1 Introduction

Non-resonant optical-intensity-dependent refractive index is very important in ultrafast laser science and technology. For example, this intensity-dependent refractive index plays a key role in ultrafast pulse propagation dynamics in transparent media, such as pulse compression down to few optical cycles [1], filamentation [2], optical soliton interactions [3], and so on. This effect originates from the third order of nonlinearity and is described as a modification of n 2 I to the linear refractive index [4], where I is the optical intensity and n 2 is nonlinear refractive index coefficient. This modification in refractive index could lead to effects like temporal chirping and spectrum broadening of optical pulse, as well as self-focusing in spatial domain. Many applications of modern lasers benefit from these effects such as Kerr-lens mode-locked Ti:sapphire oscillators [5] and atmospheric LiDAR [6]. On the other hand, there are also many situations suppressing them, like transportation of high-field laser pulses [7] to targets for light–matter interaction research under extreme conditions. For both ends, an easy way to measure the parameter n 2 under different working wavelengths and conditions will be beneficial.

Experimentally determination of n 2 for various materials has been a research interest since the 1970s. Up to now, there are many ways to measure n 2 for different materials under different conditions. Depending on the number of beams involved in the measurement, these methods can be divided into single-beam method and multi-beam method. Multi-beam method, such as degenerate four-wave mixing [8], polarization spectroscopy [9], spectral interferometry [10], spectrally, spatially resolved interferometer [11], and so on, usually needs complex experimental apparatus and requires a strict spatial overlap of different beams, compared with single-beam method. Although the multi-beam method can give very accurate results, the single-beam method can also provide adequate accuracy in most cases, such as the powerful and widely used Z-scan method [12]. However, Z-scan method requires a tight focusing geometry, which limits the light–matter interaction region to <1 cm. This is the reason that Z-scan method is only used to measure liquids and solids of large nonlinearity. Another single-beam method is based on spectral analysis of light pulses after weakly nonlinear propagation in samples [13–15]. It is suited to the fast determination of n 2 for a large variety of transparent materials, ranging from liquids, solids to gases. This method employs an unfocused beam and let it propagate freely through samples. Through measuring and analyzing the spectra before and after the samples, n 2 can be extracted. The pulse energy in this method is usually tens of millijoules, and for gas medium the sample length is several meters. Although the small-scale self-focusing during pulse propagation, which leads to error in final results, is controlled by monitoring the beam profile with CCD, estimating the B-integral and using a 1-mm-diameter diaphragm to select a small part on the beam transverse profile, it cannot be eliminated totally due to the inherit wavefront curvature from energetic CPA systems.

In this paper, we extend the principle of spectral analysis [13] by exploring pulse propagation in hollow-core fibers (HCFs) to measure n 2 of inert gases. It is based on the Kerr effect-induced spectral broadening in HCFs, taking advantage of the structure of waveguide which can accumulate small nonlinear effects over long distance. As a demonstration, the n 2 of argon at 800 nm and 1.8 μm and neon at 800 nm are determined. The rest of the paper is organized as follows. In Sect. 2, the principle and implementation of the method is described with emphasis on the differences between current work and earlier ones. Section 3 will give the measuring results and analysis of n 2 of argon and neon at different wavelengths. The paper will end with a conclusion.

2 Method

In the experiment, the near Fourier transform-limited (FTL) input pulse is focused and coupled into a standard HCF compressor [1]. The gas tube where the HCF is placed is filled with inert gas whose n 2 is to be measured. The inner diameter of HCF is several hundred micrometers and the input pulse energy is around 1 mJ level. At the output of the tube, the spectra are recorded with a spectrometer (like Ocean Optics, USB4000).

The pulse propagation in the fiber is described by nonlinear Schrödinger equation [16]. There are several terms in the equation that affect the propagation dynamics, for example, dispersion, nonlinear Kerr effect, self-steepening, and ionization effect. The relative importance of each effect can be simply judged by their corresponding characteristic lengths. According to [16], if the characteristic length of nonlinear Kerr effect is much shorter than those of other (non)linear effects, we can keep only Kerr effect term in the equation while discarding other terms for pulse propagation analysis. As a result, the pulse spectrum will be broadened only due to the self-phase modulation (SPM) which originates from Kerr effect [13]. The spectral broadening factor for a FTL Gaussian pulse is given by [16]

