Nonlinear optical properties of 6H-SiC and 4H-SiC in an extensive spectral range

Xiao Guo; Xiao Guo; Zeyu Peng; Zeyu Peng; Pengbo Ding; Pengbo Ding; Long Li; Xinyi Chen; Haoyun Wei; Zhen Tong; Liang Guo

doi:10.1364/OME.415915

1. Introduction

Silicon carbide (SiC), as the primary representative of the third-generation semiconductors, has a colossal number of advantages such as wide band gap, high saturation drift velocity, and high thermal conductivity [1–3]. Due to these merits, SiC can be used for high voltage, high frequency, and extreme-environment electronic devices [4–6]. Nevertheless, high hardness, high brittleness, and superior chemical stability of SiC bring great challenges to fabrication of SiC devices. Femtosecond laser has become the preferred implement for SiC precision manufacturing due to its virtues such as high instantaneous power, non-contact nature, and ability to process inside the sample [7–10]. Because of the wide band gap of SiC, processing of SiC by femtosecond laser, which is available typically in the visible and the near-infrared range, usually relies on multi-photon absorption. In order to better guide femtosecond laser processing of SiC, it is important to acquire the accurate values of the nonlinear absorption coefficient and the nonlinear refractive index coefficient of SiC. On the other hand, there have been many examples of using SiC for nonlinear photonic applications such as microresonators [11] and waveguides [12]. Further exploration of SiC for such applications also requires knowledge of the nonlinear optical properties of SiC.

To date, available data of the nonlinear absorption coefficient and the nonlinear refractive index coefficient of SiC are scattering in literatures. At the times of ruby laser, the nonlinear absorption coefficient of 6H-SiC was measured at 694 nm through Z-scan measurement [13]. Later, Ti:sapphire femtosecond laser was applied to measure the 2PA coefficient and the third-order nonlinear susceptibility of 6H-SiC near 800 nm by Z-scan measurement [14,15]. In addition, self-phase modulation was also used to measure the nonlinear refractive index coefficient of 4H-SiC [16]. In order to better guide femtosecond laser processing of SiC and fully realize the potential of SiC for photonic applications, it is crucial to reveal the wavelength dependence of the nonlinear optical properties. Although a theoretical work has been conducted to predict the dispersion of 2PA and third-order refraction properties of SiC [17], to the best of our knowledge, there is still a lack of measurement over a broad spectral range to provide comprehensive information of the nonlinear absorption coefficient and the nonlinear refractive index coefficient of SiC. In this work, the nonlinear optical properties of SI 6H-SiC and SI 4H-SiC, two important SiC polytypes, are measured in an extensive spectrum from 400 nm to 1000 nm by Z-scan technique. In particular, we perform a detailed study about the wavelength ranges for the occurrence of 2PA and 3PA.

2. Z-scan measurement

The samples investigated in this work are SI 6H-SiC and SI 4H-SiC (TankeBlue, Beijing), both of which are semi-insulating and polished on both sides with surface perpendicular to the [0001] orientation. We have used X-ray diffraction (XRD) and scanning electron microscope (SEM) to check the quality of the samples, which show high crystallinity and smooth surface (see Figs. S1 and S2 in Supplement 1). The thicknesses are 258 $\mu$m and 498 $\mu$m for SI 6H-SiC and SI 4H-SiC respectively, and the impurity concentrations are both smaller than $1\times 10^{16}\textrm {cm}^{-3}$. In order to obtain the band gap values which will facilitate distinguishment of 2PA and 3PA, we measured the linear absorption spectrum. The linear absorption coefficient $\alpha =-\textrm {ln}(\frac {T}{(1-R)^2})/L$. Here, $T$ is the transmittance measured by an UV/VIS/NIR spectrometer (Lambda 750 S, PerkinElmer) at room temperature, $R=\frac {(n_\textrm {A}-n_\textrm {B})^2}{(n_\textrm {A}+n_\textrm {B})^2}$ is the reflectance at the front surface and rear surface where $n_\textrm {A}$ and $n_\textrm {B}$ are the refractive indices of the two mediums forming the interface (the incident light is normal to the sample surface), and $L$ denotes the sample thickness. In our calculation, $n_\textrm {A}$ is taken as 1, the refractive index of air, while $n_\textrm {B}$ is taken from [18]. Due to the indirect-band gap nature of SI 6H-SiC and SI 4H-SiC, the band gaps $E_\textrm {g}$ can be extrapolated by calculating the intercepts of the curves of $(\alpha h\nu )^{1/2}$ in the linear region versus $h\nu$ with the $h\nu$ axis [19] as shown in Fig. 1, where $h$ is the Planck constant, and $\nu$ is the frequency of light . In this way, the indirect band gaps of SI 6H-SiC and SI 4H-SiC are extracted to be 2.99 eV and 3.25 eV respectively, which are close to the reported results [20]. Also, we have calculated the band structures of SI 6H-SiC and SI 4H-SiC employing the density functional theory (DFT) and the results are shown in Fig. S3 in Supplement 1. According to the calculated results, indirect band gaps correspond to the energy gap from $\Gamma$-point in the valence band to M-point in the conduction band. The calculated values of the band gaps agree with the extracted values from measurements.

