1. Introduction

Whispering gallery mode (WGM) is a high-Q optical resonance that has proven its capability in functions such as sensing [1], filtering [2], switching [3], lasing [4], and the study of quantum electrodynamics [5]. WGMs in a microsphere resonator can be categorized into various families that are differentiated by the mode numbers λ, q, and m which indicate the angular, radial, and azimuthal order respectively. Depending on the applications, higher-order WGMs in the resonator need to be selectively suppressed or enhanced.

On one hand, a clean spectrum without higher-order modes is favored for filtering and many nonlinear applications where mode competition demands that power of the desired resonances are maximized. Usually, the higher-order azimuthal modes can be eliminated by optimizing resonator geometry, such as the microtorus [6] and microtoroid resonators [7]. Modes of a higher radial order (q>1) are difficult to eliminate without changing the resonator structure. Efforts have been made to clean up spectrum of a microbottle resonator by scarring the resonator surface [8]. Mode filtering on a disk resonator has been demonstrated with the help of a prism mode damper [9]. However, a non-destructive mechanism that does not rely on additional structures for suppression of unwanted resonances is still lacking.

On the other hand, higher order WGMs prove to be useful in several aspects. Bio-chemical sensing that is carried out in microfluidic samples and is confined by capillaries depends on the higher radial order WGMs for higher sensitivity [10]. The greatest challenge of such applications is the thick capillary wall against mode penetration: the wall thickness needs to be reduced in order for the mode to penetrate and interact with the sample [11]. In addition to wall thickness reduction, radially gradient refractive index profile is a new proposal of the solution [12, 13]. However, a precise gradient index profile is hard to achieve in practice. In this case, higher radial order WGM for deeper penetration is considered as a promising solution. In the domain of nonlinear optics, high order modes are usually desired by applications that require rather large interaction volume and that are not so stringent on Q-factor requirement [14, 15]. Moreover, for WGM resonators in general, the higher radial order WGMs are desired to avoid surface defects of the resonator so that higher Q-factors can be achieved [13]; burying the mode inside the resonator also bears the potential benefit of achieving critical coupling at mechanical contact [13].

In this work, we investigate an angle-incident subwavelength metallic diffraction grating coupler that selectively enhances or suppresses higher radial order WGMs. The metallic grating has a wide coupling bandwidth [16] and is able to couple efficiently to the targeted WGM [17]. A small incident angle gives rise to an unequal excitation of two counter-propagating waves inside the resonator. By phase-matching designs, the grating is able to suppress or enhance modes of different orders and selectively send them into two propagation directions.

2. Working principle

As depicted in the 3-D schematic of Fig. 1(a), the investigated structure consists of a one-dimensional gold grating patterned on the end face of a single mode fiber (SMF) and a microsphere placed vertically above the grating. The SMF is angle-cleaved so that input beam is incident upon the grating at a small angle. The schematic shows a normal cleaved end face with a tilted input beam, which is an equivalent case of the angle-cleaved SMF. The grating is able to excite multiple diffraction orders that bear transverse momentum to couple to the WGM of the microsphere [16]. At 0° incident angle, the diffracted beams consist of the zeroth and first orders while the higher orders are negligible. This is still the case when the incident angle is small (θ<10°). Figure 1(a) shows that, when the light is input at an angle θ, it would emerge from the grating as a tilted zeroth order diffraction that shoots through the sphere and the ± 1st order diffractions that then curve inside the sphere as the counter-clockwise (CCW) and clockwise (CW) propagating WGMs.

 

Fig. 1 (a) 3-D schematic illustration of the metallic grating coupled silica microsphere with a small incident angle; (b) 2-D illustration of the x-y cross section of the 3-D structure.

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Since the 3-D WGMs of fundamental azimuthal orders are equivalent to those of a 2-D microdisk, we use 2-D finite difference time domain (FDTD) method to study the WGMs coupled by a gold diffraction grating at a small incident angle. Figure 1(b) shows the 2-D schematic of the configuration, which is the x-y cross section of the 3-D structure in Fig. 1(a) when z = 0. Figure 2(a) presents the power flux of light (termed the “power transmission through monitor”) integrated along Monitor 1 that is inserted near the surface of a silica microdisk (radius = 15μm). The line monitor is able to detect the propagation direction of waves: it indicates the CCW propagating mode as positive transmission and the CW propagating mode as negative transmission. The incident angle is 7° and the grating pitch size is Λ = 1200nm. The grating has a filling factor (gold/air) of (50/50) and a thickness of 100nm. The microdisk is placed from the top of the grating to maintain an air gap of 100 nm, so that critical coupling for the fundamental WGM can be achieved [16, 17]. In this work, the transverse electric (TE) polarization is investigated (along the z-axis) given that the transverse magnetic (TM) waves behave similarly.

