1. Introduction

The optical Kerr effect due to a third-order nonlinear term in the dielectric polarization of the fiber material is responsible for distortions and performance degradation in fiber transmission systems. On the other hand a rich variety of nonlinear wave propagation phenomena with potential applications is easily observable with the strongly localized light fields in glass fiber waveguides after long interaction lengths [1]. Nonlinear modifications of pulse propagation have been investigated and demonstrated experimentally since the first fibers became available with solitons [2], self-phase modulation (SPM) [3] and modulation instability (MI) [4]. All of these Kerr-induced effects have been so well investigated and understood during the last 40 years that the three mentioned reference examples could easily be extended by hundreds of others. A good documentation is found in standard textbooks on nonlinear fiber optics [1,5]. The field still develops as shown by recent experiments on more advanced solutions of the nonlinear Schrödinger equation (NLSE) [6,7]. For example, nonlinear pulse propagation in fibers serves as basis for experimental investigations of optical breathers with the promise for an extension of our knowledge on extreme events in wave propagation and rogue waves [8].

For a theoretical description and optimization of experiments a best possible value for the nonlinear third-order susceptibility $\chi ^{(3)}$ as responsible material parameter is required. A measurement of the cubic nonlinearity is mostly based on the mentioned typical Kerr-induced effects, with nowadays very detailed and excellent descriptions in textbooks [1,9]. However, the situation is not so simple. Different physical processes with different dynamics and time constants contribute to $\chi ^{(3)}$ so that it is strongly frequency dependent, especially close to phonon resonances, and it changes with pulse length [10–13]. Measurements of bulk silica and silica fiber preforms with pulses from fs to ns and methods from nonlinear spectroscopy, interferometry and SPM to Z-scans investigated the $\chi ^{(3)}$ dependence on pulse length and wavelength [14–20]. Interestingly, both Z-scan experiments [18,19] show another dispersion than the other experiments and theory. The cubic nonlinearity measured in fibers using cw lasers can include a relatively slow electrostriction contribution [13,21–23]. Experiments in fibers with ps-long pulses [24–28] provide the electronic plus the nuclear-vibration contributions and from some measurements the ratio between both can be estimated [14,28]. Material brand, quality and doping affect the measured value as well [20–23]. Finally, the polarization state of the electric field along the propagation influences the effective acting $\chi ^{(3)}$ [29]. Therefore, reported nonlinearity parameters of telecom fibers vary in a wide range (see [1], chapt. 11) depending on the fiber type and the measuring technique [12,30].

In fact, at the beginning the goal was not the measurement of cubic nonlinearities in glass fibers. I used a frequency optical gating (FROG) technique [17,31] to characterize SPM and measure $\chi ^{(3)}$ in waveguides on crystals with a separation of electronic and second-order cascading contributions. For a scaling check of the setup glass fibers were measured. In contrast to classic SPM experiments the complex pulse envelope was measured directly with a FROG technique. The fibers with lengths of 0.5 to 5 m are very short. The polarization state of the electrical field is under control and does not change statistically. The short nonlinear interaction length requires pulse peak powers of up to a few thousand Watt. For pulse lengths between 1 and 5 ps slower physical contributions like electrostriction do not contribute to $\chi ^{(3)}$ so that the result is a relatively frequency-independent susceptibility valid in the important ps regime with only constant electronic and nuclear-vibration or Raman contributions. Using the FROG an observation of the nonlinear phase shift following directly the pulse power is possible yielding directly the nonlinearity factor $\gamma$ of the fiber. For the used ps-long pulses dispersion has only a small and controllable effect in m-long fibers. The more indirect SPM characterization through nonlinear spectral broadening was performed to confirm the results. The basic material of modern standard single mode fibers is synthetic silica glass of highest purity [32]. Germanium doping in the fiber core leads to the index increase for light guiding but also increases $\chi ^{(3)}$. The mode fields were measured with the far-field technique [9] yielding the refractive-index and doping profile. The dependence of $\chi ^{(3)}$ on dopant concentration is taken from literature [21,22,24]. With the electric mode field and the radius-dependent nonlinearity the overlap integrals for the conversion of $\gamma$ to the bulk silica glass susceptibility $\chi ^{(3)}$ are obtained [22].

Because the direct measurement with minimized polarization disturbances is very accurate with a reliable error analysis, because my SPM experiment works under exactly opposite conditions (short fiber, high power) than the previous SPM experiments (long fiber, low power), and because the result agrees well with most of the reported comparable measurements of the cubic nonlinearity in bulk silica [17] and in silica fibers [12,13,20,22,24–29] it could be valuable to share the results in this manuscript.