Fig. 1. The plot of $(\alpha h\nu )^{1/2}$ versus $h\nu$ for SI 6H-SiC and SI 4H-SiC.

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The Z-scan measurement system used in this work is illustrated in Fig. 2. In this experiment, the laser source is an Yb:KGW regenerated amplifier (PHAROS-10W, Light Conversion) operating at a repetition rate of 50 kHz, with a pulse duration of 290 fs and a central wavelength of 1030 nm. The wavelength of the laser is tuned to the desired wavelength between 400 nm and 1000 nm by an optical parametric amplifier (OPA, ORPHEUS-F, Light Conversion). The beam from the OPA is focused by a convex lens whose focal length is 400 mm and propagates through the sample, causing nonlinear optical effect in the sample. The system can characterize nonlinear absorption by open aperture measurement as shown in Fig. 2(a) as well as nonlinear refraction by closed aperture measurement as shown in Fig. 2(b). Considering the different configurations of the OPA for output at different wavelengths and dispersion due to optics, for each wavelength, a laser beam profiler (LBP2-HR-VIS2, Newport) is used to characterize the spot size at the focus and an autocorrelator (GECO, Light Conversion) is used to measure the pulse duration. During pulse duration measurement, we used glass plates with the same compositions and thickness to replace the focal lens (we consider the center thickness), ensuring the same dispersion as that in Z-scan measurement. The power values of both the incident light and the transmitted light are measured by a power meter (1936-R, Newport) with the detector (918D-SL-OD3R, Newport), D in Fig. 2. Since the parameters of the OPA output vary with wavelength, the details of the experiment parameters are listed in Table 3 in the appendix.

Fig. 2. Z-scan (a) open aperture and (b) closed aperture measurement apparatuses. L represents a convex lens with a focal length of 400 mm, D denotes a power meter detector, and A stands for an aperture.

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For a Gaussian beam propagating along the $+z$ direction with a beam waist radius of $w_0$, the electric field distribution within the light can be described by

(1)$$E(r,z,t) = E_0(t)\frac{w_0}{w(z)}\textrm{exp}\left(-\frac{r^2}{w^2(z)}+\frac{ikr^2}{2R(z)}\right)e^{i\phi(z,t)},$$

where $w(z)=w_0(1+z^2/z_0^2)^{1/2}$ is the radius of beam, $R(z)=z(1+z_0^2/z^2)$ is the curvature radius of the wave front, $z_0=kw_0^2/2$ is the Rayleigh length of the beam, $k=2\pi /\lambda$ is the wave vector, $\lambda$ is the laser wavelength, $E_0(t)$ is the electric field amplitude at the focus containing the time envelope of the pulse, and $e^{i\phi (z,t)}$ is phase factor during spatial propagation and temporal evolution. In this theoretical frame, the zero of $z$ axis is at the location of the focus. Regarding the spatiotemporal phase, there are two conventions as $(kz-\omega t)$ and $(\omega t-kz)$, the selection of which further affects the sign in front of $\frac {ikr^2}{2R(z)}$, the sign within the complex refractive index, the signs in Eqs. (7) and (10), and the sign of the Gouy phase shift in Eq. (12). Therefore, this selection must be clarified to avoid misleading. In this work, the former convention is chosen as in [21]. The intensity distribution of the incident light is as following

