Direct visualization of phase-matched efficient second harmonic and broadband sum frequency generation in hybrid plasmonic nanostructures

Second harmonic generation and sum frequency generation (SHG and SFG) provide effective means to realize coherent light at desired frequencies when lasing is not easily achievable. They have found applications from sensing to quantum optics and are of particular interest for integrated photonics at communication wavelengths. Decreasing the footprints of nonlinear components while maintaining their high up-conversion efficiency remains a challenge in the miniaturization of integrated photonics. Here we explore lithographically defined AlGaInP nano(micro)structures/Al2O3/Ag as a versatile platform to achieve efficient SHG/SFG in both waveguide and resonant cavity configurations in both narrow- and broadband infrared (IR) wavelength regimes (1300–1600 nm). The effective excitation of highly confined hybrid plasmonic modes at fundamental wavelengths allows efficient SHG/SFG to be achieved in a waveguide of a cross-section of 113 nm × 250 nm, with a mode area on the deep subwavelength scale (λ2/135) at fundamental wavelengths. Remarkably, we demonstrate direct visualization of SHG/SFG phase-matching evolution in the waveguides. This together with mode analysis highlights the origin of the improved SHG/SFG efficiency. We also demonstrate strongly enhanced SFG with a broadband IR source by exploiting multiple coherent SFG processes on 1 µm diameter AlGaInP disks/Al2O3/Ag with a conversion efficiency of 14.8% MW−1 which is five times the SHG value using the narrowband IR source. In both configurations, the hybrid plasmonic structures exhibit >1000 enhancement in the nonlinear conversion efficiency compared to their photonic counterparts. Our results manifest the potential of developing such nanoscale hybrid plasmonic devices for state-of-the-art on-chip nonlinear optics applications.


Experimental setup and choices of input fundamental wavelengths
shows that when the sample is excited at 1300 nm, the SHG is centred around 650 nm and does not cause re-emission of PL. For any wavelength longer than 1300 nm, we believe that they only cause SHG or SFG processes in the materials and the re-absorption of SHG and SFG at certain wavelengths.

Comsol simulation
Wave Optics module in COMSOL MULTIPHYSICS 5.2a simulation package was used to simulate the SHG in frequency domain. To accurately model the SHG intensity collected by the objective in the far field, a 3D model was used. A 3D Gaussian beam (beam waist w0 is set the same value as /2NA,  the fundamental wavelength) was applied as background field to approximate the excitation beam. The SHG process is entered by the nonlinear polarization vector (discussed in details in section 11) and the frequency domain stationary solutions were obtained for a series of simulation parameters (see Fig. S8 for the examples of E field distribution at the FW and Fig. S4 for the examples of E field distribution at the SHG frequency). In order to reduce the calculation time and yet still capture the essential physics, the simulations were run with scattering boundary condition and without perfect matching layers. For the calculations of resonant modes in a cavity at the FW, the eigenfrequency solver was used.

Beat patterns observed on 113 nm  580 nm  8 µm AlGaInP waveguide released on Al2O3/Ag at 45 input polarization
As mentioned in the main texts, the period of the beat pattern observed on 113 nm  584 nm  8 µm AlGaInP waveguide released on Al2O3/Ag at 45 input polarization is not sensitive to the wavelength from 1300 nm to 1500 nm. This effect comes from the dispersion relationship of TM0 and TM1 mode at the fundamental wavelengths, as shown in the figure below. The periodicity of the beam pattern generated by the combination of TM0 and TM1 modes can be approximated by 2/(kTM0-kTM1) = 0/(nTM0-nTM1). From 1300 nm to 1500 nm, the periodicity varies from 2.35 µm to 2.45 µm, corresponding to only a 4.3% change. This is the reason we called them 'insensitive'. The effective refractive indices in Fig. S3 is obtain by COMSOL eigenfrequency solver.

