Lithium niobate (LN) is attractive for investigations due to its intriguing electro-optical, acousto-optical and optical nonlinear properties^[1,2]. In addition, LN crystal has an optical transparency window with a large bandwidth from 350 to 5200 nm, which offers a wide range of phase-matching possibilities^{[3–6" target="_self" style="display: inline;">–6]}. Furthermore, single-crystal LN thin film has attracted intensive research interest because of the rapid commercial development of chip-scale fabrication techniques^[7,8]. Recently, a vast number of integrated optical components based on single-crystal LN thin film, such as waveguides^[9], whispering gallery mode cavities have been successfully designed and fabricated^{[1012" target="_self" style="display: inline;">–12]}.

Due to the expensive and time-consuming fabrication of LN chip-scale devices^[13,14], the theoretical simulations and predictions become really important, especially for researchers. Based on LN thin film, there are some works on second-harmonic generation (SHG) in strip waveguides^{[1517" target="_self" style="display: inline;">–17]}. In this Letter, we tried to study the intrinsic quadratic phase-matching conditions in the single-crystal LN thin film as a natural planar waveguide structure. We predicted different mode phase-matching conditions for three-wave mixing in single-crystal LN thin films of different thicknesses by theoretical simulation and then at 1018 nm pump light we observed a green SHG light of 509 nm with the highest nonlinear coefficient d33^[18,19], corresponding to one of the mode phase-matching points. By analyzing and calculating the dispersion relations of waveguides in LN thin films with different thicknesses, we found that for the guiding wave confined in the film, the larger the thickness is, the larger the mode size and effective refractive index are. Specifically, we gave full possibilities of types I and II mode phase-matching of SHG in a 900 nm single-crystal LN thin film planar waveguide, where type I phase matching of SHG includes e+e→e, o+o→e, o+o→o and type II SHG includes e+o→o and e+o→e. The spontaneous parametric downconversion (SPDC) was also analyzed by studying type II SHG at 1.55 μm for the e+o→o case.

During the three-wave mixing in LN, the energy and momentum conservation must be satisfied^{[1921" target="_self" style="display: inline;">–21]}. They can be expressed by ω1+ω2=ω3,(1)Δk→=k3→−(k1→+k2→),(2)where ki=niωi/c(i=1,2,3), ωi and ki→ represent the frequency and wave vector, respectively. When k1→ and k2→ are equal, the three-wave mixing represents the SHG. The second-harmonic (SH) waves can be efficiently generated after satisfying the phase-matching condition Δk→=0, in which the refractive indices of the fundamental wave (FW) n1 and SH wave n3 must be the same. Generally, the condition of n1=n3 cannot be satisfied because of the intrinsic material dispersion.

A schematic diagram of LN on insulator (LNOI) is shown in Fig. 1(a); 1.9 μm silica is sandwiched by LN thin film and the LN substrate^[22]. Figure 1(b) illustrates the propagation path of guiding mode waves. nf, nc, and ns represent the refractive indices of the thin film layer (guiding layer), air (cladding layer), and silica layer (substrate layer), respectively. h is the thickness of the guiding layer, which can be processed ranging from 300 to 900 nm. Figure 1(c) is the two different mode field distributions in the film with the thicknesses of 300 and 900 nm, respectively. The mode field distribution under different thicknesses has different areas, which is shown in Fig. 1(d). It represents the relationship between the mode size and the thickness of the film, which is approximately linear. For the guiding wave confined in the film, the larger the thickness is, the smaller the mode field mismatch is, and the higher the coupling efficiency is. It is worth noting that we use Z-cut 5% MgO-doped LN thin film at 25°C. nf of LN satisfies the Sellmeier equation^[23,24], and ns of silica is uniformly set as 1.45. Assuming that the wave propagates along path ABCD in Fig. 1(b), the total horizontal (Z axis direction) phase shift is: 2k0nfcosθ·h+φfc+φfs.(3)

Fig. 1. (a) Schematic diagram of LN thin film. The blue light line represents the light beam at 1018 nm propagating in the planar waveguide. (b) A sectional view of the waveguide and propagation of guiding mode waves. (c) Mode field distributions when the thicknesses of the film are 300 and 900 nm, respectively. (d) The mode size versus different thicknesses of the film.

