## Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells

Optics Express, Vol. 15, Issue 16, pp. 10303-10317 (2007)

http://dx.doi.org/10.1364/OE.15.010303

Acrobat PDF (2063 KB)

### Abstract

Cell surgery based on ultrashort laser pulses is a fast evolving field in biophotonics. Noninvasive intra cellular dissection at sub-diffraction resolution can be performed within vital cells with very little hazardous effects to adjacent cell organelles. Microscope objectives of high numerical aperture (NA) are used to focus ultrashort pulses to a small spot. Due to the high order of nonlinearity, plasma formation and thus material manipulation is limited to the very focus. Nonetheless nonlinear plasma formation is generally accompanied by a number of additional nonlinear effects like self-focusing and filamentation. These parasitic effects limit the achievable precision and reproducibility of applications. Experimentally it is known that the intensity of these effects decreases with increasing NA of the focusing optics, but the process of nonlinear plasma formation at high NA has not been studied numerically in detail yet. To simulate the interaction of ultrashort laser pulses with transparent materials at high NA a novel nonlinear Schrödinger equation is derived; the multiple rate equation (MRE) model is used to simultaneously calculate the generation of free electrons. Nonparaxial and vectorial effects are taken into account to accurately include tight focusing conditions. Parasitic effects are shown to get stronger and increasingly distortive for NA <0.9, using water as a model substance for biological soft tissue and cellular constituents.

© 2007 Optical Society of America

## 1. Introduction

1. N.T. Nguyen, A. Saliminia, W. Liu, S.L. Chin, and R. Valle, “Optical breakdown versus filamentation in fused silica by use of femtosecond infrared laser pulses,” Opt. Lett. **28**1591–1593 (2003). [CrossRef] [PubMed]

2. A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. GaiŽauskas, D. Faccio, and P. Di Trapani, “Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water,” Appl. Phys. B **84**439–446 (2006). [CrossRef]

3. C.L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B **80**247–253 (2005). [CrossRef]

4. C.B. Schaffer, A.O. Jamison, and E. Mazur, “Morphology of femtosecond laser-induced structural changes in bulk transparent materials,” Appl. Phys. Lett. **84**1441–1443 (2004). [CrossRef]

5. A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, “Nonlinear side effects of fs pulses inside corneal tissue during photodisruption,” Appl. Phys. B **74**419–425 (2002). [CrossRef]

6. A. Heisterkamp, I.Z. Maxwell, E. Mazur, J.M. Underwood, J.A. Nickerson, S. Kumar, and D.E. Ingber, “Pulse energy dependence of subcellular dissection by femtosecond laser pulses,” Opt. Express **13**3690–3696 (2005). [CrossRef] [PubMed]

7. K. König, I. Riemann, and W. Fritzsche, “Nanodissection of human chromosomes with near-infrared femtosecond laser pulses,” Opt. Lett. **26**819 (2001). [CrossRef]

8. M.F. Yanik, H. Cinar, A.D. Chisholm, Y. Jin, and A. Ben-Yakar, “Neurosurgery: Functional regeneration after laser axotomy,” Nature **432**822 (2004). [CrossRef] [PubMed]

2. A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. GaiŽauskas, D. Faccio, and P. Di Trapani, “Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water,” Appl. Phys. B **84**439–446 (2006). [CrossRef]

3. C.L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B **80**247–253 (2005). [CrossRef]

9. Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, and C.R. Thomson, “Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses,” IEEE J. Quantum. Electron. **33**127–137 (1997). [CrossRef]

10. W. Liu, O. Kosareva, L.S. Golubtsov, A Iwasaki, A. Becker, V.P. Kandidov, and S.L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in *H*_{2}*O*,” Appl. Phys. B **76**215–229 (2003). [CrossRef]

11. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. **20**1598–1600 (1995). [CrossRef] [PubMed]

12. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D **157**112–146 (2001). [CrossRef]

13. M. Kolesik, J.V. Moloney, and M. Mlejnek, “Unidirectional Optical Pulse Propagation Equation,” Phys. Rev. Lett. **89**283902-1-4 (2002). [CrossRef]

14. M. Kolesik and J.V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E **70**036604-1-11 (2004). [CrossRef]

*P*is greater than the critical power for self-focusing

*P*.