(2)$$I_\textrm{i}(r,z,t)=\frac{1}{2}cn\varepsilon_0|E(r,z,t)|^2=\frac{1}{2}cn\varepsilon_0E_0^2(t)\frac{w_0^2}{w^2(z)}\textrm{exp}\left(-\frac{2r^2}{w^2(z)}\right),$$

where $c$ is the speed of light, $n$ is the refractive index, $\varepsilon _0$ is the vacuum dielectric constant, and $I_0(t)$ is the intensity at the focus with $I_0(0)=I_\textrm {0,max}$.

When the thickness of the sample is much smaller than the Rayleigh length (3-13 mm in this work), the change of the beam radius inside the sample is very small so that it can be neglected. The nonlinear absorption of SiC in this experiment involves the effect of 2PA and 3PA. The light intensity at a certain position $z^{'}$ (coordinates inside the sample whose zero is at the front interface of the sample with direction the same as $z$) satisfies the following differential equations (Eq. (3) illustrates 2PA and Eq. (4) illustrates 3PA)

(3)$$\frac{dI}{dz^{'}}={-}\alpha I-\beta I^2,$$

(4)$$\frac{dI}{dz^{'}}={-}\alpha I-\gamma I^3,$$

where $\alpha$ is the linear absorption coefficient, $\beta$ is the 2PA coefficient, and $\gamma$ is the 3PA coefficient. Equations (3) and (4) can be solved by integral and imposing boundary conditions [22,23]. With the solved intensity $I$ at the back surface of the samples, the normalized transmittance expressions (divided by linear absorption transmittance) considering 2PA and 3PA [22–26] are as following

(5)$$\begin{aligned}T_\textrm{2PA}(z)&=\frac{\int_{-\infty}^{\infty}dt\int_{0}^{\infty}2\pi rI_\textrm{e,2PA}dr}{\int_{-\infty}^{\infty}dt\int_{0}^{\infty}2\pi r(1-R)I_\textrm{i}(r,z,t)\textrm{exp}(-\alpha L)dr} \\ &=\frac{\int_{-\infty}^{\infty}\textrm{ln}\left[1+qe^{{-}x^2}\right]dx}{\sqrt{\pi}q}, \end{aligned}$$

(6)$$\begin{aligned}T_\textrm{3PA}(z)&=\frac{\int_{-\infty}^{\infty}dt\int_{0}^{\infty}2\pi rI_\textrm{e,3PA}dr}{\int_{-\infty}^{\infty}dt\int_{0}^{\infty}2\pi r(1-R)I_\textrm{i}(r,z,t)\textrm{exp}(-\alpha L)dr} \\ &=\frac{\int_{-\infty}^{\infty}\textrm{ln}\left[pe^{{-}x^2}+\sqrt{1+p^2e^{{-}2x^2}}\right]dx}{\sqrt{\pi}p}, \end{aligned}$$

where

\begin{aligned}q&=\beta (1-R)I_\textrm{i}(r=0,z,t=0)L_\textrm{eff}=\beta\frac{(1-R)I_0(0)}{1+\tau^2}L_{\textrm{eff}}, \\ p&=\sqrt{2\gamma [(1-R)I_{\textrm{i}}(r=0,z,t=0)]^2L_{\textrm{eff}}^{'}} \\ &=\sqrt{2\gamma L_{\textrm{eff}}^{'}}\frac{(1-R)I_0(0)}{1+\tau^2}, \\ L_\textrm{eff}&=(1-e^{-\alpha L})/\alpha, \\ L_{\textrm{eff}}^{'}&=(1-e^{{-}2\alpha L})/2\alpha, \\ \tau&=z/z_0, \\ x&=t/\left(\frac{t_\textrm{FWHM}}{2\sqrt{\textrm{ln2}}}\right), \end{aligned}

where $t_\textrm {FWHM}$ is the full width at half maximum (FWHM) pulse duration. Compared with the previous literatures [14,23], the reason why we choose to keep the integral is that our experimental conditions cannot meet the corresponding assumptions, i.e. $q<1$ and $p<1$ (the values of $p$ and $q$ at certain wavelengths are greater than 1).

