1. Introduction

Optical gyroscopes are widely used as key components in many industrial and military applications such as aircrafts, satellites, and remote control devices. These applications call for gyroscopes with high sensitivity, compact size, and low power consumption. Over the past few decades, the gyroscope based on both waveguide and fiber has been intensively studied, and commercial products have become available [1–4]. However, recent developments in photonic media and structure reveal the remaining potential for important improvement of the gyroscope’s performance.

The well-known Sagnac effect was associated with the “slow light” phenomenon in some research. This concept was originally proposed by Leonhardt and Piwnicki [5]. They claimed that the “slow light” property induced by electromagnetically induced transparency (EIT) and coherent population trapping (CPT) may greatly boost the gyroscope’s sensitivity by as much as light slows. Later, a series of work utilized this concept for improving the gyroscope’s performance. Steinberg et al. analyzed the effect in photonic crystal defect cavities [6,7], and Matsko et al. studied the gyroscope based on whispering-gallery-mode cavities [8]. Recently, Scheuer and Yariv investigated another coupled-resonator waveguide structure for the same purpose [9]. All these studies indicated the possibility of using a highly dispersive structure to construct an ultrasensitive gyroscope.

However, there had been a historical dispute on the interplay between the Sagnac effect and the Fresnel drag effect [10] since the first demonstration of an operating fiber optic gyroscope was reported by Vali and Shorthill in the 1970’s [11]. It is finally recognized that dispersion can in no way influence the magnitude of the Sagnac effect [2,12]. Leonhardt and Piwnicki used an equivalent time-space metric in moving media to deal with the Sagnac effect [5]. The premise of their theory is Doppler detuning. Since the Sagnac effect in its pure form is in no way related to the Doppler effect [13], as Peng et al. pointed out [14], the dispersive medium such as EIT cannot be utilized to enhance the absolute rotation-induced Sagnac phase shift, but the dispersive structure with slow light property is still beneficial to the enhancement of the Sagnac effect, since the structure’s response is susceptible to the phase perturbation that was induced by pure Sagnac effect.

On the other hand, Smith et al. demonstrated that an EIT-like effect, called coupled resonator induced transparency (CRIT), can be established in coupled optical resonators due to classic destructive interference [15]. The analogy between the coupled optical resonator and the atomic resonance system was made. In addition, it was shown that the coupled optical resonator exhibits behavior similar to that of an EIT medium, which has a highly dispersive property with low absorption at resonant frequency [15]. From this perspective, the perturbation of optical parameters induced by the Sagnac effect is equivalent to the perturbation of atomic parameters, and the structure is expected to achieve high-sensitivity rotation sensing as its highly dispersive property is considered.

In this letter, we propose the use of a highly dispersive coupled-resonator structure with an EIT-like property for the purpose of detecting a rotation. In Section 2, we follow the derivation of Smith et al. [15] and analyze the configuration of the CRIT. In Section 3, we investigate the Sagnac effect in the CRIT structure, and we give an explicit expression of the Sagnac phase shift. In Section 4, issues that ought to be considered for practical purpose are discussed, such as the possibility of fabrication, linewidth, shot-noise-limit sensitivity, and integration. Finally, we provide our conclusion.

2. Analysis of the configuration with CRIT

 

Fig. 1. Typical-energy-level diagram for the observation of degenerated three-level EIT.

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Fig. 2. CRIT configuration with N coupled optical resonators.

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First, let us briefly review the EIT in an atomic three-level system, as illustrated in Fig. 1. The interference occurs between the coherent atomic states. As a result, the medium exhibits a transparent property with high dispersion at resonant frequency, thus the light can be slowed because of the high dispersion, and the “slow light” phenomenon happens. This EIT medium is much different from the ordinary dispersive medium whose dispersion is always accompanied with absorption. The transition rate for the absorption of an arbitrarily detuned probe beam is given by [16]

where Ω_p and Ω_c are Rabi frequency of the probe and control field, Δ is the angular frequency detuning of the probe field, and Γ is the decay rate from level ∣2> to levels ∣1> and ∣3>. Equation (1) gives a split Lorentzian line shape that is determined by these atomic parameters.