_{crit}*P*. Thus self-focusing is of minor importance. Conversely the interaction with the generated dense plasma, namely plasma-defocusing, can be very strong. Indeed it was shown that the side-effect of streak formation accompanying optical breakdown is mostly due to plasma-defocusing [3

_{crit}3. C.L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B **80**247–253 (2005). [CrossRef]

**80**247–253 (2005). [CrossRef]

*x*-polarized into the

*z*-polarization direction. The transversal focal intensity distribution becomes asymmetric at the same time [15

15. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. A **253**358–379 (1959). [CrossRef]

16. A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissue,” Appl. Phys. B **81**1015–1047 (2005). [CrossRef]

16. A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissue,” Appl. Phys. B **81**1015–1047 (2005). [CrossRef]

16. A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissue,” Appl. Phys. B **81**1015–1047 (2005). [CrossRef]

**81**1015–1047 (2005). [CrossRef]

*ρBd*=10

^{21}cm

^{-3}was used as criterion for the occurrence of optical breakdown.

## 2. Theory of pulse propagation and nonlinear plasma formation at high NA

17. F. Williams, S.P. Varma, and S. Hillenius, “Liquid water as a lone-pair amorphous semiconductor,” J. Chem. Phys. **64**, 1549–1554 (1976). [CrossRef]

*E*⃗=(

*ε*

_{0}

*n*

^{2})

^{-1}∇

*P*⃗

_{NL}was used.

*E*⃗(

*x,y,z*)=(

*E*) is the electric field vector,

_{x}(x,y,z),E_{y}(x,y,z),E_{z}(x,y,z)*k*the linear wave number and n is the medium’s linear refractive index; ε0 is the vacuum permittivity and

*P*⃗

_{NL}(

*x,y,z*) is the nonlinear polarization vector, ∇

^{2}=

*∂*+

^{2}_{x}*∂*+

^{2}_{y}*∂*and ∇=(

^{2}_{z}*∂*) are the Laplace and the Nabla operators, respectively. Note that the wave number

_{x},∂_{y},∂_{z}*k*has different representations depending on whether Eq. (1) is written in frequency space or in normal space. In frequency space it is simply

*ω*is the angular frequency. In normal space

*k*has a time dependent representation that will be introduced later.

*P*can be split into fractions induced by the Kerr-effect ∂

_{NL}*K*and by the presence of a generated plasma ∂

_{err}*P*:

_{fe}*c*is the vacuum speed of light and

*n*

_{2}=2×10

^{-16}cm

^{2}/W [10

10. W. Liu, O. Kosareva, L.S. Golubtsov, A Iwasaki, A. Becker, V.P. Kandidov, and S.L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in *H*_{2}*O*,” Appl. Phys. B **76**215–229 (2003). [CrossRef]

*λ*=780 nm. It is defined in the scalar limit by the simplified expression

*n(I)*=

*n*

_{0}+

*n*

_{2}

*I*for the intensity dependent refractive index, where

*I*is the intensity of the incident electric field. The factor

*γ*in Eqs. (3) specifies the physical origin of the Kerr effect. Here

*γ*=1/2 for the Kerr effect induced by nonresonant electrons is considered; other possible origins like molecular orientation are to slow to be of importance for ultrashort pulses. The susceptibility

*χ*contains all effects concerning the generation and the presence of generated free electrons of density

_{fe}(ρ)*ρ*on the medium’s dielectric function.

*ε*

_{0}is the vacuum permittivity,

*m**

_{e}is the electrons reduced mass and e is the electron charge. If

*ω*>

_{P}*ω*, the plasma density exceeds the critical density

*p*≈1.6×10

_{crit}^{21}cm

^{-3}. The first term in Eq. (5) describes the refractive effect of the generated plasma and thus accounts for plasma defocusing. The second term describes the absorption of laser power in the generated plasma. To calculate the density of free electrons, the more sophisticated multiple rate equation (MRE) model [18, 19

19. B. Rethfeld, “Free-electron generation in laser-irradiated dielectrics,” Phys. Rev. B **73**035101–6 (2006). [CrossRef]

*ρ*as compared to the Drude model. Here

*ħ*is Planck’s constant and

*ε*is the critical energy for impact ionization, which will be discussed later.

_{crit}*K*is the number of photons to be absorbed by a free electron to overcome

*ε*. The last term in Eq. (5) is due to nonlinear photo ionization, where Δ is the material’s bandgap potential and

_{crit}*W*is the Keldysh rate for nonlinear photo ionization [20].

_{PI}*x*and

*y*directions are scaled in

*x*

_{0}, which is chosen to be roughly the focal beam waist (1/

*e*). The propagation direction

*z*is scaled in diffraction lengths

*L*=

_{Df}*k*

_{0}

*x*

^{2}

_{0}, where

*ω*

_{0}and

*f*=(

*k*

_{0}

*x*

_{0})

^{-1}is introduced as a small parameter. Using the substitutions (6) in Eq. (1) and some algebra gives the nonlinear wave Eq. for each vector component

*A*:

_{i}*f*can be understood as a nonparaxial or vectorial parameter. For relatively large focal beam diameter (2

*x*

_{0}≫λ/2) at low numerical aperture,

*f*≪1 and terms of order

*O*(

*f*

^{2}) and higher can generally be neglected. Nonetheless, at high numerical aperture and a beam waist of about the diffraction limit, the nonparaxial parameter

*f*is of the order of unity (2

*x*

_{0}≈

*λ*/2⇒

*f*≈0.5). Terms of order

*O*(

*f*

^{2}) then have to carefully be examined.