For the nonlinear refraction, we only consider the third-order process. The nonlinear refraction will cause a shift of light electric field phase $\phi (z,t)$ defined in Eq. (1), $\Delta \phi (z,t)$. The shift of phase can be described by the equations [27]

(7)$$\frac{d\Delta\phi(r,z^{'},t)}{dz^{'}}=\Delta n(I)k,$$

(8)$$\frac{dI}{dz^{'}}={-}\alpha I,$$

where $\Delta n=n_2I$ ($n_2$ is the nonlinear refractive index coefficient), and we only consider linear absorption while calculating the nonlinear refractive index coefficient. Combining the two equations, we can get the expression of the phase shift as following

(9)$$\Delta\phi(r,z,t)=\Delta\phi(r=0,z,t)\textrm{exp}\left(-\frac{2r^2}{w^2(z)}\right),$$

where $\Delta \phi (r=0,z,t)$ is the phase shift at the focus, we can calculate the nonlinear refractive index coefficient with the following equation

(10)$$\Delta\phi(r=0,z,t)=\frac{kn_2(1-R)I_0(t)L_\textrm{eff}}{1+z^2/z_0^2}.$$

Combined with Taylor series, the electric field distribution at the exit of the sample can be expressed as

(11)$$\begin{aligned}E_\textrm{a}(r,z,t)&=E_\textrm{i}(r,z,t)e^{-\alpha L/2}e^{i\Delta\phi(r,z,t)}=E_\textrm{i}(r,z,t)e^{-\alpha L/2}\sum_{m=0}^{\infty}\frac{[i\Delta\phi(r,z,t)]^m}{m!} \\ &=E_\textrm{i}(r,z,t)e^{-\alpha L/2}\sum_{m=0}^{\infty}\frac{[i\Delta\phi(r=0,z,t)]^m}{m!}\textrm{exp}\left(-\frac{2mr^2}{w^2}\right). \end{aligned}$$

We can regard the series as a summation of Gaussian beams, the electric field at the aperture can be derived as

(12)$$\begin{aligned}E_\textrm{a}(r,z,t)&=E_\textrm{i}(r=0,z,t)e^{-\alpha L/2} \\ &\cdot\sum_{m=0}^{\infty}\frac{[i\Delta\phi(r=0,z,t)]^m}{m!}\frac{w_{m0}}{w_m}\textrm{exp}\left(-\frac{r^2}{w_m^2}+\frac{ikr^2}{2R_m}-i\theta_m\right). \end{aligned}$$

The distance between the sample and the aperture is defined as $d$. The parameters in Eq. (12) are as following

\begin{aligned}w_{m0}^2&=\frac{w^2(z)}{2m+1}, \\ d_m&=\frac{kw_{m0}^2}{2}, \\ \frac{w_{m0}}{w_m}&=\frac{1}{\sqrt{g^2+\frac{d^2}{d_m^2}}}, \\ R_m&=d\left[1-\frac{g}{g^2+d^2/d_m^2}\right]^{{-}1}, \\ \theta_m&=\textrm{tan}^{{-}1}\left[\frac{d/d_m}{g}\right], \\ g&=1+d/R(z). \end{aligned}

The normalized transmittance for third-order nonlinear refraction is calculated by dividing the transmittance by the linear absorption transmittance and the expression is as shown in Eq. (13). Because $|\Delta \phi (r=0,z,t=0)|\ll 1$ is not satisfied in this experiment, we choose to keep 10 terms for convergence in the numerator, which is proved in Section 3 in Supplement 1.

