Universal frequency engineering tool for microcavity nonlinear optics: multiple selective mode splitting of whispering-gallery resonances Download: 520次
1. INTRODUCTION
Parametric nonlinear optics based on the second-order () or third-order () nonlinear interaction of light has been studied in a variety of optical materials in both nonresonant geometries (bulk crystals, fibers, waveguides) and resonant structures (Fabry–Perot cavities, millimeter-size cavities, and microcavities). Efficient nonlinear mixing based on these processes requires intense light with frequency and phase matching [1,2]. Among these structures, optical microcavities that confine light in whispering-gallery modes (WGMs) with high quality factors and small mode volumes and therefore enhance light–matter interactions in both time and space have been particularly successful [3,4]. However, such an enhancement requires that the interacting WGMs be frequency matched, that is, their resonance frequencies must satisfy energy conservation with an accuracy approximately given by their linewidths. Realization of frequency matching requires consideration of both the specific material platform and the targeted nonlinear optical process.
WGM microcavities have been extensively studied for parametric nonlinear optics in a variety of material platforms including, but not limited to, silicon [5], silicon nitride [6
In contrast, direct control of targeted WGM frequencies, without influencing other WGMs, would be a desirable tool for frequency engineering in nanophotonics. One approach towards this end is to couple the clockwise (CW) and counterclockwise (CCW) propagating modes of the resonator and split the cavity resonances. Such mode interactions have often been introduced by random scattering (e.g., by sidewall roughness) in a microcavity [37]; however, this random scattering is difficult to control. Later on, the mode interaction introduced by a foreign point scatterer to a microcavity was studied [38], and is particularly useful for particle sensing [39]. More controllable frequency splitting has been induced using interacting degenerate modes from two [40] or multiple [41] adjacent microcavities. Within one microcavity, the coupling between CW and CCW modes can be realized by integrated distributed Bragg grating reflectors [42], integrated phase shifters [43], and coherent modulation of the boundary, termed selective mode splitting (SMS) [44]. In particular, SMS allows accurate frequency control of a selected cavity mode and has been used to help frequency matching in spontaneous four-wave mixing [44], identify the azimuthal mode numbers of WGMs [9], and offset nonlinear frequency shifts in microcomb generation [45]. Here, we show that its potential for frequency engineering is much greater; it can be controllably applied to multiple modes simultaneously, while leaving untargeted modes nearly unaffected. Such control is of high value to frequency mixing in nonlinear optics, where multiple modes naturally interact.
We introduce our approach, termed multiple selective mode splitting (MSMS), in the following sections. We first develop a basic model of MSMS using perturbation theory (Section
2. THEORY
A rotationally-symmetric (i.e., without azimuthal modulation) microring resonator supports cavity modes propagating in opposite directions, CW and CCW, which are degenerate in frequency , where is the azimuthal mode number. A modulation of the microring geometry, for example, of its ring width as shown in Fig.
Fig. 1. Illustration of MSMS. (a) Example of microring ring width modulation targeting and modes, with 20% and 10% of the nominal value (dashed line), respectively. This modulation selectively frequency-splits the (d) and (e) modes, while behaving as a normal microring with rotational symmetry for all other modes. We note that the number and modulation amplitude are quite different than that implemented in real devices, for illustration purposes. (b) The modulation of the ring width for the MSMS device is plotted versus azimuthal angle in the top panel and has two frequency components in this case. The middle panel shows the modulation component targeting the mode with 20% modulation of the ring width, and the bottom panel shows the modulation component targeting the mode with 10% modulation of the ring width. (c) In the device transmission (top), this MSMS device introduces frequency splitting of the and modes, and the amplitude of the splittings linearly depends on their modulation amplitudes. This transmission trace is equivalent to the two individual SMS transmission traces (middle and bottom) together. (d) For the mode, the MSMS device is equivalent to the SMS device described by the (6, 0.2) modulated boundary (blue). Degenerate CW and CCW modes are renormalized to two standing-wave modes. The standing-wave mode that always experiences a wider ring width has a larger resonance wavelength, and therefore a redshifted resonance frequency. The other mode always experiences a narrower ring width and is blueshifted. (e) For the mode, the MSMS device behaves as the SMS device described by the (8, 0.1) boundary (red). (f) For other modes ( ), for example, , the MSMS device does not induce coherent scattering, and therefore behaves similarly to a rotationally symmetric ring (dashed line).
For transverse-electric-field-like (TE) modes or transverse-magnetic-field-like (TM) modes, the above equation can be further simplified because the term containing either or dominates the integral in the numerator. Take TE-like modes for example, with the proposed boundary modulation on the ring width: The standing-wave mode with a larger frequency has a dominant displacement field . Equation (
3. EXPERIMENTS
In the previous section, we showed how a simple theory predicts that MSMS should achieve our goals, namely, selective frequency splitting of targeted modes with a controllable splitting amplitude. Next, we examine how well this technique works in practice. Of particular interest is our ability to experimentally create a sufficiently faithful modulation pattern on the inner surface of the microring to realize the mode-selective, adjustable frequency splitting we desire. For this purpose, we use a fabrication process (see Appendix
3.2 A. Single-Mode SMS
We first examine the control of the frequency splitting of a single azimuthal mode, which we term single-mode selective mode splitting (1SMS). Here, we target a fundamental TE mode with , which has resonance around 1550 nm in a device with nominal parameters of a 25 μm outside ring radius, a 1.1 μm ring width, and a 500 nm thickness. Such a structure supports only fundamental radial modes in both TE and TM polarizations. When there is no modulation, as shown in the top panel of Fig.
