The success of nonlinear optics has positioned it as a pivotal component within the optical industry: Spontaneous parametric downconversion (SPDC) has been widely employed in the generation of quantum entangled light resources^[1]. The optical frequency comb (OFC)^[2], which is capable of producing broadband light output with an equidistributed frequency interval, has found remarkable applications in the astronomical spectrograph^[3]. The optical parameter oscillator (OPO) also holds a prominent position within the laser field, benefiting from its extensive tuning ability and narrow bandwidth. Many nonlinear optics materials, such as KDP^[4] and BBO^[5], have been investigated to achieve various nonlinear optical processes. Lithium niobate (${\mathrm{LiNbO}}_3$) and lithium tantalate (${\mathrm{LiTaO}}_3$) stand among these materials, due to their high nonlinear coefficient and broad transition window within the mid-infrared range^[6]. Consequently, a vast amount of research has been reported on bulk LN and LT crystals over the past two decades^[7,8]. However, with the rapid development of microfabrication and integrated circuits, bulk LNO and LTO crystals face a dilemma because of their large physical volume and high light threshold. Researchers have invested significant effort into reducing both the size and threshold of LNO and LTO crystals, with the aim of making them suitable for integration into microscale photonic devices.

The whispering gallery resonator (WGR) has attracted a great deal of researchers’ attention since its initial proposal^[9]. Due to its small mode volume, low threshold, and high-quality factor ($Q$ factor), WGR is considered a feasible method for the miniaturization of certain conventional photonic devices. Consequently, many researchers are striving to achieve nonlinear processes in WGR, including the second-harmonic generation (SHG)^[10–12], the sum-frequency generation (SFG)^[13], and OPO processes^[14–16]. Moreover, the quasi-phase matching (QPM) technique, which has been extensively applied to bulk nonlinear optical crystals, can also be employed in WGR^[17–19]. By manually designing the inversion domain, QPM exhibits a frequency tuning ability that is far beyond that of conventional natural phase matching (NPM)^[10]. Furthermore, two different domain inversion periods can be introduced in QPM, thereby triggering the cascade process and achieving a richer variety of output frequencies^[17,18]. In previous studies, researchers have successfully demonstrated the cascade process between different nonlinear processes in a single WGR, including OPO and SHG^[20], OPO and SFG^[17,18], or SHG and SFG^[21,22]. However, few reports focus on the cascade OPO process in WGR, although much work has been reported on the bulk crystal^[23–25].

In this Letter, we design two distinct domain structures in WGR to achieve a cascade process through the same nonlinear process, i.e., the cascaded OPO. With our carefully designed domain structure and distorted QPM structure, diverse cascade OPOs are attained with the same idler light output, whose wavelengths are able to reach 4802.9 nm. In addition, the cascade OPO operates under an effective pump power of 32.7 mW continuous wave (CW) at almost room temperature (36°C), while other previous works operate under high power pulsed laser input and high temperature. Our results offer innovative approaches to the development of low-power cascade OPOs and facilitate the miniaturized production of such devices.

Prior to the design of the QPM structure, it is essential to ascertain the modes that are involved in the WGR. Only when understanding the specific phase conditions of each mode can we calculate the requisite phase compensation between different modes. In WGR with major radius $R$ and minor radius $r$, modes are discriminated by three integers, namely, $p,q$, and $m$^[26]. The integers $p$ and $q$ serve to differentiate the number of modes along the polar and radial directions, respectively. Once the integers $p$ and $q$ have been specified, the azimuthal mode $m$ along the equatorial circumference is determined in accordance with the geometric dispersion^[26,27]. Subsequently, the value of $m$ differs for different wavelengths when the same $p$ and $q$ modes are considered. According to the value of $m$, the phase difference among light modes can be confirmed. Inspired by the QPM technique, the discrepancy of mode number $m$ among different light modes can be rectified by modifying the sign of the second-order nonlinear coefficients.