On the other hand, Smith et al. indicated that the EIT-like phenomenon can occur through a classical mean in a coupled resonator structure due to classical destructive interference, and the analogy of optical parameters to atomic parameters was made [15]. The configuration is constructed by a waveguide with N ring resonators coupled together, as shown in Fig. 2. The response of this coupled-resonators structure can be described by the transfer function τ͂_j, which is obtained following the iterative approach [17]:

where r_j and $t_j=\sqrt{1-r_j^2}$ are the reflection and transmission coefficients of the coupler, ϕ_j = β_j L_j is the single-pass phase shift, and a_j = e ^-α_jL_j is the attenuation factor for the j th ring. α_j, β_j, and L_j are the absorption coefficients, propagating constant, and length of the j th ring, respectively. Moreover, τ͂₀ = 1, ϕ₀ = 0.

The transfer function τ͂_j represents the complex transmission property of the coupled resonator; hence, the effective phase shift for the j th ring can be obtained as ϕ_j ^(eff) = arg(τ͂_j), and the transmittance across the j th ring is written as T͂_j ≡ ∣τ͂_j∣². Additionally, the absorptance of the whole structure is [15]

where ${\tilde{A}}_j^{\left(\mathit{env}\right)}({\varphi }_{j-1},{\varphi }_{j-2},\dots ,{\varphi }_1)=\frac{\left(1-r_j^2\right)\left(1-a_j^2{\mid {\tilde{\tau }}_{j-1}\mid }^2\right)}{{[1-r_j{a}_j\mid {\tilde{\tau }}_{j-1}\mid ]}^2}$ is an envelope function, and ${\tilde{F}}_j({\varphi }_{j-1},{\varphi }_{j-2},\dots ,{\varphi }_1)=\frac{4r_j{a}_j\mid {\tilde{\tau }}_{j-1}\mid }{{[1-r_j{a}_j\mid {\tilde{\tau }}_{j-1}\mid ]}^2}$ is a function related to finesse.

The analogy with EIT is made by the two-rings configuration with identical optical path length (analogous to a degenerate atomic system); hence, ϕ₁ = ϕ₂ = ϕ and assuming a ₁ = 1 (analogous to the dipole transmission disallowed between ∣1> and ∣3>). For small detuning cosϕ₂ ≈ 1-(ϕ² ₂/2) , and sufficiently weak coupling between resonators (1 - r₁)² << r ₁ϕ² ₂, Eq. (3) can be rewritten as [15]

Here γ = 2(1-r ₂ a ₂)/√ητ_R is related to the linewidth,η = r ₂ a ₂/r ₁,τ_R is the single-ring round-trip time, $\mathit{\Delta \omega }=\frac{\mathit{\Delta \varphi }}{{\tau }_R}=\frac{2\sqrt{2\left(1-r_1\right)}}{{\tau }_R}$ is the frequency difference between the split modes, and δ = ϕ₂/τ_R is the detuning. It is noted that Eqs. (4) and (1) are identical; hence, analogies between CRIT and EIT are Δ → δ, Γ → γ, Ω_c → Δω,K(Δ) → κ(δ). Notably, Eq. (4) displays a transparence (minimum) at the single-ring resonances (ϕ mod 2π = 0), as shown in Fig. 3, and the associated strong dispersion results in light that is considerably slowed.

 

Fig. 3. Two-identical-rings CRIT configuration, analogous to the degenerated two-level EIT: a₁ = 0.9999, a₂ = 0.88; r₁ = 0.999, r2 = 0.8; and four-identical rings CRIT configuration, analogous to the multilevel EIT:

a₁ = a₂ = a₃ = 0.9999, a₄ = 0.88; r ₁ r ₂ = r ₃ = 0.999, r ₄ = 0.8.

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As Smith et al. pointed out [15], this EIT-like effect is the result of the internal coupling between the resonators. From this perspective, the optical parameters are analogous to the atomic parameters, and the structure exhibits an EIT-like property through a classical mechanism. The two-rings-configuration, in which the rings are of the same size, is analogous to the degenerate three-level EIT; and the two-rings-configuration, in which the rings are of different size, is analogous to non-degenerate three-level EIT. Furthermore, the larger number of equal-size resonator is analogous to the multilevel EIT.

