1. Introduction

The nonlinear effects of light in the medium depend greatly on the timescale of the propagation pulses. Under continuous wave (CW) (or quasi-CW) pumping, the variation of light transmission in the medium is set by Brillouin scattering [1]. When femtosecond pulses with high peak power are employed, a variety of nonlinear effects such as self-focusing, self-phase modulation, pulse-front steepening, generation of optical shocks, multiphoton absorption, ionization, and free electron plasma generation dominate [2]. The nonlinear properties of the waveguides are distinctly modified when compared to their bulk counterparts, primarily owing to the substantial confinement of light by its structure. This pivotal alteration in nonlinear behavior directly results from the waveguide’s inherent ability to control and manipulate light tightly. Investigating nonlinear properties within waveguides has profound research implications across a spectrum of applications reliant on waveguide technology. These applications encompass but are not limited to, the development of waveguide lasers, advanced waveguide integrated components, and a multitude of other innovative functionalities.

The depressed cladding waveguides prepared by ultrafast laser inscription (ULI) has many compelling advantages. These attributes encompass minimal loss and the simplicity of fabrication, enabling the flexible design of structural dimensions. In most crystalline materials, they additionally exhibit the capability to support two distinct polarization modes, namely transverse electric (TE) and transverse magnetic (TM) modes. These waveguides present a novel paradigm for manipulating ultrashort optical pulses, affording opportunities for the modulation of spatial, temporal, and spectral properties. As such, they have emerged as a focal point of interest [3–6].

The depressed cladding waveguides on crystals have higher laser damage threshold than conventional optical fibers made of glass [7], rendering them a suitable platform for exploring intricate nonlinear phenomena. The choice of sapphire is underpinned by its excellent properties in terms of the transmission range, cubic nonlinearity(about 1.5 factor higher than in fused silica) [8], and optical damage threshold [9]. Current investigations into sapphire have predominantly concentrated on high-input pulse energy capable of generating supercontinuum spectra. Previous studies have mainly revolved around sapphire bulk [10,11] and sapphire optical fibers [12]. The enhancement of the generation efficiency of the supercontinuum spectrum in the sapphire waveguides fabricated via femtosecond laser was investigated previously [13]. However, the nonlinear properties at lower input pulse energies, specifically those slightly below the threshold necessary for supercontinuum generation, need further investigation. To the best of our knowledge, most of the waveguide structures currently used to study nonlinear phenomena related to the third harmonic and photoluminescence are ridged [14–16], and there have been few studies related to the depressed cladding waveguides. Furthermore, there are relatively few studies on nonlinear losses (NLs), with the majority of investigations primarily centered on optical fibers. For instance, Ferraro et al. elucidated the nonlinear transmission characteristics of femtosecond pulses in multimode fibers, emphasizing the key role played by spatial self-imaging (SSI) in asymptotic refractive index fibers for enhancing NLs [17]. However, studies related to NLs in femtosecond laser-prepared waveguides are still scarce.

In this study, we have conducted a comprehensive investigation into nonlinear phenomena within both the depressed cladding waveguides and the bulk. Our findings show that the nonlinearity within depressed cladding waveguides undergoes a significant enhancement, a phenomenon evidenced by the enhancement of the NLs and the increase of the third harmonic (TH) blue shift. In addition, we simulated and analyzed the TH evolution process in the waveguides by incorporating nonlinear effects such as self-phase modulation (SPM), cross-phase modulation (XPM), and group delay. The results of our simulations illuminate the predominant role played by the phase mismatch between the TH and the pump pulses for the asymmetric broadening and blue shift of the TH spectrum. Our findings demonstrate the unique propagation properties of high-peak-power near-infrared femtosecond pulses in waveguides, which provide new possibilities for the modulation of spatiotemporal and spectral properties of ultrashort laser pulses. Furthermore, it carries profound implications for diverse applications rooted in waveguide technology.

