Measurement of nonlinear refractive index and ionization rates in air using a wavefront sensor

Jens Schwarz; Patrick Rambo; Mark Kimmel; Briggs Atherton

doi:10.1364/OE.20.008791

1. Introduction

The optical Kerr effect gives rise to many of the phenomena in nonlinear optics. Its associated nonlinear index of refraction n₂ leads to self-focusing [1] in high power laser beams through index variation n(I) = n₀ + n₂I over the spatially varying intensity profile I. This in turn leads to a plethora of effects such as filamentation [2, 3], self-phase modulation [4], spectral broadening [5], and self-compression [6, 7]. Measurements of n₂ in air have been made but these values vary significantly, in part due to differences in wavelength and pulse duration (see table 1). The variations due to wavelength are the result of dispersion, with n₂ decreasing at increasing wavelength [8]. Variations due to pulsewidth stem from the combination of pure Kerr effect (dominant at short pulsewidths ≤ 100 fs) and molecular alignment from rotational Raman effects (noticeable at a few 100 fs [9]), causing n₂ to increase with pulsewidth. As seen from table 1, wide scatter in measured data can mask these effects. A direct measurement of the index of refraction is therefore highly desirable for a given wavelength, pulse duration, and pressure. We employ the self-focusing of a high power ultrashort pulse laser beam in order to measure the nonlinear index of refraction. Previous work by [9,10] in a focusing beam geometry used the focal shift at P > P_crit to calculate the critical power for self focusing and hence the nonlinear index of the medium. For Liu et al [9], as the laser beam power increases, the focal shift with respect to the geometrical focus is determined by measuring the spatial shift of the plasma fluorescence in air. The power at which a shift is first observed is then identified with the critical power. A fundamental drawback of this method is the fact that the plasma will have an effect on the focusing behavior which is not taken into account (as pointed out by [11]). In fact, as we show, plasma effects occur at much lower powers than the n₂ based P_crit. Hence the onset of self-focusing can be difficult to isolate in observations. In this paper, we have appropriately modeled plasma de-focusing in conjunction with self-focusing and as a result we can extract both laser ionization rates and n₂ from our measurements. While n₂ measurements are readily available at 800 nm for sub-picosecond pulses (see Table 1), little n₂ data exists in the literature for our wavelength (1054 nm) and pulsewidth (540 fs) of interest. Even though the present paper discusses measurements at 1054 nm only, its technique should be applicable to a broad range of wavelengths and pulsewidths.

Table 1. Comparison of various values of n₂ found in the literature. All values are in units of 10⁻²³m²/W and correspond to atmospheric pressure. Note, reference [17] uses a 20 fs beam chirped to 200 fs.

View Table

2. Experimental setup

We propose a new method for measuring n₂ and ionization rates which allows us to detect beam shifts at much lower power (P ≪ P_crit). In our setup (see Fig. 1), a wavefront sensor is used to directly sense a shift in focal length as opposed to indirectly measuring shifts via a secondary effect such as plasma fluorescence. This is critically important because plasma induced focal shifts occur well before plasma fluorescence can be detected.

Fig. 1 Setup to measure the self-focus shift using a wavefront sensor (WFS). The blue lines represent the low peak power case while red lines represent powers sufficient for self-focusing.

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A short pulse laser at low power P ≪ P_crit is focused with a focal element M1 (focal length f1) into a gas ionization cell and re-collimated with a second focal element M2 (focal length f2). A wavefront sensor (WFS) is placed at a distance d behind M2 and a flat wavefront is recorded as a reference (see Fig. 2(a)) at minimum powers. Finally, to avoid any nonlinear effects in the WFS filters, the front surface reflections of two uncoated wedges were used to attenuate the beam energy prior to any filters. For the same reason, great care was taken to not use any transmissive optics in the short pulse beam train. Energy was adjusted through a wave-plate/polarizer combination in the temporally stretched part of the laser system. Furthermore, only spherical mirrors were used in the focusing and re-collimation of the laser beam (see M1 and M2 in Fig. 1) in order to avoid nonlinear phase accrual in the lens material which may compromise data. The use of spherical mirrors in down-collimation requires a small off-axis angle which could induce astigmatism. As a result of our large f/#, the angles were kept to less than 1°, yielding a negligible measured astigmatism with the wavefront sensor.