(13)$$\begin{aligned}T(z)&=\frac{I_\textrm{a}}{I_\textrm{a}|_{\Delta\phi(r=0,z,t)=0}}=\frac{|E_\textrm{a}(r=0,z,t=0)|^2}{\left|E_\textrm{a}(r=0,z,t=0)|_{\Delta\phi(r=0,z,t)=0}\right|^2} \\ &=\frac{\left|\sum_{m=0}^{\infty}\frac{\left[i\Delta\phi(r=0,z,t=0)\right]^m}{m!}{(g^2+d^2/d_m^2)}^{{-}1/2}\textrm{exp}({-}i\theta_m)\right|^2}{\left|{(g^2+d^2/d_0^2)}^{{-}1/2}\textrm{exp}({-}i\theta_0)\right|^2}. \end{aligned}$$

3. Results and discussion

First, taking the curves of SI 4H-SiC as examples, Figs. 3(a) and 3(b) show the results of nonlinear absorption measurement at 490 nm and 940 nm in which the fitting curves are plotted based on Eqs. (5) and (6), respectively. Figures 3(c) and 3(d) show the results of nonlinear refraction measurement at 490 nm and 940 nm in which the fitting curves are plotted based on Eq. (13). In Figs. 3(a) and 3(b), there is a valley when the sample is at the focus, which indicates the transmittance decreases due to nonlinear absorption. In Figs. 3(c) and 3(d), a valley appears before the focus ($z<0$) and a peak appears behind the focus ($z>0$), which corresponds to the self-focusing property.

Fig. 3. The open aperture Z-scan curves and fitting curves of SI 4H-SiC at (a) 490 nm and (b) 940 nm; the closed aperture Z-scan curves divided by open aperture Z-scan curves and fitting curves of SI 4H-SiC at (c) 490 nm and (d) 940 nm.

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Theoretically, only three-photon absorption (or even more-photon absorption) can occur when the photon energy is smaller than the half of the band gap. The nonlinear absorption coefficients of the samples within 400 nm-1000 nm spectral range are shown in Fig. 4. The error bars are the standard deviations of the results from two experiments and they have been amplified by a factor of 5 for the convenience of vision, which, therefore, indicate small statistical fluctuations in the measurements. The overall trend of the 2PA coefficient decreases with the increase of the laser wavelength. This trend is attributed to the indirect nature of the band gaps of both SI 6H-SiC and SI 4H-SiC, which generally results a gradual absorption increase with increasing photon energy above the band gap. When the laser photon energy is smaller than the half of indirect band gap, the value of the 2PA coefficient is too small to be considered. The results are in positive agreement with the calculation results in [17] which only considers indirect transition. Also, the calculation of the cross sections of 2PA and 3PA is discussed in Section 4 in Supplement 1. With the measured nonlinear absorption coefficients, we estimated the temperature excursion of the samples after heating by a femtosecond laser pulse. Due to the high thermal diffusion coefficients of both SI 6H-SiC and SI 4H-SiC, the femtosecond laser with 50 kHz repetition rate does not cause severe heat accumulation.

Fig. 4. (a) 2PA coefficients of SI 6H-SiC from 400 nm to 830 nm and SI 4H-SiC from 400 nm to 760 nm; (b) 3PA coefficients of SI 6H-SiC from 840 nm to 1000 nm and SI 4H-SiC from 770 nm to 1000 nm. (The error bars have been amplified by a factor of 5 for clarity)

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For the nonlinear refraction in this work, we only consider the third-order nonlinear process. The third-order nonlinear refractive index coefficients from 400 nm to 1000 nm for SI 6H-SiC and SI 4H-SiC are shown in Fig. 5. The error bars are the standard deviations of the results from two experiments and they have been amplified by a factor of 5 for the convenience of vision, similar to the case for nonlinear absorption. Different from the 2PA coefficients, the nonlinear refractive index coefficients stay roughly on the same order across this spectrum.

