Coupled-resonator-induced transparency (CRIT) has been widely studied since it was discovered in coupled whispering-gallery-mode (WGM) resonator structures.^[1–3] CRIT renders a coupled-resonator structure transparent over a narrow spectral range because of the classical destructive interference, analogous to electromagnetically-induced-transparency (EIT). CRIT creates extreme dispersion within the transparency window, which leads to a high group index and slow light. Experimental observations of CRIT have been reported in various WGM resonators, such as fiber ring resonators, microspheres, and integrated micro-ring resonators. It has potential applications in various fields, such as slow light, optical buffers, and optical sensors.^[4,5]

Coupled resonator optical waveguide (CROW) structures used as angular velocity sensors have attracted much interest in recent years, because their large structural dispersion may enhance the Sagnac phase shift.^[6] Zhang et al. confirmed that the gyroscope sensitivity has been enhanced in the slow light structure in experiment.^[7] Researchers proposed different kinds of integrated CROW structures, the transmission loss of waveguide is much larger than that of the fiber, because of the material and process limitation of the waveguide. In this paper, we proposed a compact CRIT structure composed by two coupled silica microspheres, which can be used in ultra-low transmission loss CRIT gyro devices.

Microspheres are often used as a WGM resonator in experimental devices owing to its ultra-high sensitivity factor (up to 10¹¹) and convenience in preparation.^[8] Microspheres are generally fabricated by melting the tip of an optical fiber in a hydrogen–oxygen flame; they possess ultra-high surface smoothness because of the surface tension.^[9] The most effective way of coupling to a microsphere is using a tapered fiber, where the coupling efficiency can reach 99%.^[10] In our study, we melted a single fiber to obtain silica microspheres as WGM resonators, and placed them on three-dimensional (3D) adjustment stages to control their coupling gap accurately. CRIT phenomena were observed on a test system.

We analyzed a model of coupled microspheres, and obtained the theoretical angular velocity sensitivity enhancement factor of the structure. For a passive microsphere resonator, the phase difference between the counter-propagating light is given by the Sagnac effect, that is

For a single microsphere resonator, the theoretical limit δΩ can be expressed as^[6]

In Ref. [11], the authors analyzed the rotation sensitivity of two coupled resonator structure, the two resonators of which have the same radius. In this bidirectional structure, the light travels in opposite directions in the two resonators, the phase shifts induced by Sagnac effect have opposite signs, the two Sagnac contributions cancel out. However, if the sizes of the two resonators are slightly different, and the same bias and coupling optimization is applied again, the sensitivity increases dramatically. That is because when the resonators are no longer identical, the Sagnac phases in the two resonators no longer have the same amplitude, and they no longer cancel as strongly. In Ref. [12], the authors compared the sensitivity of the CRIT structure with that of a usual passive single resonator. When the parameters of the two are respectively the same, the CRIT structure is more dispersive than the single resonator. The highly dispersive structure is more susceptible to the phase shift induced by the Sagnac effect. And a folded configuration was proposed, in this structure, light travels in the same direction in the two resonators, thus enhancing the sensitivity of rotation.

Figure 1 shows a stacked system comprising a microsphere coupled to another microsphere coupled to a tapered fiber. A light wave enters from the left end of the tapered fiber. The transfer function of the coupled microspheres is expressed as

Fig. 1.

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	Fig. 1. Two coupled microspheres, coupled to a tapered fiber. The input and output fields are E1 and E2; E3, E4, E5, E6, E7, and E8 are optical fields at several points inside the system.

Let k₁ and t₁ be the coefficients for coupling and transmission of the tapered fiber and the first sphere, respectively; similarly, let k₂ and t₂ be the coefficients for coupling and transmission of the first sphere and second sphere, respectively. A coupling matrix formalism introduced by Poon et al.^[13] is used in the analysis. The relationship among the mode amplitudes can be expressed as

In Eqs. (10) and (11), ϕ_i = 2πn_effL_i/λ (i = 1,2) is the round-trip phase for each microsphere. The optical pathways through the first sphere and the second sphere are illustrated in Fig. 1. The ϕ = ϕ₁ − arg (T₂) is the phase difference between two optical pathways.

Fig. 2.

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	Fig. 2. Results from the simulation model of the two coupled microspheres with t1 = 0.9, t2 = 0.999: a1 = 0.88, a2 = 0.999: (a) transmission, (b) effective phase shift, and (c) group delay.

In the simulation calculations, the impact of the coupler insertion loss was ignored and the resonator attenuation factors a₁ = 0.88 and a₂ = 0.999 were maintained. Set t₁ = 0.9 and let the first microsphere and the tapper in the under-coupled state. Gradually change the coupling factor t₂ between the two microsphere resonators and let the coupling of two microspheres change from the over-coupled state to the under-coupled state. Mathematically calculated resonance spectra under different t₂ are shown in Fig. 3.

Fig. 3.

