1. Introduction

The tendency for miniaturizing modern devices has stimulated the research of nanomaterials such as nanoparticles [1–4], nanoneedles [5,6] and nanowaveguides [7] (NWs). From all types of nanomaterials, NWs stand out for their ability to confine and guide light that has already been demonstrated in several materials such as semiconductors like Si [7], ZnO [8], CdS [9], SnO₂ [10], and also in polymers [11]. Thanks to this guiding ability, NWs are one of the essential components of the future optical devices that involve light transfer. For example, NWs have been applied to develop sophisticated optical applications such as lasing [12,13], all-optical switching [9], localized excitation [10] and sensing [14,15].

The NW application can be further developed by involving nonlinear optical effects such as second-harmonic generation for upconverting the light [16–18]. By spectrally separating the incident and the generated light a natural contrast mechanism occurs, which can provide numerous advantages to control light with light [19,20]. Recently, generation and waveguiding of the second-harmonic (SH) signal was demonstrated in semiconductor nanomaterials like CdS [17], and perovskite nanomaterials such as KNbO₃ [18], [21], NaNbO₃ [21] and LiNbO₃ [21–24]. This nonlinear property has been used for developing applications such as localized imaging [18] and all-optical switching [17].

Further development of nanowaveguide-based applications can be complemented by controlling the directionality of the guided light, which is important for increasing the numerical aperture of the emitted light for imaging applications [10], [18], [21] or engineering light-emitting diodes [25–27]. This directionality can be controlled by shaping the end facet of the NW, which is crucial for light delivery and it has been mostly studied theoretically [11]. Alternatively, NWs with different cross-sections would also lead to different modes, thus bringing different directionalities [28]. Here, we propose to control the directionality by using the properties of nonlinear optical phenomena in nanowires to change the guided mode and, thus, change the directionality. We take advantage of the phase-matching (PM) condition between the fundamental (FH) and SH modes. The PM condition selects a single SH mode in our sub-micron nanowaveguides. We use this SH mode to characterize the SH light emission from the NW.

Several techniques for studying the directionality of light have been developed. The directionality of light can be studied with a scanning detector technique [29–31], with a far-field technique [32,33] and with optical Fourier transformation [34–39]. The scanning detector technique consists on an angular detection, which is realized by rotating a detector around a waveguide output and is applied to study light directionality of macroscopic waveguides only [29–31]. On the other hand, the far-field technique relies on direct imaging of the NW facet. When the direct image is defocused, the far-field pattern is obtained. The main disadvantage of this technique is the calibration in terms of the light emitting angles. Thus, the most appropriate technique is the optical Fourier transformation. It is realized by a set of optical lenses located at a certain distance from each other and does not require any scanning procedure. This technique is already applied for studying directionality of the guided signal in gold NWs [34–36] and scattering of luminescent light by InP [37,38] and GaAs NWs [39]. The obtained image maps the directionality of the emitted light because the light spatial spectrum is observed at the back-focal plane of the collection objective, which provides information about the orientation of the light with the normal and azimuthal angles.

In this work, we investigate the directionality of second-harmonic guided signal in lithium niobate (LiNbO₃) NWs. LiNbO₃ is a suitable material choice because it is transparent from the ultraviolet to the far infrared (0.33 μm to 5.5 μm) and shows a relatively high second-order nonlinearity in terms of the χ⁽²⁾ 2nd order susceptibility tensor components [40,41]. We show that the directionality of the nonlinear emitted light from the LiNbO₃ NWs can be controlled via a selection of SH modes under PM conditions. The mode switching mechanism is led by varying the incident laser wavelength. We can also use PM to determine the mode and apply it for a stronger directionality with the optical Fourier transformation technique. Then, this enhanced SH mode shapes and defines the nonlinear light distribution.