Fig. 5. Nonlinear refractive index coefficients from 400 nm to 1000 nm for (a) SI 6H-SiC and (b) SI 4H-SiC. (The error bars have been amplified by a factor of 5 for clarity)

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The comparison of the nonlinear absorption coefficients and the nonlinear refractive index coefficients with previous results [14–16] is shown in Tables 1 and 2. They agree reasonably well with each other and the deviations may be induced by the assumptions of Gaussian-shape temporal pulse shape and spatial beam profile, which may not be true for real femtosecond laser.

Table 1. Comparison of 2PA coefficients between our results and results in previous literatures.

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Table 2. Comparison of nonlinear refractive index coefficients between our results and results in previous literatures.

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Due to high hardness, brittleness and chemical inertia of SiC, femtosecond laser has been one of the few feasible processing tools for SiC, relying on its capability of inducing multi-photon absorption. Nowadays, besides Ti: sapphire femtosecond laser with wavelength tunability in the range of 690-1040 nm, Yb-doped gain medium allows access to 515 nm and 520 nm femtosecond laser by second-harmonic generation. With the measured nonlinear absorption coefficients in this work, proper processing fluences with new wavelength laser can be estimated based on empirical fluences at other wavelengths, therefore facilitating optimization of femtosecond laser processing parameters for SiC.

The measured nonlinear optical properties could also provide insights for design of photonics. One such example is realization of ultrafast all-optical switching with modulation rate far exceeding that of electronic switching. There have been a few works utilizing nonlinear absorption in semiconductors to achieve ultrafast all-optical switching [28,29]. Although the nonlinear absorption coefficient for non-degenerate 2PA, which is typically used in optical switching, is not the same as that for degenerate 2PA studied in Z-scan experiment, they are generally on the same order [28,29]. It has been found that GaP, with 2PA coefficients on the order of $10^{-11}\ \textrm {m/W}$ in the range of 610 nm to 980 nm, can be used to realize sub-30 fs all-optical switching in this red part of the visible spectrum [28]. From the results in Fig. 4(a), the 2PA coefficients of SI 6H-SiC and SI 4H-SiC are on the similar order in the blue part of the visible range, so SiC may have the potential to realize ultrafast switching covering the blue spectrum. As a demonstration, the ultrafast optical modulation based on the SI 6H-SiC is characterized by pump-probe technique in this work. A modulating light (pump) with wavelength at 860 nm, fluence at 26.67 $\textrm {J/m}^{2}$, and pulse width 120 fs, irradiates the sample with an incidence angle of 2.75° to modulate the sample optical properties. When the modulating pulse and the modulated pulse overlap in time, non-degenerate 2PA sharply increases absorption of the modulated light and decreases its transmission. The relative transmission change (-$\Delta T$/$T$) for a modulated light (probe), which is perpendicular to the sample with wavelength from 430 nm to 675 nm and pulse width about 140 fs, is measured. The modulating and the modulated light have the same polarization direction. The evolution signals of (-$\Delta T$/$T$) of the modulated light at 430 nm, 490 nm, 600 nm, and 675 nm are presented in Fig. 6. The circles are the experimental results and the solid lines are the Gaussian fitting curves. The FWHMs of the fitting curves are 379.6 fs, 268.6 fs 191.9 fs and 216.1 fs for the modulated light at 430 nm, 490 nm, 600 nm, and 675 nm, respectively. Figure 6 clearly illustrates the potential of SI 6H-SiC in the application of sub-picosecond all-optical switching based on nonlinear absorption with modulation depth on the order of 1$\%$. Especially, at 430 nm, the modulation depth is close to 10$\%$. With compressed pulse width and enhanced peak intensity, the modulation depth can be improved further.

Fig. 6. The evolution of (-$\Delta T$/$T$) for modulated light with wavelength at 430 nm, 490 nm, 600 nm, and 675 nm. The circles are the experimental results and solid lines are the Gaussian fitting curves.