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	Fig. 3. The transmission change curves with the transmission coefficient of sphere 2 when t1 = 0.9.

As seen in Fig. 3, with the change of t₂, the shape of the resonance peaks does not change and only the resonance peaks shift, while two microspheres are in over-coupled state (t₂<a₂); while when t₂>a₂, the shape of the resonance peaks changes, as the two microspheres are in under-coupled state most of the light propagates in the first microsphere.

When this structure is rotated around an angular velocity Ω, rotating axis perpendicular to the plane of propagation of light in the spherical cavities, the transmitted power of the output is changed with the Sagnac phase shift. The sensitivity reaches the maximum at the CRIT transparent peak. The total complex phase shift experienced by the signal while it travels once around the first microsphere and the second microsphere are given by

Fig. 4.

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	Fig. 4. Rotation-dependent transmitted power P as a function of angular velocity Ω.

The experimental setup is schematically shown in Fig. 5. Silica microspheres were fabricated by melting the end of an optical fiber. The tapered fiber was used as a waveguide to excite the WGMs using a tunable laser with the wavelength of 1550 nm and the spectral linewidth of 300 kHz. The function generator was used to linearly tune the laser frequency by setting a triangular voltage signal to the laser. To obtain the CRIT spectrum of the two microspheres, a 3D adjustment stage was used to couple a microsphere to the taper, adjusting the gap between the microsphere and the taper, making them in the critical coupling state. The image of microsphere coupled-resonator-induced transparency structure was captured by a CCD as shown in Fig. 5. The resonance spectrum of the single microsphere was observed on an oscilloscope, and then we adjusted the other microsphere towards the first sphere. In this process, a transparency peak (see Fig. 6) arose at the WGM resonance point in the absorption spectrum of the single microsphere. This resonance peak was split into two modes, the peaks on the right hand side of Fig. 6 are high-order modes of the first microsphere, because the sphere is a 3D structure resonator. We show the emergence of the CRIT spectrum as the second microsphere approaches and couples to the other.

Fig. 5.

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	Fig. 5. Schematic diagram of the two-coupled microsphere CRIT test system.

Fig. 6.

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	Fig. 6. Test results of the experiment system shown in Figs. 4(a)–4(c). Transmission, effective phase shift, and group delay with two coupled microspheres, with diameters of two spheres of d1 = 1000 μm and d2 = 800 μm. (d1)–(d4) In drawing near one microsphere to couple to the other, the WGM resonance spectrum is split into two modes.

After the CRIT spectrum obtained, we divided the input light into two equal beams by a 3 dB coupler C1, one beam through the two-coupled microsphere resonator structure, the other through a LiNbO₃ phase modulator. Two beams of light through the fiber length are equal to offset the phase shift caused by the fiber length. The phase modulator applied a phase offset π/2 on the second beam. Two beams through 3 dB coupler C2 interfere to form a fiber MZ interferometer,^[16] the expressions of optical fields inside the upper and the lower arms can be written as

As shown in Figs. 6(d1) and 6(d4), the transmission spectrum of a single microsphere cavity is an absorption peak, a transparent peak emerges in the absorption peak, due to the destructive interference of light when the other microsphere cavity approaches, which is the CRIT effect. The relative position of the transparent peak and the absorption peak could be tuned by adjusting the relative position of the two microsphere resonators. The effective phase of this CRIT effect could be measured by the experiment system shown in Fig. 6, and the result is shown in Fig. 6(b). Normal dispersion appears at the CRIT transparent peak, the group velocity of light is slowed down, and anomalous dispersion emerges at both sides of the CRIT peak, fast light phenomenon occurs. At CRIT transparent peak, the slow light effect is most evident, the phase change caused by rotation has the greatest impact on the output. From Eq. (12), we obtain the group delay curve of the structure (see Fig. 6), which shows the coupled induced transparency point (wavelength = 1550 nm). The group delay of the structure has a maximum peak at 3.4 ns. The group velocity of light in the structure is calculated to be V_g = 1.662 × 10⁶ m/s, the group index of the structure is n_g = 180.46 giving a relative index of n_g/n₀ = 124.46.

Fig. 7.

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	Fig. 7. Schematic diagram of the two-coupled microsphere phase shift test system.

Compared with that of a single microsphere cavity, the group index of the two-coupled microsphere structure is much larger than the refractive index of the microsphere resonator mainly because of the structure dispersion, the theoretical sensitivity of this structure can reach 8°/h.

We analyzed a theoretical model of two coupled microspheres, and derived the CRIT transfer function, the effective phase shift, and the group delay. We melted a tip of a single fiber and prepared microsphere resonators, built a precise testing platform, and obtained the CRIT spectrum of the paired microspheres, from which the group velocity and group index could be calculated, giving values V_g = 1.662 × 10⁶ m/s and n_g = 180.46, while λ = 1550 nm. This low loss and compact CRIT structure can be used in the research of slow light structure gyroscopes.