*x*-axis, the dimensionless amplitudes satisfy the following relations [12

12. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D **157**112–146 (2001). [CrossRef]

*x*-polarized. The

*y*-polarization direction can always be neglected, since the power is much smaller than both the

*x*- and

*z*-polarization direction. Thus the system of Eqs. (7,8,9) is simplified by considering only the Eq. for the

*x*-polarization

*A*and by dropping all terms of order higher

_{x}*O*(

*f*

^{2}).

*O*(0), Eq. (11) contains three more terms that scale either with power or with the density of generated free electrons

*ρ*or with the paraxial parameter

*f*. If the numerical aperture of the focusing optics is high, these terms cannot easily be dropped. However, if the pulse power or the density

*ρ*are low, the contributions of (

*P*̃

_{Kerr})

*i*or (

*P*̃

_{fe})

*i*respectively are small. The basic theory of self-focusing provides a critical laser power

12. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D **157**112–146 (2001). [CrossRef]

*P*is exceeded, simple scalar and paraxial propagation models fail to describe the nonlinear propagation. Due to self focusing the beam spatially collapses and forms a singularity. However, if the beam power is small compared to the critical power (

_{crit}*P*≪

*P*), the propagation is only slightly nonlinearly modified. This is generally the case for pulse energies required to induce optical breakdown with ultrashort pulses at high numerical aperture. For NA=1 in pure water, pulse duration

_{crit}*τ*=150 fs (FWHM) and central wavelength λ=780 nm, the pulse energy to induce optical breakdown is well below 5 nJ and the pulse peak power is below 1% of the critical power

*P*. In this case almost all terms of the Kerr-effect in Eq. (11) can be neglected, even when the nonparaxiality is high. There is also no considerable vectorial coupling between the amplitude components

_{crit}*A*due to the Kerr effect. A detailed investigation of which terms in Eq. (11) must be kept, depending on the relative power

_{i}*P/P*and the nonparaxial parameter

_{crit}*f*, was performed using a two-dimensional model for cw-laser beams.

11. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. **20**1598–1600 (1995). [CrossRef] [PubMed]

**157**112–146 (2001). [CrossRef]

*O*(0) are kept except the nonparaxiality. Equation (11) then reads:

*∂*

_{z̃}-operator can be found by cancelling out the amplitude

*A*and solving for

_{x}*∂*

_{z̃}. The

*∂*

_{z̃}-operator is once again applied to the amplitude

*A*.

_{x}13. M. Kolesik, J.V. Moloney, and M. Mlejnek, “Unidirectional Optical Pulse Propagation Equation,” Phys. Rev. Lett. **89**283902-1-4 (2002). [CrossRef]

14. M. Kolesik and J.V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E **70**036604-1-11 (2004). [CrossRef]

22. J.E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. **17**583–585 (1992). [CrossRef] [PubMed]

23. Y.M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperture beams,” J. Opt. Soc. Am. A **21**2135–2145 (2004). [CrossRef]

*k*are the transversal wavenumbers in frequency space with the transformations

_{x},k_{y}24. P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. **17**172–174 (1992). [CrossRef] [PubMed]

22. J.E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. **17**583–585 (1992). [CrossRef] [PubMed]

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

*n(ω)*≈

*n*

_{0}was used. The expression transforms back to normal space by using

*k*=

*k*

_{0}+1/

*v*(

_{g}*ω-ω*

_{0})+

*k*

*″*/2(

*ω-ω*

_{0})

^{2}+…), Eq. (15) is identical with the SEWA introduced by Brabec and Krausz, except for the influence of generated free electrons, which was not included in the original work [25

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

*O*(

*f*

^{2}). The terms of order

*O*(

*f*

^{2}) neglected in deriving the nonparaxial version of Eq. (11) are again added to Eq. (13), yielding the nonparaxial and nonlinear propagation Eq. (16) for the

*x*-polarization component

*A*of the vector field A⃗. It is linearly exact and includes nonlinear effects and nonlinear nonparaxial mixed-terms to order

_{x}*O*(

*f*

^{2}).

