Rotation-induced evolution of far-field emission patterns of deformed microdisk cavities

A critical issue that hinders the development of chip-scale optical gyroscopes is the size dependence of the Sagnac effect, which manifests as a rotation-induced phase shift or frequency splitting between two counterpropagating waves or resonances, and is proportional to the size of the optical system. We show numerically and theoretically that the far-field emission patterns (FFPs) of optical microdisk cavities depend strongly on rotation and can therefore provide an alternative approach. At low rotation speed where resonant frequencies barely shift with rotation (i.e., a negligible Sagnac effect), the FFPs already exhibit a significant rotation-induced asymmetry, which increases linearly with the rotation speed. We further identify the basic requirements to maximize this effect, including distinct output directions for the clockwise and counterclockwise waves in a cavity mode, as well as a vanishing frequency splitting between one such mode and its symmetry related partner mode. Based on these requirements, we propose several microcavity shapes that display orders of magnitude enhancement of the emission sensitivity to rotation and could stimulate a new generation of optical gyroscopes with small footprints and on-chip integrability. © 2015 Optical Society of


INTRODUCTION
Optical gyroscopes have revolutionized precision measurement of rotation thanks to their scientific ingenuity, affordability, long-term reliability, and compact size [1][2][3][4][5].They have been widely utilized for both civilian and military aircraft as well as satellites, rockets, and nautical navigation.Meanwhile, optical microcavities have also found a broad range of applications since their debut two decades ago [6,7], from coherent light sources in integrated photonic circuits to cavity quantum electrodynamics, single-photon emitters, and biochemical sensors.Due to their small footprints and on-chip integration capability, microcavity-based gyroscopes [8][9][10][11] can play an important role in reducing the equipment cost in space missions and open the possibility for a new generation of on-chip optical gyroscopes [12].
One obstacle to miniaturization [13][14][15][16][17][18][19] is imposed by the current measurement scheme of optical gyroscopes, which is based on the Sagnac effect [1] and has barely changed in the last 50 years since its introduction.The Sagnac effect manifests as a rotation-induced phase shift in a nonresonant structure (e.g., an optical fiber) or a frequency splitting in a resonant cavity, between two counterpropagating waves or resonances; it is proportional to the size of the cavity, which puts optical microcavities at a serious disadvantage in terms of sensitivity when compared with macroscopic cavities used in current optical gyroscopes.Therefore, to make optical microcavities a viable option for rotation sensing, a new detection scheme must be developed.Previous studies [11,20,21] indicate that the quality (Q) factor of resonant modes also displays a rotation-induced variation, and its relative change can be higher than that of the resonant frequencies.This enhancement, however, is still not sufficient to compensate for the small size of microcavities, with sensitivity far below the Sagnac effect in macroscopic cavities.
In this work we investigate rotation-induced changes of the far-field emission patterns (FFPs) of microdisk resonances.We find a surprisingly strong dependence of FFPs on rotation speed, which may be used as a measurable signature of rotation.This strong FFP sensitivity to rotation is achieved by satisfying three basic requirements.The first one is that the emission is nonisotropic so that the output direction may be changed by rotation.This requirement can be realized by deforming the disk shape from a circle [22][23][24][25][26][27].The second requirement is that the clockwise (CW) and counterclockwise (CCW) waves in the nonrotating cavity should have distinct FFPs.Rotation changes the relative weight of CW and CCW waves in a cavity resonance, resulting in a strong change in the FFP when this requirement holds.Finally, the relative weight of CW and CCW waves should be extremely sensitive to rotation even at very low rotation speed.This requirement can be satisfied by minimizing the frequency splitting between a pair of quasi-degenerate resonances in the nonrotating cavity.By implementing these requirements, we show that a dramatic enhancement of the FFP sensitivity to rotation can be achieved.

