## Abstract

Optical spectra broadening as a result self-phase modulation in a channel waveguide fabricated on a high quality tantalum pentoxide (Ta_{2}O_{5}) film by using RF sputtering is measured. The full-width at half maximum of the optical spectra for transverse electric (TE)/transverse magnetic (TM) polarizations of 42.5/31.7 nm is obtained using pulses of 10 nm at a wavelength of 800 nm with a peak-coupled power of 43.77 W. The nonlinear Kerr coefficients of 2.14 × 10^{−14} cm^{2}/W and 1.92 × 10^{−14} cm^{2}/W for TE and TM polarizations, respectively, are then extracted from the experiments using a theoretical model based on the method of moments. The obtained results on the nonlinearity further suggest that Ta_{2}O_{5} is a promising material to develop nonlinear waveguide devices for integrated photonics.

© 2016 Optical Society of America

## 1. Introduction

Nonlinear waveguide applications have been widely used in integrated optical circuits for various functions, including optical generation [1–3], all-optical switching [4–6], and signal processing [7], and have been realized using nonlinear optical waveguides. Due to the high optical nonlinearity of Si, many nonlinear waveguide applications of Si waveguides, including wavelengths conversion based on four-wave-mixing/self-phase modulation (FWM/SPM) and all-optical modulation based on Kerr effect/cross-phase modulation (XPM) have been reported [8–10]. However, due to the small bandgap energy of Si, the inevitable two-photon absorption (TPA) and TPA-induced free-carrier absorption (FCA) effects in Si degraded the nonlinear figure of merit at high power operation [11]. Furthermore, the modulation speed of the Si-based all-optical modulator is limited by the relatively long free-carrier lifetime [12, 13]. Therefore, the large bandgap of materials without nonlinear absorption has been regarded as the most important property for developing nonlinear waveguide applications.

Recently, supercontinuum generation and a four-wave-mixing based optical parametric oscillator (FWM-based OPO) have been successfully realized using silicon nitride (SiN) based optical waveguides [14]. Similarly, large bandgap materials such as AlN/ZnO have also been utilized to demonstrate optical frequency combs/white-light generation by using micro-ring resonators/waveguides [15, 16]. Recently, by using RF sputtering, a high optical quality tantalum pentoxide (Ta_{2}O_{5}) film demonstrating the behavior of FWM optical parametric amplification was reported [17–19]. Ta_{2}O_{5} is a large bandgap material with energy bandgap similar to that of Si_{3}N_{4} or AlN. However, there is less or even no report of linear and nonlinear absorption in Ta_{2}O_{5} from the visible to infrared regions. SPM and FWM in Ta_{2}O_{5} waveguides have been demonstrated in previous reports [17, 18]. A nonlinear refractive index as high as 10^{−14} cm^{2}/W was obtained. As compared to the reported value of SiN or AlN [19], the larger nonlinear refractive index of Ta_{2}O_{5} indicates that it is a suitable optical material for developing nonlinear waveguide applications. However, the mode areas of previously developed Ta_{2}O_{5} waveguides were several square micrometers, which supported several transverse optical modes [17]. The large mode area as well as the multimode supported waveguide design would decrease the peak intensity in the waveguide, and thus would degrade the nonlinear optical effect in the Ta_{2}O_{5} waveguides. In this work, a low-loss submicron Ta_{2}O_{5} based channel waveguide has been demonstrated. The effect of SPM in the Ta_{2}O_{5} channel waveguide has been realized. The SPM effects of two polarizations modes, transverse electric (TE) and transverse magnetic (TM), in the Ta_{2}O_{5} channel waveguide are analyzed. In addition, spectral broadening of the coupled pulse is simulated using the nonlinear Schrodinger equation (NLSE) with the influence of chromatic dispersion.