2. Sample preparation

In order to fabricate perovskites structures with micron sizes, there are fabrication methods that are efficient in terms of time and cleanroom resources [42]. In our case, since we aim to fabricate LiNbO₃ NWs with sub-micron cross-sections, the NWs are fabricated in a LiNbO₃ wafer by the ion beam enhanced etching method [43] [Fig. 1], which is more complex than previous methods. The first step of the fabrication flow is the deposition of fused silica by plasma enhanced chemical vapor deposition, chromium by physical vapor deposition and the e-beam resist by spin-coating on the LiNbO₃ wafer in this chronological order. In the deposited layers, we write a mask that defines the position and dimensions of the to-be-fabricated NWs. First, we write the mask in the e-beam resist by e-beam lithography and we further transfer it into the layers of chromium and fused silica by reactive ion etching.

 

Fig. 1 (a) SEM image of the entire LiNbO₃ NW half-suspended in air. (b) SEM image of the NW output facet with the corresponding crystal structure. The rectangular NW cross-section has 626 ± 31 nm in width and 664 ± 50 nm in height.

Download Full Size | PDF

After the e-beam resist is removed, argon ions irradiate the LiNbO₃ wafer to amorphize the wafer regions that are not protected by the chromium mask. The layers of chromium and fused silica are then removed and the wafer is irradiated by the He ions. Thanks to the lighter mass of the He ions, they penetrate deeper into the LiNbO₃ wafer to create a buried amorphous layer at a certain depth. The depth is controlled by adjusting the fluence and acceleration voltage of the He ions. The argon-ion irradiation was performed at different energies: first at the energy of 600 keV with the fluence of 7.2·10¹⁴ cm⁻² and further at the energies of 350 keV, 150 keV and 60 keV with the fluence of 1.38·10¹⁴ cm⁻². The helium-ion irradiation was performed at the energy of 285 keV with the fluence of 1.38·10¹⁴ cm⁻² [43]. Further, the wet etching is applied to the LiNbO₃ wafer. Since crystalline LiNbO₃ has a high chemical resistance, only the amorphized regions of the wafer are etched away and, as a result, LiNbO₃ NWs are formed on a membrane.

The NWs are fabricated in an x-cut LiNbO₃ crystal. Thus, the x-axis of the crystal structure is perpendicular to the substrate, the z-axis is parallel to the substrate and the y-axis lies along the NW longitudinal axis.

The LiNbO₃ NWs are transferred to the SiO₂ substrate by a micromanipulator. The micromanipulation technique relies on electrostatic forces between the NWs and the SiO₂ microtips. The micromanipulator allows one to move the NW and precisely place it in a desired position. In Fig. 1(a), we show one of the studied NWs that lies at the edge of the SiO₂ substrate and is partially suspended in air. In this configuration, scattering of the guided light by the substrate is avoided during the out-coupling of the SH signal at the end of the NW [Fig. 1(b)]. The NW output faces the collection objective and the SH signal is recorded from the output of the NW.

3. Experimental setup

In Fig. 2, we show the schematic of the experimental setup that consists of two parts.

 

Fig. 2 (a) Schematic of the experimental setup for the optical Fourier transformation technique measurements, which includes a control part for the pump beam and an imaging part for the SH detection (both real and spatial spectrum images). (b) The real image of the output facet is obtained with the Image Camera. (c) The spatial spectrum image is obtained with the Fourier camera, which collects the SH light emitted under 53.13° (graph with radial steps of 10°). Inset: Basic principles of the optical Fourier transformation technique in NWs. See the full description in the text.

Download Full Size | PDF

The first part is a homemade transmission microscope that is used for coupling the laser light into a NW.

The laser beam from a femtosecond tunable Ti:sapphire laser (with repetition rate of 79.5 MHz and pulse duration of 100 fs at the laser output) is steered perpendicularly to the sample substrate and is focused with a lens L₁ onto the input facet o f a NW. To control the coupling process, we collect the transmitted laser beam and the sample image with a 20x objective Obj₁ (0.40 NA) behind the sample. The collected light is further imaged on the control camera [Fig. 2(a)]. The lens L₁ and the sample holder are placed on 3D stages to control the coupling process. We use a half-wave plate at the setup input to change the polarization of the laser and maximize the power of the generated and guided SH signal.