ROTATION-DEPENDENT FAR-FIELD PATTERN
We consider a deformed semiconductor microdisk [22][23][24][25][26][27] with thickness much less than the radius.Due to the strong index guiding of light in the disk plane, the microdisk can be treated as a two-dimensional (2D) cavity with an effective refractive index n.Here we focus on a cavity with at least one symmetry axis, chosen to be along θ 0, 180°in the polar coordinates, and asymmetric cavities will be discussed elsewhere.The disk boundary is described by ρθ, and the symmetry requires ρ−θ ρθ.When the cavity does not rotate, the resonant modes generally form quasi-degenerate pairs k even , k odd , whose wave functions have even [ψ even r; θ ψ even r; −θ] and odd [ψ odd r; θ −ψ odd r; −θ] symmetries about the cavity axis.Therefore, they consist of an equal amount of CW and CCW waves, denoted by ψ m<0 and ψ m>0 , where m is the angular momentum number [see Eq. ( 2)].Their FFPs I FFP θ are both symmetric about the symmetry axis, i.e., I FFP −θ I FFP θ.
As a microcavity rotates, a pair of such quasi-degenerate ψ even , ψ odd resonances couple to each other and gradually become a pair of CW (ψ cw ) and CCW (ψ ccw ) resonances [8].The CW and CCW waves experience opposite frequency shifts by rotation, leading to an increase of their frequency splitting (Sagnac effect).However, such an increase is significant only when the rotation speed exceeds a critical value Ω c .Below it the frequency splitting barely changes with rotation, which is referred to as a "dead zone" in deformed microcavities [8][9][10].However, within the dead zone, the balance between CW and CCW waves in a resonance is already broken by rotation.If the CW and CCW waves have different output directions, the FFP may start changing even within the dead zone.As will be shown in the next section, the unbalance between CW and CCW waves introduces an asymmetry in the FFP, which increases linearly with rotation speed in the "dead zone" and hence displays a much stronger dependence on rotation speed than the Sagnac effect.
Without loss of generality, we focus on transverse magnetic (TM) resonances whose electric field is perpendicular to the disk plane.We take the angular velocity Ω to be a constant and perpendicular to the cavity plane, with the convention that Ω > 0 indicates a CCW rotation.When jRΩ∕cj ≪ 1, the resonances of an optical microcavity are determined by the modified Helmholtz equation [8] ∇ to the leading order of Ω in the rotating frame.Here R is the average radius of the cavity, n⃗r is the refractive index, ω is the complex resonant frequency of mode ψ⃗r, and c is the speed of light in vacuum.To avoid the confusion with the rotation speed, we use the wave vector k ≡ ω∕c instead of ω henceforth.We assume that the detecting apparatus is integrated on the same chip as the microcavity; thus it measures the FFP in the rotating frame where the microcavity is stationary.
To find the optical resonances in a rotating microcavity and their FFPs, one can use the finite-difference time-domain method adapted to the rotating frame [20,28].Here we employ a frequency-domain method that is grid-free-the scattering matrix method [21].In this approach the wave function of a resonance is decomposed in the angular momentum basis, i.e., ψ⃗r P m0;1;2;… A m re imθ , where Here H m are the Hankel functions of the first and second kind, describing outgoing and incoming waves.km ≡ nk 2 − 2mk Ω∕R

A. Case Study: Symmetric Limaçon Cavity
The first example we analyze is the limaçon cavity, whose boundary is given by ρθ R1 ε cos θ, where ε is the deformation parameter.Over a wide range of ε, the cavity supports resonances with high Q-factor and directional emission [24].Figure 1(a) shows a ψ even resonance at ε 0.41.This whispering-gallery (WG) like resonance has the normalized frequency k even R ≃ 33.78, which has dominant angular components m 101 and corresponds to a vacuum wavelength λ ≈ 930 nm if we take R 5 μm.The FFP in Fig. 1(b) shows that the emission is predominantly in the forward direction θ 0.
As mentioned previously, the cavity symmetry ρθ ρ−θ determines that the CW and CCW waves in its resonances at rest have the same magnitude.The CW wave of this ψ even resonance has two main peaks in the FFP, one near θ 0 and a slightly weaker one at θ cw ≈ −138° 222°[Fig.1(d)]; they are attributed to the chaotic diffusion of optical rays inside the cavity [24,29], which can be seen in the logarithmic-scale intensity plot in Fig. 1(c).The mirror image of these patterns about the symmetry axis gives the FFP of the CCW wave, with the secondary peak located at θ ccw ≈ 138°.The constructive interference of the CW and CCW waves enhances the emission near θ 0 of the ψ even resonance.
As the cavity rotates, the initial balance between the amplitudes of the CW and CCW waves is broken, which is similar to the finding in closed billiards [8].In this ψ even resonance the CW wave becomes stronger with rotation speed, and the intensity peak at θ cw increases with respect to the ones at θ ccw and θ 0 [see Fig. 2(a)], leading to an asymmetric FFP.The opposite takes place in the corresponding odd-symmetry resonance ψ odd , with the CCW wave becoming prevailing and an increasing intensity peak at θ ccw (not shown).Consequently, we find that the CW and CCW waves ψ m<0 , ψ m>0 in the ψ even , ψ odd resonances at rest give good approximations of the CW and CCW resonances (ψ cw , ψ ccw ) at high rotating speed, i.e., ψ cw ≈ ψ m<0 and ψ ccw ≈ ψ m>0 .
For the FFP evolution to have a strong dependence on rotation, obviously we require ψ cw , ψ ccw to have very different FFPs from ψ even , ψ odd .It is clear from the discussion above that this criterion can be directly evaluated in a cavity at rest, by requiring that ψ m<0 and ψ m>0 have very different FFPs.In a limaçon cavity the FFP peaks of ψ m<0 at θ cw and ψ m>0 at θ ccw satisfy this requirement.Utilizing this difference, the evolution of the FFP I FFP θ of the ψ even resonance shown in Fig. 2(a) can be quantified by the asymmetry where σ is the angular detection window of each peak and taken to be 15°.We note χ Ω 0 0. In Figs.2(b) and 2(c) we show how the Sagnac effect, given by the real part of the dimensionless frequency splitting Δ k cw − k ccw R of the ψ cw , ψ ccw resonances, and the FFP asymmetry χ evolve with rotation speed.The "dead zone" for Δ lies below a critical speed Ωc ∼ 10 −9 (or equivalently, Ω ∼ 6 × 10 4 rad∕s), within which Δ barely changes.In   contrast, χ displays a linear increase with the rotation speed Ω in the "dead zone," based on which the rotation speed can be detected.The minimum detectable Ω scales linearly with the smallest asymmetry that can detected in the FFP measurement.For example, if we assume that minχ ∼ 10 −4 , the lowest rotation speed that can be measured is Ω ≈ 10 −13 , or equivalently, Ω ∼ 6 rad∕s.This performance is comparable to commercial optical gyroscopes based on the Sagnac effect in macrocavities, whose sensitivity would be 10 4 times lower when simply scaled down to microcavities.