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Here, k0 represents the wave vector in a vacuum, θ represents the reflection angle, and φfc and φfs represent the total reflection phase shift of point B and point C, respectively. It should be noticed that the TE- and TM-polarized definitions of φfc and φfs are different. They are given bytanφfc2={−nf2sin2θ−nc2nfcosθTE−polarized,−nfnf2sin2θ−nc2nc2cosθTM−polarized.(4)tanφfs2={−nf2sin2θ−ns2nfcosθTE−polarized,−nfnf2sin2θ−ns2ns2cosθTM−polarized.(5)

Generally, when light propagates along the X axis in a Z-cut LN thin film, TE and TM guiding modes represent the ordinary (o) and extraordinary (e) waves, respectively.

In order to form a guiding mode in the guiding layer, Eq. (3) has to satisfy the eigenvalue equation^[25]2k0nfcosθ·h+φfc+φfs=2mπ,(6)where m(m=0,1,2…) is the mode number. A finite number of m means the reflection angle is also limited. The effective refractive index neff(neff=nfsinθ) represents different modes for a specific waveguide structure. Incident angles of the FW that can be constrained in the waveguide are also discrete. That means that different modes correspond to different θ. neff is determined by nf, θ, and m for a specific waveguide structure (fixed values of nc, ns, and h) from Eq. (6). A searching method called stepwise extension is used to solve the eigenvalue equation and to obtain the effective refractive index of the planar waveguide. In order to generate an SH wave, we chose an LN thin film planar waveguide as a specific structure corresponding to different wavelengths that could satisfy the phase-matching condition neff−FW=neff−SH (neff−FW and neff−SH stand for the effective refractive indices of the FW and SH, respectively).

First, we analyzed the relationship between the wavelength λ and the effective refractive index neff for specific thicknesses of the layer h and polarizations. Figures 2(a) and 2(b) represent a dispersion relation of zero-order TE and TM mode， which are constrained in a thin film waveguide with different thicknesses for FW and SH waves, respectively. It shows that neff has a linear decrease trend with increasing λ, and neff of the TE-polarized wave is larger than for the TM-polarized wave for both FW and SH waves. We also found that the larger thickness, the larger neff.

Fig. 2. (Color online) Dispersion relation of TE- and TM-polarized modes for (a) FW and (b) SH waves in LN thin film with different thicknesses at 25°C. h1, h2, h3, h4, h5, and h6 represent thicknesses of the layers, which are 500, 600, 700, 800, 900, and 1000 nm, respectively.

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Then, we analyzed two types of SHG that both meet the phase-matching conditions. Type I means two incoming waves of the FW have the same polarizations, which includes four cases: e+e→e, e+e→o, o+o→e, and o+o→o. Type II means their polarizations are orthogonal with respect to each other, which includes types e+o→o and e+o→e.

In Fig. 3, we presented three type I mode phase matchings for 900 nm thin film. Here, in definition, the case of o+o→o means that both FW and SH waves are TE-polarized. The phase matching of e+e→o is invalid because in LN (point group 3 m) there is no corresponding nonlinear tensor element^[19]. The lines in different colors represent the effective indices neff of different modes, where solid and dashed lines represent FW and SH waves, respectively. The intersections between different lines are phase-matching points. We found that the overlap integrals of the interaction waves were 0.0681 for the 0th to 2nd modes and only 0.0020 for the 0th to 1st modes^[26]. Taking those into consideration, some phase-matching conditions cannot be used to achieve the effective SHG because of the poor overlap of the space-mode field distributions with each other, such as the overlap between odd and even mode fields. We marked the bad overlap with black points. On the contrary, the red points have a good overlap where the SH wave can be generated efficiently. In Fig. 3(c), point A satisfies the phase matching and space-mode fields matching simultaneously, where SHG could exist near 1018 nm in the 900 nm thickness of the LN thin film. Then we show this SHG experiment around 1018 nm in the experimental part, since it utilizes the largest nonlinear coefficient d33.

Fig. 3. (Color online) Three-mode phase-matching diagrams in 900 nm LN thin film: (a) o+o→o; (b) o+o→e; (c) e+e→e. The solid and dashed lines represent FW and SH waves, respectively, and the black, red, blue, and green lines represent the 0th, 1st, 2nd, and 3rd orders of the space guiding modes. The black points on the intersection stand for nonoverlapping of the space-mode field distribution for the FW and SH waves. The red points stand for the simultaneous satisfaction of the space-mode overlapping and phase-matching modes of the FW and SH waves. (d) The wavelength of the phase-matching point between the 0th FW and 2nd SH waves varies almost linearly with the thickness of the thin film.