2. Experiments and results

2.1. Waveguide fabrication and characterization

2.1.1 Waveguide inscription

A c-axis sapphire crystal (E//c-axis) measuring 2${\times}$4${\times}$5 mm served as the experimental substrate. A femtosecond laser with a central wavelength of 1036 nm was used for waveguide preparation. A 40${\times}$microscope objective (NA = 0.6) focused the laser beam approximately 150 μm beneath the sample’s surface. The scanning velocity was 1 mm/s along the 5-mm edge. Pulse duration, repetition rate, and pulse energy were set to 450 fs, 300 kHz, and 0.4 μJ, respectively. Polarization was linear and perpendicular to the tracks. The cross-section image of the waveguide taken under an optical microscope (KEYENCE VH-Z500) is shown in Fig. 1(a).

 

Fig. 1. (a) End view of the depressed cladding waveguide composed of 16 parallel tracks with depressed refractive index. The horizontal distance between the tracks is 4 µm. The unmodified core has a width and height of 30 µm and 18 µm, respectively. And the length of the waveguide is 5 mm. (b) and (c) are the unsaturated and saturated state mode diagrams of the waveguide, respectively. (d) Illustrates the schematic diagram of the nonlinear observation experiment.

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2.1.2 Mode profiles and loss

The light was coupled into the waveguide through a lens with a focal length of 5 cm (NA = 0.126). We employed a 10× microscope objective with an NA of 0.25 for mode measurements. Images were captured using a CMOS camera (MAKO U-130 pixels), as shown in Fig. 1(b) and (c), which represents the optimal coupling condition with the lowest loss. To facilitate a more detailed observation of modal distribution, waveguide mode diagrams of both the unsaturated state and the saturated state are presented in Fig. 1(b). The waveguide core was located in the spatial locations of the laser-induced tracks, marked by dashed lines. The effective mode area of the waveguide was calculated to be ${A_{\textrm{eff}}} \approx 488$μm² by using ${A_{eff}} = {\left( {\int\!\!\!\int_S |E(x,y){|^2}dxdy} \right)^2}/\int\!\!\!\int_S |E(x,y){|^4}dxdy$ [18,19]. The normalized frequency ($\textrm{V} = 2\pi r \times NA/\lambda $) of an optical waveguide is commonly used to assess its mode transmission characteristics. A waveguide is considered to have single-mode transmission when V ≤ 2.4048 [20]. After substituting the parameters of the prepared waveguide into the calculation, it was determined that the normalized frequency of the waveguide is between 4.574 and 7.6235, which is much larger than 2.4048. Therefore, the waveguide is a typical multimode waveguide. We also performed relevant simulations to analyze the multimode properties of the waveguide in Supplement 1. The loss measurement was conducted using the method described in Ref. [13], and the test light source was a He-Ne laser. The measured loss for the waveguide was determined to be 2.3 dB/cm.

2.2 Nonlinear properties of the waveguide and the bulk

The AQ6370D (wavelength coverage range 600 nm-1700nm, with a resolution of about 0.1 nm) and the Avaspec-ULS2048CL-EVO-RS miniature fiber-optic spectrometer (wavelength coverage range 191.7-763.2 nm, with a resolution of about 0.299 nm) were used in our research. In the NLs experiments, the same femtosecond laser as in the waveguide preparation was used, and the experimental setup was essentially the same as in our previous studies [13]. The output light from the waveguide or bulk was collected into a power meter (S322C) by a 10${\times}$(NA = 0.25) microscope objective. A half-wave plate and a polarizing beam splitter were used to tune the laser power continuously.

Figure 1(d) illustrates the nonlinear observation experiment. In the experiment conducted in the transmission direction, the outgoing light from the waveguide/bulk passed through a short-wave pass filter (FGB37S) to eliminate the pump light before it was collected into the Avaspec-ULS2048CL-EVO-RS miniature fiber-optic spectrometer through a lens. Different spectral integration times were used for the measurements. A 500 ms integration time was selected for low input pulse energies to ensure clear observation of spectral changes. Conversely, a 10 ms integration time for high input pulse energies was preferred to prevent spectral intensity saturation. Additionally, changes of the fundamental spectra were measured by AQ6370D. During the side-observation experiments, a Nikon-D5100 digital camera captured side images of both the waveguide and bulk at various input pulse energy levels. A short-wavelength pass filter (FGB37S) was positioned in front of the camera lens to exclude the pump light. However, for subsequent experiments involving spectra measurements from the side, the filter was omitted as the luminescence intensity on the waveguide and bulk sides was minimal. Notably, the position exhibiting the highest luminescence intensity was selected for subsequent measurements and analysis.