Optical intensity evolution of FW and SHG along propagation distance simulation
The optical intensity evolution is simulated following reference 1,2 , where FW and SH are calculated by solving the coupled-mode equations: Here and are the mode amplitudes at FW and SH, respectively. and are the electric field loss coefficients at FW and SH, respectively. Z is the propagation direction and ∆ the phase mismatch constant. The nonlinear coupling efficiency (NCC) 1,2 is calculated using the following equations 1 : where and are the normalized electric field of modes at FW and SH, respectively.
Assuming | |<<| |, which is the case in current experimental condition, the 2 nd term on the right side of Eq. (S1) is ignored. If we further assume the phase matching condition is satisfied, the solution to Eq. (S1) and (S2) can then be obtained analytically 2 : The change of SHG amplitude along z direction is then essentially determined by the two loss constants. The maximum occurs at location where = 0, with expression given in Eq.
(1) of the main texts. Fig. S4(a) gives the maximum as a function of , at two different values. Fig. S4(b) shows the SH electric field |ESH| distribution at 670 nm, where the input FW is polarized along the long axis of the waveguide (90) and edge coupled into the waveguide from the left side, with (I) of 8 µm in length and (II) 4 µm in length.

Calibration of the collection system
In all experiments, we used the same CCD (AVT Prosilica GC1290) to take images. By comparing the quantum efficiency data provided by the company with the SFG spectrum ( Fig.  5(c)), we understand that the SFG spectrum is mostly centred around 680 nm to 710 nm, where the quantum yield of the CCD is between 34% to 29%.
To calibrate the SFG output power, we used a band pass filter (740 nm  10 nm) to filter out a narrow band input light at 740 nm (quantum yield 23.9) from the supercontinuum source. We then used the power meter to measure the laser powers before entering the microscope and after the objective (on the sample) to establish the ratio of light being delivered to the sample. We know from other experiments that the reflectivity of the sample is around 0.9. We then focused the laser around 740 nm onto our sample (on the smooth region) and the focused light spot is recorded by the CCD camera. By integrating the output reading over the region of light spot on the CCD image, we can further establish the relationship between the power of light being delivered to the sample and the total integrated reading from the CCD. We use this relationship to approximate the power of collected SFG signal from the CCD image. This calibration method, however, overestimates the collected SFG power by (34+29)/23.9/2-1 = 31.8%, which is considered in the final conversion efficiency.
The SHG experiments were carried out by the OPO system. In this case, we use the 660 nm red laser pointer. Similar relationship of input laser power at 660 nm with the integrated light spot on the CCD image was obtained. In SHG cases, the calibrated SHG power is normalized against the quantum efficiency difference at various wavelengths of the CCD camera. One point that needs to be noted is that as shown in Fig. S16(b), the objective only collects about 1/5 of the generated SHG. In both SFG and SHG cases, the calibrated results only account for the 'collected' signal, not the really emitted total signal.

Conversion efficiency of waveguides
In the waveguide case, the conversion efficiency is defined as is the waveguided input peak power, − , the waveguided output SHG peak power and L the length of the waveguide. To measure the coupling efficiency of a waveguide, we first measured the reflected power of input light on a smooth region of the substrate, which we call value a. We then measured the reflected power of input light when the light spot was focused on the middle of the waveguide, which we call value b. The value (a-b) approximates the scattered light due to the presence of the waveguide. In the 3 rd step, we positioned the input light spot at the coupling end of the waveguide and measure its reflected power, which we call value c. We approximate the input light that was coupled into the waveguide by (a-c)-(a-b)/2 =a/2+b/2-c and the coupling efficiency would be ((a+b)/2-c)/a. This method, however, still overestimates the input light that couples into the waveguides, meaning that we counted more input light than that was really coupled into the waveguide, but this is the closest value we can obtain from the experiments. The measured input coupling efficiency varies with the input polarization for the 113 nm  584 nm  8 µm waveguide, with 18.6% for TM0 FW mode, 9.3% for FW TM1 mode and 12.1% for 45 polarization. For input fundamental wave polarized along the long axis of the waveguide of 113 nm  584 nm  8 µm, we obtained time-averaged 0.36 nW waveguided SHG signal in the far field from 7.6 mW input power at 1340 nm using OPO, giving a , = 12% −1 −2 . For a shorter waveguide of 113 nm  584 nm  4 µm, the expected , can be further increased to 96% −1 −2 . For the broad band SFG, we measured SFG signal of 9.1 pW at 1.8 mW input power using SC laser, giving a , =42% −1 −2 . For the 113 nm  250 nm  8 µm waveguide, the input coupling efficiency is only 8.3% for TM0 FW mode, with 0.60 pW at 1.8 mW input power, with the , =14% −1 −2 .

