## 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

The nonlinear interaction of ultrashort laser pulses with transparent materials obeys a vast number of different effects such as self-focusing, self-phase modulation, nonlinear ionization, super continuum generation, plasma-defocusing, filamentation and optical breakdown. The actual mode of nonlinear interaction is a function of many parameters like power, intensity, pulse length and spatial focusing. The generation of optical breakdown generally competes with nonlinear propagation effects. In the case of weak spatial focusing, filamentation [1, 2] and streak formation [3] as consequences of the interplay of self-focusing and plasma-defocusing are likely to occur instead of optical breakdown, or can occur as accompanying side-effects to optical breakdown. These effects are of parasitic nature to applications utilizing nonlinear plasma formation for micro structuring of materials[4] or biological tissue [5], limiting precision and reproducibility. Thus applications of ultrashort pulses have recently evolved towards tight focusing at high numerical aperture (NA) to reduce pulse energy, enhance precision, and limit nonlinear side effects. Exemplary is the ultrashort pulse based laser cell surgery [6]. Manipulating cells and single cell organelles in combination with simultaneous multiphoton imaging, proved to be a powerful tool enabling physicists and biologists to study many aspects of the functionality of single cells and microorganisms [7, 8].

There is a multitude of publications concerning the numerical simulation of the nonlinear interaction of ultrashort laser pulses with the bulk of transparent media at weak spatial focusing [2, 3, 9, 10]. In contrast, the nonlinear interaction taking into account both nonlinear pulse propagation and nonlinear plasma formation at strong spatial focusing has not been numerically studied in detail yet. Indeed aspects of high numerical aperture like nonparaxiality and vectorial self-focusing were considered before [11, 12, 13, 14], but mostly as possible limiting processes to catastrophic self-focusing, when the pulse power *P* is greater than the critical power for self-focusing *P _{crit}*.

Self-focusing and connected effects, such as filamentation, are sensitive to laser power. The generation of optical breakdown on the other hand is sensitive to intensity. At tight focusing conditions, the intensity for optical breakdown is easily achieved at a pulse peak power much lower than the critical power *P _{crit}*. 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]. Plasma-defocusing severely distorts the pulse’s spatio-temporal profile and strongly influences the density, size and shape of the generated plasma [3].

The objective of this paper is to numerically investigate nonlinear plasma formation induced by ultrashort laser pulses focused by high NA microscope objectives. A numerical model containing nonparaxial and vectorial effects is developed to describe the interaction of ultrashort laser pulses with transparent materials, such as biological soft tissue or fused silica, at tight focusing conditions. The model is based on a nonlinear Schrödinger equation that is introduced in a general manner. A number of approximations and assumptions is made to particularly suit high NA focusing. The initial electric field at the focus used for the simulations is most important at tight focusing. Nonparaxiality alters the focal field distribution and focal depolarization transforms a noticeable amount of power from a pulse initially linearly *x*-polarized into the *z*-polarization direction. The transversal focal intensity distribution becomes asymmetric at the same time [15]. Thus, a nonparaxial and vectorial diffraction integral was used to calculate focal fields for all three polarization directions.

We concentrate on parameters usually found in applications of cell surgery and use water as a model substance for cellular constituents. There are two different methods for femtosecond nanoprocessing of biomaterials [16]. One uses long trains of oscillator pulses of high repetition rate, usually 80MHz, and pulse energy well below the optical breakdown threshold of a single pulse. The dissection is mediated by free-electron-induced chemical decomposition [16]; The other method uses pulses of low repetition rate and pulse energy slightly above the threshold for optical breakdown. In this regime the dissection relies on the thermoelastic induced formation of cavitation bubbles [16]. The simulations presented in this letter are performed for the second regime. In the first regime only a negligible density of free electrons is generated per pulse and nonlinear propagation effects are not likely to occur. Although Vogel pointed out that the free electron density sufficient for bubble formation at tight focusing conditions might be lower [16], the usually accepted breakdown density of *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

A (3+1)-dimensional Schrödinger equation is derived to numerically describe the spatial and temporal nonlinear propagation of ultrashort laser pulses in dielectric media at high numerical aperture. Water, which can in good approximation be treated as an amorphous semi conductor with a band gap energy Δ=6.5 eV [17], is used as model substance. The nonlinearity is induced by both the Kerr-effect and the presence of generated free electrons. The Kerr-effect is responsible for self focusing and self phase modulation and the generated plasma accounts for plasma defocusing and strong absorption. At weak focusing the numerical description of nonlinear ultrashort pulse propagation is governed within the well known framework of the scalar and paraxial approximations. However, when the beam diameter gets small either due to critical self-focusing or usage of high NA focusing optics, the scalar and paraxial approximations fail to describe the propagation properly. A wave Eq. for this regime must essentially be more complex, taking into account nonparaxial effects and nonlinear vectorial coupling. The derivation of the nonlinear wave Eq. starts with the vector Helmholtz-equation for a non-conducting isotropic Kerr medium, Eq. (1).