$$ \frac{{\Updelta \omega_{\text{out}} }}{{\Updelta \omega_{\text{in}} }} = \left( {1 + \frac{4}{3\sqrt 3 }\phi_{ \hbox{max} }^{2} } \right)^{1/2} , $$
(1)

where ∆ω out and ∆ω in are spectral widths of the output and input pulses, \( \phi_{ \hbox{max} } \) is the maximum phase shift of the pulse after propagation and is given by \( \phi_{\hbox{max} } = \gamma P_{0} L_{\text{eff}} \). P 0 is the input pulse peak power and \( L_{\text{eff}} = {{\left[ {1 - \exp \left( { - \alpha L} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {1 - \exp \left( { - \alpha L} \right)} \right]} \alpha }} \right. \kern-0pt} \alpha } \) is the effective length of the fiber incorporating the effect of power loss along the fiber, L is the fiber length and α is power attenuation. The nonlinear parameter γ is defined as \( \gamma = n_{2} \omega_{0} /cA_{\text{eff}} \), where ω 0 is the central angular frequency of the input pulse, c is light speed in vacuum and A eff is the effective area of the fiber mode. For the HCF with radius a, \( A_{\text{eff}} = 0.48\pi a^{2} \) for fundamental mode HE11.

As a result, n 2 can be calculated according to Eq. (1) after the spectral broadening factor is obtained. The spectral width is calculated as the RMS spectral width for complex spectral shapes, which is defined as

$$ \Updelta \omega_{\text{rms}}^{2} = \langle (\omega - \omega_{0} )^{2} \rangle - \langle (\omega - \omega_{0} )\rangle^{2} , $$
(2)

where the angle brackets indicate a weighted average over the spectrum as

$$ \left\langle {(\omega - \omega_{0} )} \right\rangle^{n} = \frac{{\int_{ - \infty }^{\infty } {(\omega - \omega_{0} )^{n} I(\omega ){\text{d}}\omega } }}{{\int_{ - \infty }^{\infty } {I(\omega ){\text{d}}\omega } }} , $$
(3)

for which I(ω) is the measured spectrum.

The criterion on whether the above analysis is valid is that the characteristic length of Kerr effect is less than one-tenth of the characteristic lengths of other effects. The definitions of characteristic lengths of different effects are summarized in Table 1, where T 0 is pulse width at 1/e 2 intensity and β 2 is the GVD coefficient of the gas. The integration in plasma effect characteristic length L I is on normalized transverse coordinate r in the fiber core, J 0(u 1 r) is zero order of Bessel function with the first zero point u 1, ρ is density of electron and \( \eta = {{q_{e}^{2} } \mathord{\left/ {\vphantom {{q_{e}^{2} } {2m_{e} \varepsilon_{0} c\omega_{0} }}} \right. \kern-0pt} {2m_{e} \varepsilon_{0} c\omega_{0} }} \) is a constant. q e and m e denote the electron charge and mass, \( \varepsilon_{0} \) is dielectric permittivity in vacuum. The electron density is calculated according to PPT model described in [17]. It should be noted that the effect of plasma is overestimated by taking the density value at the tail of the pulse. By doing so, we can save the effort of considering the accumulation effect of plasma and meanwhile reserve enough margin for judging the plasma influence.

Table 1 Definitions of characteristic lengths of different processes taking place during the propagation
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In this method, initial chirp in the input pulse should be as small as possible. This is usually guaranteed by the high-quality commercial laser systems. By adjusting the grating spacing in the lasers or time delays in OPA module to maximize the output spectral broadening, a near-FTL input pulse is obtained. If there is nontrivial chirp in the input pulse, the above analysis is not accurate and an iterative phase retrieval algorithm should be used to extract the nonlinear refractive index [18]. In this paper, a near-FTL input pulse is used.

Due to the focusing geometry of this method, there are small regions before and after the ends of the fiber where light intensity is as high as in the fiber. Therefore, spectral broadening in these areas should be taken account into the data analysis. The total length of these areas is about 2–4 cm beyond which the light intensity drops quickly and propagation can be considered linear. For a 1-m-long fiber, this amounts to 2–4 % overestimate in the final results. Since the windows of the gas tube are far from the fiber end and the thickness is only 1 mm in total, the nonlinear phase contribution from them is negligible.