2. Pulse propagation and nonlinear phase shift in SPM

The electric field of a pulse propagating along a waveguide in $z$-direction can be expressed as

The asterisk indicates the complex conjugate and ’m.f.’ stands for ’minus frequency’. Because the envelope function varies slowly with time, $a(\omega -\omega _0,z)$ is nonzero only in a narrow frequency band near $\omega _0$. The frequency dependence of the mode fields in the narrow band can be neglected. The coupled mode theory provides the evolution equation for the development of the positive-frequency spectrum $a(\omega -\omega _0,z)$ in propagation direction $z$ when the linear waveguide propagation is disturbed by a nonlinear polarization $\vec {P}^{\rm NL}$ [33,34]:

The positive-frequency spectrum is expressed by a convolution integral [34,35] with the frequency condition $\omega =\omega _1+\omega _2+\omega _3$:

Here a linear polarization of the electric field in $x$-direction is assumed and the tensor product is evaluated without further marking the vector components. The third-order nonlinear susceptibility $\chi ^{(3)}$ is the $xxxx$ tensor component and is in general a function of the frequencies $\omega _i$. However, the electronic part of the two contributions to $\chi ^{(3)}$ is instantaneous for ps-long pulses. In time domain the susceptibility becomes a $\delta$-distribution, in frequency domain it can be considered a frequency-independent value $\chi ^{(3)}(\omega _0,\omega _0,-\omega _0)$ at the center frequency. That can also be assumed for the Raman contribution as the pulse bandwidth is much smaller than the Raman resonance frequency of 13 THz [36]. The now frequency-independent $\chi ^{(3)}$ can be extracted from the integral together with the mode fields. A separation of the fiber specific transverse-coordinate dependence and the spectral pulse evolution in the right side of Eq. (4) is the convenient result:

For the narrow-band excitation with ps-long pulses the self-steepening term $\omega$ in front of the right side can be replaced with a constant $\omega _0$. The overlap integral with the susceptibility and the constants are defined as nonlinearity factor of the fiber in the widely used form:

With an expansion of the propagation constant in a Taylor series up to the quadratic term $\beta (\omega )=\beta _0 + \beta ' (\omega -\omega _0) + \frac {1}{2!} \beta '' (\omega -\omega _0)^2$ with frequency derivatives $\beta '$ and $\beta ''$, after a transformation into a system moving with the linear group velocity of $v_{g}=\frac {1}{\beta '}$, by using $\xi =z$ and $\tau =t-\frac {z}{v_{g}}$ and extracting the fast oscillations along $\xi$ with the ansatz

The simulations were based on the integro-differential Eqs. (7) and (10) in frequency domain.

The inverse Fourier transform of Eq. (10) yields the well known NLSE for the complex pulse envelope $A(\tau,\xi )=|A(\tau,\xi )| \exp (j\varphi (\tau,\xi ))$ in the moving coordinates $(\tau,\xi )$:

The pulse power is $p(\tau,\xi )=|A(\tau,\xi )|^2 p_0$. Note that in contrast to the textbooks [1,5] where the pulse power is $p=|A|^2$ the envelope here is dimensionless with the normalized mode power $p_0=1 {\rm W}$.

The typical case of SPM is observed when the dispersion term $\beta ''$ can be neglected. The resulting equation

The envelope $|A(\tau,\xi )|=|A_0(\tau )|$ and the pulse power do not change on propagation. Only the phase acquires a power-dependent nonlinear shift $\varphi _{\rm NL}(\tau,\xi )= -(\varphi (\tau,\xi )-\varphi (\tau,0))= \gamma p_0 |A_0(\tau )|^2 \xi$ proportional to the momentary pulse power. When the complex pulse envelope Eq. (13) can be measured, a plot of the phase shift versus momentary pulse power gives a straight line with the slope proportional to the nonlinearity factor $\gamma$ of the fiber.

3. Extraction of the nonlinear susceptibility and the nonlinear refractive index from the nonlinearity factor

Two standard single mode fiber types were investigated: SMF28e-fibers fabricated by Corning in the last 15 years and Thorlabs 1550BHP-fibers from Coherent. These fibers have a Germaniumoxid (GeO$_2$) doped core with an influence of the radius-dependent doping on $\chi ^{(3)}$. Therefore $\chi ^{(3)}$ changes with radius and cannot simply be extracted from the overlap integral Eq. (8). The correct evaluation of the overlap is beside a good pulse characterization a critical part of serious nonlinearity measurements of waveguides [28]. The fiber parameters are proprietary of the manufacturer, so I measured the mode field $e(x,y)$ with the far- and near-field methods [37,38]. Figure 1(a) compares the mode intensity for both fiber types from the far-field measurements. The near-field measurement (or just imaging) gives slightly wider modes due to a convolution with the transfer function of the imaging objective. Because of less systematic errors in the far-field measurement these results were used. The refractive index profile was evaluated from the fields and the mode equation and is shown in Fig. 1(b).