The second part of the experimental setup is used for collecting and processing the guided signal from the NW output facet. To collect the guided signal, we use a 100x objective Obj₂ (0.80 NA) that is placed co-axially with respect to the NW and is focused on its output facet. We use the lenses L₃ and L₄ that have the same focal lengths of f and are placed at a distance of 2f from each other. Lens L₃ builds a real image of the output signal in its focal plane, which coincides with the back-focal plane of L₄. In turn, lens L₄ performs an optical Fourier transformation of the collected signal and images its spatial spectrum on the Fourier camera [Fig. 2(c)]. Each point of the spatial spectrum corresponds to a particular set of polar ϑ and azimuthal ϕ angles [see inset of Fig. 2(a)]. Each set of these angles describes the direction of a particular wave-vector of the studied signal. Thus, the image of the spatial spectrum provides information on the directionality of the guided light. Since the numerical aperture of the collection objective Obj₂ is 0.80, the polar angle ϑ is limited by 53.13°. Starting from the center of Fig. 2(c), each circular dashed line represents an increment of 10° in the polar angle ϑ. The azimuthal angle ϕ covers the full range from 0° to 360°. In the focal point of the lens L₃ and back-focal point of lens L₄, we place an aperture to block any noise and transmit only the signal that is collected from the output facet. Therefore, the measured spatial spectrum only relates to the Fourier spectrum of the out-coupled SH signal.

There is also the second path in the collection part of the setup with the lens L₅. In this path, we perform the back-Fourier transformation of the spatial spectrum and obtain the real image of the collected signal at the image camera [Fig. 2(b)]. A flip-mirror enables to image either the spatial spectrum or the real image of the collected signal.

We use bandpass filters to block the laser light and only transmit the SH signal by placing them in front of all cameras.

4. Results and discussion

To show mode switching, we use a LiNbO₃ NW with a non-rectangular cross-section of 573 ± 10 nm in width and varying heights of 522 ± 15 nm and 745 ± 21 nm [see inset of Fig. 3(a)] and length of 52 μm. We sweep the laser wavelength from 790 nm to 997 nm with steps of 3 nm and measure the power of the SH response at the output facet at each laser wavelength [Fig. 3(a)]. The SH power is obtained by integrating the entire SH output signal [Fig. 2(b)]. The SH signal from the NW output facet is recorded with the image camera of the experimental setup [Fig. 2(b)]. In Fig. 3(a), we show the obtained curve of the SH power versus the laser wavelength. The curve reveals two peaks at 862 nm and 924 nm which arise from the PM mechanism.

 

Fig. 3 (a) SH nonlinear spectrum for a LiNbO₃ NW with a non-rectangular cross-section. The NW width is 573 ± 10 nm and the height varies from 522 ± 15 nm to 745 ± 21 nm . It contains two PM peaks. The nonlinear light distributions of the NW at different wavelengths were taken. (b), (c), (d), (e), (f) are the spatial spectra at the wavelengths λ_FH = {831, 862, 893, 924, 955} nm, respectively. The intensity of all the spatial spectra are normalized with respect to the maximal signal, at λ_FH = 924 nm. The light is better confined at the PM peak because the signal-to-noise ratio is the highest among all Fourier images. Also, different light distribution are obtained at the two PM peaks.

Download Full Size | PDF

Two PM peaks appear where the conversion efficiency between the fundamental and the second-harmonic modes is maximized. In the PM regime, the effective refractive indices n_FH and n_SH are matched and the overlap integral between the FH and SH mode profiles at that specific wavelength is maximized (the full method is described in previous works [44–46]). Thus, the PM behavior strongly depends on the shape and the size of the NW cross-section.

Since at each PM wavelength different SH modes are enhanced in the process, these modes can be switched by shifting the laser wavelength from one PM peak to another. Hence, it is expected that the directionality of the guided SH light changes with wavelength [28]. To demonstrate the change of the directionality, we record spatial spectra of the guided SH signal at the laser wavelengths of 831 nm, 862 nm, 893 nm, 924 nm and 955 nm. In Fig. 3(b)-3(f), we show the recorded images of the spatial spectra at the respective wavelength. All images are normalized by the maximum of the spatial spectrum at 924 nm [Fig. 3(c)].

