1. Introduction

Lithium niobate (LiNbO₃, LN) is one of the most attractive synthetic dielectric crystals due to its excellent properties of electro-optic, acousto-optic, piezoelectric and nonlinearity, which are already widely used in many functional optical waveguide devices. To realize the preparation and control of LN optical waveguides, various techniques have been employed so far, such as titanium in-diffusion [1] or proton exchange [2], in which mask etching technique is required for reliable fabrication of microstructures and devices are always limited to a planar geometry. In more recent years, there has been a rapid progress of femtosecond laser writing method, owing to its simple fabrication process, fast realization of three dimensional patterns, and non-clean room requirement, which may open up an era of optical device fabrication [3]. Various integrated optical devices such as wavelength converters [4] splitter [5] couplers [6] modulators [7] and Bragg gratings [8, 9] in LN have been demonstrated.

To achieve efficient nonlinear optical frequency conversion, especially for the second harmonic generation (SHG), ferroelectric domain poling technology provides an effective way to implement the quasi-phase-matched (QPM) principle. Therefore, QPM waveguide in periodically poled material has been considered as the most attractive way to enhance frequency doubling process. Two different types of waveguide structure have been identified in periodically poled LN (PPLN) crystal depending on the inscribed laser energy. In the first type of waveguide, guiding property occurs in the adjacent region rather than focal volume primarily from high pulse energy and double-line processing is employed to enable guiding with a mode matched cross section [4, 10, 11]. Although high confinement is observed, temporary structural change because of stress relaxation between two lines has to be involved [12]. In the other type of waveguide, guiding occurs in a focal modification region [13], and multiscan technique is usually applied to broaden waveguide width [14]. Meanwhile, using low repetition rate and high repetition rate laser, different micro-structures in LN or PPLN can be obtained [15–18].

At the same time, because of tight confinement of light in waveguides, high intensity induces high order nonlinearity. Of these nonlinear effects, Kerr nonlinearity including the self-phase modulation (SPM) and cross-phase modulation (XPM) effect is typically dominant and leads to nonlinear phase shift over much shorter interaction length, which gives rise to valuable phenomena such as spectral broadening [19–23]. However, there has been no research on nonlinear formation process and spectral broadening characteristics of filamentary PPLN waveguides.

In this paper, we report on the first observation of significant broadband radiation of FF and SH wave in the filamentary PPLN waveguide. The filamentary formation dynamics was numerically analyzed based on the modified spatiotemporal Schrödinger equation. The enlongated structures are formed by the combined effects of optical diffraction and self-focusing, and pulse intensity is believed to be responsible for the nonlinear process. Large spectral broadening spanning of FF wave to 230.15 nm and that of SH wave to 47 nm at the power of 19.5 mW was observed under 1550 nm excitation. The spectral evolution with the increased pump powers was obtained by solving the partial differential coupling equations. Analysis of the characteristic peaks by Kerr nonlinearity inside the waveguide provides a reasonable agreement with the experimental results.

2. Experiment and simulation of waveguide fabrication

To fabricate the waveguide, a Ti: sapphire laser (HP-Spitfire, Spectr-Physics Inc.) was used to generate linearly polarized pulses with duration of 50 fs at a repetition of 1 kHz, which is shown in Fig. 1. The laser pulse energy is adjusted by using a variable neutral density (ND) filter. The laser beam was focused via 50 × microscope objective (NA = 0.55) at a certain depth below the surface of sample, which was mounted on a computer-controlled motorized translation stage. The waveguide was formed by translating the sample along the x axis perpendicular to the laser beam propagation direction. The z-cut PPLN waveguide is 10-mm long and 0.5-mm thick, which contains a uniform domain period of 18.6 µm. The effective waveguide cross section area is $S_{eff}=2.3\times {10}^{-10}{m}^2$. The NA is to be 0.17 with the refractive change of 10⁻³, and the effective nonlinear coefficient for the first order QPM is $d_{eff}=2d_{33}/\pi =16pm/V$ [24].

 

Fig. 1 Schematic diagram of the waveguide fabrication setup using femtosecond laser pulses. Z-Y represents the end facets plane.