Here Δ*E*⃗=(*ε*
_{0}
*n*
^{2})^{-1}∇*P*⃗_{NL} was used. *E*⃗(*x,y,z*)=(*E _{x}(x,y,z),E_{y}(x,y,z),E_{z}(x,y,z)*) is the electric field vector,

*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 $k\left(\omega \right)=\frac{\omega}{c}n\left(\omega \right)$, where

*ω*is the angular frequency. In normal space

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

The nonlinear polarization ∇*P _{NL}* can be split into fractions induced by the Kerr-effect ∂

*K*and by the presence of a generated plasma ∂

_{err}*P*:

_{fe}Here *c* is the vacuum speed of light and *n*
_{2}=2×10^{-16} cm^{2}/W [10] is the nonlinear refractive index at *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 _{fe}(ρ)* contains all effects concerning the generation and the presence of generated free electrons of density

*ρ*on the medium’s dielectric function.

Here ${\omega}_{P}={\left(\frac{\rho {e}^{2}}{{\epsilon}_{0}{n}^{2}{m}_{e}^{*}}\right)}^{1\u20442}$ is the plasma frequency, where *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 ${\rho}_{\mathrm{crit}}=\frac{{\epsilon}_{0}{n}^{2}{m}_{e}^{*}}{{e}^{2}}{\omega}^{2}$ and the plasma gets highly reflective and absorbing. For the parameters used, the critical density is

*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] is used instead of the Drude model. The Rethfeld model takes into account a varying kinetic energy distribution of free electrons in the conduction band. The plasma absorption is hence higher by a factor of $\frac{\hslash \omega}{{\epsilon}_{\mathrm{crit}}}{\left({2}^{1\u2044K}-1\right)}^{-1}$ at the same density

*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}To separate the fast oscillations from the slowly varying amplitude and to simultaneously transform Eq. (1) to a dimensionless form, the following substitutions are introduced:

$${\tilde{\nabla}}_{T}^{2}={\partial}_{\tilde{x}}^{2}+{\partial}_{\tilde{y}}^{2},\phantom{\rule{.9em}{0ex}}\tilde{z}=\frac{z}{2{L}_{\mathrm{Df}}},\phantom{\rule{.9em}{0ex}}{L}_{\mathrm{Df}}={k}_{0}{x}_{0}^{2},\phantom{\rule{.9em}{0ex}}f=\frac{1}{{k}_{0}{x}_{0}},$$

$${\tilde{\overrightarrow{P}}}_{\mathrm{Kerr}}=\frac{2}{3}\left[\left(\overrightarrow{A}{\overrightarrow{A}}^{*}\right)\overrightarrow{A}+\frac{1}{2}\left(\overrightarrow{A}\overrightarrow{A}\right){\overrightarrow{A}}^{*}\right],\phantom{\rule{.9em}{0ex}}{\tilde{\overrightarrow{P}}}_{\mathrm{fe}}=\frac{1}{{f}^{2}}\frac{{\chi}_{\mathrm{fe}}}{{n}_{0}^{2}}\overrightarrow{A},{\phantom{\rule{.9em}{0ex}}\tilde{\overrightarrow{P}}}_{\mathrm{NL}}={\tilde{\overrightarrow{P}}}_{\mathrm{Kerr}}+{\tilde{\overrightarrow{P}}}_{\mathrm{fe}}$$

The *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 ${L}_{\mathrm{Df}}={k}_{0}{x}_{0}^{2},\phantom{\rule{.2em}{0ex}}\mathrm{where}\phantom{\rule{.2em}{0ex}}{k}_{0}=\frac{{\omega}_{0}}{c}n\left({\omega}_{0}\right)$ is the wave number at the pulse’s center angular frequency

