Highly confined optical modes in micro/nanowaveguides have broad applications spanning from atomic systems, particle trapping, high-bit-rate optical communication to optical sensing^[1–10]. So far, a variety of techniques have been developed for modal inspection of micro/nanowaveguides. One common method is direct imaging of modal profiles at the output end^[11]. However, the method is not applicable for the presentation of the mode evolution process. Another approach is using near-field scanning optical microscopy (NSOM) to detect the evanescent field^[12–14]. For the method, complex operating systems are generally required. Modal inspection based on Rayleigh scattering has also been reported, which is limited by a rather weak signal collection from geometrically uniform fibers^[15,16]. Recently, direct observation of multimode interference in rare-earth-doped micro/nanofibers has been demonstrated^[16]. However, since the approach is based on up-conversion luminescent interference, it is still challenging to observe multimode interference directly in passive micro/nanowaveguides.

Benefitting from attractive features such as high scalability and compatibility, transverse second-harmonic generation (TSHG) and transverse third harmonic generation (TTHG) in optical waveguides have been attracting continuous attentions for developing integrated photonic circuits and devices^[17–21]. In 2016, an optical correlator was reported based on the TSHG effect in a single CdTe nanowire^[22]. Benefitting from high nonlinearity, the input energy went down to a fJ/pulse level.

In this paper, we demonstrate in-situ modal inspection based on the TSHG effect inside a single CdS nanobelt (NB). With 1064 nm pumping light coupled in, transverse second-harmonic (TSH) interference patterns are observed in the direction perpendicular to axis of the NB. Modal superposition is analyzed using the fast Fourier transform (FFT) method. A few modes, including fundamental mode and several high-order modes, are extracted from FFT results of the TSH interference patterns, which are in agreement with calculated results. With the assistance of the refractive index difference ($\mathrm{\Delta }{n}_{\mathrm{eff}}$) between different optical modes, we have also investigated the period of TSH patterns in theory, revealing the characteristic of modal dispersion. It is worth mentioning that this method can be, in principle, operated for many varieties of nonlinear micro/nanowaveguides (e.g.,  ZnO, CdS, CdTe, GaAs, ${\mathrm{LiNbO}}_3$)^[23–25].

The CdS NBs with widths ranging from a few to a couple of micrometers and a height of a few hundred nanometers were synthesized by a thermal evaporation process^[26,27]. Electron and optical microscope characterizations confirm a uniform dimension and smooth surface, as shown in Fig. 1, indicating a rather low optical propagating loss^[28].

The experiment setup is schematically illustrated in Fig. 2. A CdS NB is placed across a slit of two ${\mathrm{MgF}}_2$ substrates. The continuous wave (CW) pumping light is split into two parts equally and coupled into the NB from both ends using fiber tapers. As counter-propagating light overlaps with each other in the NB, the TSH signal can be collected in the direction perpendicular to the axis of the NB using an object lens, as required by the wave-vector matching condition^[18,19].

Supposing that, there are two counter-propagating modes with propagating constants of ${\beta }_{\omega }$ and ${\beta }_{\omega }^{\prime }$, respectively, inside the NB, as shown in Fig. 3. Determined by the phase matching condition, the emitting angle ($\theta$) of TSH light is given by$$\cos \theta =\left|\frac{{\beta }_{\omega }-{\beta }_{\omega }^{\prime }}{k_{2\omega }}\right|,$$(1)where $k_{2\omega }$ is the wave vector of TSH light. Since the propagating constants for different optical modes at the same wavelength differ with each other, $\theta$ is slightly different from 90° for the TSHG process involving different modes.