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Formation of photon-induced structure involves nonlinear propagation of intense femtosecond laser pulses with strong coupling of temporal and spatial effects, which makes it difficult to handle the complex nonlinear phenomena. Considering the laser parameters used in the experiment, we used a modified nonlinear Schrödinger equation (NLSE) [25] to describe the nonlinear effects occurred in LN, such as diffraction, group velocity dispersion (GVD), and Kerr nonlinearity, as shown in Eq. (1):

where, $E(z,r,t)$ is the complex amplitude of the laser field in the rest frame moving towards the direction of z axis. The radial Laplacian is given by ${\nabla }_{\perp }^2=\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}$. The wave vector is $k=2\pi n_0/\lambda$ with $n_0$ being the linear index of refraction at the centre wavelength $\lambda$. $\beta$ represents the dispersion coefficient, and equates the second derivative of frequency. $n_2$ is nonlinear index of refraction dependent on the material properties. These physical quantities we used are as follows. $n_0=2.175$, $\lambda =800nm$, $\beta =9.725\times {\text{10}}^{\text{-26}}{s}^2\cdot m^{-1}$, and $n_2=1.5\times {10}^{-15}c{m}^2/W$ [17].

The optically induced region is thought to be the result of a combination of self-focusing and diffraction effects. While the power of the beam exceeds the critical power, self-focusing prevails over diffraction, and leads to the collapse of the beam on itself. The critical power described for Gaussian pulses is $P_{crit}=3.77{\lambda }^2/8\pi n_0{n}_2$ [26]. For an 800 nm pulse in LN, it amounts to 0.3 MW. To obtain the transverse spatial intensity profiles, Eq. (1) was solved using Split-Step Fourier Method (SSFM), given the initial field as $E(z=0,r,t)=E_0\exp (-\frac{r^2}{2{\omega }_0^2})\sec h(\frac{t}{t_p})$, and the parameters ${\omega }_0$ and $t_p$ are 50 µm and 50 fs, respectively.

Waveguide morphology under different pulse power are represented in Figs. 2(a)-2(h) corresponding to the incident peak power of 0.1 P_crit, 0.2 P_crit, 0.5 P_crit, 0.83 P_crit, 3.75 P_crit and 5.8 P_crit. With the increase of peak power, the laser-induced region becomes narrower and longer. Self-focusing overcomes diffraction and leads to collapse only if the input peak power exceeds the critical threshold. It should be noted that optimal intensity could enhance the structural aspect ratio, which is useful for photonic devices especially for the integration with fiber technology. Laser-induced waveguide end-facets with the increased powers are experimentally illustrated in Figs. 2(a1)-2(f1). The elliptical structures are due to the combined action of optical diffraction and self-focusing, which is dependent on the writing power.

 

Fig. 2 (a)-(f) Numerical simulation of femtosecond pulses propagation under initial diffraction and self-focusing nonlinearity in LN obtained via solving Eq. (1). Transverse spatial intensity profile with incident peak power below (a)-(d) and beyond (e)-(f) critical power. Figures 2(a1)-2(f1). Laser-induced waveguide end-facets corresponding to the peak powers shown in Figs. 2(a)-2(f), i. e, 0.1 P_crit, 0.2 P_crit, 0.5 P_crit, 0.83 P_crit, 3.75 P_crit and 5.8 P_crit, respectively.

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The evolution of the pulse propagation shown in Fig. 2 is summarized in Fig. 3, where the spatial and the peak intensity for the field were depicted. From these data we see that higher intensity, the nearer modified position to the sample’s surface, and the higher optical intensity occurred in the focused regime. The field undergoes strong self-focusing with the transverse width decreasing from the initial 50 μm to about 12 μm.

 

Fig. 3 (a) Transverse width and (b) optical intensity versus propagation distance under different input pulse intensity.

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Laser-written channel width and depth of end facet with the dependence on laser energy and translation velocity were shown in Fig. 4. It can be seen that under certain translation velocity, channel width and depth increased with the increase of laser energy, and then decreased. Additionally, written depth is much larger than channel width, which results in the enlongated structure. Under constant laser energy, channel width and depth decreased with the increase of translation velocity. A depth to width aspect ratio (AR) can be controlled by optimizing the processing parameters. Typically, laser induced channel (AR less than 3:1) fabricated with the pulse energy of 2 µJ and the translation velocity of 0.4 mm/s was used to spectral broadening. Notably, the longer computed filaments than the measured modifications are due to neglecting the intensity loss in the simulation.

 

Fig. 4 Different laser-written channel width and depth with the dependence on laser energy and translation velocity.