When analyzing the obtained spatial spectra images, we make two observations. First, the spatial spectra significantly vary for each wavelength. We refer this observation to the fact that, at each laser wavelength, both coupling of the laser light into the guided FH modes and generation of the SH modes by each FH mode are different [33], [44]. Since the waveguide modes define the directionality of the guided light, the light directionality varies at each wavelength. Thus, the spatial spectra at the phase-matched wavelengths also vary. The SH light at 431 nm (pump wavelength at 862 nm) propagates more in the coaxial direction of the NW [strong red color in the center of Fig. 3(b)]. Whereas the SH light at 462 nm (pump wavelength at 924 nm) propagates in terms of three lobes with an angle to the optical axis of the NW, i.e., a non-coaxial signal [Fig. 3(c)]. Thus, each guided mode has its own light distribution. In the case of low order SH modes, the spatial spectra obtained by a single image, without any scanning procedure, can be used to distinguish the guided SH modes, as it will be shown later. Since the light distributions of the PM peaks in Fig. 3(b)-3(c) are different due to the different excitation of SH modes, mode switching of the SH modes is achieved in a single NW by tuning the pump wavelength. As a conclusion one can engineer the directionality of the guided light by tuning the laser wavelength.

Second, both PM peaks have intensities in the same order of magnitude [Fig. 3(a)]. In the rest of the spatial spectra, though, the intensity drops between 1 and 3 order of magnitudes [Fig. 3(f) and Fig. 3(d), respectively], since no SH mode is enhanced. Therefore, the SH light distribution is a SH multimode superposition of non-enhanced SH modes. As a result of this multimode behavior, the light distribution can even be almost homogeneously distributed (Fig. 3(d) at λ_FH = 831 nm, for instance), but with a very low signal intensity. Therefore, the light distribution cannot be attributed to a single SH mode, like in the PM case. However, between the two PM peaks (Fig. 3(e) at λ_FH = 893 nm) it can be observed how the nonlinear light distribution partially retrieves each PM peak distribution. Indeed, a coaxial signal as well as lobes are present on Fig. 3(e). In this case, the two modes responsible of PM at different wavelengths (λ_FH = 862 nm and 924 nm) are mismatched. With the rest of the modes, they form a superposition of SH modes and, as a result, the out-coupled signal presents a lower intensity than in the PM peak cases.

To find out which mode is phase-matched, we develop a semi-analytical model that can fit the SH power spectra. Mode simulations were performed using the finite element method (FEM). In the simulations, three FH modes, i.e., TM₀₀, TE₀₀ and TM₁₀, coupled to 30 SH modes are considered. We assume that the three FH modes are coupled into the NW with the same power. The conversion efficiencies between each FH mode with each SH mode vary. Then, the conversion efficiency is maximized for the phase-matched modes [44].

In Fig. 4(a), we show the SH response of the NW shown in Fig. 1, with typical cross-section dimensions of 626 ± 31 nm in width, 664 ± 50 nm in height and length of 43 μm and the corresponding simulated values. Contrary to the NW displayed in the inset of Fig. 3(a), the NW cross-section is uniform along its length, which allows one to perform accurate SH mode simulations. The simulations are performed for all cross-sections within the uncertainty range of the SEM measurement. The best match is obtained at the width of 615 nm and height of 660 nm.

 

Fig. 4 (a) Measured SH and semi-analytical simulation of the phase-matching response of the LiNbO₃ NW in the upper VIS-NIR range spectrum, normalized with respect to the maximum conversion efficiency. (b) Enhancement of the phase-matched modes at the pump wavelengths of 906 and 946 nm.

Download Full Size | PDF

Two experimental PM peaks are again observed in the considered spectrum range (790-982 nm). The two PM peaks are located at the SH wavelengths 456 nm and 474 nm (912 nm and 948 nm at the FH wavelength, respectively) [Fig. 4(a)]. Due to the coupling configuration that we use, we do not control the modes at the fundamental harmonic (FH) that are coupled into the NW. We also do not control and cannot estimate how much power each FH waveguide mode carries. However, the power of the guided FH modes define the height of the phase-matching peaks. Thus, to match the experimental and simulation results, we adjust the height of the phase-matching peaks [multiplying factors x4, x10 and x1 in Fig. 4(a)].