## 2. Waveguide properties and experimental setup

Figure 1(a) shows the illustration of the Ta_{2}O_{5} channel waveguide. The fabrication of Ta_{2}O_{5} channel waveguide is described as follow. First, the Ta_{2}O_{5} thin film with a thickness of 400 nm is deposited on the Si substrate with a 3-μm thick thermal oxide. The material properties of the sputtered Ta_{2}O_{5} thin films are similar to those reported in our previous work [20]. Then, E-beam lithography is use to define the waveguide pattern. The waveguide width is set at 700 nm, and the waveguide length is 5 mm. Cr is deposited on the developed sample to serve as a hard mask. After etching of Ta_{2}O_{5} using the CHF_{3} plasma, the Ta_{2}O_{5} waveguide is formed with a waveguide height of 400 nm. SiO_{2}, with a thickness of 2 μm deposited using plasma-enhanced chemical vapor deposition (PECVD), serves as the upper cladding of the Ta_{2}O_{5} channel waveguide. Finally, the Ta_{2}O_{5} waveguide is diced and polished at both ends. The SEM cross-sectional view of the Ta_{2}O_{5} channel waveguide is shown in Fig. 1(b). The waveguide width and height are 750 nm and 400 nm, respectively. The sidewall angle is ~82°. The propagation loss of the Ta_{2}O_{5} channel waveguide is ~2.4 dB/cm at 800 nm, which is measured using the cutback technique.

To demonstrate the SPM effect in the Ta_{2}O_{5} channel waveguide, a free-space coupling system is applied, as shown in Fig. 1(c). The Ti:sapphire pulse laser at 800 nm serves as the pumping source to trigger the strong SPM effect in the Ta_{2}O_{5} channel waveguide. The pulsewidth and repetition rate of the pumping laser are 200 fs and 82 MHz, respectively. The half-wave plate is utilized to control the polarization of the pumping laser, and the polarization controlled pumping pulse is injected into the Ta_{2}O_{5} waveguide by using a 100X objective lens with a numerical aperture of 0.9. The total coupling loss between the objective lens and the waveguide facets is ~29 dB. Finally, the collected photons are analyzed using an optical spectrum analyzer.

## 3. Results and Discussions

Figure 2(a) shows the power-dependent optical spectra of the Ti:sapphire laser passing through the 5-mm long Ta_{2}O_{5} channel waveguide with TE polarization. The full-width at half maximum (FWHM) of the optical spectra for TE polarization almost linearly increases from 15.9 nm to 42.5 nm as the peak coupled power increases from 4.69 to 43.77 W. At a peak coupled power of ~43 W, the optical spectra of the injected laser broadens to 42.5/31.7 nm for TE/TM polarizations [referred to Fig. 2(b)]. Linewidth broadening is defined as the linewidth difference between the input and output spectra. The linewidth broadening of the TM mode is ~21.7 nm, whereas that of the TE mode is ~32.5 nm, which is much larger than that of the TM mode. Linewidth broadening for different polarizations originates from the different nonlinear coefficients of guided modes in the Ta_{2}O_{5} channel waveguide. The linewidth broadening induced by the SPM effect is directly proportional to the effective nonlinear coefficient *γ* = 2π*n*_{2}/*λA _{eff}*. Here,

*n*

_{2}is the optical Kerr nonlinear coefficient, λ is the central wavelength of the optical pulse, and

*A*is the effective mode area in the Ta

_{eff}_{2}O

_{5}channel waveguide. The mode confinement of the TE mode is much better than that of the TM mode, and the effective mode areas of the TE/TM modes are calculated to be 0.283/0.353 μm

^{2}, which is in agreement with the mode profiles of the Ta

_{2}O

_{5}channel waveguide simulated using the full-vector beam propagation method (BPM), as shown in Fig. 2(c). Thus, the large effective mode area (small nonlinear coefficient) of the TM mode will result in degradation of the SPM effect as compared to the waveguide operating in the TE mode with the same injected power. Notably the circular and triangular marks and the error bars in Figs. 2(b) and 2(d) are the mean and the standard deviation of the measured data of 10 repeats.