Fig. 2 (a) Flat reference/initial wavefront recorded for the f/120 geometry at a beam energy of 0.3 mJ. The associated “defocus” term was measured at $R_{0}^{2} = - 0.005$ waves. (b) Wave-front curvature recorded at 5.4 mJ energy with $R_{0}^{2} = 0.05$ waves “defocus” for the same experimental configuration as (a).

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As the laser beam power increases, the beam focus will shift by a distance Δ toward M1. Correspondingly, the light emanating from the new self-focused position f1 – Δ or f2 + Δ will not be collimated any more but rather be imaged a certain distance D behind M2. This distance D is just the radius of curvature R of the wavefront (as measured by the wavefront sensor; see Fig. 2(b)) plus the distance d of the sensor to the lens M2: D = R+d. The focal shift Δ can then be calculated from the thin lens equation:

\begin{matrix} \frac{1}{f 2} = \frac{1}{f 2 + Δ} + \frac{1}{R + d} & \Leftrightarrow & Δ = \frac{1}{\frac{1}{f 2} - \frac{1}{R + d}} - f_{2} . \end{matrix}

The wavefront radius R at position d can be calculated from the measured defocus Zernike term $R_{2}^{0}$ (see Fig. 2(b)). In our case $R_{2}^{0} = 2 a^{2} - 1$ , where a is the pupil radius. For a measured pupil of radius a on the wavefront sensor and a defocus of $R_{2}^{0}$ in waves one calculates:

R = \frac{a^{2}}{4 R_{0}^{2} λ},

where λ is the wavelength of the laser.

The validity of this approach was cross-checked by ZEMAX [12] modeling. A test measurement with the following parameters was taken: f1 = 1 m, f2 = 0.3 m, d = 0.3 m, and a = 1.75 mm. A defocus term of $R_{2}^{0} = 0.07$ waves was recorded which corresponds to a focus shift of Δ = 9 mm. This result has been corroborated using a ZEMAX model where M1 was modeled to have a focal length of f1 – Δ = 99.1 cm. All other parameters mirrored the experimental setup. The model shows indeed a defocus of 0.068 waves when the modified M1 focal length was used. One can conclude that the above method should be valid for determining the focal shift as a function of laser beam power.

3. Experimental results

A laser beam at λ = 1054 nm with a pulsewidth of τ = 540 fs FWHM and a spatial first order Gaussian with a 1/e² waist radius of w_i = 6.2 mm was focused by spherical concave mirrors of f/120 and f/40. A larger f/# (greater than f/120) was used as well but led to the onset of filamentation at low laser energies. Each data point represents an average of 200 measurements at 10 Hz on the wavefront sensor (Phasics SID-4, see http://www.phasicscorp.com/) as well as the energy meter. The peak to valley “defocus” Zernike term $R_{2}^{0}$ was then fitted to the averaged wavefront and converted to focal shift as discussed earlier. The error bars in Fig. 3 represent the standard deviation of the energy spread over 200 shots and the achievable focal shift accuracy is based on the sensitivity of the wavefront sensor. Note that the wavefront sensor amplitude maps (beam near fields) were concurrently measured with the phase maps. Care was taken to assure that the amplitude maps remained Gaussian and that the near field diameter remained self-consistent throughout a scan. Measurements were stopped once changes to amplitude profile or beam size were detected.

Fig. 3 Plot of focal shift Δ versus laser input energy. The scatter plots show experimental data for f/120 and f/40 focusing geometries. The solid lines represent theoretical fits based on the pulse propagation equation [Eq. (9)] (see below) and the dashed lines depict the theoretical behavior without ionization present.

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Figure 3 shows the measured focal shift versus laser input energy for the two f/# mentioned earlier. One can see that, depending on f/#, the onset (elbow in the graph) of the focal shift is different for any given focus geometry. Two main processes contribute to focal shift:

Nonlinear self-focusing: As the laser beam power P becomes comparable to the critical power P_crit = 3.77λ/(8πn₀n₂), the nonlinear index of refraction n₂ leads to additional focusing in beams with Gaussian (and also other) intensity profiles. For a collimated input beam entering a nonlinear medium and an unchanged amplitude profile during self-focusing, one can write the self-focusing distance z_sf due to the nonlinear index n₂ as [13]: $z_{s f} = \frac{π w_{i}^{2}}{λ} {(\frac{P}{P_{crit}} - 1)}^{- 1 / 2},$ where w_i is the initial beam waist at the entrance of the nonlinear medium. In a focusing geometry, the new effective focal length f_new can then be found as a combination of the geometrical focal length f_g through the lens transformation formula: f_new = f_gz_sf /(f_g + z_sf). One can see that this will lead to a shift of the minimum beam waist (focus) toward the input focusing lens.
Based on Eq. (3), a focal shift can only occur at P ≥ P_cr which would be independent of focus geometry (see dashed lines in Fig. 3). Liu et al [9] uses this argument when he identifies this elbow as the onset of P = P_cr in order to calculate n₂. According to measured data in Fig. 3, the elbow is clearly dependent on beam focusing geometry and hence one cannot consider the effects of nonlinear self-focusing alone.
Ionization: It is clear that ionization must occur as the laser beam propagates towards the minimum beam waist. This ionization will be highest around the focal region and will profoundly affect the self-focusing behavior [11]. As the beam collapses towards the focus the electron density increases as a result of ionization due to increased beam intensity. This leads to plasma de-focusing before the geometrical focus is reached. Hence ionization will lead to a focal shift toward the input focal lens as will nonlinear self-focusing due to the nonlinear index n₂ (see Fig. 4).

It can be seen that one has to consider both processes in order to adequately explain the results presented in Fig. 3. One should note that for f/40, where ionization is dominant, the focal shift begins at lower power but shows smaller shifts at larger powers as compared to the regime where nonlinear self-focusing is dominant (f/120).

Fig. 4 Plot of the modeled laser beam waist versus propagation distance for the f/120 focusing geometry at various laser input energies. It can be seen that the waist moves towards the input optic as the laser beam energy increases. The geometrical focus is marked by the black dashed line. This is consistent with the experimental observation in Fig. 3. One can also see that the beam waist initially decreases as a function of self-focusing and later increases as plasma de-focusing sets in. The model parameters were as follows:w_i = 6.2 mm, τ = 540 fs, λ = 1054 nm, N_O₂ = 4.6 × 10²⁴ m⁻³ (accounting for 20% of O₂ in the atmosphere and 85% of atmospheric pressure at the altitude of Albuquerque, NM), σ⁽^K⁼¹¹⁾ = 3 × 10⁻¹⁹¹m²²W⁻¹¹s⁻¹, and n₂ = 2.6 × 10⁻²³ m²/W.

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4. Experimental analysis

In order to extract values for both n₂ and the ionization rates, one has to model the beam waist as a function of laser power P and laser induced ionization. We used the semianalytical model employed in [4,11,14] and it was shown that it adequately reproduces the beam waist evolution until the minimum waist is reached [11] which is the regime of interest. The laser beam waist w(z,t) as a function of time and propagation distance z can be written as:

\frac{\partial^{2} w (z, t)}{\partial z^{2}} = \frac{4}{k^{2} w {(z, t)}^{3}} (1 - \frac{P (t)}{P_{crit}}) + \frac{2 K}{{(K + 1)}^{2} n_{0} N_{crit} w (z, t)} \times N_{e} (z, t),

where the first term describes diffraction and self-focusing and the last term describes the defocusing due to ionization. In this equation k is the wavenumber, P(t) is the laser power, P_crit = 3.77λ²/8πn₀n₂ is the critical power for self-focusing for a Gaussian beam, and n₂ is the nonlinear index of the gas under study (air). K is the ionization order of the multi-photon ionization (MPI) process. For an ionization potential of W_O₂ = 12.06 eV for oxygen and a photon energy of 1.18 eV for a photon at 1054 nm one calculates: K = ⌈12.06eV/1.18eV⌉ = 11. n₀ is the index of air and

N_{crit} = ɛ_{0} m_{e} ω_{l}^{2} / e^{2}

is the critical plasma density for the laser. Here, ɛ₀ is the dielectric constant in vacuum, m_e is the mass of the electron, ω_l is the angular frequency of the laser, and e is the charge of the electron. Including saturation of ionization, the initial electron density N_e(z,t) is defined as [15]:

\frac{d N_{e} (z, t)}{d t} = (N_{O_{2}} - N_{e} (z, t)) σ^{(K)} I {(z, t)}^{K},

where N_O₂ is the initial density of air (oxygen), I(z, t) is the laser intensity, and σ⁽^K⁾ is the Kth order ionization coefficient. For N_e(z,t) ≪ N_O₂ Eq. (5) simplifies to:

\frac{d N_{e} (z, t)}{d t} = N_{O_{2}} σ^{(K)} I {(z, t)}^{K} .