 

Fig. 2 (a) Power transmission through a line monitor that is inserted near the surface of the silica microdisk with a radius of 15μm. The grating has a pitch of Λ = 1200nm and an incident angle of 7°. Insets show typical power density distribution of the 1st, 2nd, and 3rd radial-order WGM; (b) Power spectrum of the CW- and CCW- propagating waves that are coupled out by a fiber taper and measured at its two ends respectively.

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In Fig. 2(a), altogether three radial orders of WGM resonances (indicated by mode number q) are observed: the fundamental mode (q = 1, or the 1st) is coupled by the negative first-order diffraction and as a result, travels in the CW direction; the second and third radial-order WGMs (q = 2, 3, or the 2nd, 3rd) are coupled by the positive first-order diffraction and travels in the CCW direction. All of the resonances are indicated by their radial orders in the graph and three power density distributions are shown as insets for the typical 1st, 2nd, and 3rd radial-order WGMs at a wavelength of 1515.4nm, 1513nm, and 1590.7nm respectively. To verify the traveling direction of the excited WGMs, a tapered silica fiber coupler is placed on top of the microdisk to measure the out-coming spectra. The tapered fiber is set at a waist diameter of 1μm and is positioned to create an air gap of 300nm between the taper and resonator. Line monitors 2 and 3 [Fig. 1(b)] record the spectra of waves that are propagating in the negative x (CCW) and positive x (CW) directions respectively. Figure 2(b) shows the measured spectra, where the CCW-propagating waves are presented in negative values for clear recount of spectral details. At the CW propagating end, only resonances of the 1st radial-order are observed. At the other end that couples only the CCW propagating waves, the 2nd and 3rd radial-orders are present along with a Fabry-Perot background generated by the zeroth-order diffraction.

The probability of one WGM being coupled to a specific diffraction order is determined by the phase-matching conditions, which can be understood as numerical matching of the effective mode indices (n_eff). The effective mode index of a WGM (n_WGM) is determined by the results of Reference [18] for the case of spherical resonator. On the other hand, the effective index of the diffraction mode’s transverse component is given by the grating equation:

 

Fig. 3 (a) Calculated effective mode indices of the WGM and diffraction orders for the structure investigated in Fig. 2; Enlarged resonances from spectra presented in Fig. 2(a) at wavelengths of around (b) 1.534μm; (c) 1.552μm; (d) 1.570μm; (e) 1.590μm.

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From Fig. 3(a), it is observed that the fundamental WGM is prone to couple with the negative first–order diffraction. The 2nd and the 3rd radial-order WGMs have smaller n_eff and are more likely to couple with the positive first-order diffraction. Hence, the WGM propagation directions agree with the results of Fig. 2(a). As the wavelength increases, the n_eff of the first-order diffractions increases while that of the WGMs decreases. In this case, the 1st radial-order WGM parts away from the negative diffraction while the 2nd and 3rd radial-orders grow closer to the positive diffraction. This is why in Fig. 2(b), the peak intensity of the 1st orders decreases and that of the 2nd and 3rd orders increase with the wavelength. Moreover, during this process, the n_eff of the 1st radial-order WGM grows closer to the positive first-order diffraction, making it more probable for the WGM to couple to the positive diffraction. Mode splitting is observed as a result of intermodal coupling between the CW- and CCW-propagating modes in small resonators where ratio of modal to resonator volume is large [19]. Due to the presence of different diffraction orders, waves of both propagation directions are excited. In this case, the occurrence of mode splitting at a particular wavelength is determined by the relative field strength of the counter-propagating waves, which is in turn controlled by the phase-matching conditions. Figure 3(b-e) zoom in on four resonances of Fig. 2(a) to show the evolution of mode splitting. It is observed that splitting does not appear till the wavelength of 1.552μm in Fig. 3(c). As the wavelength increases, the field strength of the CCW-propagating 1st radial-order WGM increases and intermodal coupling grows stronger.