Fig. 2. 1SMS. (a) Transmission scans for four devices with different modulation amplitudes, where the labels ( , ) represent the azimuthal mode number and the (nominal) modulation amplitude in nanometers. In the top panel, the control device without SMS shows a cavity transmission spectrum without frequency splitting for any of the modes. From the second panel down, the devices exhibit a transmission spectrum with increasing frequency-splitting for the targeted mode, with a splitting proportional to the prescribed modulation amplitude. Besides the TE1 modes under investigation, another set of modes appears, which is verified to be fundamental transverse-magnetic modes (TM1). The appearance of these modes is due to wavelength-dependent polarization rotation in the fibers used, which could be resolved by using polarization-maintaining fibers. (b) The top panel shows that the amount of mode splitting is linearly proportional to the modulation amplitude and reaches a value of . The bottom panel shows that the adjacent four modes have much smaller splitting ( of the targeted mode, that is, below the black line) than the targeted modes. In both panels the uncertainties in the mode splitting, determined from nonlinear least squares fits to the data, are smaller than the data point size. (c) Transmission spectrum and fit trace of the singlet resonance without mode splitting, (163, 0), where is the intrinsic optical quality factor. (d) Transmission spectrum for the doublet resonance (163, 3), with an observable mode splitting that is a couple of times the mode linewidths. From these linewidths, we determine and , the intrinsic optical quality factors for the blueshifted and redshifted modes, respectively. The differences in the coupling and intrinsic Q s for these two modes are potentially due to the different localization of the two standing-wave modes with respect to the ring-waveguide coupling region and possible point defects, respectively. (e) and (f) Two doublet resonances (163, 9) and (163, 21) that are well separated (262 pm and 591 pm splitting, respectively). See Appendix C for fitting methods.
In Fig.
Importantly, the collateral splitting, i.e., the splitting for untargeted modes, is comparatively small. As shown in the bottom panel of Fig.
A natural question is whether the introduced modulation adversely affects the modal quality factors. Other works utilizing single-mode splitting [9,45] have observed that high can be maintained, and in this work we consistently observe when large-mode splittings are realized. In fact, to our surprise, with a larger mode splitting, the are not decreased but are actually increased compared to the zero-splitting or small-splitting case (see Appendix
3.3 B. Multiple SMS
We now set up the modulation of the inner ring boundary to target multiple modes, considering both three modes (3SMS) and five modes (5SMS). As the results are qualitatively quite similar (other than the number of targeted modes), we focus on the 5SMS cases in the main text, with the 3SMS case considered in Appendix
For 5SMS, we consider five devices with different modulation configurations, and their measured transmission traces are shown in Fig.
Fig. 3. 5SMS. (a) Cavity transmission traces of five devices with different configurations of 5SMS. Each split mode is labeled by ( , ), with in units of nanometers. (b) The measured mode-splitting amounts clearly correspond to the expected modulation pattern, where and ring width modulation amplitudes correspond to and mode splittings, while the untargeted modes have collateral splitting. The slopes (splitting divided by the modulation amplitude) are and , respectively, close to the 1SMS slope in Fig. 2 (b). The uncertainties in mode splitting are smaller than the data point size. (c) Typical example of the cavity transmission for a targeted mode with 408 pm mode splitting; (d) typical example of the cavity transmission of a targeted mode with 140 pm mode splitting; (e) the average of the intrinsic optical quality factors of all the modes in these 5SMS devices is, surprisingly, more than 1 standard deviation higher than the average in the 1SMS devices (red open circles). for individual modes are shown as solid blue dots. We note that the of 1SMS modes are generally higher with modulation amplitude (see Appendix A ). The 3SMS devices (where three modes are targeted for mode splitting) have that are in-between the 1SMS and 5SMS cases (see Appendix B ).
Clearly, selective frequency control of multiple cavity modes is of most benefit if the cavity remain high. Counterintuitively, we have observed that the complicated modulation patterns that split five cavity modes do not decrease the of the devices, but increase them instead. This trend is displayed in Fig.
3.4 C. Microring-Waveguide Coupling
Aside from the change in intrinsic optical introduced by MSMS, we also observe a change in the extrinsic (coupling) optical , which is related to the microring-waveguide coupling rate by . In theory, traveling-wave modes and standing-wave modes should have different coupling rates [46], as illustrated in Fig.