An initial rough estimate of the 11th order ionization coefficient σ⁽^K⁼¹¹⁾ can be calculated based on the weak field approximation of the Keldysh theory [16]:

σ^{(K)} = ω_{l} {(\frac{W_{O_{2}}}{\bar{h} ω_{l}})}^{1.5} \times {(\frac{e^{2}}{4 c ɛ_{0} m_{e} ω_{l}^{2} W_{O_{2}}})}^{K} \times e^{K} = 3.2 \times 10^{- 181} m^{22} W^{- 11} s^{- 1} .

For a square temporal pulse shape of full width τ, Eq. (5) can be integrated to yield:

N_{e} (z) = N_{O_{2}} [1 - Exp (- σ^{(K)} I_{0}^{K} {(\frac{w_{i}}{w (z)})}^{2 K} τ)],

where we have used the relation

I (z, t) = I_{0}^{K} {(\frac{w_{i}}{w (z)})}^{2 K}

. I₀ is the initial laser intensity at z = 0, τ is the laser pulsewidth, and w_i is the initial beam waist at z = 0.

Staying with a square temporal pulsewidth assumption, one can combine Eqs. (4) and (8) to yield a propagation equation that is solely dependent on z:

\frac{\partial^{2} w (z)}{\partial z^{2}} = \frac{4}{k^{2} w {(z)}^{3}} (1 - \frac{P_{0}}{P_{crit}}) + \frac{2 K N_{O_{2}} (1 - e^{- σ^{(K)}} I_{0}^{K} {(\frac{w_{i}}{w (z)})}^{2 K} τ)}{{(K + 1)}^{2} n_{0} N_{crit} w (z)},

where P₀ is the initial power for a square temporal pulse of full width τ. Equation 9 can be solved numerically with initial conditions: w_i = w(z = 0) and dw(0)/dz = −w_i/f. Note that Eq. (9) now accounts for saturation effects by using the integrated solution of Eq. (5) as inserted into Eq. (4). The assumption of a square pulse has allowed the generation of Eq. (9) in a relatively simple analytical form. A Gaussian pulse profile would be more realistic but yields a complex form for the integrated solution to Eq. (4) which causes numerical integration issues in Eq. (9).

Figure 4 depicts an example for a calculated laser beam waist versus propagation distance for a f/120 focusing geometry at various laser input energies. It can be seen that for a low energy of 2 mJ the geometrical focus coincides with the minimum beam waist. For larger energies/powers, the focus moves toward the input optic as a result of self-focusing and ionization and the minimum waist grows as a result of laser generated plasma de-focusing. Based on Eq. (9), one can now calculate the position of the minimum beam waist (and hence the focal shift Δ) as a function of laser energy/power and compare the model to the measured data in Fig. 3.

Since there exist two data sets and two unknowns (n₂ and σ⁽^K⁾), one should be able to find a unique solution that fits the measurement. This is accomplished rather easily since the physics in the f/40 focusing geometry is largely dominated by ionization which sets the limit for σ⁽^K⁾. Conversely, self-focusing plays a dominant role in the f/120 focusing geometry which leads to a determination of n₂. The higher influence of ionization in the f/40 case is evidenced by the fact that the onset of a focal shift (elbow) occurs at much lower powers.

The importance of ionization for the correct estimation of n₂ and σ⁽^K⁾ is illustrated by the dashed lines in Fig. 3. They are based on theoretical curves that do not include ionization. As expected, those curves are shifted to the right because the focal shift contribution from the laser plasma is omitted. If only one f/# measurement is considered (as is the case in [9]), one is inclined to identify the elbow in the curve as P_crit, leading to an erroneously low P_crit (or conversely high n₂).

The solid lines in Fig. 3 show the calculated focal shift versus energy for n₂ = (2.6 ± 0.2) × 10⁻²³ m²/W and σ⁽¹¹⁾ = (3 ± 1.5) × 10⁻¹⁹¹m²²W⁻¹¹s⁻¹. While the value for n₂ compares well with other data from the literature (see Table 1), one should note that, at our pulsewidth of 540 fs, the measured n₂ is a combination of pure optical Kerr effect and Raman contributions. This is also evidenced by the fact that the 1 ns result for 1053 nm in table 1 is larger than our number due to its greater Raman contribution. On the other hand, one should be aware that the measured ionization coefficient for the 11th order process is 10 orders of magnitude lower than expected from Eq. (7). The reason for this discrepancy will become apparent at the end of this section.