 

Fig. 1. (a) Mode intensity of the two measured fibers; (b) refractive index increase $\delta n$ in the core region of both fibers. The oscillations are probably an artefact due to a limitation of the measurable space frequencies in the far-field scan.

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The increase of $\chi ^{(3)}$ with the GeO$_2$ concentration is approximately linear [21,22,24]. The refractive index increase $\delta n$ in the core is also proportional to the GeO$_2$ concentration [39]. The combination gives the linear dependence $\chi ^{(3)}=\chi ^{(3)}_0 (1+0.2 \Delta )$ between the nonlinear susceptibility and the relative index increase in percent $\Delta =\frac {\delta n}{n_0}100 \, \%$ [23]. $\chi ^{(3)}_0$ is the nonlinear susceptibility of undoped silica and $n_0$ is the refractive index of the fiber material. The slope of 0.2 varies in literature by approximately $\pm 20\,\%$. The fiber nonlinearity factor Eq. (8) becomes a product of the susceptibility $\chi ^{(3)}_0$ and a mode overlap integral $OI$:

With the data from Fig. 1 the overlap integrals for the fibers are calculated $OI_{\rm SMF28}=3.58 \cdot 10^{15} \frac {\rm V^4}{\rm m^2} \pm 2.3\, \%$ and $OI_{\rm 1550BHP}=3.91 \cdot 10^{15} \frac {\rm V^4}{\rm m^2} \pm 2.3\,\%$. The error is estimated from the 1 % variation of different measurements plus 1.3 % due to the uncertainty of the dopant influence. Fortunately the 20 % uncertainty in the $\chi ^{(3)}$-index-increase relation yields only a small uncertainty of 1.3 % of the overlap integral because of the small core index increase $\Delta$ in the order of $0.35\,\%$. The effective area $A_{\rm eff}=\frac {(\int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \! \mathrm {d}x \mathrm {d}y |e|^2)^2} {\int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \! \mathrm {d}x \mathrm {d}y |e|^4}$ of the fibers are $A_{\rm eff\ \ SMF28}=80.87 \, \mu {\rm m}^2$ (2.5 % smaller than information from Corning) and $A_{\rm eff\ \ 1550BHP}=75.10 \, \mu {\rm m}^2$.

The measured nonlinear phase shift in Eq. (13) gives $\gamma$ which is transformed into the cubic susceptibility with Eq. (14).

An alternative parameter for characterizing the cubic nonlinearity is the nonlinear refractive index $n_2$ from the relation of the intensity-dependent refractive index $n(I)=n_0+n_2 \ I$. $n_0$ is the low-power refractive index of the material and $I$ the intensity. The relation between the cubic susceptibility and the nonlinear refractive index is [1]

4. Experimental conditions and pulse measurements

The measurements were done during the last eight years with different Pritel laser systems and second-harmonic-based FROG instruments (SHG FROG) from Southern Photonics (SP) and Coherent Solutions (CS). The setup is shown in Fig. 2. A fiber laser generated nearly transform limited pulses which were amplified in a fiber amplifier. The laser operated at wavelengths close to 1550 nm. Pulse lengths between 6.2 and 0.8 ps were available. The pulses after the amplifier were slightly up-chirped. A half-waveplate-polarizer combination was used for power control. The pulses were coupled with a microscope objective into the test fiber to be measured. With a second microscope lens the fiber output can be directed into the FROG, an optical spectrum analyzer (OSA) or an Ophir power meter. With the quarter- and half-waveplate combination the polarization is adapted to the polarization-sensitive FROG input. Fiber losses are negligible, reflections and transmissions of optical elements are separately measured and $P_{\rm out}$ is transformed into the fiber input power. With a measurement of the reference power $P_{\rm ref}$ the throughput and incoupling efficiency are monitored. The power meter was calibrated at the Physikalisch-Technische Bundesanstalt Braunschweig to an expanded measurement uncertainty $2\sigma$ of 0.6 %. The connecting fibers to the FROG and the OSA are 25-cm long and the power in the two instruments was kept low to keep nonlinear effects in the coupling fibers negligible.

 

Fig. 2. Setup for the characterization of the complex pulse envelope $A(\tau )$; Amp – fiber amplifier, HP – half-waveplate, P – polarizer, BS – beam sampler, MO – microscope objective, QP – quarter-waveplate.