Multimode interference also results in periodic TSH interference patterns along the axis of the NB. The period ($D$) can be expressed by^[29]$$D=\frac{\pi }{|{\beta }_{\omega }-{\beta }_{\omega }^{\prime }|}.$$(2)

Although the phase matching condition is not applied in the direction perpendicular to the axis of the NB, the obvious TSHG effect is also expected. For TSHG in a single micro/nanowaveguide, a coherent length caused by phase mismatch in the direction perpendicular to the axis of the waveguide can be estimated as^[30]$$L_c=\frac{\pi }{\mathrm{\Delta }k}.$$(3)

The phase mismatch is given by$$\mathrm{\Delta }k=k_{2\omega }•\cos \theta =\frac{2\pi n_{2\omega }}{{\lambda }_{2\omega }}•\cos \theta ,$$(4)where $n_{2\omega }$ and ${\lambda }_{2\omega }$ are the refractive index and vacuum wavelength of TSH light, respectively. Considering a simplified case in which the TSHG effect is only caused by counter-propagating modes with the same propagating constant, $\theta$ should be 90° according to Eq. (1), and $L_c$ can be described by$$L_c=\frac{{\lambda }_{2\omega }}{2n_{2\omega }}.$$(5)

For CdS NBs, we obtained an $L_c$ of ∼102 nm at a pumping wavelength of 1064 nm. Therefore, a relatively strong TSH signal is expected for CdS NBs with a height of a few hundred nanometers^[22,29].

To demonstrate the TSHG effect experimentally, a CdS NB with a width of 2 µm and a height of 200 nm was used [Fig. 4(a)]. As 1064 nm pumping light with a power of 5 mW is coupled in, green TSH light was observed. As shown in Fig. 4(b), there were obvious luminescent patterns with a period of ∼9.3 µm along the axis of the NB, which is believed to be owing to optical interference caused by multimode interaction. The mode profiles and effective refractive index were analyzed by finite-difference time-domain (FDTD) simulation. For the fundamental mode, power is concentrated on the center of the NB. However, for the second-order mode, power falls into two parts, which is distributed symmetrically along the axis. Considering the calculated effective refractive indices of 1.868 and 1.812 for the fundamental mode and the second-order mode, respectively, the period of TSH patterns can be calculated to be ∼9.5 µm, agreeing well with the experimental result (∼9.3 µm). The emitting angle was calculated to be ∼88.4° using Eq. (1).

To demonstrate the influence of multimode interference on luminescent patterns, the TSH signal in a single nanowire with a diameter of $300\pm 10\mathrm{nm}$ was also collected [Fig. 4(c)]. Considering a much smaller dimension, the effective refractive index for modes inside the nanowire is much smaller. As a result of better symmetry for the cross section, the effective refractive index difference between the fundamental mode and the second-order mode inside the nanowire is also much smaller than that in an NB. There was no obvious period observed within a length of ∼100 µm, as shown in Fig. 4(d). It is reasonable, considering a calculated interference period over 500 µm is extracted from a rather small $\mathrm{\Delta }{n}_{\mathrm{eff}}$ of ∼0.001 between the fundamental mode and the second-order mode inside the nanowire.

The green signal was analyzed by a spectrometer (iHR550, HORIBA Inc.) after passing through a 1064 nm blocking notch filter (Edmund, Inc.). As shown in Fig. 5(a), within a broad spectral range, there are no other peaks except for two peaks corresponding to the pumping light at 1064 nm and the TSH light at 532 nm. The TSH signal intensity with different input power is shown in Fig. 5(b), confirming a second-order nonlinear relationship.