Due to the different beam profiles (effective mode areas) of TE/TM modes in Ta_{2}O_{5} channel waveguide, the peak coupled intensities are different for TE/TM polarizations with same coupled power. To further demonstrate the polarization-independent optical nonlinearity of Ta_{2}O_{5}, the intensity-dependent spectral broadening results for the Ta_{2}O_{5} channel waveguide are shown in Fig. 2(d). Under the same peak coupled intensity in the Ta_{2}O_{5} waveguide with TE/TM polarizations, the spectral broadening results for both polarizations are in good agreement, which implies the polarization-independent optical nonlinearity characteristic of Ta_{2}O_{5} and it is even clearer when the Kerr coefficient, *n*_{2} is extracted from experimental data in Fig. 2(b) to be discussed later. In addition, to determine the nonlinear coefficient from the SPM spectrum, a well-accepted formula [i.e., Eq. (1)] describing the SPM-induced spectral broadening was proposed by Kean *et. al.* in 1987 [21, 22].

*λ*/ Δ

_{i}*λ*are the FWHM of the optical spectra at the input/output end of the waveguide,

*t*is the pulse duration,

_{p}*P*

_{0}is the coupled peak power of the optical pulse,

*A*is the transverse mode area and

_{eff}*λ*is the central wavelength.

*L*is the effective length of the waveguide,

_{eff}*n*

_{2}is the Kerr coefficient and

*c*is the speed of light in free space. In Eq. (1), the invariance of the temporal structure of a picosecond (ps)-duration optical pulse propagating in a nonlinear waveguide with very short effective length is considered. In addition, the effective length is much shorter than the nonlinear length, which is much shorter than the diffraction length [23], thus leading to a linear relation between the linewidth broadening and the coupled peak power. However, the measured SPM-induced spectral broadening effect becomes saturated as the peak intensity increases up to 18.4 GW/cm

^{2}, clearly shown in Fig. 2(d). Notably the transmittance is remained the same value when increasing the peak coupled power up to 52.1 W, indicating no nonlinear absorption occurred in the Ta

_{2}O

_{5}channel waveguide even the peak coupled intensity up to 18.4 GW/cm

^{2}. The spectral broadening induced by SPM effect would not be influenced by the nonlinear absorption (TPA and FCA effects) in the Ta

_{2}O

_{5}.

Without nonlinear absorptions i.e., TPA and induced FCA, in Ta_{2}O_{5} from the visible to infrared region, such a spectral broadening saturation phenomenon could potentially be attributed to the decrease in the peak power as a result of the chromatic dispersion in the Ta_{2}O_{5} channel waveguide. Before detailed investigations, the chromatic dispersion in the Ta_{2}O_{5} channel waveguide having a dimension of 700 nm × 400 nm, as illustrated in Fig. 3, is calculated. By fitting the effective index calculated using full vectorial BPM package, the group velocity dispersion (GVD) at a wavelength of 800 nm is estimated to be −803.7 ps/nm-km (272.4 fs^{2}/mm) and −866.6 ps/nm-km (294.3 fs^{2}/mm) for the TE and TM modes, respectively. Our experiment employed an optical pulse of 200 fs measured as 1/e half-width of the Gaussian profile, and the optical pulse experiences a dispersion length of 147 mm/137 mm for the TE/TM polarized states. With such a high peak-coupled intensity employed in our work, the chromatic dispersion cannot be neglected in contrast to that reported by C.-Y. Tai *et. al.* [17], where the low peak-coupled power of the optical pulse propagates in a large core waveguide (*A _{eff}* = 3.5μm

^{2}) with smaller waveguide dispersion.