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With the FROG instrument a spectrogram of the pulse under investigation is recorded. A correct scaling of the FROG was verified by comparing the “marginals" of the two-dimensional FROG spectrogram $I_{\rm FROG}(\omega,\tau )$ with independent measurements of the pulse spectrum with an OSA and the intensity autocorrelation in noncollinear SHG [40]. The frequency marginal $M_\omega (\omega )=\int _{-\infty }^{\infty } \! \mathrm {d} \tau \, I_{\rm FROG}(\omega,\tau )$ is obtained by integration of the FROG spectrogram over the delay axis $\tau$. It has to be proportional to the autoconvolution of the pulse spectrum $M_\omega (\omega ) \propto \frac {1}{2 \pi }\int _{-\infty }^{\infty } \! \mathrm {d}\widetilde {\omega } \ I(\widetilde {\omega }) I(\omega -\widetilde {\omega })$ [41]. The delay marginal $M_\tau (\tau )=\int _{-\infty }^{\infty } \! \mathrm {d} \omega \, I_{\rm FROG}(\omega,\tau )$ is the integral of the FROG trace along the frequency axis. For an SHG FROG trace the delay marginal should be proportional to the standard intensity autocorrelation of the pulse $M_\tau (\tau ) \propto \int _{-\infty }^{\infty } \! \mathrm {d}t \ I(t) I(t-\tau )$. A good agreement of the marginals from the FROG spectrograms $I_{\rm FROG}(\omega,\tau )$ with the autoconvolution of the OSA-measured spectra and the autocorrelator measurements confirmed the correct FROG instrument scaling. Typical results in Fig. 3 show in average a good scaling with maximum variations $< 1 \, \%$.

 

Fig. 3. Normalized marginals from SHG FROG spectrograms: (a) frequency marginal compared to the autoconvolution of the pulse spectrum, SMF28 output, input pulse length 1.7 ps, input peak power 815 W, fiber length 2 m; (b) delay marginal compared to the intensity autocorrelation of the pulse, 6.2-ps-long laser pulse.

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From the spectrogram the pulse envelope is retrieved with the FROG software Vers. 3.2.4 from Femtosoft Technologies [42]. Figure 4 shows typical measured pulses $A(\tau )=|A(\tau )| \exp (j \varphi (\tau ))$ with power $|A(\tau )|^2 p_0$ and phase $\varphi$ at the fiber output at $\xi =L$. In Fig. 4(a) the pulse shape of a 6-ps-long pulse propagating in a 5-m-long fiber is not measurably affected by dispersion. According to Eq. (13) only the phase changes dependent on power. In Fig. 4(b) a 1.7-ps-long pulse propagating in a 2-m-long fiber narrows increasingly at higher peak powers because dispersion becomes increasingly influential for a growing nonlinear chirp.

 

Fig. 4. Two sets of measured complex pulse envelopes $A(\tau )$ with different input peak power: (a) 6-ps-long pulses after a SMF28 fiber with length of $L=5.057 \, \rm {m}$, dashed curves show the phase uncertainty between different measurements; (b) 1.7-ps-long pulses after 2.058 m of SMF28 fiber. The legend shows the input peak power. For comparison directly measured fiber input pulses at $\xi =0$ are shown. The phase in the pulse center is set to zero. Input and output are linearly polarized.

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The retrieval is another critical part of the nonlinearity measurement because the hereby introduced error is only indirectly accessible. The quality of the retrieval is pulse-shape dependent. Comparing many different retrievals from different programs ([42,43], SP and CS instrument software) for these relatively simple pulse shapes an energy fluctuation of the pulse of approximately 1.5 % and a phase uncertainty of better than 0.05 radian were found. A strong indication for a correct pulse retrieval is the good agreement between retrieved output pulses from a known nonlinear dispersive system and calculated output pulses using retrieved input pulses as simulation input (see Figs. 5 to 7).

 

Fig. 5. 6.2-ps pulse propagation in a 2.058-m-long SMF28 fiber, input peak power 665 W: (a) input and output power; (b) comparison of the input power and the nonlinear phase shift; (c) nonlinear phase shift versus momentary power. Here, the power is approximated by the average of in- and output power. Dashed lines show theory.