Then, the two theoretical PM peaks are slightly shifted with respect to the experimental PM peaks of Fig. 4(a) (for the FH wavelengths, 6 nm at the 912 nm experimental PM peak and 2 nm at 948 nm). The simulated cross-section gives the best fit to the experimental PM peaks and falls within the cross-section range given by the SEM images, with a width of 626 ± 31 nm and a height of 664 ± 50 nm. Still, the small shifts at the PM peaks indicate that the NW facet has an imperfect squareness due to the etching process during fabrication, while in the simulation round edges were not considered. Even though this factor has the largest impact in the simulations, the simulated PM curve can be also influenced by other factors. For instance, the NW can present inhomogeneities along its length, i.e., its longitudinal axis. Therefore, the effective refractive index of the modes can slightly vary along the NW length. In addition, the roughness of the output surface could also modify the out-coupling of light, since the NW facet is not perfectly flat.

In our case (for this wavelength range and the cross-sections mentioned above), only one SH mode is enhanced. The quasi-TM₁₀ and the quasi-TE₀₀ FH modes (from now on all quasi-modes are referred only with the type of mode, i.e., TM₁₀ or TE₀₀, for instance) are clearly predominant at wavelengths 906 nm and 946 nm, respectively. The theoretical strongest conversion efficiency is obtained at λ_FH = 946 nm [Fig. 4(a)]. In our NWs, we mostly couple FH light within the first three FH modes. Additionally, with PM this conversion efficiency increases with the length of the NW (theoretically there is a saturation limit, not reached at micrometer lengths [40]). For these NW cross-sections, only one SH mode is phase-matched with a FH mode. The intensity of this SH coupled mode will increase if there is PM. In the case of the PM peak at 906 nm, the TM₁₀ FH mode is phase-matched with the TE₂₂ SH mode, whereas at 946 nm the TE₀₀ FH mode enhances the TE₂₀ SH mode. The intensities of the enhanced SH modes increase along the NW length [TE₂₀ and TE₂₂ SH modes in Fig. 4(b)]. The rest of the SH modes show a phase mismatch, i.e., their intensities follow a periodic pattern and are orders of magnitude below the phase-matched SH mode intensity (in Fig. 4(b) we only show the TE₀₀ SH modes at both wavelengths, but the same type of periodic pattern is obtained for all the non-enhanced SH modes). Consequently, within the PM regime they are negligible and the SH signal that is out-coupled to free space is given by the phase-matched SH mode.

Knowing the SH modes at the phase-matched wavelengths, we calculate their distributions using a finite difference time domain simulation (FDTD), and compare them with the experimental results. We choose a specific waveguide mode at specific wavelength and calculate its light distribution that represents the spatial spectrum.

In Fig. 5, we compare the calculated real and spatial spectra images for the TE₂₂ mode at the FH wavelength of 906 nm and the TE₂₀ mode at FH wavelength 946 nm with the experimental results. In the latter case, for the TE₂₀ mode the theoretical [Fig. 5(b)] and experimental [Fig. 5(c)] spatial spectra show a very good match. The theoretical spatial spectra include the SH light distribution up to 80° with the green circle in Fig. 5(b) and Fig. 5(e), whereas the red circle includes the 53.13° limit in the experimental setup from the numerical aperture of the collection objective. The experimental SH spatial spectra were obtained with the Fourier camera displayed in the experimental setup [Fig. 2(c)].

 

Fig. 5 Second-harmonic mode profiles of the phase-matched modes [left graphs, (a) and (d)] and their theoretical [middle graphs, (b) and (e)] and experimental [right graphs, (c) and (f)] spatial spectra. In all spectra, the red circles represent the experimental collection range (up to 53.13°), whereas the green circles are the simulation domain limits (up to 80°). Each white circular dashed line represents 10°, whereas the straight lines represent the azimuthal angle at 0°, 45°, 90°, 135°, 180°, 225°, 270° and 315°.