2.2.1 Nonlinear losses

The transmittance of the waveguide and bulk as a function of input pulse energy is displayed in Fig. 2(a). The inset provides an enlarged view of the input pulse energy range from 2 μJ to 5.1 μJ. The energy density on the waveguide end face was far below the damage threshold of the sapphire [21–24]. To eliminate any potential linear contributions to losses, which may arise from material absorption or waveguide coupling losses, we implemented the method described in Ref. [17], normalizing the respective transmissions at low input pulse energies to 100%. The observed transmittance of both the waveguide and bulk exhibits nonlinear behavior in response to changes in input pulse energy, a hallmark characteristic of NLs. Furthermore, it becomes evident that the transmittance of the waveguide decreases at a more rapid rate and to a greater extent when compared to the bulk. The maximum difference in transmittance can be as substantial as 10%. These findings indicate that the NLs of the waveguide are greater than that of the sapphire bulk.

 

Fig. 2. (a)Dependence of transmittance versus input energies, measured at λ=1036 nm for both a 5 mm WG and sapphire bulk. (b) and (c) are the spot obtained on the white paper for the outgoing light from the waveguide and bulk, respectively. (d) and (e) Side images of the waveguide and bulk, respectively. Note: WG stands for the waveguide.

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As the input pulse energy increases, distinctive blue spots emerge at the output of both the waveguide and the bulk, depicted in Fig. 2(b) and Fig. 2(c). Notably, the intensity of the spot of the waveguide (b) is significantly stronger than that of the bulk (c). In addition, light emitting can be observed on the side of the waveguide and the bulk, as illustrated in Fig. 2(d) and Fig. 2(e).

2.2.2 Third harmonic and photoluminescence generation

Figure 3 presents a comparison of the ultraviolet portion of the spectra of transmitted light at different input pulse energies. The legends in the figure indicate the input pulse energy. Figure 3(a) and Fig. 3(b) represent the waveguide and bulk, respectively, in the low input pulse energy range (0.2 μJ to 1.57 μJ). A 500 ms integration time was selected to ensure clear observation of spectral changes. In the high input pulse energy range (1.57 μJ to 5.1 μJ), Fig. 3(c) represents the waveguide and Fig. 3(d) represents the bulk. The spectrometer was measured with an integration time of 10 ms to prevent saturation of the spectral intensity. The near-infrared portion of the spectra of transmitted light through the waveguide (red line) at an input pulse energy of 0.52 μJ is displayed in the inset of Fig. 3(a), compared to the spectra of the laser's direct output without passing through the crystal (black line). In Fig. 3, the spectral peaks generated by the waveguide and the bulk at low input pulse energies are around 344.5 nm. Notably, the black line in Fig. 3(a)’s inset indicates that the center wavelength of the laser’s fundamental output wavelength is approximately 1036.3 nm, with its corresponding TH wavelength at 345.3 nm. In this experiment, it was observed that the input pulse energy had to reach 0.52 μJ before the UV spectrum become distinctly visible. This event led to a blueshift in the center wavelength of the fundamental wave, as shown by the red curve in the inset of Fig. 3(a), consequently resulting in a blueshift of the TH wave. In addition, the spectral peaks exhibited a consistent blueshift tendency with increasing input pulse energy. Furthermore, it is noteworthy that at the same input pulse energy, the TH intensity, the degree of blueshift, and the spectral width produced by the waveguide are larger than those of the bulk. Meanwhile, the spectra display an asymmetric broadening trend, the underlying reasons for which will be explored in the forthcoming discussion section.