In order to gain a better understanding of the validity of the measured ionization coefficient, one has to investigate the electron density contribution in some more detail. Figure 5 shows the calculated peak laser focal intensity that corresponds to each input energy in the experiment. As expected, one can see that intensity clamping occurs for each f/# almost at the same level. Furthermore, the intensity rises more strongly for the f/40 case than for f/120. This is ultimately the reason why the f/40 case is ionization dominated, because the early rise in intensity causes ionization to be more prominent at lower energy. Furthermore, one can see that the intensity clamping of I_cl = 3.5 × 10¹⁸ W/m² occurs at the elbow in both measured data sets.

Fig. 5 Plot of the laser intensity versus input energy (right axis). Experimental data are shown for comparison for the same energy range (left).

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At this clamping level, one has to consider the contribution of tunneling ionization. To see this one must have a closer look at the Keldysh (or adiabatic) parameter γ defined in Appendix A. For γ ≫ 1 MPI is the dominant ionization process; for γ ≪ 1 tunneling ionization is more important. At the above mentioned I_cl = 3.5 × 10¹⁸ W/m², γ corresponds to γ < 0.5. In order to obtain a clear picture on how MPI and tunneling ionization contribute for our intensity regime of interest, we have modeled the ionization rates for oxygen and nitrogen based on the PPT theory [4, 21–23] (see Appendix A). Figure 6 shows the ionization rate R versus laser intensity for oxygen and nitrogen. Note, that dN_e/dt = (N_O₂ – N_e) × R.

Fig. 6 Ionization rate versus intensity for oxygen and nitrogen. The dashed, vertical green line depicts the transition region between MPI and tunneling ionization, where the Keldysh parameter γ = 1 for λ = 1054 nm.

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One strong feature of the PPT theory is the fact that it is able to show the transition of ionization rates from an MPI dominated regime into the tunneling regime. To show that the model is self-consistent, we have plotted the ionization rate for oxygen at 800 nm and 1054 nm. One can see that the slope for the 800 nm case is lower than the 1054 nm case due to the fact that ionization at 800 nm requires K = 8 photons as opposed to K = 11 in the other case. Consequently, one observes that the ionization rate is higher for 800 nm (higher photon energy and lower ionization order) versus 1054 nm. In addition, both curves merge as they approach the tunneling ionization regime. This due to the fact that tunneling ionization is independent of laser wavelength (photon energy) but rather is driven by the amplitude of the electric field. Comparing ionization rates of oxygen and nitrogen at 1054 nm, one can see that the slope for nitrogen is steeper than for oxygen. This is again due to the fact that nitrogen requires K = 14 photons to ionize at 1054 nm rather than K = 11 for oxygen. In addition, we have plotted the ionization rate (dashed line) that corresponds to the Keldysh approximation in Eq. (7). As expected, for a weak field approximation, the Keldysh rate is the same as the PPT rates for weak electric fields (meaning low intensities). The excellent agreement between the Keldysh number and the PPT model at low intensities shows that both models are self-consistent. Looking at the ionization rate for oxygen in Fig. 6 one can observe that an 11th order slope (see dashed line) is only valid for very low laser intensities I ≪ 10¹⁷ W/m². However, looking at the ionization dominated f/40 focusing geometry in Fig. 5, one can see that intensities above 10¹⁸ W/m² are reached at very low laser input energies. This means that tunneling ionization is the dominant effect (see Fig. 6) and that MPI alone does not fully apply. However, for the commonly used intensity regime (≤ 5×10¹⁷ W/m²) in the semi-analytical model of Eq. (9), MPI is the electron density source term. Ideally, one should use the intensity dependent PPT model calculated ionization rates to obtain the electron density source term in Eq. (9). This however complicates matters to the point where the semi-analytical model would become unusable. As a result, we have kept the simple MPI source term in Eq. (9) which means that the measured result produces an “effective MPI coefficient” corresponding to the intensity clamped regime of 3.5× 10¹⁸ W/m² (see Fig. 5) only. We call this coefficient “effective” since MPI does not fully apply to this regime. However, this “effective coefficient” produces the correct electron density in Eq. (9) at the clamping intensity which allows one to correctly fit the data and get a reliable n₂ value. Because this value was measured in the Tunneling Ionization dominated intensity clamped regime, the “effective MPI coefficient” (dotted black curve in Fig. 6) is 10 orders of magnitude lower than a value that would be predicted by simple MPI Keldysh theory (dotted red curve in Fig. 6). This perceived discrepancy actually makes sense because Keldysh theory does not account for the effect of Tunneling Ionization at higher intensities (see dashed red curve versus solid red curve in Fig. 6). The PPT model supports this argument by exactly predicting our measured ionization rate for the clamping intensity of 3.5 × 10¹⁸ W/m² (see intersection of solid red curve with dotted black line and dashed black line in Fig. 6). Please note:

Our measured “effective MPI coefficient” should not be confused with a σ⁽^K⁼¹¹⁾ coefficient from Eq. (7). This latter coefficient is truly valid in the MPI regime whereas our measured number (the “effective MPI coefficient”) only applies to the intensity clamped regime at our specific intensity clamped value.
On a much broader note, we would like to point out that for λ = 1054 nm, only measurements taken at I ≤ 10¹⁶ W/m² can yield a true MPI coefficient. When larger intensities are used, the ionization order (slope) changes to smaller values and MPI becomes less valid. At this point, the slope of the ionization rate is not constant but varies with intensity. This implies that, in principle, each of our focusing geometries have different ionization rates. However, since the intensity clamping is almost the same for both f/# (see Fig. 5), the resulting ionization coefficient is almost the same as well. This is the reason that we can fit the same ionization coefficient to both curves in Fig. 3.
The PPT model has been validated for oxygen in ambient air at atmospheric pressure at λ = 1054 nm by showing agreement with measured data at high intensities and by demonstrating excellent agreement with the Keldysh approximation for low intensities. This model can now be used over a large range of laser intensities, e.g. for simulation purposes.

5. Conclusion

We have presented a novel method for measuring the nonlinear index of gases and their corresponding ionization rates (in this case air/O₂) under atmospheric pressure at λ = 1054 nm. It has been pointed out that measured focal shifts result from a combination of self-focusing and plasma-defocusing which has been appropriately treated in the semi-analytical model. Based on this model, we have calculated effective MPI coefficients for O₂ in the intensity regime of 10¹⁸ W/m². The measured coefficient is in excellent agreement with the PPT model presented in Appendix A. This result has broader implications, because it allows one to safely use the presented PPT curves for modeling efforts on laser propagation in air and other gases. In principle, the above technique is applicable to a broader range of wavelengths and pressures. However, one should make some initial estimates of the expected focal shift in order to verify that the wavefront sensor is sensitive enough to register the expected wavefront deformations.

Appendix A

The PPT model used here is based on the work by Perelomov et al. [21] and is presented in a similar form as in [4]. For a linearly polarized beam with electric field E, the ionization rate R(E) (measured in Hz) can be expressed as follows:

\begin{array}{l} R (E) & = & \sqrt{\frac{6}{π}} \frac{e E_{H}}{\sqrt{2 m_{e} W_{H}}} {| c_{n^{*}, l^{*}} |}^{2} f (l, m) \frac{W_{i}}{2 W_{H}} A_{m} (γ) \\ {(\frac{2 E_{0}}{E \sqrt{1 + γ^{2}}})}^{2 n^{*} - | m | - 3 / 2} exp (- \frac{2 E_{0}}{3 E} g (γ)), \end{array}

where

γ = ω_{l} \sqrt{2 m_{e} W_{i}} / (e E)

is the adiabatic parameter. For γ ≫ 1 MPI is the dominant ionization process, for γ ≪ 1 tunneling ionization is more important. Here, ω_l is the angular frequency of the ionizing field, e, m_e are the charge and the mass of the electron, and W_i is the ionization potential for the atom or molecule that is to be ionized (e.g. 15.58 eV for N₂). The remaining constants in Eq. (10) are defined as: W_H = 13.6 eV is the ionization potential of hydrogen, and E₀ = E_H (W_i/W_H)^3/2 where