Download Full Size | PDF

In Fig. 5(a)-5(c), at the SH wavelength of 473 nm (or FH wavelength of 946 nm), theory and experiments for the SH modes match accurately and show strong SH lobes on both sides, reaching the limit for the polar angle (ϑ = {40, 53}°) and on the sides for the azimuthal angle; ϕ_right = {330, 30}° and ϕ_left = {150, 210}°) and a weaker spot in the center of the spatial spectrum (ϑ = {0, 15}°, ϕ_center = {0, 360}°. This case corresponds to the theoretical strong conversion efficiency shown in Fig. 4(a)-4(b). However, for the TE₂₂ mode at the SH wavelength 453 nm (thus, FH wavelength of 906 nm), the theoretical and the experimental spatial spectra do not match. Here, the experimental spatial spectrum shows four lobes, two of them within 20° in the polar angle ϑ, whereas the simulated spatial spectrum shows four lobes at the experimental limit, plus one single lobe within 10° in the polar angle. One of the reason for the mismatch could be the high order of the SH mode that may not be resolved by our setup. The enhanced high order modes confine light close to the edges of the NW cross-section. These small lobes do not fall within our detection range (red circle in the spatial spectra of Fig. 5, which represents 53.13°) and, thus, cannot be resolved in the experimental images.

Then, the SH mode identification can only be carried out for SH low order modes [Fig. 6] because their characteristic spatial spectrum falls within 53.13°. According to simulations, the quasi-TE₃₀ mode [Fig. 6(f)] is the highest order mode that is distinguishable with the experimental limitations of our setup because it is the highest order mode that contains characteristic features within the setup limits (again, red circles in Fig. 5 and Fig. 6 with the limit of 53.13°). The quasi-TE₂₂ mode shown in Fig. 5(d)-5(f) is a higher order mode (in terms of the effective refractive index) than the quasi-TE₃₀ mode [Fig. 6(f)] and, therefore, its distinctive lobes are not fully resolved in our experiment. Using a higher numerical aperture objective would help to collect more signal and include more signal features of the modes. Apart from simply changing the free-space collection objective to one with higher numerical aperture, oil-immersion objectives would also increase this numerical aperture, but their main drawback is that they are very challenging to incorporate into our setup due to the half-suspension of the NWs [Fig. 1].

 

Fig. 6 Simulated spatial spectra of the out-coupled SH guided modes at the SH wavelength 473 nm (FH wavelength 946 nm) for a LiNbO₃ nanowaveguide with a rectangular cross-section with 660 nm in height and 615 nm in width. The displayed modes are the (a) quasi-TE₀₀, (b) quasi-TE₀₁, (c) quasi-TE₁₀, (d) quasi-TE₀₂, (e) quasi-TE₁₁ and the (f) quasi-TE₃₀ modes, respectively. The red circle represents the angular limit of the experimental setup, 53.13°. At this particular wavelength, the quasi-TE₃₀ mode is the highest order mode that is distinguishable in our experimental setup.

Download Full Size | PDF

5. Conclusions

We have presented the first Fourier study of SH in LiNbO₃ NWs at the phase-matching regime. We achieved mode switching in a single NW by tuning the excitation wavelength. By doing so, the nonlinear spectrum shows that the nonlinear light distribution is clearly confined in the presence of a phase-matched intensity peak. More interestingly, two completely different light distributions are obtained due to the selection of a different phase-matched SH mode, allowing the mode switching to be easily detected.

At a single PM peak with strong conversion efficiency we have been able to theoretically simulate and experimentally distinguish the profile of the phase-matched SH mode (Fig. 5 at the FH wavelength of 946 nm). In the case of a lower conversion efficiency (Fig. 5 at the FH wavelength of 906 nm), a high order SH mode does not reproduce the simulated single SH mode Fourier distribution because of the strong angular emission.

Further engineering of the cross-section of the NWs (shaping the output facets, as simulations have shown for linear light in [11], for instance) can better localize the SH out-coupled signal in different regions of space. This means that nonlinear signals can be spatially enhanced and, therefore, developed into several applications such as spectroscopy or local light delivery through nonlinear NWs. This provides a powerful tool for a more compact design of up-conversion light sources, since photonic circuits can benefit from this control on the SH light directionality and use a single NW as an optical switch.