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In addition, nonlinear refraction within dielectric cavities is another effective strategy to achieve ultrafast all-optical switching, which takes use of the resonance shift of cavity modes by Kerr effect [30]. In this case, nonlinear absorption, which tends to broaden the resonance spectrum and deteriorate the modulation depth, is an undesirable effect. Since nonlinear absorption and refraction are always coupled due to the Kramers-Kronig relations, a figure of merit (FOM) $n_2(\beta \lambda )^{-1}$ is proposed to evaluate whether a dielectric medium is suitable for making nonlinear refraction-based optical switching in certain spectral range [31]. The values of the FOM for SI 6H-SiC are plotted in Fig. 7 based on the measured nonlinear optical properties in this work. The FOM gradually increases as wavelength in the studied range because of the decreasing 2PA coefficients and the roughly flat nonlinear refractive index coefficients. From the FOM, SI 6H-SiC could also be explored for optical switching using nonlinear refraction in the range of 700 nm to 830 nm, in which 2PA is weak due to the indirect band gap and the FOM is over 0.5.

Fig. 7. Plot of FOM versus wavelength for SI 6H-SiC from 400 nm to 830 nm.

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4. Conclusion

In this paper, the nonlinear absorption coefficient and the nonlinear refractive index coefficient of SI 6H-SiC and SI 4H-SiC have been measured in an extensive range from 400 nm to 1000 nm for the first time. We use the half of the band gaps as the dividing lines between 2PA and 3PA. The values of the nonlinear absorption coefficients and the nonlinear refractive index coefficients can guide the processing of SiC by femtosecond laser and promote the optoelectronic applications of SiC. The ultrafast all-optical switching in the visible spectrum especially in the blue region by 6H-SiC based on nonlinear absorption is demonstrated and its potential for ultrafast switching by nonlinear refraction is predicted.

Appendix

The experimental parameters we used in this paper are shown in Table 3, where $\lambda$ is wavelength, $t_\textrm {FWHM}$ is pulse duration, $w_0$ is beam waist, and $P$ is power. The relative root mean square error (RRMSE, RRMSE= $\sqrt {\frac {1}{n}\sum _{i=1}^{n}(1-e_i)^2}$ where $e_i$ is the ratio of the corresponding points on the measured curve and the fitting curve in Fig. 3) is used to judge the quality of the fitting. The absorption model RRMSEs for SI 6H-SiC and SI 4H-SiC are shown in Fig. 8. The refraction model RRMSEs for SI 6H-SiC and SI 4H-SiC are shown in Fig. 9. We can find RRMSEs are all less than 0.1, which proves that the fitting keeps a great quality.

Fig. 8. Plots of RRMSE versus wavelength (a) from 400 nm to 830 nm with 2PA model and from 840 nm to 1000 nm with 3PA model for SI 6H-SiC, (b) from 400 nm to 760 nm with 2PA model and from 770 nm to 1000 nm with 3PA model for SI 4H-SiC.

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Fig. 9. Plots of RRMSE with the third-order refraction model versus wavelength from 400 nm to 1000 nm for SI 6H-SiC and SI 4H-SiC.

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Table 3. Experiment parameters for SI 6H-SiC and SI 4H-SiC from 400 nm to 1000 nm.

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Funding

National Natural Science Foundation of China (51806094); Natural Science Foundation of Guangdong Province (2019A1515010745); the Shenzhen Science and Technology Program (KQTD20170810110250357); Characteristic Innovation Project of the Department of Education of Guangdong Province (2018KTSCX202).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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	2PA coefficient (m/W)
Wavelength (nm)	Ref.	This Work
780 (SI 6H-SiC)	6.4 $\times 10^{- 13}$ [14]	4.6 $\times 10^{- 13}$
800 (SI 6H-SiC)	3.46 $\times 10^{- 13}$ [15]	3.95 $\times 10^{- 13}$

	Nonlinear refractive index coefficient ( $m^{2} / W$ )
Wavelength (nm)	Ref.	This Work
532 (SI 4H-SiC)	1.88 $\times 10^{- 19}$ [16]	3.19 $\times 10^{- 19}$ (530 nm)
780 (SI 6H-SiC)	4.75 $\times 10^{- 19}$ [14]	3.69 $\times 10^{- 19}$
800 (SI 6H-SiC)	1.678 $\times 10^{- 19}$ [15]	3.88 $\times 10^{- 19}$