In ${\mathrm{OPO}}_1$, the $p$, $q$ mode numbers are the same ($p=0$, $q=1$) for the pump light ${\nu }_p$, the signal light ${\nu }_{s1}$, and the idler light ${\nu }_{i1}$. Furthermore, the azimuth mode numbers ($m_p$, $m_{s1}$, and $m_{i1}$) of the three light modes can be confirmed. The difference among the three mode numbers allows us to ascertain the phase compensation provided by adjusting the nonlinear coefficients. Therefore, the phase-compensated relationship is given by^[28]: $$m_p=m_{s1}+m_{s2}+M_1.$$(1)

NPM is fulfilled when $M_1=0$^[10,14]. With regard to the radial QPM structure, the compensation quantity can be expressed as ${\theta }_1=2\pi /M_1$, which is particularly useful for organizing the distribution of different types of QPM structures. Accordingly, a similar approach is employed to determine the compensation quantity $M_2$ or ${\theta }_2=2\pi /M_2$ in ${\mathrm{OPO}}_2$.

In ${\mathrm{OPO}}_2$, the signal light ($s1$) from ${\mathrm{OPO}}_1$ serves as a pump light, producing the same frequency of idler light, as $i2=i1$ and $m_{i1}=m_{i2}$. Similarly, the azimuthal mode number of the signal light ($s2$) is identified as $m_{s2}$, as was the case with ${\mathrm{OPO}}_1$. The relationship between the compensation quantity $M_2$ and the mode number is therefore established as $$m_{s1}=m_{s2}+m_{i2}+M_1.$$(2)

The compensated quantities $M_1$ and $M_2$ for the cascaded OPO have thus far been obtained. The subsequent stage is to devise the distribution of the two QPMs. The issue of modifying the distributions can be transformed into the task of identifying the quantities $n_1$ and $n_2$ of sectors $M_1$ and $M_2$ that will occupy a full circle, which can be expressed as $$\min |n_1{\theta }_1{-n}_2{\theta }_2|,$$(3)$$\text{s.t.}\{\begin{array}{c}{n}_1{\theta }_1{+n}_2{\theta }_2=2\pi \\ {0<n}_1{<M}_1\\ {0<n}_2{<M}_2\\ n_1{,n}_2\in N\end{array}.$$(4)

To guarantee the same interaction length of the two desired OPO processes, the quantities are obtained from Eq. (4) that $n_1=225$ and $n_2=205$ for $M_1=450$ and $M_2=410$, respectively. Figure 1(a) shows the ideal radial QPM structure and the distorted QPM structure with an offset between the disk center and the domain pattern center. The distorted QPM structure leads to multiple $M$ excitations^[18]. Alternative modulation $M$ in a distorted QPM structure can be calculated numerically using the fast Fourier transform (FFT) algorithm. FFT is an algorithm used to calculate the reciprocal vectors and their intensities in a structure, i.e., the modulation $M$ ’s intensity. Therefore, it is able to calculate the period in the eccentric structure.

Without the offset, few modulation values are revealed in the resonator. However, diverse modulation values arise when the offset increases, as shown in Fig. 1(b). The larger the offset, the lower the intensity of the center $M_{1,2}$, along with more $M$ values around the center being excited. Temperature tuning curves with specific modulation $M$ and mode are obtained, according to the temperature and wavelength dependence of the refractive index of CLT^[6].

In the ideal QPM structure with $M_1=450$ and $M_2=410$, the 1064 nm wavelength pump input results in cascaded OPO outputs of idler light, which is illustrated in Fig. 2(a). When the disk’s temperature changes, the wavelength difference between the two idler lights can be calculated, indicating that there are two temperature points that can satisfy the coincident output of the idler light, as illustrated in the subplot of Fig. 2(a). Inevitably, however, machining errors can cause distortions in the QPM structure. Therefore, it is necessary to identify alternative temperature points, mode numbers, and compensated quantities ($M_1,M_2$) that will result in the output of the idler frequency becoming coincident. Three points of coincident light were identified in the vicinity of room temperature, as illustrated in Fig. 2(b). The subplot shows the corresponding idler light around 4.8 µm. The distorted QPM structure offers a greater range of potential for the detection of the coincident output of the idler light.