Fig. S5
Diagram demonstrating of SHG, with two photons of the same frequency, and SFG processes with two photons of different frequencies.
7. SFG spectrum at the end of 113 nm  584 nm  8 µm waveguide  Once the width of the waveguide is further decreased down to 190 nm ( Fig. S7(b)), waveguided SFG signal can only be achieved at the polarization > 45.

Fig. S9
Polarization dependence SHG image at FW of 1320 nm.

Second order susceptibilities tensor for input wavelengths from 1300 nm to 1600 nm
The second order susceptibilities tensor d are obtained from ref 3 . As detailed in ref. 3 , the naturally cleaved facets of ALGaInP thin film grown on AlGaAs/GaAs are (110) and (1 ̅ 10) facets. The fabricated waveguides are therefore aligned either along [110] or [1 ̅ 10] direction, which is later determined by the polarization dependant second harmonic generation (SHG) intensity study on the circular AlGaInP/GaInP/AlGaInP disks released on Al2O3/Ag. According to ref. 3 , we identify our lab frame second order susceptibilities tensor as: 0d gives the value used in simulation with the unit of C/V 2 . The nonlinear polarization vector P is then written as: This vector is entered in the COMSOL for SHG simulations as discussed in section 2. To obtain an analytical expression for the polar pattern, all three components of electric field at the FW need to be considered. In current experimental setup, the input beam is focused on the sample with mostly x and y components, where we can define E1x = E0cos(-0) and E1y = E0sin(-0). Due to the excitation of the hybrid plasmonic mode at the FW, z component cannot be ignored in the calculation, where we define E1z = E0 and  is the input geometric factor. Once these three components are plugged into Eq. (S3), the nonlinear polarization vector P can be obtained and also has three components. The total SHG emission power PSH is determined by 2 At fundamental wavelengths that are not 1500 nm, we scale all dij values against those obtained at 1500 nm through a calibration sample, a thin AlGaInP structure released on glass. As shown in Fig. S10(a), the second harmonic generation (SHG) is measured through the calibration sample in transmission by a photomultiplier tube at different fundamental wavelengths while the input power is fixed. The lateral dimension of the structure was chosen to be over 5 μm  30 μm, much larger than the beam diameter so the size effect can be ignored.
The measured SHG signal difference is therefore attributed to the dij value change at various wavelengths. To remove the contribution from the interference through the thin AlGaInP layer, COMSOL simulation is run on the structure of 115 nm thick to obtain the SHG values with fixed dij values (at 1500 nm) to show the effect due to fixed thickness at various wavelengths ( Fig.   S10(a)). The square root of ratio of the measured SHG/simulated SHG gives the contribution from the dij values at various wavelengths, as shown in Fig. S10(b). Here, the d ratios are

Input and output responses in SHG and SFG for the disks
The overall SHG can be considered as the combination effects of input response funnction finput with = ∭ , and output response function foutput, SHG with , = ∬ ⃗ • ⃗ (∭ , ) 2 . The designed thickness of the AlGaInP structures is 110 nm. However, the last etching step before releasing the structures to Al2O3/Ag substrate could have slightly over etched or under etched the structures. To find out the simulation results that fits the best to the experiments, various thicknesses around 110 nm were examined by simulations. Fig. S12 shows the simulated finput ( Fig. S12(a)) and foutpu,SHG (Fig. S12(b)) respectively of a 1 μm diameter AlGaInP disk released on the Al2O3/Ag/SiO2/Si substrate at various disk thickness from 90 nm to 125 nm. Using this method, we can evaluate the contributions from the fundamental input beam and that from the SHG separately. The overall conversion efficiency for a cavity is , which is plotted in Fig. S12 (c). We can see from the plots that when the thickness increases, there is a trend showing resonant peaks emerging in output response factor foutput. These peaks, however, are located at different wavelengths relative to the input response. As a result, the overall conversion efficiency response become a broader band with multiple peaks and a 50% to 60% increase in overall conversion efficiency. By comparing the simulation results with our experimental data, we identify that the thickness around 105 nm gives the best fit to the experimental results. Fig. S12(d) and (e) shows the |E| field distribution of the resonant mode at 1320 nm at the thickness of 105 nm. Similarly, we can develop the finput and foutput for SFG processes as well. In this case, finput is defined the same as that in SHG case, while foutput, SFG is defined as ) , which includes two fundamental wavelengths FW1 and FW2.