To justify the argument that the saturation behavior of spectral broadening is attributed to the GVD, we simulate the optical pulse evolution in the Ta_{2}O_{5} channel waveguide by using a semi-analytical treatment to the wave equation. It also provides a more precise estimation of the nonlinear refractive index of Ta_{2}O_{5} by using the SPM measurement. Typically, although the bandwidth broadening from SPM [refer to Eq. (1)] already gives a feasible estimation of the pulse broadening process, its usefulness is, however, limited. Moreover, the spectral broadening is exaggerated, and the nonlinear coefficient is underestimated. The NLSE can be used to describe the pulse evolution in an optical waveguide [23],

*U*is the complex envelope of the electrical field at a center frequency

*ω*= 2π

*c*/

*λ*,

*P*

_{0}is the peak power of the optical pulse,

*z*and

*t*are variables related to the propagation distance in the waveguide and time, respectively.

*β*

_{2}refers to GVD and

*γ*= 2π

*n*

_{2}/

*λA*is the effective nonlinear coefficient. Numerical studies based on a more generalized NLSE including waveguide loss and TPA are discussed in various literatures, which show that an initially transform-limited pulse can produce maximum spectral broadening [22]. To get more physical insight about nonlinear dynamics, we herein adopt a semi-analytical technique, moment method [23], to obtain spectral broadening as a result of SPM. By assuming the solution of the chirped Gaussian form as,

_{eff}*Z*subject to the length of the effective nonlinear waveguide

*L*, i.e.,

_{eff}*z*=

*ZL*and the temporal unit ${T}_{0}=\sqrt{\left|{\beta}_{2}\right|{L}_{eff}}$. The parameter

_{eff}*γ*=

_{PL}*γP*

_{0}

*L*denotes the effective nonlinearity, and sgn() is the sign function.

_{eff}*T*is the pulse duration, and

_{p}*C*is a dimensionless linear chirp parameter. By obtaining the Fourier transformation of the chirped Gaussian pulse, one can also obtain its FWHM spectral width, which is given byFor the optical pulse at a wavelength of 800 nm in a sub-micrometer Ta

_{p}_{2}O

_{5}waveguide, dispersion is normal, i.e., (

*β*

_{2}>0). On substituting ${C}_{p}={T}_{p}\cdot \left(d{T}_{p}/dZ\right)$ from Eq. (5) into Eq. (6), a nonlinear ordinary differential equation can be obtained,

*T*(

_{p}*Z*) =

*T*(0) +

_{p}*δT*(

_{p}*Z*), in which $\delta {T}_{p}\left(Z\right)\ll {T}_{p}\left(0\right)$. It can then be shown thatwherein $\Gamma =\frac{1}{{T}_{p}{\left(0\right)}^{3}}+\frac{{\gamma}_{PL}}{\sqrt{2}}\frac{1}{{T}_{p}\left(0\right)}$ and $\Xi =\frac{{\gamma}_{PL}}{\sqrt{2}}\frac{2}{{T}_{p}{\left(0\right)}^{2}}+\frac{3}{{T}_{p}{\left(0\right)}^{4}}$. The solution to Eq. (8) can be subjected to the initial conditions

Then, we use Eqs. (6)–(10) to compute the Kerr nonlinear coefficient. In our experiment, the pulse duration is 200 fs, and the effective length of the waveguide is 4.59 mm. The *L _{eff}* is defined as

*L*= (1-exp[-

_{eff}*αL*])/

_{WG}*α*, where

*L*is the waveguide length and

_{WG}*α*is the linear propagating loss [21]. For the TE modes, the time scale in the normalized form of Eqs. (3)–(10) is $\sqrt{\left|{\beta}_{2}\right|{L}_{eff}}\approx $35.4 fs, and the initially normalized pulse duration

*T*(0) is 5.64. The time scale and normalized initial pulse duration for the TM mode are 36.8 fs and 5.43, respectively. Since the incident pulse is slightly positively chirped due to the optics before the waveguide, by assuming that the initial chirp is negligible, a simple criterion is used to check the validity of the approximation, which is given as follows

_{p}*δT*(1)≪

_{p}*T*(0). This criterion yields that

_{p}*γ*<17.88 in our experiments when the chromatic dispersion induced change in pulse duration is within 20% of its initial value. The Kerr coefficients,