The basic material is a 3-inch CLT $z$-cut wafer with a thickness of 500 µm. The mask pattern is designed with $M_1=450$ and $M_2=410$. Ultraviolet (UV) photoresist is coated to the $+z$-face of the wafer, and the pattern on the mask is transferred to the photoresist by exposure to UV light. Following the exposure, the wafer deposited a 200 nm thick chromium film using an electron beam evaporator (EBE). In the absence of a photoresist, the metal film is in direct contact with the wafer. Two QPM structures are implanted in the wafer by applying an external pulsed field^[29]. The wafer was then immersed in hydrofluoric acid (HF), which revealed the domain structure, as illustrated in Figs. 3(a) and 3(b). The pattern on the CLT wafer is then cut into a chip using a dicing machine. Subsequently, the chip, affixed to the brass rod with high-strength UV glue (NOA61), is ground into a disk using a high-speed rotating spindle and sandpaper. During this time, the sidewall is shaped. The resonator is obtained by polishing the microdisk with grinding paste of different particle sizes [see Fig. 3(c)]. Furthermore, the bottom of the copper rod is configured to accommodate a negative temperature coefficient (NTC) thermistor to detect the temperature of the disk. The major radius $R$ and minor radius $r$ of the disk are approximately 2 and 0.3 mm, respectively.

The experimental setup is shown in Fig. 4(a). The loaded $Q$-factor measured at approximately 1064 nm is around $3.2\times {10}^7$ with an undercoupling condition when a photodetector (PDA015C, Thorlabs, Inc) replaces power meter $P_p$ in Fig. 4(a). The transmission spectrum and $Q$-factor measurement of the disk are illustrated in Figs. 4(b.i) and 4(b.ii), respectively. A linearly polarized CW ytterbium doped fiber laser (YDFL), injected from the tuned seed source, pumps the disk while the output beam is focused on the coupling surface of the rutile prism. After four lights generated from the disk pass through the prism and lens, the dichroic mirror (DM1, 45°HR@1064 nm, HR@1400–1850 nm, HT@2500–4450 nm) separates the pump light, while the remaining light passes through a beam splitter (BS) made by ${\mathrm{CaF}}_2$ and enters the Fourier transform optical spectrum analyzer (FTOSA, OSA207C, Thorlabs, Inc). Power meter $P_p$ measures pump power. The reflected light from the BS passes through another dichroic mirror (DM2, 45°HR@1400–1850nm, HT@2500–4500 nm), separating signa1 light ($S_1$) from the remaining lights, and the corresponding power is measured separately. The sum power of two lights, i.e., signal light ${\nu }_{s2}$ and ${\nu }_i$ in ${\mathrm{OPO}}_1$ and ${\mathrm{OPO}}_2$, is measured by $P_{s2}+P_i$; $S_1$’s power is measured by $P_{s1}$.

By adjusting the coupling angle and distance^[30], different modes of pump light produce signal light and idler light corresponding to different wavelengths under different domain structures. The pump power can be ascertained when applying the power meter (S121C, Thorlabs, Inc). The cascade OPO process is observed with a coupling efficiency of about 19.6% in the definition of coupling efficiency^[28,31], which corresponds to the pump power on the coupling face of the prism being 163.3 mW. Nevertheless, the ${\mathrm{OPO}}_1$ threshold is about 4.32 mW with 21.6 mW on the face of the prism, while the experimental setup remains unchanged. If a single period modulates the OPO process, the threshold of ${\mathrm{OPO}}_1$ will probably further decrease^[18,32,33]. Figure 5(a) shows the output light detected by the FTOSA with an effective power of 58.9 mW. As shown in Fig. 5(a), ${\mathrm{OPO}}_1$ pumped by 1063.8 nm with 36°C of WGR generates two lights with wavelengths of 1566.6 nm ($S_1$, green) and 3314.6 nm ($I_i$, green), which represents $M_1=444$. $S_1$ pumps ${\mathrm{OPO}}_2$ to generate lights with wavelengths of 2970.4 nm ($S_2$, cyan) and 3314.6 nm ($I_i$, green) representing $M_2=411$. This is consistent with the result in Fig. 2(b). The power of $S_1$ in ${\mathrm{OPO}}_1$ (1566.6 nm) is 3.93 mW when the effective power is 58.9 mW on the prism’s face and the WGR temperature is at 36°C. $S_2$ and $I_i$’s power are 2.82 mW in Fig. 5(a), marked with an orange line. The conversion efficiency is 11.5%. Compared with the other result in Ref. [28], the conversion efficiency is lower because our disk is undercoupled and almost half the interaction length is compared with the full circumference^[33]. We speculate that the cascade OPO is weak from the spectra. According to the FTOSA’s power meter function, $S_2$’s power is probably 26.0 µW. We propose that the low power is caused by the wavelength deviation between the generated light and the resonant peak of the disk. As mentioned above, it is difficult to match four lights simultaneously^[21,33], and the deviation between the light and the resonance will increase the threshold of OPO.

