 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 67, 063820 (2003). [CrossRef]
- T. Wulle and S. Herminghaus, �??Nonlinear Optics of Bessel Beams,�?? Phys. Rev. Lett. 70, 1401-1403 (1993). [CrossRef] [PubMed]
- R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, �??Traveling wave optical parametric generator pumped by a conical beam,�?? Opt. Commun. 146, 253-256 (1998). [CrossRef]
- M.A. Porras, A. Parola, D. Faccio, A. Dubietis and P. Di Trapani, �??Nonlinear Unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,�?? Phys. Rev. Lett. 93, 153902 (2004). [CrossRef] [PubMed]
- H. Schroeder and S.L. Chin, �??Visualization of the evolution of multiple filaments in methanol,�?? Opt. Commun. 234 399-406 (2004). [CrossRef]
- T. Brabec and F. Krausz, �??Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,�?? Phys. Rev. Lett. 78, 3282-3285 (1997). [CrossRef]
- A. M. Perelomov, V. S. Popov and M. V. Terent�??ev, �??Ionization of atoms in an alternating electric field,�?? Sov. Phys. JETP 23, 924-934 (1966).
- A. Dubietis, E. Gaižauskas, G. Tamošauskas and P. Di Trapani, �??Light filaments without self channeling,�?? Phys. Rev. Lett. 92, 253903 (2004). [CrossRef] [PubMed]
- W. Liu, S.L. Chin, O. Kosareva, I.S. Golubtsov, V.P. Kandidov, �??Multiple refocusing of a femtosecond laser pulse in a dispersive liquid (methanol),�?? Opt. Commun. 225, 193-209 (2003). [CrossRef]
- M. Erdelyi, Z. L. Horvath, G.Szabo, Zs. Bor, F.K. Tittel, J.R. Cavallaro and M.C. Smayling, �??Generation of diffraction-free beams for applications in optical microlithography,�?? J. Vac. Sci. Technol. B 15, 287-292 (1997). [CrossRef]
- J. Amako, D. Sawaki and E. Fujii, �??Microstructuring transparent materials by use of nondiffracting ultrashort pulse beams generated by diffractive optics,�?? J. Opt. Soc. Am. B 20, 2562-2568 (2003). [CrossRef]

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