 

Fig. 3. (a) and (b) are the ultraviolet portion of the spectrum of transmitted light passing through the waveguide and bulk, respectively. These spectra were measured at a spectrometer integration time of 500 ms (input pulse energies of 0.2 μJ-1.57 μJ). The near-infrared portion of the spectra of transmitted light through the waveguide (red line) at an input pulse energy of 0.52 μJ is displayed in the inset of Fig. 3(a), compared to the spectra of the laser's direct output without passing through the crystal (black line). (c) and (d) are the ultraviolet portion of the spectrum of transmitted light passing through the waveguide and bulk, respectively, measured at a spectrometer integration time of 10 ms (input pulse energies of 1.57 μJ-5.1 μJ).

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Figure 4(a) shows the variation of spectral integrated intensity as a function of input pulse energy, measured within the 300-400 nm range when the integration time of the spectrometer is 10 ms. The inset is the spectral integrated intensity measured in the same wavelength range with the input pulse energy of 0.2 μJ-1.57 μJ and a spectrometer integration time of 500 ms. It can be seen that the TH intensity of both the waveguide and the bulk starts to rise at about 0.20 μJ, a point coinciding with the input pulse energy when the transmittance starts to decrease. Notably, the TH intensity of the waveguide consistently exceeds that of the bulk over the whole range of the input pulse energy variations. The left and right axes in Fig. 4(b) show the variation of the TH peak position and the corresponding spectral width with the input pulse energy (0.20 μJ - 5.10 μJ) for the waveguide and the bulk. It is worth noting that the peak position change was compared at a spectrometer integration time of 10 ms, while the spectral width was compared at 500 ms. Since saturation easily occurs under high input pulse energy conditions, the traditional half-height full-width method can not be applied, so the spectral width at the position where the spectral intensity first drops to zero was used as the spectral width index.

 

Fig. 4. (a) Spectral integrated intensity versus input pulse energy measured in the 300-400 nm range when the integration time of the spectrometer is 10 ms. The inset is the variation of the spectral integrated intensity measured in the same wavelength range with the input pulse energy of 0.2 μJ-1.57 μJ and a spectrometer integration time of 500 ms. The left and right axes in (b) show the variation of the TH peak position and the corresponding spectral width with the input pulse energy (0.20 μJ - 5.10 μJ), respectively. It is worth noting that the peak position change was compared at a spectrometer integration time of 10 ms, while the spectral width was compared at 500 ms.

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This figure makes it more intuitive to see that the spectrum of the waveguide is wider than that of the bulk for the same input pulse energy. For instance, the waveguide’s spectrum is approximately 11.65 nm wider than the bulk at an input pulse energy of 5.10 μJ. Additionally, the waveguide’s TH blueshift is more pronounced than the bulk. Specifically, the TH of the waveguide undergoes a blueshift from ∼344.42 nm to 341.80 nm, while the TH of the bulk blueshift from ∼344.42 nm to 342.97 nm. At an input pulse energy of 5.10 µJ, the TH peak positions of the waveguide and the bulk differ by about 1.17 nm.

Figure 5 displays side images captured for both the waveguide and the bulk at varying input pulse energies, revealing striking disparities. In the input pulse energy range of 3.2 μJ-5 μJ, there is only a single emitting point at the beginning of the bulk, whereas an array of equally spaced emitting points in the waveguide. Moreover, the waveguide produces significantly higher side emitting intensity than the bulk.

 

Fig. 5. Side images of the waveguide and bulk for the input pulse energy range of 3.2 μJ-5.1 μJ. The enlarged image of the waveguide at the input pulse energy of 3.9 μJ is shown in the red box.

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The side-emitting spectra measurements of the waveguide are shown in Fig. 6. However, it is worth noting that due to the limited measurement sensitivity of the spectrometer, the side-emitting spectra produced by the bulk are too weak to be measured. Figure 6(a) shows the side-emitting spectra recorded for the waveguide under the different input pulse energies, while Fig. 6(b) illustrates the relationships between the positions of the two peaks and the input pulse energies. Figure 6(c) is a log10-log10 plot of the emitting intensity as a function of the input-beam peak power. The experimental data (circles) are fitted with solid lines, and the corresponding slope (n) is indicated in the legend. Figure 6(d) shows the spectral integrated intensity in the range of 280-340 nm over the input pulse energies of 0.20 μJ-5.10 μJ, measured at 500 ms of the integration time of the spectrometer. The inset within Fig. 6(d) offers a localized enlargement of the curve in the range of the red box.

