## Abstract

This paper theoretically investigated type-I optical parametric oscillations (OPOs) in lithium niobate (LN) ellipsoidal microcavities. We calculated the dependence of signal (idler) wavelength on temperature and found that multiple OPO channels are universal phenomenon in LN whispering gallery mode (WGM) microcavities. Additionally, We discovered OPOs in LN WGM microcavities, instead of having a lower-temperature limit, have an upper-temperature limit beyond which no OPO takes place. The dependence of the OPO tuning behaviors on the cavity geometry and pump wavelength was discussed systematically. Our investigations provide new conceptions for OPOs in LN WGM microcavities and may benefit the research on tunable laser sources or high-order entangled photon sources.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Lithium niobate (LN) crystal is a well-known versatile material referred to as “silicon of photonics”. It has birefringence, a wide transparent window and giant nonlinearity. Whispering-gallery-mode (WGM) microcavities with high quality factors can dramatically boost light intensity inside by confining light within a small mode volume for a long time. They have the ability to realize nonlinear optical effects under low-power pump. LN WGM microcavities combine the advantages of both lithium niobate and WGM microcavities providing an excellent platform for nonlinear optical investigations. A variety of nonlinear optical effects, including second harmonic generation [1–6], sum-frequency generation [7–10], optical parametric oscillation (OPO) [5,11–13], were demonstrated in LN WGM microcavities under weak continuous pump with power on the order of or less than mW. By contrast, observation of nonlinear optical effects in bulk materials generally requires focused pulse laser with high intensity as the pump.

Compared with Fabry-Perot and photonic crystal microcavities, WGM microcavities can exhibit high quality factors in a very broad frequency window due to total internal reflection [14, 15]. Such a window is only limited by the transparency of the WGM cavity material. The transparency window of LN spreads from 0.4 to 5 *μ*m. The refractive index of LN can vary with applied electric field, temperature and even mechanical stress. Benefiting from such tuning properties of LN refractive index, LN WGM optical parametric oscillators show great active tuning behavior [11–13]. The signal and idler wavelengths for LN OPO can be thermally or electrically tuned in a large range by changing device temperature or applying an external voltage across LN crystal.

According to the polarizations of the involved light waves, OPO processes are classified into three types. Type 0: signal, idler and pump have equal polarizations; Type I: signal and idler have the same polarization, which is perpendicular to that of the pump; Type II: signal and idler have perpendicular polarizations. Generally, the dependence of the signal and idler wavelengths on the device temperature for type-0 and type-I OPOs has a parabolic shape, and the tuning curve has infinite slope at the frequency degenerate point [11, 12, 16]; by contrast, the tuning curves for type-II OPOs cross each other and have finite slope at the cross point [13]. Type-0 tuning behavior was reported in LN WGM microcavities; The wavelength-temperature curve is a parabola going to the right [11, 16]. It indicates that OPO effects take place only when device temperature is higher than a lower temperature limit. In periodically poled lithium niobate (PPLN) WGM microcavities both type-0(I) and type-II like tuning behaviors were observed in type-0 OPO process using WGMs with distinguishing quantum numbers to relieve the mode degeneracy [13]. Note that the type-0-like tuning behavior in PPLN WGM microcavities has an upper temperature limit rather than a lower temperature limit as in monocrystalline ones.

In this paper, we theoretically investigate type-I OPOs in LN WGM microcavities. A 3-shaped temperature-wavelength tuning curve for type-I OPO associated with only fundamental WGMs with radial quantum mode numbers equal to 1 was observed for the first time in LN WGM microcavities. We find that OPO associated with particular WGMs are only observable when the sample temperature is under instead of over a temperature limit. Additionally, one pump photon can turn into signal and idler photons through one, two or four possible channels. The upper temperature limit and multi-channel effects are universal for OPOs associated with signal and idler modes of either equal or different radial quantum numbers.

## 2. Principle for OPOs in WGM microcavities

This paper concentrates on OPOs in LN WGM microcavities with diameters of tens of *μ*m and edge thicknesses of hundreds of nm, which is the general size of LN WGM microcavities on a chip. To save time in calculation, we investigate OPOs in ellipsoidal LN WGM microcavities whose resonance frequencies can be analytically calculated [17]. Figures 1(a) and 1(b) show the side and top views of an ellipsoidal microcavity, respectively. The geometry of an ellipsoidal microcavity is exclusively defined by the radius *R* and curvature radius *ρ* in a cross section consisting of the rotational axis. The geometrically rotational axis was set parallel to the axis of LN crystal for simplicity. The eigen frequencies *ν* of such an ellipsoidal microcavity are approximately given by

*m*,

*q*and

*l*are the azimuthal, radial and polar quantum number for WGMs, respectively.