*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}$$+i{f}^{2}{\partial}_{x}^{2}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{x}+i{f}^{2}{\partial}_{\tilde{x}}{\partial}_{\tilde{y}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{y}+i\frac{{f}^{3}}{2}{\partial}_{\tilde{x}}{\partial}_{\tilde{z}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}-f{\partial}_{\tilde{x}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}$$

$$+i{f}^{2}{\partial}_{\tilde{x}}{\partial}_{\tilde{y}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{x}+i{f}^{2}{\partial}_{y}^{2}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{y}+i\frac{{f}^{3}}{2}{\partial}_{\tilde{y}}{\partial}_{\tilde{z}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}-f{\partial}_{\tilde{y}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}$$

$$-{f}^{2}{\partial}_{\tilde{z}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}+i\frac{{f}^{3}}{2}\left({\partial}_{\tilde{x}}{\partial}_{\tilde{z}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{x}+{\partial}_{\tilde{y}}{\partial}_{\tilde{z}}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{y}\right)+i\frac{{f}^{4}}{4}{\partial}_{\tilde{z}}^{4}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}$$

The small parameter *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.

Fibich et al. showed that, if the electric field is mainly linearly polarized along the *x*-axis, the dimensionless amplitudes satisfy the following relations [12]:

These relations are in good agreement with the initial fields provided by diffraction theory. Even at tight focusing, the z-polarized power is considerably smaller than the *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 _{x}* and by dropping all terms of order higher

*O*(

*f*

^{2}).

$$-\underset{\mathrm{Vector}\phantom{\rule{.2em}{0ex}}\mathrm{Coupling}}{\underbrace{{f{\partial}_{\tilde{x}}\left({\tilde{P}}_{\mathrm{NI}}\right)}_{z}}}+\underset{\mathrm{Nonlinear}\phantom{\rule{.2em}{0ex}}\mathrm{Diffraction}}{\underbrace{i{f}^{2}{\partial}_{x}^{2}{\left({\tilde{P}}_{\mathrm{NI}}\right)}_{x}}}+O\left({f}^{3}\right)$$

Additionally to the nonlinearity which is of order *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 ${P}_{\mathrm{crit}}\approx 1.86\frac{{\lambda}^{2}}{4\pi {n}_{0}{n}_{2}}$ [12] known to distinguish strong and weak nonlinear propagation regimes. If *P _{crit}* 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 (

*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.

Due to it’s linearity the nonparaxial term in Eq. (11) becomes most important in the case of high numerical aperture and low peak power. It is well known that nonparaxial effects severely alter beam propagation at high numerical aperture [21] and thus most attention must be paid to a proper description of the nonparaxiality. From the numerical point of view, the nonparaxial term in Eq. (11) is complicated to deal with. The second order derivative in propagation direction transforms the initial value problem into a boundary value problem, generally much more complicated to solve. A common method of dealing with the nonparaxiality is to approximate the ∂2ẑ -operator by transverse operators that can be calculated more easily [11, 12]. However, a different approach is used here. To find the nonparaxial version of Eq. (11), only terms of order

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

The expression for the *∂*
_{z̃}-operator can be found by cancelling out the amplitude *A _{x}* and solving for

*∂*

_{z̃}. The

*∂*

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

*A*.

_{x}Equation (13) cannot be numerically integrated straight away, because of the mix of linear and nonlinear terms combined under the square root. While the linear terms can easily be calculated in frequency space, the nonlinear part must be evaluated in normal space. Dropping the nonlinear terms, the 3-dimensional Fourier transform of the square root is the exact linear nonparaxial wave propagator in frequency space (Eq. 14), known as K-operator or spherical wave operator [13, 14, 22, 23]. The additional nonlinear terms under the square root can be understood as nonlinear modification to the linear propagator.

Here *k _{x},k_{y}* are the transversal wavenumbers in frequency space with the transformations $\frac{k}{{k}_{0}}\approx \frac{1}{{k}_{0}}\frac{{n}_{0}}{c}\left(\left(\omega -{\omega}_{0}\right)+{\omega}_{0}\right)=\left(1+\frac{\omega -{\omega}_{0}}{{\omega}_{0}}\right)$. Note that the common nonlinear propagation Eqs. for ultrashort pulses like SVEA [24, 22] and SEWA [25] can be obtained by applying different approximations to Eq. (13). The SEWA can be derived from Eq. (13) by expanding the square root to first order. The term $\frac{k}{{k}_{0}}$ is evaluated in frequency space by $\frac{k}{{k}_{0}}\approx \frac{1}{{k}_{0}}\frac{{n}_{0}}{c}\left(\left(\omega -{\omega}_{0}\right)+{\omega}_{0}\right)=\left(1+\frac{\omega -{\omega}_{0}}{{\omega}_{0}}\right)$, where

*n(ω)*≈

*n*

_{0}was used. The expression transforms back to normal space by using $\left(\omega -{\omega}_{0}\right)\stackrel{{\mathrm{FFT}}^{-1}}{\to}i{\partial}_{t}$:

When the wave number is Taylor expanded in frequency space (*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].