 

Fig. 6. (a) shows the side-emitting spectra of the waveguide under different input pulse energies, and (b) shows the positions of the two peaks versus the input pulse energies. (c)A log10-log10 plot of the PL intensity as a function of the input-beam peak power. The experimental data (circles) are fitted by solid lines, the slope n of which is indicated in the legend. (d) The spectral integral intensity in the range of 280-340 nm, spanning input pulse energies from 0.20 μJ-5.10 μJ (the spectrometer integration time is 500 ms, and the inset is a localized enlargement of the curve in the range of the red box).

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In Fig. 6(a), the side-emitting spectra primarily consist of two distinct peaks, denoted as A and B. As depicted by the red line in Fig. 6(b), the position of the peak A consistently undergoes a significant blueshift with the increasing input pulse energy, mirroring the trend observed in Fig. 4(b). Meanwhile, the peak B (∼332.5 nm, 3.73 eV), exhibits only minor fluctuations within a narrow wavelength range, independent of the input pulse energy, as illustrated by the black line in Fig. 6(b). According to Ref. [25], it is known that the luminescence peak in the range of 3.83 ± 0.17 eV (ca. 310-338 nm) is mainly associated with the 1B$\to $1A electric-dipole-allowed transition of the F⁺ center (this range is due to the temperature). The F + center generally has three absorption bands of 4.8 eV, 5.4 eV, and 6 eV, associated with the luminescence peak near 3.8 eV [26]. For pump light with a center wavelength of 1036 nm (∼1.2 eV), it mainly corresponds to 4- and 5-photon absorptions. The absorption band at 6 eV, corresponding to 5 photon absorption, plays a comparatively weaker role in generating this luminescence peak [27]. By fitting to Eq.${I_{PL}} = \alpha P_P^n$, n equal to 4.3 can be extrapolated. ${I_{PL}}$ is the intensity of the photoluminescence(PL), ${P_\textrm{P}}$ is the input peak power, and n is the average number of photons involved in the MPA process exciting the PL [28,29]. This indicates that peak B of about 332.5 nm (3.73 eV) in this experiment is a PL peak mainly caused by 4 photon absorption. Furthermore, according to Fig. 6(d), it becomes evident that the PL intensity of the waveguide is significantly enhanced from the input pulse energy of about 1.57 μJ. It is worth noting that the initial enhancement threshold may be lower than this value because the precise determination is constrained by the limited sensitivity of the spectrometer.

3. Discussion

As depicted in both Fig. 4 and Fig. 5, the waveguide consistently generates more TH and PL for equivalent input pulse energies. This observation shows the higher nonlinear conversion efficiency exhibited by the waveguide compared to the bulk. Typically, a key factor in this enhanced nonlinear performance is the nonlinear coefficients $\gamma $: larger coefficients tend to correspond to stronger nonlinear effects and more efficient nonlinear transformation. Due to the refractive index modulation by the intense light, the transmission phase shift of the optical field can be expressed as $\phi = {\phi _L} + {\phi _{NL}}$. At low power levels, the maximum nonlinear phase shift mainly originates from the SPM effect, which can be expressed as ${\phi _{\max }} = {L_{eff}}/{L_{NL}} = \gamma {P_0}{L_{eff}}$.The number of spectral peaks is related to the magnitude of the nonlinear phase shift [30]. By comparing the number of spectral peaks observed in both the waveguide and the bulk at the same input pulse energy, we can qualitatively assess the strength of the nonlinearity and $\gamma $ in sapphire bulk and the waveguide. As shown in Fig. 7, two spectral peaks can be clearly observed in the waveguide at the input pulse energy of 1.14 μJ. With the same pulse energy, the second peak in bulk is just beginning to emerge, and a further increase of the pulse energy is required to attain comparable spectral peaks as those observed in the waveguide. This observation indicates that the nonlinearity in the waveguide is significantly stronger than that in the bulk, which stems mainly from the structural confinement that effectively confines and sustains high-intensity pulses over longer distances within the waveguide [31].