*c*is the speed of light in vacuum,

*n*stands for refractive index and

*p*=

*l*−

*m*= 0, 1, 2, · · · is for simplification.

*α*

_{q}is the

*q*th root (

*α*

_{q}> 0) of Airy function. Polarization-dependent factor

*χ*is unity for TE modes and 1/

*n*

^{2}for TM modes. The refractive index

*n*of LN crystal empirically given by Sellmeier equation [18] is a function of wavelength, temperature and polarization. It is known that LN is a birefringent crystal with extraordinary refractive index

*n*

_{e}smaller than the ordinary one

*n*

_{o}. Only type-I OPO is supported when no quasi-phase matching is employed. For this reason, we discuss type-I OPO in LN WGM microcavities.

The energy distribution for a WGM in ellipsoidal microcavities can be calculated using both analytical or numerical methods. Fig. 1(c), calculated by COMSOL, shows the energy distributions for WGMs with radial quantum number *q* equal to 1, 2 and 3, respectively. In consideration of the conversion efficiency, the investigated OPOs in LN WGM microcavities in this paper were limited in the WGMs shown in Fig. 1(c) with *q* ≤ 3 and *p* = 0.

When all the waves involved in OPOs are resonant with the WGM microcavity,

where*n*

_{eff}indicates effective refractive index of the WGM, and

*λ*stands for the resonance wavelength. Light intensity in a WGM microcavity can be enhanced due to constructive interference thereby reducing the requirement on pump power. To obtain OPO in WGM microcavities, the laws of conversation of both energy and momentum need to be fulfilled. They are written as

*p*,

*s*, and

*i*indicate pump, signal and idler waves, respectively. Note that

*m*

_{p,s,i}are determined by the effective refractive index

*n*

_{eff}of the corresponding WGMs other than

*n*of the bulk material. Therefore, we can adjust geometric parameters such as radius

*R*and curvature radius

*ρ*of the WGM microcavity to manipulate dispersion and thus to fulfill momentum conservation.

We calculated a series of resonance wavelengths in LN transparent window [19] for interested WGMs of different polarizations in a LN ellipsoidal microcavity at a given temperature based on Eq. (1). Then verify whether there are a set of resonance wavelengths which satisfy energy and momentum conservations (Eqs. (3) and (4)) simultaneously. When the difference between the eigen wavelength of the WGM and the wavelength that satisfy Eq. (3) is less than the linewidth of the microcavity, we find a solution. The linewidth was set as 10^{−2} nm that corresponds to a quality factor on the order of 10^{5}, which is a typical value for on-chip LN microcavities. If a solution was found, we repeat the calculation process for other temperatures while fixing azimuthal mode number *m*_{p} of the pump beam. By connecting the data points of signal and idler wavelengths for different temperature, we get the wavelength-temperature tuning curve in LN microcavities. Here, we are only interested in the resonance wavelengths within the transparent window of LN [19]. Beyond the transparent window, serious material absorption reduces the quality factor of the microcavity and thus the accumulation of the light energy. In this case, nonlinear optical effects are weak.

## 3. Wavelength tuning behaviors of OPOs in LN microcavities

We systematically investigated the dependence of signal and idler wavelengthes on temperature in LN WGM microcavities based on the aforementioned theoretical model. The geometry, temperature and quantum mode numbers impact OPOs through the resonance wavelength and effective refractive index of WGM in LN microcavities. Firstly, we focused on the OPO processes among fundamental modes, i. e., *q*_{p,s,i} = 1. We calculated the wavelength-temperature curves for ellipsoidal LN microcavities with various curvature radii *ρ*, radii *R* and pump wavelengths, respectively, while fixing other parameters.

The dependence of signal/idler wavelengths on device temperature for ellipsoidal LN microcavities with different geometries was calculated. Figures 2(a) and 2(b) are the corresponding results obtained when *ρ* and *R* vary, respectively. The fixed cavity geometric parameters for Figs. 2(a) and 2(b) are *R* = 60 *μ*m and *ρ* = 1100 nm, respectively. During calculation, the pump wavelength *λ*_{p} was set to be around 775 nm. The curves in Fig. 2 have a shape similar to “3” with upper temperature limits. Taking the black curve as an example, the red dotted line indicates the corresponding upper temperature limit, beyond which no OPO takes place. This phenomenon is significantly different from the reported results [16] for millimeter-sized LN WGM microcavities.