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In Fig. 3(d), we draw the whole natural phase-matching “map,” which represents the relationship between the phase-matching wavelength and the thickness of the thin film. As shown in Fig. 3(d), the wavelength of the phase-matching point increases with the increase of thickness in three effective type I phase-matching situations: e+e→e, o+o→o, and o+o→e.

The type II SHG shown in Fig. 4(a) means that the FW of 1.55 μm has the orthogonal polarizations, and includes the two possible phase-matching cases: e+o→o and e+o→e. We just checked one case of e+o→o. When the FW of 1.55 μm with both the TM- and TE-polarized is incident into LN thin film (as shown in Fig. 1), SH should be TE-polarized. The black solid line stands for the FW with TM- and TE-polarized of zero-order mode fields, while black dashed line stands for TE-polarized SH wave with of the zero-order mode field.

Fig. 4. (Color online) (a) Type II mode phase matching for SHG: e+o→o, where the FW is 1.55 μm with both the TE- and TM-polarized modes, SH is 0.775 μm with the TE-polarized mode. (b) A schematic of SHG(e+o→o) and SPDC(o→e+o).

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In Fig. 4(a), the two line intersects in point A (839 nm, 1.6373), which means SH generates in the case when the thickness of LN thin film is 839 nm and the effective refractive indices of the FW and SH are both equal to 1.6373. Point A stands for the condition in which SHG could happen. Oppositely, point B cannot generate SHG efficiently due to the lack of space-mode field overlap, as we mentioned above. In particular, the phenomena of the SHG and SPDC could be inverse to each other in nonlinear optics, which is shown in Fig. 4(b). It has been proven that SPDC is one of the most effective methods of generating single-photon pairs, including polarization entanglement photon-pair sources, which plays an important role in quantum communication^[27,28].

Experimentally, we chose the predicted-mode phase-matching condition of point A in Fig. 3(c) to generate a SH wave by using the largest nonlinear coefficient d33. The experimental setup is shown in Fig. 5(a). At a room temperature of 25°C, a picosecond pulse laser with 2.4 ps at 1018 nm was focused into the single-crystal 5% MgO-doped congruent LN thin film sample of 900 nm thickness (supplied by NANOLN). The first polarizer behind the laser source was used to keep the fundamental light as an extraordinary light. The first lens with a 100 mm focal length (Daheng Optics, GCL-010129) was used to focus and couple the incident beam into the LN thin film, which was placed on a five-dimensional adjustable platform. The second 20× objective lens with NA=0.40 (BOCIC, OMO103) was to focus the pump and green light after the crystal into the corresponding detectors. The second polarizer was used to measure the polarization of the SH wave. A prism before the detector could separate the FW and SH waves. The detectors are placed to measure the light intensity and the spectrum.

Fig. 5. (Color online) Mode phase matching SHG experiment. (a) The sketch of the experimental setup. (b) The top view of the green light field captured by the camera. (c) The spectra of the input pulse laser and the output SH light at 509 nm (for the 1018 nm pump light), respectively. (d) The quadratic intensity relationship between the TM-polarized SH and FW waves.

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We generated an SH wave with a wavelength of 509 nm successfully. Figure 5(b) shows the top view of the green light captured by camera. The spectra of the pump and the SH waves are shown in Fig. 5(c). The generated SH light in single-crystal LN film is located at 509 nm. The central wavelength of the input picosecond pulse laser is 1018 nm. It was in agreement with one of our theoretical predictions. We also verified the quadratic relationship of the light intensity between the pump and generated SH green light, as shown in Fig. 5(d). The conversion efficiency of SHG is about 0.12%/(W·cm2)^[29], which is limited by the coupling efficiency. Integrating the grating structure or using the endface-coupling method might be worthwhile methods to improve the coupling efficiency^[30,31].

In this Letter, we introduce a method of the eigenmode equation for a planar waveguide to analyze the mode phase-matching conditions. The natural types I and II phase-matching conditions are achieved by calculating the effective refractive indices with different polarization modes for single-crystal LN thin films with different thicknesses less than 1 μm. For the SHG of incident waves with parallel polarizations, the type I phase-matching includes three cases: e+e→e, o+o→e, and o+o→o; and we also verify the mode phase matching of e+e→e by the SHG experiment with d33 at 1018 nm. As for the type II phase matching with orthogonal polarizations, we show one-mode phase matching in the case of e+o→o. Type II nonlinear interaction may have significant applications in generating polarization-entangled photon-pair sources by SPDC when combined with a whispering-gallery-mode resonator in LN thin film. The diversified natural mode phase-matching processes in single-crystal LN thin film can find some interesting applications in on-chip photonic integrated devices.