 

Fig. 7. Comparison of fundamental spectra of the bulk and the waveguide when the input pulse energy is 1.14 μJ.

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The multiple modes supported by the waveguide are transmitted according to their respective propagation constants. This propagation characteristic leads to a dynamic alteration in the beam waist during transmission. The excitation of MPA primarily occurs at the point where the beam waist reaches its minimum, accompanied by the highest intensity. This phenomenon gives rise to the formation of light spots in the transmission direction with a period of ${Z_s}$ [32,33], presenting the SSI phenomenon as shown in Fig. 5. According to Ref. [32,33], the SSI period ${Z_s}$ is related to the core-cladding index difference by ${Z_s} = \pi {r_c}/\sqrt {2\Delta } $, in which $\Delta = ({\textrm{n}_0^2 - \textrm{n}_1^2} )/2\textrm{n}_0^2$. We take the average value ${r_c} \approx 12$μm of the waveguide length-short axis as the radius of the waveguide core. The refractive index of the waveguide core is ${n_0} = 1.7551$. The refractive index of the cladding is ${n_1} = {n_0} - \Delta n$, and $\Delta n$ is the amount of refractive index change caused by the laser processing. The SSI period is ${Z_s} \approx 0.78$ mm can be extrapolated according to Fig. 5. Based on these parameters, $\Delta n \approx{-} 2 \times {10^{ - 3}}$ can be calculated, consistent with the amount of laser-induced refractive index change reported previously [34–36].

In addition, there are obvious asymmetric broadening and blueshifting of the TH wave, as shown in Fig. 3, 4(d) and 6(b). This phenomenon has also been observed in multiple contexts, including microstructured fibers [37], tapered fibers [38], and air [39]. These references suggest that the primary driver for the spectral broadening of the TH is the cross-phase modulation (XPM) exerted by the pump pulse. As shown in Fig. 8(a), with the increase of the input pulse energy, the pump pulse shows a significant spectral broadening due to nonlinear effects such as SPM, pulse splitting, self-steepening, and ionization [2]. Among these, SPM(caused by the Kerr effect and plasma generation) plays a key role in the spectral broadening of the pump pulse [40,41]. The spectral broadening of the pump pulse is subsequently imprinted onto the TH pulse by XPM, resulting in the spectral broadening of the TH. Since the intensity of the TH is weak, its own SPM is negligible. The XPM induced by the pump pulse is the main reason for TH spectral broadening [42,43]. It is important to note that a group delay exists between the TH and the pump pulses.

 

Fig. 8. (a) Comparison of the pump-pulse spectra after passing through the waveguide and the bulk with the output spectra of the laser for the input pulse energy of 5.1 μJ. (b)The effective wave-vector mismatch(curves 1,2; left axis) and the dimensionless frequency deviation of the pump pulse ωp(curves 3,4; right axis) versus the propagation coordinate Z, under low input pulse energy(0.2 μJ; solid line) and high input pulse energy(5.1 μJ; dashed line).(c), (d) and (e), (f) are the time-domain and frequency-domain curves of the pump pulses and TH pulses involved in the THG, SPM and XPM processes for input pulse energies of 2 μJ and 5.1 μJ, respectively.

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According to the theories in Ref. [38,44] [45,46], the evolution of the waveguide-generated TH pulse in time- and frequency-domain has been analyzed by taking into account SPM and XPM. The influence of group delay associated with the pump and the TH pulses has also been included (see Supplement 1 for details). The study combines these theoretical principles with carefully chosen parameters that align with actual experimental observations (selected parameters used for the simulations are shown in Tab. 1). The results are shown in Fig. 8(b)-Fig. 8(f).