They have a lower temperature limit and a parabolic wavelength-temperature curve similar to the curve in the dashed box in Fig. 2(a) if only fundamental modes get involved in OPOs. Additionally, we observed two OPO channels in a temperature window, in which two couples of photons with two different sets of wavelengths might be generated in a microcavity. For the black curve, the two-channel window is the region highlighted by the light green background. Such multi-channel effects were theoretically demonstrated in periodically poled LN micro disk resonators for OPO associated with WGMs with distinguished quantum numbers. However, here it is observed in OPO associated with only fundamental modes. The wavelength-temperature curve has a trend of right shift when *R* or *ρ* increases while keeping other geometric parameters unchanged, which helps us to realize OPOs in room temperature by adjusting cavity geometry.

For different pump wavelengths, we get the curves of signal and idler wavelengths versus temperature shown in Fig. 3. This figure is calculated when *R* = 60 *μ*m and *ρ* = 1100 nm. In Fig. 3, the exact shapes of the wavelength-temperature curves are changed obviously for different pump wavelengths. When the pump wavelength is shorter, the temperature range in which two-channel OPO occurs is broader. At the same time, the wavelength range covering the possible wavelengths of the idler and signal photons expands as well. In this case, the idler photon with wavelength out of the transparent window of LN will be absorbed strongly. Therefore, it is difficult to observe OPO associated with two separate channels. On the contrary, when the pump wavelength becomes shorter, it is much easier to obtain OPO effect due to the broadening of the temperature window between the upper temperature limits and the conventional lower temperature limit where the signal and idler wavelengths are equal. By choosing different geometric parameters and pump wavelengths, it is accessible to obtain a proper result, in which the signal and idler photons will not be absorbed so strongly and simultaneously the temperature range for two-channel OPOs is wide enough.

The above investigation on the wavelength-temperature tuning curve only covers the fundamental WGMs in LN ellipsoidal microcavities. Hereafter, we will study OPO with high order WGMs (in the view of radial quantum numbers *q*) participated. Figure 4(a) shows the wavelength-temperature curve for OPO with pump belonging to both the fundamental or high-order WGMs (*q*_{p} ≥ 1) and signal and idler being fundamental modes. The calculation parameters are *R* = 60 *μ*m, *ρ* = 1100 nm, and *λ*_{p} ∼ 775 nm. The OPO processes are denoted as *q*_{p} to *q*_{s,i}, *q*_{i,s}. According to this regulation, the expression 2 to 1, 1 in the legend of Fig. 4(a) indicates *q*_{p} = 2 and *q*_{s} = *q*_{i} = 1, respectively. We find from Fig. 4(a) that when the radial quantum number *q*_{p} of pump wave increases the shape of the wavelength-temperature curve is hardly changed, however, the curve shifts significantly to the high temperature range.

The corresponding results for OPO with signal and idler being distinguished WGMs is demonstrated in Fig. 4(b). Similar to the situation in PPLN WGM microcavities [13], the wavelength-temperature curves have finite slope when the wavelengths of the signal and idler waves are identical (see the cross point of the solid and dashed curves of the same color). As shown in the black curve, the upper temperature limit indicated by the red dotted line and the two-channel OPO highlighted by the light green background were observed. Interestingly, in the red and black curves, we find that OPO can take place via four separate channels in a very narrow temperature window. As an illustration, the four-channel region for the black line is marked by using red background. Such multi-channel OPO may find applications in entangled photon sources.

## 4. Explanation of the temperature tuning behaviors

The unexpected temperature tuning behaviors in ellipsoidal LN microcavities, including multi-channel and upper-temperature-limit effects, that seem contradict to the results in millimeter-sized LN cavities, are attributed to the relatively strong geometry dispersion of the micrometer-sized LN microcavities. The geometry dispersion plays a more important role than the material dispersion in OPOs when the idler wavelength is comparable to or larger than the size of the LN microcavity.

As described by Eqs. (2–4), to realize OPO effects in a WGM microcavity, energy conversation, angular momentum conversation, and multiple resonances need to be satisfied simultaneously. For a pump with a fixed frequency, there are countless photon pairs that match energy conversation. So angular momentum conversation and resonance conditions determine the OPO channel actually. Angular momentum conversation *m*_{p} = *m*_{s} + *m*_{i} can be expressed as *n*_{p}/*λ*_{p} = *n*_{s}/*λ*_{s} + *n*_{i}/*λ*_{i} as well, where *n*_{p,s,i} are the effective refractive indices for the pump, signal, and idler wavelengths, respectively. The effective refractive indices of WGMs, that depend on both wavelength (owing to both material dispersion and geometry dispersion) and temperature, determine the possible OPO channels. This is the universal principle for OPO effects in WGM resonators of any sizes and shapes.