$λ$	$t_{FWHM}$	$w_{0}$	$P$	$λ$	$t_{FWHM}$	$w_{0}$	$P$
(nm)	(fs)	( $μ$ m)	(mW)	(nm)	(fs)	( $μ$ m)	(mW)
400	180.7	44	$\sim$ 5	710	187.0	43	$\sim$ 29
410	184.1	38	$\sim$ 7	720	185.2	42	$\sim$ 35
420	180.1	36	$\sim$ 5	730	187.8	41	$\sim$ 36
430	173.8	36	$\sim$ 3	740	187.5	40	$\sim$ 35
440	175.1	34	$\sim$ 5	750	188.4	41	$\sim$ 37
450	173.7	32	$\sim$ 4	760	188.8	41	$\sim$ 53
460	169.1	32	$\sim$ 4	770	185.0	40	$\sim$ 46
470	175.9	30	$\sim$ 3	780	178.9	39	$\sim$ 43
480	183.0	31	$\sim$	790	176.4	38	$\sim$ 40
490	172.5	32	$\sim$ 7	800	176.1	40	$\sim$ 44
500	152.4	27	$\sim$ 2	810	176.4	40	$\sim$ 44
520	137.7	32	$\sim$ 9	820	175.7	40	$\sim$ 42
530	122.7	32	$\sim$ 10	830	171.4	39	$\sim$ 37
540	122.7	36	$\sim$ 18	840	168.1	39	$\sim$ 35
550	139.6	33	$\sim$ 16	850	165.1	37	$\sim$ 30
560	144.3	33	$\sim$ 13	860	162.2	36	$\sim$ 27
570	132.4	31	$\sim$ 12	870	162.0	37	$\sim$ 29
580	140.1	31	$\sim$ 11	880	162.1	37	$\sim$ 27
590	136.4	30	$\sim$ 12	890	160.5	37	$\sim$ 51
600	132.6	29	$\sim$ 10	900	161.6	37	$\sim$ 53
610	132.9	29	$\sim$ 10	910	159.5	38	$\sim$ 55
620	131.7	29	$\sim$ 10	920	160.6	36	$\sim$ 53
630	139.7	36	$\sim$ 15	930	168.7	35	$\sim$ 54
640	152.4	38	$\sim$ 39	940	170.7	35	$\sim$ 48
650	154.0	38	$\sim$ 41	950	160.3	36	$\sim$ 57
660	167.3	42	$\sim$ 63	960	153.8	36	$\sim$ 60
670	180.7	43	$\sim$ 81	970	191.8	36	$\sim$ 52
680	185.3	44	$\sim$ 83	980	160.4	37	$\sim$ 72
690	185.8	43	$\sim$ 89	990	176.5	36	$\sim$ 73
700	186.1	43	$\sim$ 30	1000	131.7	35	$\sim$ 28

	2PA coefficient (m/W)
Wavelength (nm)	Ref.	This Work
780 (SI 6H-SiC)	6.4 $\times 10^{- 13}$ [14]	4.6 $\times 10^{- 13}$
800 (SI 6H-SiC)	3.46 $\times 10^{- 13}$ [15]	3.95 $\times 10^{- 13}$

	Nonlinear refractive index coefficient ( $m^{2} / W$ )
Wavelength (nm)	Ref.	This Work
532 (SI 4H-SiC)	1.88 $\times 10^{- 19}$ [16]	3.19 $\times 10^{- 19}$ (530 nm)
780 (SI 6H-SiC)	4.75 $\times 10^{- 19}$ [14]	3.69 $\times 10^{- 19}$
800 (SI 6H-SiC)	1.678 $\times 10^{- 19}$ [15]	3.88 $\times 10^{- 19}$

Nonlinear optical properties of 6H-SiC and 4H-SiC in an extensive spectral range

Abstract

1. Introduction

2. Z-scan measurement

3. Results and discussion

4. Conclusion

Appendix

Funding

Disclosures

Data Availability

Supplemental document

References

Supplementary Material (1)

Data Availability

Cited By

Figures (9)

Tables (3)

Equations (15)

Optical Materials Express