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More than thousand pulses in 12 fibers were measured. Measurements of nearly undisturbed SPM according to Eq. (13) were possible with 6 and 6.5-ps-long pulses in 1-, 2- and 5-m-long fibers and with 2.6-ps-long pulses in 1- and 2-m-long fibers. Propagation of the stronger chirped 1.7-ps-long pulses in 0.5- and 1-m-long fibers shows small dispersive pulse narrowing but can be still described approximately with a linear phase-power dependence. The nonlinear phase shift $\varphi _{\rm NL}(\tau )$ accumulated along the fiber is determined as difference of the phase of the high-power output pulse and a reference phase, ideally the low-power output phase. The phase of the output pulse comes directly from measurements as shown in Fig. 4. It is most important to find a good reference phase. The first option was a direct measurement of the laser output by removing the test fiber. Another option with less new alignment is the replacement of the test fiber with a very short reference fiber at low power. The such measured input pulses were indeed practically identical. These input pulses are later important to simulations for comparing to the theory. In the case of negligible dispersion the input pulse phase can be used as a good reference phase. However, when dispersion in the test fiber comes into play, the linear dispersive phase change needs to be taken into account for finding the reference phase. A very-low-power measurement of the reference phase through the test fiber would be the choice in the dispersive case. In Fig. 4(b) the small influence of dispersion is observable as the difference between the input phase and the low-power phase at 21.95 W. This measurement is limited by a minimum needed power for recording good FROG spectrograms (peak power $>10$ W). All three possibilities to find good reference phases were applied.

A very clean SPM scenario is shown in Fig. 5 for propagation of a $6.2$-ps pulse with input peak power of 665 W in a 2.058-m-long SMF28 fiber. A dispersive narrowing of the input pulse during the propagation is below the measurement resolution and not observable in Fig. 5(a). Because of the narrow spectra of 6-ps-long pulses, the FROG produces slightly bumpy pulses as evident already in Fig. 4(a). The bumps are different in repeated measurements and would average out. In Fig. 5(b) the curve of the nonlinear phase shift $\varphi _{\rm NL}=-(\varphi _{\rm out}-\varphi _{\rm in})$ follows directly the pulse power and the plot of the nonlinear phase shift versus the pulse power is a straight line in Fig. 5(c). The bumpy pulse shape is responsible for the deviations from straight. A slope of $0.00244 \,\frac {\rm rad}{\rm W}$ gives in Eq. (13) a fiber nonlinearity factor $\gamma =1.186 \cdot 10^{-3} \frac {1}{\rm m\ \,\ \,\ W}$ for this specific measurement. The measured results are reproduced very well by theory with the measured averaged nonlinearity from the next section. Figure 5 shows a very typical and easily reproducible result.

A still relatively clean SPM scenario is shown in Fig. 6 for propagation of a $2.6$-ps pulse with input peak power of 1421 W in a 1.135-m-long SMF28 fiber. The dispersive narrowing of the input pulse during the propagation shown in Fig. 6(a) is small but observable. In Fig. 6(b) the curve of the nonlinear phase shift $\varphi _{\rm NL}=-(\varphi _{\rm out}-\varphi _{\rm in})$ follows the pulse power well and the plot of the nonlinear phase shift versus the pulse power is a straight line in Fig. 6(c). A slope of $0.00133 \,\frac {\rm rad}{\rm W}$ gives in Eq. (13) a fiber nonlinearity factor $\gamma =1.17 \cdot 10^{-3} \frac {1}{\rm m\ \,\ \,\ W}$ for this specific measurement. The measured results are reproduced very well by theory.

 

Fig. 6. 2.6-ps pulse propagation in a 1.135-m-long SMF28 fiber, input peak power 1421 W: (a) input and output power; (b) comparison of the input power and the nonlinear phase shift; (c) nonlinear phase shift versus momentary power. Here, the power is approximated by the average of in- and output power. Dashed lines show theory.

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Stronger dispersive propagation like the example with results in Fig. 4(b) provide the nonlinearity after a little more comprehensive fitting procedure. Figure 7 shows propagation of a $1.7$-ps pulse with input peak power of 1063 W in a 2.058-m-long SMF28 fiber. The input pulse narrows significantly down to 530 fs. In Fig. 7(a) the nonlinear phase shift is compared to have a shape in between the input and the output pulse. The deviation from a linear relation between the nonlinear phase shift and the pulse power is shown in Fig. 7(b) with different power axis. Fitting the measured results to simulations with variable nonlinearity yielded the same nonlinearity factor as it was measured with clean SPM in Figs. 5 and 6. With the so determined nonlinearity pulse narrowing, nonlinear phase shift versus power in all variations and the pulse spectra in Fig. 7(c) are calculated in very good agreement to the experiment. The procedure of linear fitting of the relation $\varphi _{\rm NL}$ versus the averaged in- and output power $\frac {P_{\rm in}+P_{\rm out}}{2}$ shows to be a simple alternative to obtain a reasonable approximation for the nonlinearity factor $\gamma$ when 20 % at the upper part of the curve are omitted.