In millimeter-sized lithium niobate WGM cavities, the OPO temperature tuning curve for fundamental WGMs is a parabolic curve toward the right. It is mainly due to the material dispersion of LN crystal at different temperature. However, in a WGM microcavity with a relatively small size, the geometry dispersion begins to play an important role for more light trends to distribute in the surrounding atmosphere. Figure 5(a) demonstrates the dispersion property of the fundamental WGM in an elliptical LN microcavity with a 60-*μ*m *R* and a 1-*μ*m *ρ*. Figure 5(b) indicates the first derivative of the curve in Fig. 5(a). From Figs. 5(a) and 5(b), we can clearly see that the reduction of the effective refractive index of the WGM accelerates when resonance wavelengths are larger than approximately 1500 nm due to the increase of the ratio of energy distributed in air and in LN. The strong dispersion from 500 nm to 800 nm is a result mainly of the material dispersion. When OPOs take place near the degenerate point, the wavelength of the pump is about 775 nm and those of the signal and idler are in 1550-nm band. They are all short compared with the thickness of the LN microcavities at which the WGMs locate. In this case, the material dispersion rather than the geometry dispersion dominates the OPO processes, the OPO temperature tuning curves in micrometer-sized LN cavities are similar to those in millimeter-sized LN cavities. When the temperature increases, the idler wavelength increases and the signal wavelength decreases. With the increase of the idler wavelength the geometry dispersion plays a more important role in OPO effects, which makes the OPO temperature tuning behaviors in micrometer-sized LN cavities deviate from those in the millimeter-sized. The deviation manifests as the acceleration of the temperature tuning behavior and the increase of the slope of the wavelength temperature curve. The upper temperature limit, beyond which no OPO takes place, is the temperature corresponds to infinite slopes in wavelength temperature curves except for the degenerate down conversion point. If the idler wavelength increases further, temperatures lower than the upper-temperature limit are required to satisfy angular momentum conversation. It is the origin of the multi-channel effects for OPO processes in micrometer-sized LN microcavities.

Following the aforementioned theoretical analysis, the temperature tuning behaviors shown in Figs. 2(a) and 2(b) can be explained reasonably. The idler wavelengths corresponding to the upper temperature limits tend to go up with the increase of *ρ* (see Fig. 2(a)) while remain about the same with the increase of *R*. That is because the larger *ρ*, the thicker of the LN microcavities leading to a larger idler wavelength from which the geometry dispersion begin to dominant the dispersion properties of a WGM. On the other hand, because the initial *R* equal to 60 *μ*m is tens of times larger than the wavelength involved in the OPO precess, slight change in *R* does not seriously affect the dispersion properties of the LN microcavities. Therefore, the idler wavelength corresponding to the upper temperature limit keeps approximately constant when *R* varies slightly.

The explanation of the temperature tuning curves indicated in other figures can be explained in the similar manner. Because the WGM dispersion properties mainly depend on the geometric size of the cavity instead of the exact shape, the tuning behaviors observed in elliptical LN microcavities should be universal in WGM microcavities of other shapes such as cylindrical and wedge-shaped resonators. The 3-shaped temperature tuning behavior for OPOs associated with fundamental modes was confirmed in cylindrical LN microcavites with a 50 *μ*m radius and a 2 *μ*m thickness.

## 5. Conclusions

In conclusion, taking ellipsoidal LN microcavities as examples, we investigated the wavelength-temperature tuning behavior of type-I OPOs in LN WGM microcavities. Compared with the reported tuning curves of OPOs in both LN and PPLN WGM microcavities, a series of new phenomena were observed. (1) 3-shaped tuning curves other than the parabolic were demonstrated in LN WGM microcavities for OPOs with signal and idler being fundamental WGMs; (2) An upper temperature limit instead of a lower temperature limit was observed in LN WGM microcavties. Upper temperature limits of OPOs were reported in PPLN WGM microcavities but not in LN cavities; (3) Double-channel effects were observed in OPOs in LN microcavities regardless of the radial number of WGMs. Such effect was theoretically reported in PPLN microcavities when signal and idler have unequal quantum numbers; (4) Four OPO channels were demonstrated in LN WGM microcavities when either signal or idler is not a fundamental WGM. The unexpected tuning behaviors were attributed to the impact of the geometry dispersion that begins to have a significant impact when the resonance wavelenth is comparable to or larger than the size of the WGM cavity. These new conceptions for OPOs in LN WGM microcavities help us to optimize the microcavity geometry made from nonlinear optical materials to realize tunable laser source or to generate entangled photons.

## Funding

NSFC (11734009, 11674181, 11774182 and 11674184), the 111 Project (B07013), PCSIRT (IRT_13R29), the National Science Fund for Talent Training in the Basic Sciences (J1103208) and CAS Interdisciplinary Innovation Team.

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