B. Further Enhancement of Far-Field Sensitivity to Rotation
To enhance the FFP sensitivity to rotation, we conduct a quantitative analysis with the coupled-mode theory.The increase of the FFP asymmetry χ with the rotation speed can be attributed to the mixing of one resonance with others by rotation.Since ψ even and ψ odd are quasi-degenerate, their mutual coupling is much stronger than the coupling with other resonances farther away in frequency [30,31].Therefore, we can approximate ψ Ω ≈ a even Ωψ even a odd Ωψ odd [8][9][10]21], which gives the frequency splitting Here Δ 0 ≡ Δ Ω 0 and g is the dimensionless coupling constant between ψ even and ψ odd , which are approximately real for high-Q resonances [21].In a deformed microcavity, Δ 0 ≠ 0 in general, and the dead zone is determined by Ωc ≡ jΔ 0 j∕g.When Ω ≫ Ωc , Δ approaches its asymptote Δ Ω ≈ g Ω and displays the familiar linear scaling of the Sagnac effect.Below Ωc , the rotation-induced splitting (g Ω is much smaller than the intrinsic splitting (Δ 0 ), and the Sagnac effect becomes very weak as shown in the "dead zone" in Fig. 2(b).Using g 21.45 − 0.004i and Δ 0 2.29 0.90i × 10 −8 from the scattering matrix calculation, Eq. ( 4) gives a good approximation of Ωc ≈ 1.07 × 10 −9 .
A high rotation sensitivity of the FFP asymmetry requires a rapid increase of the mixing ratio ξ Ω ≡ a odd Ω∕a even Ω with rotation speed, in addition to very different FFPs for ψ m<0 and ψ m>0 at rest.Deep in the dead zone ( Ω ≪ Ωc ), the mixing ratio ξ Ω in the initially ψ even resonance is approximately from which we see immediately that the key quantity is a small Ω c , or equivalently, a small frequency splitting Δ 0 at rest and a large coupling g between ψ even , ψ odd resonances.g is proportional to the optical path length and hence limited by the small size of microcavities.Δ 0 , however, can be reduced by using microcavities with more than one symmetry axis [9].One example is the microcavity with spatial symmetry described by the dihedral group D 3 , ρθ R1 ϵ cos 3θ [9], which we will simply refer to as the D 3 cavity (see Fig. 3).Idealistically Δ 0 can be entirely eliminated for the resonances whose angular momenta m in Eq. ( 2) are not integer multiplies of 3, giving a linear increase of the frequency splitting Δ with the rotation speed Ω [Fig.3(d)].In practice, there is always inherent surface roughness introduced unintentionally during the fabrication process, which breaks the exact D 3 symmetry and lifts the degeneracy of ψ even , ψ odd at rest slightly.Thus a "dead zone" is created, but its size is expected to be much smaller than the intrinsic one, e.g., for the limaçon cavity.The resulting small Ωc greatly enhances the rotation dependence of the FFP asymmetry at low speed, given that the ψ m<0 and ψ m>0 waves also have very different FFPs here [Fig.3(b)].
To visualize this expected enhancement, we plot χ by constructing ψ Ω ∝ ψ even ξ Ωψ odd :ψ even , ψ odd are directly obtained by the scattering matrix method, and ξ Ω can be  4).(e) Rotationinduced FFP asymmetry χ of the ψ even resonance as a function of Ω. θ cw 254°, θ ccw 226°(equivalent to the one at −254°), and σ 15°are used in calculating χ Ω from Eq. (3).calculated from Eq. ( 5), using g ≈ 20.62 again from the scattering matrix method and taking Ωc 10 −14 , which is 10 5 smaller than that in the limaçon case.The result is shown in Figs.3(c) and 3(e), and a rotation sensitivity of χ is enhanced by roughly the same factor, i.e., 10 5 , at low speed when compared with the limaçon cavity shown in Fig. 2(b): now χ 10 −4 corresponds to Ω ≈ 10 −18 instead of 10 −13 .