 

Fig. 7. 1.7-ps pulse propagation in a 2.058-m-long SMF28 fiber, input peak power 1063 W: (a) nonlinear phase shift compared to normalized input and output power; (b) nonlinear phase shift versus momentary input power, output power and averaged power $\frac {P_{\rm in}+P_{\rm out}}{2}$; (c) OSA and retrieved FROG spectra. Dashed lines show theory.

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I preferred not to mix dispersion and nonlinearity measurements as proclaimed in [31] and separated low-power dispersion measurements and high-power nonlinearity measurements as good as possible. In longer fibers low-power spectral phase measurements with the widest available spectra (shortest pulses) yielded the group velocity dispersion of both fibers: $\beta ''_{\rm SMF28}=-2.103 \cdot 10^{-26} \frac {\rm s^2}{\rm m}$ and $\beta ''_{\rm 1550BHP}=-2.294 \cdot 10^{-26} \frac {\rm s^2}{\rm m}$. This corresponds to dispersion parameters $D_{\rm SMF28}=16.5 \frac {\rm ps}{\rm nm\ \,\ km}$ and $D_{\rm 1550BHP}=18.0 \frac {\rm ps}{\rm nm\ \,\ km}$. Figure 8(a) shows the spectral phase increase due to dispersion. Nonlinear spectral broadening is negligible. In Fig. 8(b) the corresponding dispersive pulse modification is presented. The measured dispersion numbers are 7 to 9 % smaller than the calculated values from the fiber profile measurements in Fig. 1.

 

Fig. 8. 0.8-ps pulse propagation in a 5.057-m-long SMF28 fiber, input peak power 10 W: (a) equal OSA and retrieved input and output spectral intensity, only the phase at the output is modified by dispersion; (b) dispersive pulse-shape and phase changes.

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All measurements were repeated with forward and backward propagation to check for good incoupling and fiber surfaces. For each pulse length and fiber, input pulses with different peak power were compared. The patch cables were terminated with FC/PC connectors which enabled very reproducible coupling. The fibers were aligned without stress such that the output polarization state equals the input polarization as close as possible. Linear polarization as well as circular polarization states did not deteriorate significantly for fiber lengths up to 2 or even 5 m.

5. Results: the nonlinearity factor, the nonlinear susceptibility and the nonlinear refractive index

The nonlinearity factor $\gamma$ was determined from the slope of the nonlinear phase shift versus power. The nonlinearity factor is converted with the overlap integral $OI$ of the fiber with Eq. (14) to the nonlinear cubic susceptibility $\chi ^{(3)}_0$ and with Eq. (15) to the nonlinear refractive index $n_2$. In Fig. 9(a) the nonlinear refractive index from all my measurements can be plotted in one figure versus a normalized power axis $P_{\rm peak} \cdot L$. $P_{\rm peak}$ is the peak power of the input pulse for the measurement. $L$ is the fiber length. The $n_2$ values vary due to the measuring uncertainties of the power of 2.2 % (0.4 % fiber end facet and alignment to photo detector, 0.3 % detector, 1.5 % FROG intensity retrieval). The phase could be obtained to an accuracy of 0.03 to 0.05 radian. Because the nonlinear phase shift is calculated as difference of two phase measurements (high-power phase and low-power phase), the measurement error becomes very large for small nonlinear phase shifts. The solid lines show the error range for $n_2$ estimated with the above given measurement uncertainties in very good agreement to the observation.

 

Fig. 9. Measured fiber nonlinearity: (a) the nonlinear refractive index of the fiber material silica glass measured in different fibers with pulses with different peak power and different pulse length. The colored circles show measurements with linear input and output polarization. The white circles show the decay of the effective acting $n_2$ for circular polarized input (circ pol). The solid lines show the uncertainty range. (b) Distribution of the measured $n_2$ values for linear polarization with a fit to a normal distribution (red line) with a standard deviation of $\sigma =3.7\,\%$.