To integrate Eq. (13) numerically, the square root has to be split into a linear and a nonlinear part. It is hence expanded to formerly infinite high order and succeedingly recombined to form both a linear and a nonlinear square root. Nonlinear-transversal mixed-terms are kept to order *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 _{x}* of the vector field A⃗. It is linearly exact and includes nonlinear effects and nonlinear nonparaxial mixed-terms to order

*O*(

*f*

^{2}).

$$-i\frac{{f}^{2}}{4}\frac{{k}_{0}}{k}\left[{\tilde{\nabla}}_{T}^{2}\left({\mid {A}_{x}\mid}^{2}+{f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}^{2}}\right)+\left({\mid {A}_{x}\mid}^{2}+{f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}^{2}}\right){\tilde{\nabla}}_{T}^{2}\right]{A}_{x}$$

$$+i\frac{{k}^{2}}{{k}_{0}^{2}}\left(\frac{2}{3}{\mid {A}_{z}\mid}^{2}{A}_{x}+\frac{1}{3}{A}_{z}^{2}{A}_{x}^{*}\right)+i{f}^{2}{\partial}_{\tilde{x}}^{2}\left({\mid {A}_{x}\mid}^{2}+{f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}^{2}}\right){A}_{x}$$

$$-f{\partial}_{\tilde{x}}\left(\frac{2}{3}{\mid {A}_{x}\mid}^{2}{A}_{z}+\frac{1}{3}{A}_{x}^{2}{A}_{z}^{*}+{f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}^{2}}{A}_{z}\right)+O\left({f}^{3}\right)$$

Within this work Eq. (16) is applied at pulse peak power much lower than the critical power for self focusing (*P/P _{crit}*≪1). Thus from the Kerr-effect only terms induced by the

*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.

The nonlinear propagation Eq. including all previous approximations reads:

$$+i2{L}_{\mathrm{Df}}{k}_{0}\left(-1+\sqrt{1+{f}^{2}{\mid {A}_{x}\mid}^{2}+\frac{{\chi}_{\mathrm{fe}}}{{n}_{0}^{2}}}\right){A}_{x}$$

$$-i\frac{{f}^{2}}{4}\left[\left({\tilde{\nabla}}_{T}^{2}-4{\partial}_{\tilde{x}}^{2}\right)\left({\mid {A}_{x}\mid}^{2}+{f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}_{0}^{2}}\right)+\left({\mid {A}_{x}\mid}^{2}+{f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}_{0}^{2}}\right){\tilde{\nabla}}_{T}^{2}\right]{A}_{x}$$

$$-i{f}^{2}{\partial}_{\tilde{x}}\left({f}^{-2}\frac{{\chi}_{\mathrm{fe}}}{{n}_{0}^{2}}{\partial}_{\tilde{x}}{A}_{x}\right)+O\left({f}^{3}\right)$$

Equation (17) is expressed in a coordinate systems moving with the pulse center at the group velocity *v _{g}*. Time

*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.

When the intensity in the focus grows sufficiently high, free electrons are generated by ultrashort pulses via two processes. Nonlinear photo ionization, divided into multiphoton and tunnel ionization, transfers electrons from the valence band to the bottom of the conduction band (Fig. 1). Although electrons in the conduction band are formally speaking only quasi-free, they will be denoted as free throughout the rest of this letter. Once an electron is transferred to the conduction band, it can directly gain energy from the laser field by sequential one-photon absorption, climbing a virtual energy ladder, which is equally spaced in photon energy *ħω* (Fig. 1). The process is often also termed inverse bremsstrahlung [16, 28, 29, 30]. When a free electron has gained energy *ε _{K}* greater than the critical energy

*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}The generation of free electrons is numerically performed using the MRE model recently introduced by Rethfeld [18, 19]. It describes the process of plasma formation more properly than the classical Drude model [28, 29, 30] often used in similar studies [3, 9, 10, 16, 31, 32]. The Rethfeld model combines the complexity of kinetic approaches to nonlinear ionization [33] with the mathematical simplicity of rate Eqs. It is the first numerically simple model that takes into account a varying kinetic energy distribution in the conduction band.