Fig. 4. Microring-waveguide coupling. (a) In an MSMS device, targeted modes become standing waves that equally comprise a CCW (red) part and a CW (black) part. Assuming a undirectional waveguide input as shown in the diagram, the CCW part can be coupled to at a rate , while the CW part is not directly coupled to the waveguide because of the large momentum mismatch from the input light ( ). The MSMS modes are therefore expected to have an overall coupling rate of in theory. (b) Considering a 5SMS device in which the modes are targeted, the measured (blue) is clearly larger for the split modes (each indicated by a black dashed line) than for other modes, and therefore as a function of azimuthal mode order shows an oscillatory behavior. Dividing for the split modes by 2 yields inferred values of their CCW coupling (red), which are similar to those of the unsplit modes. This trend is consistent with the prediction in (a), as . Error bars are 1 standard deviation values resulting from nonlinear least squares fits to the data.
We indeed observe this coupling behavior in experiment. For the 5SMS device targeting azimuthal mode orders with a 5 nm modulation amplitude each [top panel in Figs.
4. APPLICATION SCENARIOS
In previous sections, we have shown that MSMS can be a tool to split multiple cavity modes with a prescribed configuration of mode numbers and splitting amplitudes while retaining optical . The induced modes are standing wave in nature and thus have a reduced microring–waveguide coupling rate relative to traveling-wave modes. Although some aspects of the MSMS approach require further investigation and optimization, we believe that the demonstrated performance would already be of utility for many nonlinear optical applications. Here, we consider an example that showcases this capability. In comparison to a typical microring cavity [Fig.
Fig. 5. Nonlinear optics applications for the two-mode selective mode splitting (2SMS) case. (a) In the case of a conventional WGM microcavity (left column), realizing an efficient parametric nonlinear process relies on finding a device geometry whose global dispersion profile results in frequency matching for the modes of interest for a given nonlinear optical process, e.g., (I) intraband and (II) interband frequency conversion via FWM-BS, and (III) photon-pair generation. (b) In contrast, by using a 2SMS device (right column), frequency matching can be achieved without any specific consideration of the global dispersion profile (and hence the resonator cross section), so that any of the displayed nonlinear processes can be achieved. For intraband FWM-BS in a conventional microcavity [I(a)], global dispersion engineering typically leads to an unwanted conversion channel (dashed arrow) along with the targeted channel (solid arrow). But in an MSMS cavity, FWM-BS naturally occurs for only a single set of modes [solid arrows in I(b)]. Moreover, MSMS modes can be used in flexible ways, either exclusively MSMS modes [I(b)1], or combined with unsplit modes [I(b)2]. This MSMS cavity can also be applied to interband FWM-BS (II) and nondegenerately pumped photon pair generation (III) in similar ways, not only relaxing the frequency engineering for the targeted process (solid arrows), but also enabling suppression of the unwanted process [dashed arrows in I(a)–III(a)].
We first consider FWM-BS, which is a theoretically noiseless process for frequency conversion down to the single-photon level [50]. Recently, this process has been demonstrated in silicon nitride microrings [10], with intraband conversion (few nanometer spectral shifts) demonstrated for a quantum dot single-photon source [14] and a spontaneous four-wave mixing source [15]. This intraband FWM-BS process involved two spectral bands that are widely separated from each other, with the pump modes, shown in blue in Fig.
Similarly, the same MSMS scheme can be used in photon pair generation. In previous work, for example, telecom-band pair sources [13] and visible-telecom pair sources [9], dispersion had to be tailored to phase- and frequency-match targeted modes (solid arrows), and unwanted photon pair channels (dashed arrows) need to be carefully suppressed, as shown in Fig.
Finally, we note that the standing-wave nature of the MSMS modes must be carefully considered in the context of the applications being discussed. These standing waves are intrinsically a superposition state of CW and CCW modes, a property that could be useful for helping in the generation of orbital angular momentum in single-photon sources [51] and for enabling CW/CCW path entanglement [52]. On the other hand, in many cases we want the output signal to exit the cavity in one direction, which will not be the case for standing-wave modes. However, we note that the MSMS approach can still be valuable in such situations. In particular, frequency splitting may only be applied to pump modes, or MSMS may be used exclusively to frequency mismatch unwanted processes, while the process of interest is still accomplished using the typical traveling-wave modes.
5. CONCLUSIONS
In summary, we demonstrate a nanophotonic frequency engineering tool, MSMS, which manipulates multiple targeted cavity mode frequencies with individually controlled amounts of splitting. Such mode shifts can be large in amplitude (up to ), small in cross talk (less than 7.5% for untargeted modes), and high- (). The MSMS tool enables unique frequency engineering capabilities beyond traditional techniques, can be universally applied regardless of materials and geometries, and will have broad applications in microcavity nonlinear photonics.
Article Outline
Xiyuan Lu, Ashutosh Rao, Gregory Moille, Daron A. Westly, Kartik Srinivasan. Universal frequency engineering tool for microcavity nonlinear optics: multiple selective mode splitting of whispering-gallery resonances[J]. Photonics Research, 2020, 8(11): 11001676.