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Because of the large error in the low-power measurements, only data with normalized power $P_{\rm peak} \cdot L > 750 \,{\rm Wm}$ are used. The distribution of the measured $n_2$ values in Fig. 9(b) is approximated by a normal distribution with a standard deviation of $0.0833 \cdot 10^{-16} \frac {\rm cm^2}{\rm W}$ which corresponds to $\sigma =3.7\,\%$. Therefore, the measured value for the nonlinear refractive index is $n_2=2.22 \cdot 10^{-16} \frac {\rm cm^2}{\rm W} \pm 6.0 \, \%$. The standard deviation of 3.7 % of the data plus 2.3 % possible systematic error of the overlap integral is simply used as measure for the uncertainty of $\pm 6.0\,\%$. The corresponding value of the cubic susceptibility is $\chi ^{(3)}_0=1.64 \cdot 10^{-22} \frac {\rm m^2}{\rm V^2} \pm 6.0 \, \%$. It means that 68 % of the measured values fall in the given tolerance band. Alternatively using the expanded measurement uncertainty of $2\sigma$ plus 2.3 % as tolerance, 95 % of all measurements fall in the range between $n_2=2.22 \cdot 10^{-16} \frac {\rm cm^2}{\rm W} \pm 9.7\,\%$.

Extracting the overlap integrals the fiber nonlinearity factors $\gamma _{\rm SMF28}= 0.00119 \frac {1}{\rm Wm} \pm 3.7 \, \%$ and $\gamma _{\rm 1550BHP}= 0.00129 \frac {1}{\rm Wm} \pm 3.7 \, \%$ of both fiber types are found. The accuracy is 2.3 % better because the overlap-integral uncertainty does no longer contribute.

A clear pulse-length dependence could not be found within the measurement accuracy.

6. Polarization dependence of the cubic nonlinearity

A unique definition or measurement of a nonlinear phase shift of the entirety of a pulse is only possible for pure linear or circular polarization of the light in the fiber. Arbitrary polarization can always be decomposed into right- and left-hand circular polarized waves. They experience two different nonlinear phase shifts when they have different power [35], resulting in a power dependence of the output polarization. In addition to a non-unique nonlinear phase shift the FROG input is polarization sensitive and can not be adjusted for all the changing polarizations within the pulse.

For the measurements with linear polarization the input was linearly polarized with an extinction ratio larger than 100000:1 before the incoupling lens. Due to unavoidable intrinsic and stress induced fiber birefringence the polarization state deteriorates along the fiber. I have not performed a detailed polarization state investigation as I do not know the exact birefringence distribution and orientation in the fibers. I simply checked that the effect is negligible. The fiber output polarization ratio was typically $>500:1$ with no polarization rotation. With a simulation based on Eqs. (16) it was estimated that the nonlinear phase shifts of the two main polarization axis do not deviate measurably from that of a perfect linear polarized wave for a degradation of the polarization ratio down to $100:1$. Therefore, the fiber birefringence is negligible for linear polarization and does not influence the results for $n_2$ and $\chi ^{(3)}_0$.

With a quarter-waveplate in front of the incoupling lens circular polarized light with a polarization ratio 1.02:1 could be launched into the fiber. The circular polarization state is more sensitive to birefringence and the output polarization changed to slightly elliptical with ratios between 1.2:1 up to 1.6:1. Different nonlinear phase shifts of the two main axis and a resulting rotation of the output polarization ellipse disturbs a unique FROG adjustment for the whole pulse. Despite these difficulties the measurements with circular polarized input show in Fig. 9(a) a pretty consistent reduced effective acting $n_2=1.61 \cdot 10^{-16} \frac {\rm cm^2}{\rm W}$ with slightly larger uncertainty.

In an optical isotropic glass the cubic susceptibility has three independent tensor elements $\chi ^{(3)}_{xxyy}=\chi ^{(3)}_{yyxx}$, $\chi ^{(3)}_{xyxy}=\chi ^{(3)}_{yxyx}$ and $\chi ^{(3)}_{xyyx}= \chi ^{(3)}_{yxxy}$ with the relation to the relevant susceptibility for linear polarization $\chi ^{(3)}_{xxxx}=\chi ^{(3)}_{yyyy}=\chi ^{(3)}_{xxyy}+\chi ^{(3)}_{xyxy}+\chi ^{(3)}_{xyyx}$. $\chi ^{(3)}_{xxxx}$ is the measured cubic nonlinearity $\chi ^{(3)}_0$ in the last section. For the frequency arguments $\chi ^{(3)}(\omega ;\omega,\omega,-\omega )$ relevant for the nonlinear refractive index a further reduction to two independent tensor elements follows from a space-frequency symmetry with $\chi ^{(3)}_{xxyy}=\chi ^{(3)}_{xyxy}$ [35].