DISCUSSION AND CONCLUSION
In summary, we have investigated how rotation modifies the FFPs of open microcavities.In a 2D cavity deformed from a circle and with at least one symmetry axis, the FFPs are nonisotropric and each resonance has equal CW and CCW wave components at rest.Rotation breaks the balance between CW and CCW waves, causing a significant change in the FFP, if the CW and CCW waves have distinct output directions.At low rotation speed where the resonant frequencies barely shift with rotation, the FFPs already exhibit asymmetry, which increases linearly with the rotation speed.The sensitivity of the FFPs to rotation can be enhanced by reducing the intrinsic splitting ΔΩ 0 of quasi-degenerate resonances at rest, achievable via engineering the cavity shape.For example, the D 3 symmetry can support degenerate resonances with directional emission.Using a perturbation theory [30][31][32], one can show that the small ΔΩ 0 in a D 3 cavity due to surface roughness scales inversely with the system size to the leading order, and hence a larger cavity will have a stronger sensitivity to rotation.To eliminate this residual ΔΩ 0 , one solution is using the hexagonal cavity, which also possesses the D 3 symmetry and supports degenerate resonances.Since single-crystalline GaN or ZnO disks with hexagonal cross section can be grown with atomic flat surfaces [33][34][35][36], ΔΩ 0 due to surface roughness can be greatly reduced.Finally we comment that the current study is focused on the FFP change of individual resonances in the linear regime.It would be interesting to explore the simultaneous excitation of multiple resonances and their nonlinear interactions in rotating microcavities in future work.

1 2 1 2
and km ≡ k 2 − 2mk Ω∕R are m-dependent wave vectors inside and outside the cavity, where Ω ≡ RΩ∕c is the dimensionless rotation speed.

Fig. 1 .
Fig. 1.Near-field and far-field intensity patterns of a nonrotating limaçon cavity.(a) Near-field intensity pattern in the logarithmic scale and (b) far-field intensity pattern in the polar coordinates of a symmetric resonance ψ even at k even R ≈ 33.78.The cavity deformation is ε 0.41, and the refractive index is n 3. The resonance is concentrated near the cavity boundary, similar to a whispering-galley mode.Its emission is directional, with the main direction centered in θ 0°and the secondary peaks in θ ≈ 138°and −138° 222°.The CW wave component of this resonance is separated and plotted in (c) and (d).Its mirror image about the horizontal axis gives the CCW wave (not shown).

Fig. 2 .
Fig. 2. Rotation-induced changes in resonance frequency and far-field emission pattern of a limaçon cavity.(a) Far-field intensity pattern of the resonance shown in Fig. 1 when the dimensionless rotation speed Ω 10 −8 (thick line), 10 −9 (medium line), and 10 −10 (thin line).The maximum intensity is normalized to 1 for each curve.Rotation enhances the emission peak at θ cw and reduces the one at θ ccw .(b) Dimensionless frequency splitting Δ of the quasi-degenerate resonances as a function of the normalized rotation speed Ω. Δ remains nearly constant below Ωc ≈ 10 −9 (marked by the vertical dashed line), marking a "dead zone" for the Sagnac effect.The diamonds represent the numerical data, and the solid line shows the result of the coupled-mode theory, Eq. (4).(c) Rotation-induced FFP asymmetry χ as a function of Ω. χ increases linearly with Ω inside the "dead zone."

Fig. 3 .
Fig. 3. Enhanced sensitivity of far-field emission pattern to rotation in a D 3 cavity with R 5 μm and n 3. (a) Near-field intensity plot of the CW wave in a whispering-gallery-like resonance at k even R ≈ 33.80 in the logarithmic scale.The dominant angular components of this resonance are m 94.(b) Distinct FFPs in the polar coordinates of CW (solid) and CCW (dotted) waves at rest.(c) Far-field intensity pattern at Ω 10 −13 (thick line), 10 −14 (medium line), and 10 −15 (thin line).(d) Dimensionless frequency splitting Δ as a function of the normalized rotation speed Ω, showing a linear increase due to the vanishing "dead zone".The diamonds represent the numerical data, and the solid line shows the result of the coupled-mode theory, Eq. (4).(e) Rotationinduced FFP asymmetry χ of the ψ even resonance as a function of Ω. θ cw 254°, θ ccw 226°(equivalent to the one at −254°), and σ 15°are used in calculating χ Ω from Eq. (3).