_{PL}*n*

_{2}, of the Ta

_{2}O

_{5}channel waveguide are extracted using the following scenario. First, we use the parameters in the experiments as an initial condition to compute the “standard curve” of the normalized equation by using the model with (in blue) and without (in red) GVD, as illustrated in Fig. 4. The experimental data (mean value) of linewidth broadening at high injection level is clearly seen in the perturbation method of Eqs. (6)–(10), as shown in Fig. 4. For the TE mode in Fig. 4(a), the coefficients obtained from the model without and with the GVD effect are 0.233 and 0.298, respectively. They correspond separately to the Kerr nonlinear coefficients

*n*

_{2}of 1.68 × 10

^{−14}cm

^{2}/W and 2.14 × 10

^{−14}cm

^{2}/W provided that the effective mode area in the waveguide is

*A*= 0.283 μm

_{eff}^{2}. For the TM modes having an effective area of 0.353 μm

^{2}, the Kerr nonlinear coefficients are 1.52 × 10

^{−14}cm

^{2}/W and 1.92 × 10

^{−14}cm

^{2}/W without and with the GVD effect. Comparing the Kerr nonlinear coefficients evaluated from the experimental data for the TE and TM modes, they are within a 10% difference relative to the TE case. The extracted nonlinear coefficient is approximately linearly proportional to the deviations in the spectral broadening measurement. Our experimental data of about 3 nm deviations produces 10% errors. Therefore, the differences in nonlinear coefficient for TE and TM modes are at the level within the variation of the experimental data, indicating a polarization-independence to the third-order nonlinear coefficient

*n*

_{2}, which is consistent with the observation shown in Fig. 3. Moreover, spectral broadening tends to saturate when chromatic dispersion induced GVD is considered. The fitted nonlinear coefficient is 2 to 3 time larger than the reported value in [17]. An increase above 26% in the

*n*

_{2}value can then be retrieved using the current model, which is well above the variation in experimental data. It means the inclusion of GVD is crucial for an accurate estimation of nonlinear coefficient. It is worth to mention that in the presence of higher order dispersions the extracted nonlinear Kerr coefficients can be significant changed if the duration and shape of an ultrafast pulse is a significantly distorted when it propagates. The higher order dispersions, for one of the examples, the third order dispersion

*β*

_{3}is retrieved to be 288fs

^{3}/mm in a Ta

_{2}O

_{5}channel waveguide with core dimension of 700 × 400 nm

^{2}at an wavelength of 800 nm. Using a slightly positively chirped pulse of 200 fs pulse duration, the effect to tailor the pulse shape by third order dispersion is lower than one percent of that from GVD. The SPM spectrum measured in our sample is in general symmetric, indicating a negligible influence from higher order dispersions and the validity of our analytical approaches to include only GVD in the nonlinear coefficient extraction.

## 4. Conclusion

The low propagation loss and small core area of Ta_{2}O_{5} channel waveguide has been fabricated to realize the strong SPM effect in the highly confined waveguide geometry. By injecting the Ti:sapphire laser with coupled peak power of 43 W, the FWHM of output spectra for TE/TM polarizations is broadened to 42.5/31.7 nm. Our result shows the most linewidth broadening in the Ta_{2}O_{5} by comparing to previous results under the same coupled peak power. In addition, the SPM induced linewidth broadening becomes saturated when increasing the injecting power, which is due to the decrease in the peak power as a result of the chromatic dispersion in the Ta_{2}O_{5} channel waveguide. Based on the moment method, the saturation of SPM induced linewidth broadening in Ta_{2}O_{5} channel waveguide has been simulated, and the nonlinear Kerr coefficient of ~2 × 10^{−14} cm^{2}/W at 800 nm is extracted from the experiment. Our work not only provides an easy and accurate method to estimate the nonlinear Kerr coefficient by SPM measurement but also further suggests that Ta_{2}O_{5} is a promising material in developing nonlinear waveguide devices for integrated photonics.

## Funding

Ministry of Science and Technology, Taiwan (MOST) (104-2112-M-110-001, 104-2112-M-110-012-MY2).

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