In Fig. 3(a), it is also observed that the higher radial orders WGMs are not coupled by the negative first-order diffraction at a shorter wavelength even though their effective indices are closer. This is because the gap size between resonator and grating is optimized for the critical coupling of the fundamental modes. Since the higher radial order WGMs are buried deeper inside the resonator, achieving critical coupling for these modes requires that the gap size be reduced. Hence, when phase-matching conditions are approached, the lower orders are usually excited first (for the case when its mode index is not too different from that of the coupler).

3. Controlled excitation with grating pitch size

According to Eq. (1), the incident angle determines the separation between the negative and positive first order diffractions while the pitch size determines the slope of change. Figure 4 shows the spectra coupled by a grating with 1125nm pitch size and a fixed incident angle at 7°.

 

Fig. 4 (a) Calculated effective mode indices of the WGM and diffraction orders for incident angle of 7° a grating pitch size of 1125nm; (b) Power spectrum of the CW and CCW travelling modes that are coupled out with a fiber taper and measured at the two ends respectively. Insets show the zoom-in views of the first 3rd radial-order WGM and the last two fundamental WGMs in the CCW-propagating spectrum.

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As Λ decreases, the slope of the grating index increases while the n_eff values are shifted upward for both positive and negative diffraction orders. Figure 4(a) shows that the n_eff of the negative first-order diffraction is so elevated that all n_WGM is far away. For the 3rd radial-order WGM, since its n_eff is parting with the first-order diffraction as the wavelength decreases, decreasing resonance peaks are observed in Fig. 4(b). The 3rd radial-order resonance close to 1.515μm is weak because it coincides with a low-Q Fano resonance (magnified view is shown in the inset). This Fano resonance comes from the interference of a CCW-propagating 3rd radial-order WGM of low-Q and a CW-propagating high-Q fundamental WGM that is backscattered at the taper-resonator coupling point. For the 2nd radial-order WGM, the positive diffraction intersects with the n_eff of the resonance at around 1.57μm, yielding the highest peak intensity in the taper-coupled spectrum. The 1st radial-order WGM, on the other hand, is coupled by the negative diffraction throughout the CW-propagating spectrum and is weakly coupled by the positive diffraction for the last two resonances of the spectrum. Similar to the spectra of Fig. 2(a), bi-directionality of the travelling waves gives rise to mode splitting, some examples of which are indicated by pairs of adjacent peaks appearing in the spectra of both propagation directions for the fundamental mode.

4. Controlled excitation with incident angle

As illustrated in Fig. 5(a), the n_eff of the diffraction modes are separated further apart at an incident angle of 10°. All modes are more likely to couple with the positive diffraction as the wavelength increases. Since mode of a lower radial order has a greater probability to be critically coupled when the phase-matching conditions are approached, the 2nd and 3rd radial-order resonances have comparable peak intensities [Fig. 5(b)] even when the 3rd order WGM is closer in fulfilling the phase-matching conditions. On the other hand, considerable coupling of the negative diffraction to the 1st radial-order WGM is inevitable as the gap size is optimized for its critical coupling. Similar to the case in Fig. 4(b), Fano resonances are observed in Fig. 5(b) and one example is enlarged in the inset. At longer wavelengths, the n_eff of the fundamental WGM grows closer to the positive diffraction and appears in the CCW-propagating waves. Hence, it is demonstrated that by choosing a proper incident angle, WGMs of even higher radial orders can be selectively enhanced. As it is difficult to optimize the gap size for the critical coupling of 3rd radial-order mode in such small resonator, the coupled resonances are relatively weak.

 

Fig. 5 (a) Calculated effective mode indices of the WGM and diffraction orders for incident angle of 10° and a grating pitch size of 1200nm; (b) Power spectrum of the CW and CCW travelling modes that are coupled out with a fiber taper and measured at the two ends respectively. Insets show the zoom-in views of two 3rd radial-order WGMs in the CCW-propagating spectrum.

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5. Conclusion

It has been demonstrated that the metallic diffraction grating coupler with a small incident angle is able to excite WGMs of different radial orders and separate them by coupling into opposite propagation directions. Both second and third radial-order WGM can be individually enhanced. The metallic diffraction grating coupler provides extra freedom of direction switching between the CCW and CW propagating modes. It is also able to selectively enhance or suppress higher radial order WGMs by phase-matching designs. Such structure offers a simple and practical configuration for various WGM applications including liquid sensing, band pass filtering and fiber lasers.