Compared with earlier works [13–15], there are several advantages of using HCF. First, small-scale self-focusing is absent due to the waveguide structure. This ensures a uniform reliable nonlinear effect across the beam transverse profile and saves the effort of beam quality monitor and control. Besides, the propagation distance is largely reduced from several meters to around 1 m. Second, the pulse energy is small, at least one order of magnitude less than that of propagation in free space. These advantages make such a method easy to implement in most labs.

Below, we will use this method to obtain the n 2 of argon at 800 nm and 1.8 μm and neon at 800 nm.

3 Results and discussions

We first measure the n 2 of argon at 800 nm. The input pulse from a commercial laser system (Spectra Physics) is 0.46 mJ and 50 fs (FWHM). It is coupled into a 1-m-long HCF of 250 μm inner diameter with a lens of 1 m focal length. The gas tube is evacuated (<1 mbar) and then filled with argon. The spectra under different gas pressures are recorded (Fig. 1a) to obtain the broadening factors. The gas pressure reaches up to about 1 bar. For higher pressures, the output mode will deteriorate due to mode-coupling effect. The values of n 2 are calculated according to Eq. (1) and shown in Fig. 1b. In the gas pressure range in our experiment, n 2 changes almost linearly with pressure, which is also derived from an independent measurement [11]. The red line in Fig. 1b is a linear fit of the data and the slope of the fit indicates that the n 2 of argon is about (1.11 ± 0.05) × 10−19 cm2/W bar. Considering the 2–4 % extra contribution near the input and output end of the fiber, the n 2 of argon is (1.08 ± 0.06) × 10−19 cm2/W bar which agrees well the value of 1.0 × 10−19 cm2/W bar given in [19]. Under the above conditions, the characteristic length of Kerr effect decreases from 0.29 to 0.03 m and are orders of magnitude shorter than those of dispersion, self-steepening and ionization effects as listed in Table 2. So the Kerr effect dominates the propagation dynamics and spectral broadening, and the other effects can be neglected, although the plasma effect is becoming larger with the increase of gas pressure because the characteristic lengths are already shorter than the fiber length.

Fig. 1
figure 1

a Spectra of the output of the argon-filled hollow-core fiber under different gas pressures. The inner diameter of the fiber is 250 μm and the input pulse is 0.46 mJ/50 fs and centered at 800 nm. From bottom to top, the gas pressure increases from <1 to 960 mbar. b Nonlinear refractive index coefficient n 2 (black squares) of argon obtained from spectral broadening factors of different gas pressures. The red solid line is the linear fit of n 2

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Table 2 Characteristic lengths of different processes at 800 nm in argon
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For neon, the ionization potential is so high that the Kerr effect can dominate over a larger pressure range without the influence of plasma effect. The experimental parameters are the same as those for argon, except that the pulse energy is 0.66 mJ. The gas pressure is increased to near 3 atm with negligible plasma effect. The spectra are shown in Fig. 2a. Figure 2b (black squares) shows that the n 2 values maintain an almost linear relationship with gas pressure up to 2,600 mbar, at which the characteristic length of ionization is 44 m compared with 0.1 m of Kerr effect. This time, self-steepening is second dominant effect in propagation dynamics; however, the characteristic length is 7 m, much longer than the fiber itself. Higher gas pressures were not explored because of the risk of fracturing the thin fused silica windows. Linear fitting followed by a correction results in an average n 2 of (9.30 ± 0.25) × 10−21 cm2/W bar compared with 7.5 × 10−21 cm2/W bar in [20]. The characteristic lengths of different processes are listed in Table 3 for comparison.

Fig. 2
figure 2

a Spectra of the output of the neon-filled hollow-core fiber under different gas pressures. The inner diameter of the fiber is 250 μm and the input pulse is 0.66 mJ/50 fs and centered at 800 nm. From bottom to top, the gas pressure increases from <1 to 2,600 mbar. b Nonlinear refractive index coefficient n 2 (black squares) of neon obtained from spectral broadening factors of different gas pressures. The red solid line is the linear fit of n 2

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Table 3 Characteristic lengths of different processes at 800 nm in neon
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Quite opposite to that of neon, the ionization effect can be very strong for gases of low ionization potential like krypton and xenon. If the same input parameters as those for argon are applied to measure the n 2 of krypton and xenon with the above method, detailed analysis shows that the characteristic lengths of ionization are of the same order of magnitude or even shorter than those of the Kerr effect. Thus, the calculated n 2 does not reflect the Kerr effect any more. In order to get accurate n 2, the input intensity has to be lowered until the Kerr effect dominates the propagation dynamics again.