Table 1. Selected parameters used in the simulation^a

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Figure 8(b) shows the effective wave-vector mismatch ($\Delta {k_{eff}}$, curves 1,2; left axis) and the dimensionless frequency deviation of the pump pulse (${\omega _p}$, curves 3,4; right axis) versus the propagation coordinate Z for low (0.2 μJ; solid line) and high (5.1 μJ; dashed line) input pulse energies. At low input pulse power levels where the nonlinear phase shifts are negligible, the wave-vector mismatch remains virtually constant as the TH pulse propagates through the medium (curve 2 in Fig. 8(b)). At higher input pulse power levels, the ultrashort pulses experience spectral broadening, driven by the influence of SPM and XPM. Consequently, owing to the varying group velocities and phases corresponding to different frequency components, the group velocity mismatches and the corresponding phase mismatches between the TH and the pump pulses are no longer the same for the entire pulse. Instead, only a specific part of the pulse can effectively minimize $\Delta {k_{eff}}$. The proximity of $\Delta {k_{eff}}$ to zero directly correlates with the increased generation of TH, resulting in higher TH intensity. This phenomenon, in turn, contributes to the asymmetric broadening of the spectral broadening of the TH pulse [38].

Figure 8(c), (d) and Fig. 8(e), (f) are the time-domain and frequency-domain curves of the pump pulses and TH pulses involved in the THG, SPM and XPM processes for input pulse energies of 2 μJ and 5.1 μJ, respectively. ${C_{NL}} = P_p^3$, P_p and P_h denotes the powers of the pump and TH pulses, and S_h and S_p are the spectra of the pump and TH pulses, respectively. By comparing Fig. 8(e) and Fig. 8(f), it is evident that the TH spectrum exhibits an obvious broadening and blue-shifting phenomenon with the increasing input pulse energy. This observed behavior aligns consistently with the trends observed in the experimental results. It is essential to note that the transmission loss of the waveguide is not considered in the simulation. Consequently, the frequency-domain curves in Fig. 8(f) are slightly deviate from the experimental results, but the general trend is consistent. Owing to the group-velocity mismatch, the leading edge of the TH pulse gradually walks off from the leading edge of the pump pulse. Through the joint action of group-delay effects and changing phase matching, $\Delta {k_{eff}}$ of the THG process at the trailing edge of the pump pulse is smaller than that of the leading edge of the pump pulse. This results in the maximum intensity of the generated TH being locked to the trailing edge of the pump pulse. Since the carrier frequency of the pump pulse in our case is red-shifted on the leading edge and blue-shifted on the trailing edge of the pulse, the spectrum of the TH pulse overlapping the trailing edge of the pump pulse is blue-shifted accordingly. The blue shift of the TH spectrum of the waveguide is significantly higher than that of the bulk, which is closely related to the enhancement of its nonlinearity [47–49]. The enhancement of the nonlinearity of the waveguide will lead to the enhancement of the SPM generated by the pump pulse at the same input pulse energy and influence the TH pulse through XPM. The cumulative effect ultimately manifests as an increased magnitude of the blueshift in the TH spectrum.

4. Conclusion

In conclusion, we have successfully inscribed a depressed cladding waveguide with a length of 5 mm on sapphire crystal by utilizing a femtosecond fiber laser. Subsequently, the nonlinear phenomena in the waveguide and sapphire bulk observed from the sample transmission direction and the side were investigated and compared, respectively. The experimental results demonstrate a significant enhancement of nonlinearity in the depressed cladding waveguide compared to the bulk, which is manifested by the enhancement of the NLs and the TH blueshift. Furthermore, a simulation of the third harmonic blueshift phenomenon was executed. Nonlinear effects including SPM, XPM, and group delay have been taken into account. The simulation results and the experimental phenomena demonstrated the relationship between the asymmetric broadening and blueshift of the TH spectrum and the phase mismatch between the TH and the pump pulses. Our findings reveal unique propagation properties of ultrashort pulsed lasers in femtosecond laser inscribed waveguides, which has important implications for applications such as waveguide lasers, beam delivery, medical imaging, and waveguide integration.