Due to the lack of knowledge of the impact ionization probability *W _{imp}* for an electron in the conduction band with kinetic energy exceeding

*ε*, the system of rate Eqs. was simplified in comparison to the original MRE model [18, 19]. The impact ionization probability

_{crit}*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, 33]. No reliable value of

_{imp}*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}$$\frac{\partial {\rho}_{1}\left(t\right)}{\partial t}={W}_{1\mathrm{pt}}\left(I\left(t\right)\right){\rho}_{0}\left(t\right)-{W}_{1\mathrm{pt}}\left(I\left(t\right)\right){\rho}_{1}\left(t\right)$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\vdots $$

$$\frac{\partial {\rho}_{K-1}\left(t\right)}{\partial t}={W}_{1\mathrm{pt}}\left(I\left(t\right)\right){\rho}_{K-2}\left(t\right)-{W}_{1\mathrm{pt}}\left(I\left(t\right)\right){\rho}_{K-1}\left(t\right)$$

Here *ρ _{n}(t)* is the density of electrons with kinetic energy

*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:

Where *η* is estimated from the Drude model [28, 29, 30]. In the asymptotic case of long pulse duration *τ*
_{0}≫[2^{1/K}-1)*W*
_{1pt}]^{-1}, the Drude model is included within the MRE model [18, 19]. The coefficient *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:

The value of σ is determined by the collision time *τ _{p}*, 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

*τ*to be on the order of 1 fs in almost any material [34]. Good agreement with experimental data was found for simulations using

_{p}*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]. The critical band gap potential for impact ionization is given by [18, 33]:

The first factor in Eq. (21) is due to conservation of electron momentum [35]. The electron mass in the conduction band *m _{CB}* and in the valence band

*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, 33] of field strength

*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}Electron recombination and diffusion out of the focal volume were neglected in the system of rate Eqs. (18), since their contributions are small on a femtosecond time scale [29]. The rate for nonlinear photo ionization *W _{PI}* 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, 31, 33]. In order to model the laser pulse propagation and the interaction with the generated free electrons, Eq. (17) and the system of rate Eqs. (18) are integrated simultaneously. A forth-order Runge-Kutta algorithm is used for Eqs. (18).

Since the propagation Eq. (17) is nonparaxial and vectorial in nature, the initial focal field distribution must be provided in a nonparaxial and vectorial manner. Focal fields are calculated using a variation of the Fresnel-Kirchhoff diffraction integral [23]. The FFT-based diffraction integral is nonparaxial in nature and was additionally vectorially extended to account for microscope objectives of high numerical aperture. The laser pulses are assumed to be linearly *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]. Figure (3) shows the intensity distribution of the initial field at the focus for a numerical aperture NA=1.2 in water. The sine condition, which is usually the obeyed condition for the design of commercial objectives, was used as apodization function [21]. For the sake of simplicity, only perfect focusing objectives are taken into account. It is of course easily possible to consider arbitrary aperture functions, aberrations, and phase errors.

## 3. Results and discussion

The theoretical model introduced above was used to numerically investigate the generation of optical breakdown plasmas at various numerical apertures (0.5≤NA≤1.2) in water. All simulations were performed assuming transform limited ultrashort pulses of Gaussian temporal shape and pulse duration 150 fs (FWHM) at center wavelength λ=780 nm. It is found that the pulse energy required to generate a plasma of density *ρ _{Bd}*=10

^{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], although fixated cells were used in that particular study. A pulse energy of 2.2 nJ was found to be sufficient for intracellular ablation at pulse duration of 200–250 fs and center frequency of λ=790 nm using a NA=1.4 oil immersion objective.