*P/P*≪1). Thus from the Kerr-effect only terms induced by the

_{crit}*x*-polarized part of the field (|

*A*|

_{x}^{2}

*A*) are kept; the influence from the other polarization directions is much smaller. This approximation was carefully tested. However, when higher peak power is used, all terms in Eq. (16) must be taken into account. Since the pulse duration used for the simulations was rather long

_{x}*τ*

_{0}=150

*fs*(FWHM) with a narrow bandwidth, the frequency dependence of the nonlinear terms was neglected

*k=k*

_{0}. If the spectrum considered is broader or parameters are used for effects, such as self-phase modulation or super continuum generation likely to occur, the frequency dependence must be considered. Since the linear part is evaluated in frequency space, there is no need to neglect the frequency dependence. The dispersion relation of water is well known [26], thus the wave number is not Taylor expanded, but

*k*(

*ω*) is evaluated exactly in frequency space.

*v*. Time

_{g}*t*was substituted by the retarted time

*τ*=

*t*-

*z/v*. Equation (17) is numerically integrated using a split-step method. The linear part is integrated in frequency space and the nonlinear part is integrated using a Runge-Kutta algorithm. Each nonlinear integration step of step size Δ

_{g}_{z}is embedded within two linear steps of step size Δ

_{z}/2 [27]. The step size is adaptively controlled.

*ħω*(Fig. 1). The process is often also termed inverse bremsstrahlung [16

**81**1015–1047 (2005). [CrossRef]

28. C. DeMichelis, “Laser induced gas breakdown: A bibliographical review,” IEEE J. Quantum. Electron. **5**188–202 (1969). [CrossRef]

29. P.K. Kennedy, “A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory,” IEEE J. Quantum. Electron. **31**2241–2249 (1995). [CrossRef]

30. B.C. Stuart, M.D. Feit, A.M. Rubenchik, B.W. Shore, and M.D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. **74**2248–2251 (1995). [CrossRef] [PubMed]

*ε*greater than the critical energy

_{K}*ε*for impact ionization, it can generate an additional free electron by impact ionization of atoms and molecules, resulting in two electrons at the bottom of the conduction band. Since the total number of free electrons doubles in every cascade of impact ionization, the whole process of sequential one-photon absorption and subsequent impact ionization is called cascade or avalanche ionization. As soon as cascade ionization starts, the total number of free electrons grows exponentially.

_{crit}19. B. Rethfeld, “Free-electron generation in laser-irradiated dielectrics,” Phys. Rev. B **73**035101–6 (2006). [CrossRef]

28. C. DeMichelis, “Laser induced gas breakdown: A bibliographical review,” IEEE J. Quantum. Electron. **5**188–202 (1969). [CrossRef]

29. P.K. Kennedy, “A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory,” IEEE J. Quantum. Electron. **31**2241–2249 (1995). [CrossRef]

30. B.C. Stuart, M.D. Feit, A.M. Rubenchik, B.W. Shore, and M.D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. **74**2248–2251 (1995). [CrossRef] [PubMed]

**80**247–253 (2005). [CrossRef]

9. Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, and C.R. Thomson, “Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses,” IEEE J. Quantum. Electron. **33**127–137 (1997). [CrossRef]

10. W. Liu, O. Kosareva, L.S. Golubtsov, A Iwasaki, A. Becker, V.P. Kandidov, and S.L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in *H*_{2}*O*,” Appl. Phys. B **76**215–229 (2003). [CrossRef]

**81**1015–1047 (2005). [CrossRef]

31. L. Sudrie, A. Couairon, M. Franco, B. Lammouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond Laser-Induced Damage and Filamentary Propagation in Fused Silica,” Phys. Rev. Lett. **89**186601-1-4 (2002). [CrossRef]

32. A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B **71**125435-1-11 (2005). [CrossRef]

33. A. Kaiser, B. Rethfeld, M. Vicanek, and G. Simon, “Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses,” Phys. Rev. B. **61**11437–11450 (2000). [CrossRef]

*W*for an electron in the conduction band with kinetic energy exceeding

_{imp}*ε*, the system of rate Eqs. was simplified in comparison to the original MRE model [18, 19

_{crit}19. B. Rethfeld, “Free-electron generation in laser-irradiated dielectrics,” Phys. Rev. B **73**035101–6 (2006). [CrossRef]

*W*is a controversial issue in literature. So far only values for fused silica have been published which vary more than one order of magnitude [30

_{imp}30. B.C. Stuart, M.D. Feit, A.M. Rubenchik, B.W. Shore, and M.D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. **74**2248–2251 (1995). [CrossRef] [PubMed]

33. A. Kaiser, B. Rethfeld, M. Vicanek, and G. Simon, “Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses,” Phys. Rev. B. **61**11437–11450 (2000). [CrossRef]

*W*for water is available. Nevertheless, the impact ionization rate is generally much higher than the one-photon-absorption probability (

_{imp}*W*≫

_{imp}*W*). Therefore instantaneous impact ionization was assumed for free electrons exceeding the critical energy