If the cascade process does not happen, the corresponding signal and idler light will be produced in two different period situations, i.e., the light with 1657.1 nm should be produced. However, during this experiment, the light with 1657.1 nm was not observed using the FTOSA at 36°C and according to the conservation of energy, it confirmed that the OPO of the cascade process occurred. A pair of lights with wavelengths of 1960.6 and 2325.5 nm are generated with the domain structure $M=424$ with the mode $p_{s1,i}=0$, $q_{s1,i}=1$. This structure is revealed because of the deviation of $M_2$ caused by the offset of the disk. We find a diverse cascade OPO in the same WGR, as shown in Fig. 5(b). At a temperature of 28°C, the cascade OPO exhibits a matching of $M_1=446$ and $M_2=409$ with mode number $p_{p,s1,s2,i}=0$, $q_{p,s1,s2,i}=1$. The power of $S_1$ is 5.89 mW, while the sum power of $S_2$ and $I_i$ is 2.03 mW, with an effective power of 62.4 mW. This corresponds to a total conversion efficiency of 12.6%. In addition, the wavelengths of 1991.7 and 2283.2 nm, with mode $p_{s1,i}=1$, $q_{s1,i}=1$, are generated with the same domain structure as Fig. 5(a). The conversion efficiency of the idler light around 3.3 µm at 36°C is 4.7%, representing an improvement over the efficiency of 3.2% observed at 28°C.

Similarly, the ${\mathrm{OPO}}_1$ process at 28°C generates a signal light with 1741.8 nm ($S_1$) and an idler light with 2732.9 nm ($S_2$) with $M_1=430$. Under another period’s ($M_2=474$) influence, a cascade OPO (${\mathrm{OPO}}_2$) is generated with a signal light of 2732.9 nm and an idler light of 4802.9 nm, as shown in Fig. 6. The power output of $S_1$ is 2.30 mW, while the combined power output of $S_2$ and $I_i$ is 0.76 mW. This corresponds to a total conversion efficiency of 9.2%. The absorption spectrum of CLT^[34] reveals an absorption at 4.8 µm in comparison to the peak observed at 4 µm. While it is feasible to attain an output at 4.8 µm through a single OPO process, the requisite absorbance at this wavelength necessitates the utilization of a higher pump power. In this paper, a 4802.9 nm wavelength output with low effective pump power (33.1 mW) is achieved, which has the effect of reducing the light output conditions. In this report, the pump light’s wave vector at different temperatures is dissimilar because of thermal expansion; the refractive index depends on the temperature of CLT.

In summary, we present the design of a periodic domain structure and the corresponding distribution, which together realize a cascaded OPO output. In comparison to conventional cascade OPOs, this experiment has successfully achieved a cascade OPO with a low threshold, which can be realized at conditions close to room temperature. This has the additional benefit of reducing the size of the optical device and providing a solution for miniaturization. Moreover, the occurrence of multiple cascade phenomena within a single microdisk offers the potential for the development of miniaturized broadly tuned cascade OPOs.