A last reduction to only one independent tensor element for the electronic $\chi ^{(3)}_0$ contribution $\chi ^{(3)E}_{xxxx}=3\chi ^{(3)E}_{xxyy}=3\chi ^{(3)E}_{xyxy}=3\chi ^{(3)E}_{xyyx}$ follows from Kleinman symmetry $\chi ^{(3)E}_{xxyy}=\chi ^{(3)E}_{xyxy}=\chi ^{(3)E}_{xyyx}$. In [35] it is shown analytically that the nonlinear phase shift or the effective acting $n_2$ for circular polarized light should be 66.66 % of $n_2$ for linear polarization under these symmetry rules. The measurements in Fig. 9(a) showed a smaller reduction to 72 %. This can be caused by the different symmetry rules for the Raman contribution to $\chi ^{(3)}_0$.

The following investigation of the influence of polarization on the measurement is based on a continuous-wave version of Eq. (7) for two orthogonal polarizations:

Introducing the symmetry for the electronic susceptibility contribution $\chi ^{(3)}_{xxyy}=\chi ^{(3)}_{xyyx}=\frac {\chi ^{(3)E}_{xxxx}}{3}$ in Eq. (16) the 66.66 % reduction of the effective acting $n_2$ for circular polarization is easily confirmed. For the Raman part of the susceptibilities Raman scattering cross sections from [10] propose $\chi ^{(3)R}_{xxyy}=\chi ^{(3)R}_{xyxy}=0.44\chi ^{(3)R}_{xxxx}$ and $\chi ^{(3)R}_{xyyx}=0.12\chi ^{(3)R}_{xxxx}$ [44]. A reduction of the effective acting $n_2$ to only $88\pm 2\,\%$ from the value for linear polarization is found for circular polarization introducing the symmetry rules for the Raman contribution $\chi ^{(3)}_{ijkl}=\chi ^{(3)R}_{ijkl}$ in Eq. (16).

The measured reduction factor of 72 % lies between 66 % and 88 % and is correctly simulated when $\chi ^{(3)}_0$ consists of $75\pm 5\,\%$ of a part with the symmetry rules of the electronic contribution and $25\pm 5\,\%$ of a part with the symmetry rules of the Raman contribution: $\chi ^{(3)E}_{xxxx} = 0.75 \cdot \chi ^{(3)}_0$ and $\chi ^{(3)R}_{xxxx} = 0.25 \cdot \chi ^{(3)}_0$. The composition is strongly dependent on the correct reduction value of the effectively acting $n_2$ for circular and linear polarization and makes the composition estimation not very accurate. However, that composition uncertainty should not be mistaken for the measurement uncertainty of the total measured $\chi ^{(3)}$.

7. Summary

The measured nonlinear index $n_2=2.22 \cdot 10^{-16} \frac {\rm cm^2}{\rm W}$ for silica glass falls in the range from $n_2=1.81$ to $2.77 \cdot 10^{-16} \frac {\rm cm^2}{\rm W}$ of a majority of published data from comparable experiments [12–17,19,20,22,24–29]. The range of reported data is more balanced after a few corrections for different experimental conditions. The wavelength dependence in the infrared is small [20,30] so that the cited measurements with wavelengths between 1000 and 1550 nm are comparable without adjustment. From results of measurements with standard single mode fibers and dispersion shifted fibers, $n_2$ of silica can simply be estimated by a small reduction of the measured values according to the GeO$_2$ contents of the core. A correction of the polarization influence could not be applied because the literature does not provide enough information and exact polarization states are mostly unknown. The published GeO$_2$-contents-corrected data are compared to my measurement in Fig. 10. My measured nonlinear index of glass agrees well to published values considering the uncertainties in the measurements and different techniques. The data in a relatively small range of pulse lengths do not show a pulse-length dependence. However, a clear reduction of $n_2$ for sub-ps pulses due to a diminished nuclear-vibration contribution is surprisingly also not documented in the literature. The measurements with fs-long pulses do not provide smaller $n_2$ values [14,17,19,20] than measurements with ps-long pulses so that the influence of nuclear-vibration contributions to $n_2$ needs more investigations. An improvement of the accuracy for the ratio between the electronic and nuclear-vibration contribution to the cubic susceptibility based on their different symmetry rules could not be obtained. The previous estimations of 15 to 20 % Raman contribution to $n_2$ for ps-long-pulses could be confirmed with a slightly larger value between 20 and 30 %.

 

Fig. 10. Comparison of the measured $n_2$ (red line with gray tolerance band) with values from literature. On the $x$-axis the reference numbers are shown. Filled circles are unchanged from literature from bulk and silica core fibers. Hollow circles show measurements from standard single mode fibers corrected by $-0.15 \cdot 10^{-16} \frac {\rm cm^2}{\rm W}$ and hollow squares show measurements from dispersion shifted fibers corrected by $-0.3 \cdot 10^{-16} \frac {\rm cm^2}{\rm W}$. Error bars are shown when available.

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