_{1pt}*ε*. The simplified version of Rethfeld’s original system of rate Eqs. reads:

_{crit}*ρ*is the density of electrons with kinetic energy

_{n}(t)*ε*in the conduction band,

_{n}=nħω*W*is the rate of nonlinear photo ionization calculated from the Keldysh theory [20]. The one-photon-absorption probability

_{PI}(I)*W*is the probability of shifting a free electron from energy level

_{1pt}(I)*ε*to energy level

_{n}*e*

_{n+1}in the conduction band by absorbing one photon from the laser field. It is proportional to the intensity of the applied field:

*η*is estimated from the Drude model [28

28. C. DeMichelis, “Laser induced gas breakdown: A bibliographical review,” IEEE J. Quantum. Electron. **5**188–202 (1969). [CrossRef]

29. P.K. Kennedy, “A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory,” IEEE J. Quantum. Electron. **31**2241–2249 (1995). [CrossRef]

**74**2248–2251 (1995). [CrossRef] [PubMed]

*τ*

_{0}≫[2

^{1/K}-1)

*W*

_{1pt}]

^{-1}, the Drude model is included within the MRE model [18, 19

**73**035101–6 (2006). [CrossRef]

*η*is thus linked to the electron-neutral inverse bremsstrahlung cross section

*σ*, which is the central parameter for cascade ionization in the Drude model:

*τ*, which denotes the mean free time between collisions of a free electron with heavy particles. It is a purely phenomenological constant, since neither the free electron velocity distribution is considered, nor the materials band structure, nor the plasma temperature. Bloembergen proposed

_{p}*τ*to be on the order of 1 fs in almost any material [34

_{p}34. N. Bloembergen, “Laser-Induced Electric Breakdown in Solids,” IEEE J. Quantum. Electron. **10**375–386 (1974). [CrossRef]

*τ*=3 fs yielding σ=8.97×10

_{P}^{-18}cm

^{2}. However, both the Keldysh rate for nonlinear ionization and the rate for cascade ionization are constant points of discussion [2

2. A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. GaiŽauskas, D. Faccio, and P. Di Trapani, “Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water,” Appl. Phys. B **84**439–446 (2006). [CrossRef]

33. A. Kaiser, B. Rethfeld, M. Vicanek, and G. Simon, “Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses,” Phys. Rev. B. **61**11437–11450 (2000). [CrossRef]

*m*and in the valence band

_{CB}*m*are simply assumed to be the free electron mass

_{VB}*m*. The reduced mass of the electron is then

_{e}*m**

_{e}=

*m*/2 and the first factor in Eq. (21) takes the value 3/2. The inclusion of momentum conservation increases the band gap potential for impact ionization and thus lowers the cascade ionization rate compared to other models.

_{e}*ε*is additionally enhanced by the mean oscillation energy 〈

_{crit}*ε*〉=

_{osc}*e*

^{2}

*E*

^{2}/(4

*m*

^{′}ω^{2}) of a free electron in an oscillating electric field [20, 28

**5**188–202 (1969). [CrossRef]

**61**11437–11450 (2000). [CrossRef]

*E*. The quiver energy 〈

*ε*〉 can dynamically change the number of photons

_{osc}*K*required to overcome the critical band gap

*ε*for impact ionization.

_{crit}**31**2241–2249 (1995). [CrossRef]

*W*is calculated using the complete Keldysh theory [20]. For a detailed discussion of Keldysh’s theory for nonlinear photo ionization see the original publication [20] or [3

_{PI}**80**247–253 (2005). [CrossRef]

31. L. Sudrie, A. Couairon, M. Franco, B. Lammouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond Laser-Induced Damage and Filamentary Propagation in Fused Silica,” Phys. Rev. Lett. **89**186601-1-4 (2002). [CrossRef]

**61**11437–11450 (2000). [CrossRef]

23. Y.M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperture beams,” J. Opt. Soc. Am. A **21**2135–2145 (2004). [CrossRef]

*x*-polarized when entering the focusing optics. The focusing optics transfers an incident plane wave front into a spherically converging wave front. Due to strong focusing, the linear

*x*-polarization is partially transformed into the

*z*- and y-direction (Fig. 2). Each polarization direction is defined relative to a reference sphere. The wave front is flat on this sphere, unless phase errors from the beam itself or from the focusing optics are included. The incident

*x*-polarization is not radially symmetrically transformed into

*z*- and

*y*-polarization. Thus the focal intensity distribution becomes asymmetric along the

*x*- and

*y*-direction with increasing focusing angle [15

15. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. A **253**358–379 (1959). [CrossRef]

## 3. Results and discussion

*ρ*=10

_{Bd}^{21}cm

^{-3}associated with the occurrence of optical breakdown can be as low as 2.3 nJ at NA=1.2. This is in good agreement with experimental data published by Heisterkamp et al. [6