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3. Experiment and simulation of spectral broadening

In the spectral broadening measurement, as shown in Fig. 5(a), the output of the Coherent Legend Elite was coupled into a Coherent optical parametric amplifier (OPA, TOPAS); it generates 40 fs and 2.5 mJ pulses at a 1 kHz repetition rate. The signal light with vertical polarization was used as the pump source, which was tuned to phase matching wavelength 1550 nm. Neutral density filters were used to vary the input pulse energy, and a power meter was used for output power monitoring. The beam was coupled into and out of the waveguide with 0.25 NA objectives, and then detected with an optical spectrum analyzer (OSA) covering 350 nm- 1750 nm with a resolution of 5 nm. A coupling loss of 1.2 dB /facet and a propagation loss of 1 dB/cm have been measured for the fundamental TM mode. The mode profiles were depicted in Figs. 5(b) and 5(c) for FF and SH wave, respectively. The waveguide was placed inside a temperature-controlled oven (HC Photonics), and the operation temperature is set to 150 °C, which corresponds to the QPM temperature, and meanwhile eliminates the influence of the photorefractive effect.

 

Fig. 5 (a) Experimental setup for the measurement of spectral broadening in filamentary waveguide. OPA: optical parametric amplifier; OSA: optical spectrum analyzer. (b) and (c) Mode distribution of FF and SH wave obtained at the end of the waveguide, respectively. (d) Schematic of the crystal coordinate system.

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To describe the propagation of femtosecond pulses, invoking the slowly varying amplitude approximation and considering the walk-off, group-velocity dispersion (GVD), SPM and XPM, we can obtain the coupled-wave equations with the following expression:

Where the indices 1 and 2 represent FF and SH pulses, respectively. $i$ refers to either the index 1 or 2. $E_i$is the electric field. $\kappa ={\left(\frac{2w^2}{{\epsilon }_0{c}^3{n}_{01}{}^2{n}_{02}{S}_{eff}}\right)}^{\frac{1}{2}}$ is the coupling coefficient. $d_{eff}$ is the effective nonlinear coefficient for the first order QPM. $n_{0i}$ is the refractive index, and $v_i$ is the group velocity, which equates to $1/{\beta }_i$. ${\beta }_i$ is $7.294\times {\text{10}}^{\text{-9}}s\cdot m^{-1}$ and $7.574\times {\text{10}}^{\text{-9}}s\cdot m^{-1}$ from Sellmeier equation for the FF wave and SH wave, respectively [27]. ${\beta }_{2i}$ is the group-velocity dispersion parameter, i. e, $9.725\times {\text{10}}^{\text{-26}}{s}^2\cdot m^{-1}$ and $3.841\times {\text{10}}^{\text{-25}}{s}^2\cdot m^{-1}$ for FF wave and SH wave, respectively. The device length corresponds to 3.5 (14) times the dispersion length of a 40-fs FF (SH) pulses. $\Delta k=k_2-2k_1-2\pi /\Lambda$ is the wave vector mismatch, where $\Lambda$is the domain period, and $k_i$is the wave vector. Here, $\tau =t-z/v_1$ corresponds to a frame of reference moving with the FF pulse at its group velocity. The initial field is taken to be a Gaussian pulse, and the pulse width is 40 fs.

We measured the spectral broadening characteristics of 1550 nm laser pulses, with a 50-nm FWHM of and 40-fs duration, passing through the set of waveguides in Figs. 2(a1)-2(f1); however, significant spectral broadening was observed in the waveguide shown in Fig. 2(e1). We measured the spectra of FF wave and SH wave simultaneously from the OSA with the increased average powers coupled into the waveguide, as shown with blue solid cures in Fig. 6 and Fig. 7, respectively. These spectra are compared with numerical simulations based on the Eq. (2), which are marked with the red solid curves in Fig. 6 and Fig. 7.

 

Fig. 6 (a)-(d) Measured spectral broadening of FF wave (blue solid line) at 1550 nm as a function of coupled increased average power in the waveguide, 3 mW, 6.7 mW, 14.3 mW and 19.5 mW, respectively. The red solid curves represent the simulated spectra for the same coupled powers obtained via solving Eq. (2).

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Fig. 7 (a)-(d) Experimental spectral broadening (blue solid curve) of SH wave simultaneously obtained from spectrometer, corresponding to the increased average powers in the waveguide the same with Figs. 3(a)-3(d), respectively. The red solid curves show the simulated spectra for the same coupled powers obtained via solving Eq. (2).