We further use this method to measure the n 2 of argon at 1.8 μm. The input pulse from a home-built OPA is 57 fs (FWHM) and 1.27 mJ, which can be tuned from 1.6 to 2.0 μm. The inner diameter of the 1-m-long HCF is 400 μm for higher transmission efficiency and more efficient use of pulse energy. The gas pressures recorded here are below 500 mbar and the corresponding spectra are shown in Fig. 3a. Linear fitting followed by a correction of all values of n 2 gives the average value of (6.65 ± 0.33) × 10−20 cm2/W bar while the calculated value is 9.73 × 10−20 cm2/W bar at 1.83 μm in [21]. The characteristic lengths in Table 4 show that in the above gas range the nonlinear Kerr effect still dominates in the propagation dynamics.

Fig. 3
figure 3

a Spectra of the output of the argon-filled hollow-core fiber under different gas pressures. The inner diameter of the fiber is 400 μm and the input pulse is 1.27 mJ/57 fs and centered at 1.8 μm. From bottom to top, the gas pressure increases from <1 to 414 mbar. b Nonlinear refractive index coefficient of argon obtained from spectral broadening factors of different gas pressures

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Table 4 Characteristic lengths of different processes at 1.8 μm in argon
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We have demonstrated the determination of the values n 2 of argon and neon at 800 nm and argon at 1.8 μm with HCF. The data analysis process is simple and straightforward. It should be pointed out that with the wide wavelength range covered by OPA [22], the dispersion curves of n 2 can be measured by tuning the central wavelength of OPA.

Last but not least, we discuss the error analysis of this method. As mentioned in Sect. 2, the input pulse is assumed to be Fourier transform-limited with a Gaussian profile. Input pulses with initial chirp or different profiles (such as super-Gaussian, pulse-splitting, etc.) will decrease the spectrum broadening and result in a smaller value of n 2. Data analysis of situations with complex input should use an iterative phase retrieval algorithm to extract the nonlinear refractive index [18]. The effective fiber length for spectrum broadening is also an error source due to the focusing geometry which actually increases the effective length and should be corrected. In the above cases, the error is about 3 % and has been corrected. Besides, the single-mode propagation in fiber is also needed, which is guaranteed by optimal coupling to mode EH11 [1].

The last error source originates from the residual effects of various (non)linear terms other than Kerr effect. For hollow fibers, the linear effects such as waveguide attenuation and dispersion can be ignored as the fiber length is usually too short for these effects to affect the propagation dynamics. Self-steepening effect causes extra spectrum broadening by blue-shifting and leads to overestimation of n 2. However, the plasma-induced phase modulation will counteract the Kerr effect and lead to decrease of n 2. It should be mentioned that the self-steepening effect mainly affects the rear part of the pulse while the plasma effect affects the front part. When the characteristic length becomes comparable or less than fiber length, the effect begins to affect the dynamics. In the above case for argon at 800 nm, both effects have an influence on the spectrum, but the final n 2 is quite close to the referred value [19]. This suggests a balance between the two effects. For neon at 800, self-steepening effect is more than plasma effect, which leads to overestimation of n 2 compared to [20]. For argon at 1.8 μm, the situation is reversed with an underestimation of n 2 compared to [21]. These effects cause a deviation of about ±30 % from the referred values in the above cases. In general, the accuracy of this method can be estimated by analyzing the different characteristic lengths and controlled by choosing proper input pulse parameters to adjust these characteristic lengths.

4 Conclusions

We demonstrated a simple method to measure the nonlinear refractive index coefficient n 2 of inert gases, which explores the pulse propagation in HCFs. This method based on spectral analysis takes advantage of the structure of waveguide which can accumulate small nonlinear effects efficiently to produce considerable spectral changes of gas medium at relatively low pulse energy and over short propagation distance. Through measurement of the spectra broadening, the n 2 can be obtained from the spectral broadening factor. This method together with broadband-tunable OPA femtosecond laser source can be used to obtain n 2 of noble gases at various wavelengths, which would benefit the research of high-field laser physics and nonlinear optical phenomenon.