To analyze the TSH patterns, we extracted the image intensity along the axis of the NB in the inside image of Fig. 4(b). The intensity profile is shown in Fig. 5(c), indicating an obviously strong oscillation. The corresponding FFT spectrum was obtained, as shown in Fig. 5(d). The red arrows show the first peaks located at $0.100\mathrm{\mu }{\mathrm{m}}^{-1}$, $0.225\mathrm{\mu }{\mathrm{m}}^{-1}$, $0.299\mathrm{\mu }{\mathrm{m}}^{-1}$, $0.424\mathrm{\mu }{\mathrm{m}}^{-1}$, and $0.524\mathrm{\mu }{\mathrm{m}}^{-1}$, respectively. Considering a resolution of $\sim 0.025\mathrm{\mu }{\mathrm{m}}^{-1}$^[16], the results are in good agreement with calculated values ($0.105\mathrm{\mu }{\mathrm{m}}^{-1}$, $0.181\mathrm{\mu }{\mathrm{m}}^{-1}$, $0.286\mathrm{\mu }{\mathrm{m}}^{-1}$, $0.450\mathrm{\mu }{\mathrm{m}}^{-1}$, and $0.555\mathrm{\mu }{\mathrm{m}}^{-1}$, respectively), as shown in Table 1. The calculated results are obtained with a width of 2 µm and a height of 200 nm. The results demonstrate the existence of at least four modes inside the NB. For example, the first peak located at $0.100\mathrm{\mu }{\mathrm{m}}^{-1}$ is supposed to arise from optical interference between the fundamental mode and the second-order mode, corresponding to a calculated frequency of $0.105\mathrm{\mu }{\mathrm{m}}^{-1}$.

Considering a 20 nm resolution of the scanning electron microscope image, we have also calculated modal interference with different heights. With heights ranging from 190 nm to 210 nm, the calculated results are very similar (as shown in Table 2). The results show that a change of 20 nm in height does not introduce significant difference into modal interference inside the NB.

The influence of the surrounding refractive index on TSH patterns is discussed. According to Eq. (2), D is inversely related to $\mathrm{\Delta }{n}_{\mathrm{eff}}$ (the effective refractive index difference between the fundamental mode and the second-order mode). As the surrounding refractive index increases, the effective refractive index of both the fundamental mode and the second-order mode increases, as shown in Fig. 6. A downward trend of $\mathrm{\Delta }{n}_{\mathrm{eff}}$ between the two modes with an increasing surrounding refractive index is also obtained. It is reasonable to consider a less optical confinement, which means more fields are spreading into the surrounding area from the NB, for higher-order modes^[31,32].

Similarly, the relationship between $D$ and the waveguide dimension is also investigated by FDTD simulation, as shown in Fig. 7(a). For example, as the width of the CdS NB increases from 1.0 to 2.0 µm, $\mathrm{\Delta }{n}_{\mathrm{eff}}$ changes from ∼0.25 to ∼0.05, leading to a ∼ 5 times larger $D$ for TSH patterns.

$\mathrm{\Delta }{n}_{\mathrm{eff}}$ with a different wavelength is also discussed to reveal the characteristic of modal dispersions theoretically, as shown in Fig. 7(b). For a larger pumping wavelength, there is a greater $\mathrm{\Delta }{n}_{\mathrm{eff}}$. Combining the fact that both modes have negative dispersions within the near-infrared spectral range, we can deduce that the fundamental mode has a smaller absolute dispersion than the second-order mode.

In conclusion, we demonstrate in-situ modal inspection based on direct observation of TSH interference patterns from a single nonlinear CdS NB. Benefitting from subwavelength-scale sectional dimension of the NB, a strong TSH signal is observed. Using the FFT method, TSH interference patterns are analyzed. The experimental results agree well with the calculated results, demonstrating the existence of at least four modes inside the NB. The influence of multimode interference on the TSHG effect inside a single micro/nanowaveguide is discussed in detail. The relationships among the period of TSH patterns, the surrounding refractive index, waveguide dimension, and pumping wavelength are also investigated, respectively, revealing dispersion properties for optical modes. Based on the wavelength conversion process, in-situ modal inspection of infrared propagating light has been demonstrated using a visible imaging system. Since the mechanism for transverse interference patterns can also be other nonlinear processes such as TTHG, the method can be, in principle, operated in micro/nanowaveguides with either second-order nonlinearity or third-order nonlinearity, which may find applications on multimode nanophotonic devices such as optical sensors and correlators.