E_{H} = e^{5} m_{e}^{2} / ({\bar{h}}^{4} {(4 π ɛ_{0})}^{3}) = 514

GV/m is the electric field in the hydrogen atom. ɛ₀ is the dielectric constant in vacuum and h = 2πh̄ is Planck’s constant. Taking the quantum defect into account leads to an effective principle quantum number

n^{*} = Z \sqrt{W_{H} / W_{i}}

with Z being the residual charge state of the ion (e.g. Z = 1 for a single ionized atom like Ar). Correspondingly, l^* = n^* – 1 is the effective orbital quantum number. l is the orbital quantum number and m is the magnetic quantum number (used below in Eq. (12)). The dimensionless constant |c_n^*,l^*|² is:

{| c_{n^{*}, l^{*}} |}^{2} = \frac{2^{2 n^{*}}}{n^{*} Γ (n^{*} + l^{*} + 1) Γ (n^{*} - l^{*})},

where Γ is the gamma function. f(l,m) is defined as:

\begin{matrix} f (l, m) = \frac{(2 l + 1) (l + | m |)!}{2^{| m |} (| m |)! (l - | m |)!}, & with & f (0, 0) = 1. \end{matrix}

The remaining functions are:

A_{m} (γ) = \frac{4}{\sqrt{3} π} \frac{1}{| m |!} \frac{γ^{2}}{1 + γ^{2}} \sum_{κ > ν}^{+ \infty} exp (- (κ - ν) α (γ)) Φ_{m} (\sqrt{(κ - ν) β (γ)}),

where ν = (W_i/(h̄ω_l))(1 + 1/(2γ²)) and κ is then the next highest integer number after ν. The function Φ_m above is defined as:

Φ_{m} (x) = \frac{x^{2 | m | + 1}}{2} \int_{0}^{1} \frac{e^{- x^{2} t} t^{| m |}}{\sqrt{1 - t}} d t = e^{- x^{2}} \int_{0}^{x} {(x^{2} - y^{2})}^{| m |} e^{y^{2}} d y .

Note that, when using Eq. (13), Φ_m(x) would be replaced by

Φ_{m} (\sqrt{(κ - ν) β (γ)})

. The other functions of γ are:

α (γ) = 2 (({sinh}^{- 1} γ) - \frac{γ}{\sqrt{1 + γ^{2}}}),

β (γ) = \frac{2 γ}{\sqrt{1 + γ^{2}}},

g (γ) = \frac{3}{2 γ} [(1 + \frac{1}{2 γ^{2}}) ({sinh}^{- 1} γ) - \frac{\sqrt{1 + γ^{2}}}{2 γ}] .

R(E) can easily be expressed as a function of intensity I by replacing the electric field E with

E = \sqrt{2 I / (c n_{0} ɛ_{0})}

.

The ionization rates in Fig. 6 were produced using the following parameter inputs:

For molecular O₂ and N₂, l = m = 0 [24] and Z is measured experimentally [23] to account for the difference in charge state for a molecule versus an atom.
- – N₂: W_i = 15.58 eV, Z = 0.90.
- – O₂: W_i = 12.55 eV, Z = 0.53. The ionization potential of W_i = 12.55 eV (as opposed to the more common 12.07 eV) was chosen according to [23] in order to account for the various vibrational levels in the oxygen molecule.

Acknowledgments

We would like to thank Prof. Miroslav Kolesik for fruitful discussions. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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Gas	1053 nm	800 nm	400 nm

N₂		1.1 ± 0.2 at 90 fs [18]
		2.3 ± 0.3 (Kerr), 2.3 ± 0.4 (Raman) at 120 fs [19]
		6.7 ± 2 at 200 fs [17]

O₂		1.6 ± 0.35 at 90 fs [18]
O₂		5.1 ± 0.7 (Kerr), 8.5 ± 0.2 (Raman) at 120 fs [19]

Ar		1.00 ± 0.09 at 90 fs [18]
		1.4 ± 0.2 (Kerr) at 120 fs [19]
		19.4 ± 1.9 at 200 fs [17]

air	4.1 at 1 ns [20]	3.01 at 39 fs [10]	5.36 at 39 fs [10]
		0.96 at 42 fs [9]
		1.2 ± 0.3 at 90 fs [18]
		2.9 ± 0.4 (Kerr), 3.6 ± 0.8 (Raman) at 120 fs [19]
		5.7 ± 2.5 at 200 fs [17]
		1.92 at 200 fs [9]

Measurement of nonlinear refractive index and ionization rates in air using a wavefront sensor

Abstract

1. Introduction

2. Experimental setup

3. Experimental results

4. Experimental analysis

5. Conclusion

Appendix A

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (17)

Optics Express