Previous simulations of plasma generation at low numerical aperture or weak focusing, respectively, revealed strong nonlinear interaction of the ultrashort pulse with the generated plasma in the focus, resulting in an enhanced breakdown threshold. The interaction was found to be mostly due to plasma-defocusing [3]. The generation of plasma occurs along the pulse’s temporal profile, whereas multiphoton ionization dominates the leading half of the pulse and cascade ionization rapidly increases the plasma density during the trailing half. Because it is facing a dense plasma, the trailing half suffers much worse from plasma-defocusing. The temporally asymmetric interaction results also in a strong spatial asymmetry of the generated plasma along the propagation axis. This effect was experimentally observed as streak formation in studies of refractive surgery using ultrashort pulses at low NA [5, 3], seriously complicating surgery procedures. On the other hand, at high NA the size of achieved manipulations in applications of cell surgery is reported to be below the diffraction limit [6, 16]. It can thus be supposed that the intensity of plasma-defocusing gets weaker at tight focusing.

To confirm this assumption, nonlinear plasma formation was numerically studied in both the low and high numerical aperture regime. Figure (4) shows contour plots of calculated spatial plasma distributions for various NAs at pulse energies slightly above the threshold for optical breakdown (*ρ*>*ρ _{Bd}*). 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

*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).

The effect of plasma-defocusing competes with the spatial focusing of the incident pulse. It is more pronounced at lower NA and smaller focusing angle, respectively. The tighter the ultrashort pulses are focused, the higher the density of generated free electrons required to defocus the pulse is. On the other hand, a certain density of free electrons (*ρ=ρ _{Bd}*) is sufficient to induce optical breakdown and to modify the material in the desired manner. Hence at tight focusing conditions

*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}The pulse energy threshold obtained from the simulations for all NAs considered is shown in Fig. (5). Usually optical breakdown is associated with a certain fluence obtained at the focus. For NA=1.2 a fluence *F _{th}*=1.28 Jcm

^{-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

A theoretical model capable of simulating the nonlinear interaction of ultrashort laser pulses with transparent materials was introduced. The model is based on linear and nonlinear propagation effects as well as nonlinear plasma formation and the simultaneous spatial and temporal interaction of the propagating pulse with the generated plasma. It was particularly adapted to meet the special experimental conditions found in modern applications of cell surgery. The extraordinarily precise manipulability of cells or cell organelles feasible in these applications is based on subdiffraction generation of optical breakdown plasmas by tight focusing of ultra-short pulses. To numerically investigate the processes at these experimental conditions, plasma formation was studied at various numerical apertures ranging from low to high. For low NA the size of the focus itself and the size of the generated plasma as well as the distortive effect of plasma-defocusing prevent the usage for cell surgery. For applications of highest precision, well confined submicron high density plasmas were found producible for NA≳0.9. However, for applications with less restrictive precision requirements, optical breakdown can of course also be generated at lower NA, but side-effects may occur.

Breakdown energy thresholds were calculated for all NAs considered, ranging from *E _{th}*≈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

*τ*and the rate for nonlinear photoionization

_{P}*W*. However, the calculated energy thresholds agree well with experimental data [6]. Additionally, the dependence of the intensity of parasitic side-effects such as plasma-defocusing is not expected to be strongly influenced by varying parameters for nonlinear ionization.

_{PI}The purpose of this study was to investigate nonlinear plasma formation at high NA focusing conditions as found in cell surgery. The initial fields used as starting points for the simulations were provided taking nonparaxial and vectorial effects carefully into account. Hence the initial conditions chosen were realistic, but also idealized in the way that ideal homogeneous, flat phase laser beams and ideal focusing objectives were considered. However, this study led to a primary understanding of the dependence of optical breakdown on the numerical aperture and enables to predict the start of parasitic side-effects complicating applications.

The use of diffraction integrals to provide the initial focal fields is the key element in the model to match experimental conditions even closer. It is well known that multiphoton imaging and dissection get ever more challenging the deeper the focal plane is moved into the sample. This is due to spherical aberrations, when the focal plane deviates from the designated focal depth of the microscope objective used [21], and to the poor optical properties of biological tissue resulting in scattering and phase scrambling. Spherical aberrations can easily be taken into account as an apodization function, when calculating the initial fields. Hence the thresholds for optical breakdown can also be estimated for common experimental conditions. An apodization function can be any kind of phase or amplitude filter.

Due to the availability of suited modulators, the concept of adaptive optics has recently also entered the field of cell surgery. Deformable mirrors or spatial light modulators are used, *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.

Finally, although the model was adapted for low power and high NA focusing here, nonlinear effects occurring at high pulse power such as self-focusing, single as well as multi filamentation and supercontinuum generation can also be numerically investigated taking into account slightly different approximations in the derivation of the model.

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