6. A. Heisterkamp, I.Z. Maxwell, E. Mazur, J.M. Underwood, J.A. Nickerson, S. Kumar, and D.E. Ingber, “Pulse energy dependence of subcellular dissection by femtosecond laser pulses,” Opt. Express **13**3690–3696 (2005). [CrossRef] [PubMed]

**80**247–253 (2005). [CrossRef]

5. A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, “Nonlinear side effects of fs pulses inside corneal tissue during photodisruption,” Appl. Phys. B **74**419–425 (2002). [CrossRef]

**80**247–253 (2005). [CrossRef]

6. A. Heisterkamp, I.Z. Maxwell, E. Mazur, J.M. Underwood, J.A. Nickerson, S. Kumar, and D.E. Ingber, “Pulse energy dependence of subcellular dissection by femtosecond laser pulses,” Opt. Express **13**3690–3696 (2005). [CrossRef] [PubMed]

**81**1015–1047 (2005). [CrossRef]

*ρ*>

*ρ*). It can easily be observed that for the lowest numerical aperture (NA=0.5), the generated plasma is spatially highly asymmetric. The highest density is obtained considerably before the pulse reaches the geometrical focus located at

_{Bd}*z*=0. The area of high density is surrounded by a region of lower plasma density huge in size compared to the plasmas generated at higher NAs. The plasmas get smaller and less asymmetric with increasing NA. For NA >0.9 almost no spatial asymmetry is observable in Fig. (4).

*ρ=ρ*) is sufficient to induce optical breakdown and to modify the material in the desired manner. Hence at tight focusing conditions

_{Bd}*ρ*can be obtained without considerable defocusing of the incident pulse. Thus it is possible to generate well confined, spatially symmetric, high density optical breakdown plasmas using microscope objectives of high numerical aperture (Fig. 4).

_{Bd}*F*=1.28 Jcm

_{th}^{-2}is found to be sufficient to generate a plasma density

*ρ*>

*ρ*. It is a common simplification that the occurrence of optical breakdown is limited to the focal plane and that no plasma-defocusing occurs. In this case the breakdown fluence is constant for all NAs. As can be seen from Fig. (5), this simple assumption predicts the breakdown pulse energy well for NA≥0.9, but deviates strongly for low NA. There are two distinct reasons for the deviation observed. First, due to plasma-defocusing the pulse energy to generate plasma density of

_{Bd}*ρ*has to be higher at low NA. Second, the axial length of the focal spot roughly scales by 1/NA

_{Bd}^{2}. The assumption that plasma generation is limited to the focal plane fails at low NA, because of the cigar-like shape of the focus. The fact that the size of the plasma increases not only transversally but also axially for lower NA results in an enhanced relative absorption of pulse energy, as is shown in Fig. (6). For NA=1.2 a small percentage of about 2% of the incident pulse energy is absorbed by the generation and heating of the breakdown plasma. The relative and total absorption rapidly increases as the numerical aperture decreases. For NA=0.3 a relative absorption of 40% at a breakdown threshold of about 300 nJ was found.

## 4. Conclusion

*E*≈2.3 nJ for NA=1.2 to about 16.8 nJ for NA=0.5. It has to be noted that the numerically obtained energy thresholds are of course strongly dependent on the parameters used. Especially the model for nonlinear ionization features some uncertainties, such as the collision time

_{th}*τ*and the rate for nonlinear photoionization

_{P}*W*. However, the calculated energy thresholds agree well with experimental data [6

_{PI}**13**3690–3696 (2005). [CrossRef] [PubMed]

*e.g.*to precompensate spherical aberrations or to optimize the focal spot even more by using the Toraldo concept for tailoring particular focal fields of optical superresulution, for example. Since the numerical code can simulate nonlinear plasma formation for any kind of initial field, adaptive optics can also be taken into account and thereby directly be tested numerically.

## References and links

1. | N.T. Nguyen, A. Saliminia, W. Liu, S.L. Chin, and R. Valle, “Optical breakdown versus filamentation in fused silica by use of femtosecond infrared laser pulses,” Opt. Lett. |

2. | A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. GaiŽauskas, D. Faccio, and P. Di Trapani, “Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water,” Appl. Phys. B |

3. | C.L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses,” Appl. Phys. B |

4. | C.B. Schaffer, A.O. Jamison, and E. Mazur, “Morphology of femtosecond laser-induced structural changes in bulk transparent materials,” Appl. Phys. Lett. |

5. | A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, “Nonlinear side effects of fs pulses inside corneal tissue during photodisruption,” Appl. Phys. B |