SFG process with a broadband coherent source
For a broadband light source of n different wavelengths, the nonlinear polarisation PSFG is: Here we assume the scalar form for simplicity. If assuming ( ) = 0 − and the resulted SFG frequencies are different, we have the intensity of the FW and SFG as: In this simple demonstration, we show that the | | 2 | | 2 → 2 at the same input fundamental power.
To obtain ratio larger than 2, some special condition is required for the input light source. This is shown in the equation where we allow the SFG field generated from different 1 and 2 to be summed coherently as long as 1 + 2 = and | 1 − 2 | < ∆ . This requires some level of coherence of the input light at the fundamental wavelengths over a range of frequencies, which is satisfied by the supercontinuum light source. This 'enhancement' is not enabled by the nonlinear material but by the laser source.
The best Comsol simulated SFG results for the disk are shown in Fig. 5(c), where ∆ = 3.8 × 10 13 rad • Hz, corresponding to a width of 40 nm around 1400 nm in wavelength. The simulated spectrum reproduces the shape of the experimental results and the total simulated time-averaged power over the wide wavelengths (1300 nm to 1600 nm) is summed to be a time averaged 0.065 nW with an input power of 1.2 mW, agreeing very well with that measured.

COMSOL simulation on the excitation and SHG emission of 1 µm disk/Al2O3/Ag
The main reason that the E field at the FW is significantly enhanced in the 1 µm disk released on Al2O3/Ag is attributed to the excitation of the TM mode, which we refer as the hybrid plasmonic mode in the main texts. To illustrate this effect, we have run FW/SHG simulations on the 1 µm disk/Al2O3/Ag, a large disk (diameter > 3 µm) /Al2O3/Ag and 1 µm disk/glass an AlGaInP disk. Fig. S13 below shows its electric field amplitude at the FW of three cases up on the excitation of a focused Gaussian beam for comparison. It is clear that the electric field amplitude in the large disk is similar to that of the background excitation field, where we set |Eb|=1 V m -1 , as well as to that in the disk released on glass, but significantly smaller than that in the 1 µm disk released on Al2O3/Ag.
Electric field distribution of 1 µm disk/Al2O3/Ag, large disk/ Al2O3/Agand 1 µm disk/glass at the FW Fig. S13 COMSOL simulation of Electric field |E| at the FW of 1320 nm for the 1 µm disk near metal (a), a large disk near metal (b)and the 1 µm disk on glass (c) when excited by a Gaussian beam. The E field distribution is plotted on a plane perpendicular to the substrate and across the centre of the disk. The electric field within the semiconductor material near metal is strongly enhanced, about 10 times of the value that obtained on semiconductor released on the glass. The dashed rectangles indicate the location of the semiconductor disk in (b) and (c).
In this hybrid plasmonic TM mode, the E field has a strong out of plane (perpendicular to the substrate) component, as shown in Fig. S14 below. The hybrid TM mode and its out of plane component had been discussed in detail in ref [5][6][7] . This TM mode is effectively excited in the normal incidence of the pump light when the dimension of the disk is sub-wavelength (1 µm diameter compared to the >1.3 µm FW), where the free-space photons scatter around the perimeter of the disk. The variation of propagation constants k can be estimated by the Fourier transformation xk1, where x can be approximated by the diameter of the disk and k is therefore large enough to compensate for the momentum mismatch to excite the TM mode. Most importantly, once the out-of-plane component is excited, Ez component induces dipole moment that is perpendicular to the metal surface, see Fig. S15 (a) below. The image charges induce a dipole moment that is parallel with the original one, the electric field in the semiconductor is therefore strongly enhanced. As a comparison, the effect of E component parallel to the metal substrate is demonstrated in Fig. S15(b), in which the induced dipole is opposite to the original one and therefore no enhancement in the dielectric is induced. The metal particle dimers also exhibit similar polarization dependent field enhancement 8 .

Fig. S15
Diagrams showing that a perpendicular dipole to metal will induce an image dipole parallel to the original one (a) while a parallel dipole to metal will induce an image dipole that is opposite to the original one (b).