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Figure 6 and Fig. 7 show the onset of the spectral oscillations with an increased pump power. At pulse power of 3 mW, the output spectrum is almost the same as the input spectrum expect for the decreased amplitude because a portion of energy flowed into the SH wave. At pulse power up to 6.7 mW, the spectrum of FF wave changed dramatically from a peak that similar to the input spectrum to a broad spectrum with two peaks. However, the spectrum of SH wave was subjected to a slightly asymmetric double-peak structure. When the SH wave grows up to an apparent double-peak profile, FF wave is of broad triple-peak structure at the power up to 14.3 mW. As the pulse power increased further, i. e, 19.5 mW, almost flat four peaks appeared in FF spectrum, which is much broader than that of SH wave. Special attention should be paid to the larger spectral spanning, such as an octave-level spectrum, in which different frequency bands were merged into a single broad spectrum [18]. In this case, the coupled nonlinear envelope equation fails due to the presence of the overlapping frequency bands, and should be modified by the single-envelope approach [19, 20].

The multi-peak structures of broadened spectra with the increased pump power are indicative of cubic nonlinearity, and the peaks number depends on the maximum phase shift. Double peaks demonstrate a maximum phase shift of 1.5 π at 6.7 mW. Triple peaks for 14.3 mW results from phase shift of 2.5 π, and four distinguishable peaks for coupled power of 19.5 mW indicate a maximum nonlinear phase shift of 3.5 π in the waveguide. From the observation, it can be seen that there is a significant peak shift for FF wave, and the broad FF spectrum is quite different from the SH narrow spectrum, indicating that although broad frequency components of input spectrum, only partial energy located at QPM wavelength efficiently converted into SH wave to satisfy phase matching condition, which acts as “spectral filtering”. We measured the spectral FWHM as a function of average power coupled into the waveguide, as shown in Figs. 8(a) and 8(b).

 

Fig. 8 (a) and (b) The amount of broadening of FF and SH spectra at the waveguide output, respectively.

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The results shown in Fig. 8(a) indicate that at power about 3 mW, the stronger broadening appears, and then broadening smoothly. So do the SH wave pumped at 3 mW, shown in Fig. 8(b). About 230.15 nm broadening for FF wave and 47 nm for SH wave at the coupled power of 19.5 mW were observed. The measured FWHM is smaller than the theoretical prediction, which is attributed to the slightly asymmetric muti-peak structure.

Experimental measurement of SH power with the corresponding fundamental power was shown in Fig. 9. SH power increases almost quadratically as the input power increases. At the incident average power of 19.5 mW in the waveguide, the maximum conversion efficiency of 16% was obtained. SH conversion efficiency would be enhanced greatly if longer interaction length was used. Additionally, a decrease of the SH efficiency due to high temperature stability is common to femtosecond-laser-written waveguides. This problem can be solved by using MgO-doped PPLN waveguide, which could operate at room temperature and also overcome the limitation of high optical damage threshold. Further investigation of such waveguides is currently underway.

 

Fig. 9 Measured SH power variation with fundamental power. The central wavelength of the femtosecond pump pulse is 1550 nm.

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Output spectrum from OPA system with a central wavelength of 1550 nm is shown in Fig. 10(a), and the spectral FWHM is about 58 nm. The wavelength tuning curve for the SHG is shown Fig. 10(b). The highest efficiency about 16% almost keeps unchanged in a wavelength range about 12 nm near the central wavelength of 1550 nm, and the FWHM of the tuning curve is about 36 nm. Such wide wavelength tuning range implies that SHG process of fs pulses is significantly different from that of the “long” (picosecond and nanosecond) pulses or continuous wave (CW) light. This unique characteristic is due to the broad spectrum of femtosecond pulse, and makes it possible that light intensity of phase-matching wavelength is still high enough even under large phase-mismatching condition.

 

Fig. 10 (a) and (b) Output spectrum from OPA system with FWHM about 58 nm, and relationship between SHG conversion efficiency and fundamental wavelength, respectively.

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Such high intensity in the waveguide enables large phase shift possible with reduced pump power over short interaction length. The possibility to engineer the phase shift might be particularly useful for many applications, such as to all-optical switch, all-optical signal processing and functionalities demanding spectral tailing.

4. Conclusion

In conclusion, we have demonstrated a significant spectral broadening of OPA femtosecond pulses inside filamentary PPLN waveguide. Filamentary waveguide formation in LN crystal was numerically analyzed based on the modified spatiotemporal NLSE equation. The modified structures arise from the combined effects of optical diffraction and self-focusing as a function of pulse intensity. Large spectral broadening spanning of FF wave to 230.15 nm and that of SH wave to 47 nm with the coupled power of 19.5 mW was observed under 1550 nm excitation. Analysis of the spectral characteristics by Kerr effect inside the waveguide was carried out. The spectral evolution with the increased pump powers was obtained by solving the partial differential coupling equations numerically, which is in reasonable agreement with the experimental measurements.