6. | A. Heisterkamp, I.Z. Maxwell, E. Mazur, J.M. Underwood, J.A. Nickerson, S. Kumar, and D.E. Ingber, “Pulse energy dependence of subcellular dissection by femtosecond laser pulses,” Opt. Express |

7. | K. König, I. Riemann, and W. Fritzsche, “Nanodissection of human chromosomes with near-infrared femtosecond laser pulses,” Opt. Lett. |

8. | M.F. Yanik, H. Cinar, A.D. Chisholm, Y. Jin, and A. Ben-Yakar, “Neurosurgery: Functional regeneration after laser axotomy,” Nature |

9. | Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, and C.R. Thomson, “Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses,” IEEE J. Quantum. Electron. |

10. | W. Liu, O. Kosareva, L.S. Golubtsov, A Iwasaki, A. Becker, V.P. Kandidov, and S.L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in |

11. | S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. |

12. | G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D |

13. | M. Kolesik, J.V. Moloney, and M. Mlejnek, “Unidirectional Optical Pulse Propagation Equation,” Phys. Rev. Lett. |

14. | M. Kolesik and J.V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E |

15. | B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. A |

16. | A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissue,” Appl. Phys. B |

17. | F. Williams, S.P. Varma, and S. Hillenius, “Liquid water as a lone-pair amorphous semiconductor,” J. Chem. Phys. |

18. | B. Rethfeld, “Unified model for the free-electron avalanche in laser-irradiated dielectrics,” Phys. Rev. Lett. |

19. | B. Rethfeld, “Free-electron generation in laser-irradiated dielectrics,” Phys. Rev. B |

20. | L.V. Keldysh, “Ionization in the Field of a Strong Electromagnetic Wave,” Sov. Phys. JETP |

21. | M. Gu, “Advanced optical imaging theory,” Springer Series in Optical Sciences, Springer Berlin, Heidelberg, New York (2000). |

22. | J.E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. |

23. | Y.M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperture beams,” J. Opt. Soc. Am. A |

24. | P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. |

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

26. | The International Association for the Properties of Water and Steam, “Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure,” (1997). |

27. | G.P. Agraval, “Nonlinear Fiber Optics,” Academic Press, San Diego, London, Boston, New York, Sydney, Tokyo, Toronto (1995). |

28. | C. DeMichelis, “Laser induced gas breakdown: A bibliographical review,” IEEE J. Quantum. Electron. |

29. | P.K. Kennedy, “A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory,” IEEE J. Quantum. Electron. |

30. | B.C. Stuart, M.D. Feit, A.M. Rubenchik, B.W. Shore, and M.D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. |

31. | L. Sudrie, A. Couairon, M. Franco, B. Lammouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond Laser-Induced Damage and Filamentary Propagation in Fused Silica,” Phys. Rev. Lett. |

32. | A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B |

33. | A. Kaiser, B. Rethfeld, M. Vicanek, and G. Simon, “Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses,” Phys. Rev. B. |

34. | N. Bloembergen, “Laser-Induced Electric Breakdown in Solids,” IEEE J. Quantum. Electron. |

35. | L.V. Keldysh, “Kinetic theory of impact ionization in semiconductors,” Sov. Phys. JETP |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(140.3440) Lasers and laser optics : Laser-induced breakdown

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(270.4180) Quantum optics : Multiphoton processes

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: June 12, 2007

Revised Manuscript: July 23, 2007

Manuscript Accepted: July 24, 2007

Published: July 31, 2007

**Virtual Issues**

Vol. 2, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, "Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells," Opt. Express **15**, 10303-10317 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10303

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### References

- N. T. Nguyen, A. Saliminia, W. Liu, S. L. Chin, and R. Valle, "Optical breakdown versus filamentation in fused silica by use of femtosecond infrared laser pulses," Opt. Lett. 28, 1591-1593 (2003). [CrossRef] [PubMed]
- A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. Gaižauskas, D. Faccio, and P. Di Trapani, "Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water," Appl. Phys. B 84, 439-446 (2006). [CrossRef]
- C. L. Arnold, A. Heisterkamp, W. Ertmer and H. Lubatschowski, "Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses," Appl. Phys. B 80, 247-253 (2005). [CrossRef]
- C. B. Schaffer, A. O. Jamison, and E. Mazur, "Morphology of femtosecond laser-induced structural changes in bulk transparent materials," Appl. Phys. Lett. 84, 1441 - 1443 (2004). [CrossRef]
- A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, "Nonlinear side effects of fs pulses inside corneal tissue during photodisruption," Appl. Phys. B 74, 419-425 (2002). [CrossRef]
- A. Heisterkamp, I. Z. Maxwell, E. Mazur, J. M. Underwood, J. A. Nickerson, S. Kumar and D. E. Ingber, "Pulse energy dependence of subcellular dissection by femtosecond laser pulses," Opt. Express 13, 3690-3696 (2005). [CrossRef] [PubMed]
- K. König, I. Riemann, and W. Fritzsche, "Nanodissection of human chromosomes with near-infrared femtosecond laser pulses," Opt. Lett. 26, 819 (2001). [CrossRef]
- M. F. Yanik, H. Cinar, A. D. Chisholm, Y. Jin, and A. Ben-Yakar, "Neurosurgery: Functional regeneration after laser axotomy," Nature 432, 822 (2004). [CrossRef] [PubMed]
- Q. Feng, J. V. Moloney, A. C. Newell, E. M. Wright, K. Cook, P. K. Kennedy, D. X. Hammer, B. A. Rockwell and C. R. Thomson, "Theory and Simulation on the threshold of water breakdown induced by Focused Ultrashort Laser Pulses," IEEE J. Quantum. Electron. 33, 127-137 (1997). [CrossRef]
- W. Liu, O. Kosareva, L. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl. Phys. B 76, 215-229 (2003). [CrossRef]
- S. Chi and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995). [CrossRef] [PubMed]
- G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001). [CrossRef]
- M. Kolesik, J. V. Moloney and M. Mlejnek, "Unidirectional Optical Pulse Propagation Equation," Phys. Rev. Lett. 89, 283902-1-4 (2002). [CrossRef]
- M. Kolesik and J.V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604-1-11 (2004). [CrossRef]
- B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. London, Ser. A 253, 358-379 (1959). [CrossRef]
- A. Vogel, J. Noack, G. Hüttman and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissue," Appl. Phys. B 81, 1015-1047 (2005). [CrossRef]
- F. Williams, S. P. Varma and S. Hillenius, "Liquid water as a lone-pair amorphous semiconductor," J. Chem. Phys. 64, 1549-1554 (1976). [CrossRef]
- B. Rethfeld, "Unified model for the free-electron avalanche in laser-irradiated dielectrics," Phys. Rev. Lett. 92, 187401-1-4 (2004).
- B. Rethfeld, "Free-electron generation in laser-irradiated dielectrics," Phys. Rev. B 73, 035101-6 (2006). [CrossRef]
- L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307 (1965).
- M. Gu, Advanced Optical Imaging Theory, Springer Series in Optical Sciences, (Springer Berlin, Heidelberg, New York 2000).
- J. E. Rothenberg, "Pulse splitting during self-focusing in normally dispersive media," Opt. Lett. 17, 583-585 (1992). [CrossRef] [PubMed]
- Y. M. Engelberg and S. Ruschin, "Fast method for physical optics propagation of high-numerical-aperture beams," J. Opt. Soc. Am. A 21, 2135-2145 (2004). [CrossRef]
- P. Chernev and V. Petrov, "Self-focusing of light pulses in the presence of normal group-velocity dispersion," Opt. Lett. 17, 172-174 (1992). [CrossRef] [PubMed]
- T. Brabec and F. Krausz, "Nonlinear Optical Pulse Propagation in the Single-Cycle Regime," Phys. Rev. Lett. 78,3282-3285 (1997). [CrossRef]
- The International Association for the Properties of Water and Steam, "Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure," (1977) http://www.iapws.org/relguide/rindex.pdf.
- G. P. Agraval, Nonlinear Fiber Optics, Academic Press, (San Diego, London, Boston, New York, Sydney, Tokyo, Toronto, 1995).
- C. DeMichelis, "Laser induced gas breakdown: A bibliographical review," IEEE J. Quantum. Electron. 5, 188-202 (1969). [CrossRef]
- P. K. Kennedy, "A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory," IEEE J. Quantum. Electron. 31, 2241-2249 (1995). [CrossRef]
- B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses," Phys. Rev. Lett. 74, 2248-2251 (1995). [CrossRef] [PubMed]
- L. Sudrie, A. Couairon, M. Franco, B. Lammouroux, B. Prade, S. Tzortzakis and A. Mysyrowicz, "Femtosecond laser-induced damage and filamentary propagation in fused silica," Phys. Rev. Lett. 89, 186601-1- 4 (2002). [CrossRef]
- A. Couairon, L. Sudrie, M. Franco, B. Prade and A. Mysyrowicz, "Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses," Phys. Rev. B 71, 125435-1-11 (2005). [CrossRef]
- A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, "Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses," Phys. Rev. B. 61, 11437-11450 (2000). [CrossRef]
- N. Bloembergen, "Laser-induced electric breakdown in solids," IEEE J. Quantum. Electron. 10, 375-386 (1974). [CrossRef]
- L. V. Keldysh, "Kinetic theory of impact ionization in semiconductors," Sov. Phys. JETP 37, 509-518 (1960).

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