3. Sagnac effect in the CRIT structure

Leonhardt and Piwnicki proposed that the EIT medium, because it is extremely susceptible to Doppler detuning, could be used in an ultrasensitive gyroscope [5]. Although it was finally shown that the Sagnac effect in its pure form is in no way related to the Doppler effect [13], the additional phase perturbation induced by the Sagnac effect still causes detuning in the CRIT structure.

The Sagnac-effect-induced phase shift in an arbitrary closed light path is given by [1]

where ω is the optical (angular) frequency, c is the speed of light in vacuum, and A is the area enclosed by the light path. Apparently, in every resonator of the CRIT structure, the single-pass phase shift ϕ_j is perturbed by the Sagnac effect. The analogy of EIT reveals that ϕ_j is related to the detuning; therefore an extra detuning is caused by the Sagnac effect. Because the CRIT structure is highly dispersive and susceptible to detuning as an EIT medium, it is expected that the Sagnac effect should be enhanced in this structure.

When we assume the device is mounted in a rotating frame, the most straightforward way to evaluate the relative phase shift induced by the Sagnac effect is to follow the effective phase shift ϕ͂^(eff) _j = arg(τ͂_j), as

where Δϕ_j, is the phase perturbation induced by the Sagnac effect in the j th resonator, and two directions are notated as subscript “±”. As for the N-rings configuration, in which the rings are of the same size, Δϕ_j is of the same value in quantity but different in sign, because the beams propagate in different directions according to the respective number of resonator, odd or even. Under the “direction requirement” proposed by Peng et al. [14], viz., the beams always propagate in the same direction, the Sagnac phase shift can be “summed up” and will probably achieve the maximal effect, and the calculation can directly follow the dispersion relation of structure as

The normalized phase shift is defined as the ratio of the effective Sagnac phase shift between input and output ports and the Sagnac-effect-induced phase shift in the single resonator. It represents the enhancement of the Sagnac effect contributed by the CRIT structure. As for the two-identical-rings configuration, the results that follow the dispersion relation of the structure (dashed curve) and straightforward numerical calculation from transfer function (solid curve) are plotted in Fig. 4. This simulation proves that a highly dispersive resonator structure is beneficial for the enhancement of the Sagnac effect. Besides, we notice that the two curves have similar shape, especially near resonance. For a planar-structure CRIT device, the solid curve represents the actual situation. The contributions of even and odd number resonators are of different signs, so they may cancel each other and cannot achieve the maximum effect suggested by the structure’s dispersion relation. Therefore, we can find the solid curve is slightly below the profile of the dashed curve. However, we believe, as Fig. 4 indicates, that the dispersion relation of the structure still gives a good evaluation for the general case, and high dispersion is a prerequisite for a Sagnac-effect sensitive structure.

 

Fig. 4. Sagnac-effect-induced normalized phase shift versus single-pass phase shift for the two-rings configuration: a₁ = 0.9999, a₂ = 0.88; r₁ = 0.999, r₂ = 0.8. The results that follow the dispersion relation of the structure (dashed curve) and straightforward numerical calculation from transfer function (solid curve) are plotted. The Sagnac-effect-induced normalized phase shift versus single-pass phase shift for the single-resonator configuration: a₁ = 0.9999, r₁ = 0.8(red, dotted curve).

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Moreover, in Fig. 4 the maximum effective Sagnac phase shift is displayed on the single-ring resonance, which is about 200 times of the Sagnac phase shift in the single ring, as much as the light slows in this dispersive structure. In the resonance region, the CRIT structure exhibits the maximum dispersion with transparency, and this property enhances the Sagnac effect.

It is also necessary to compare the performance of the CRIT structure with a usual passive single resonator, under the condition that the linear geometrical size, as well as the minimal optical attenuation, should be identical. With the same procedure of the CRIT structure, the single resonator’s phase shift response is calculated and plotted in Fig. 4 (red dot curve). As we expected, the single resonator is less sensitive, and the maximum normalized phase shift is only about 10. This is mainly because the single resonator is less dispersive than the CRIT structure when the size and attenuation of the two are respectively the same. As discussed in the previous section, the CRIT possesses large dispersion at resonance. The highly dispersive structure is more susceptible to the Sagnac-effect-induced phase perturbation, as the phase perturbation can also be understood as frequency modulation.

To analyze the Sagnac effect in the CRIT structure quantitatively, the explicit expression of the phase shift is required, and we can start with the total differentiation of the transfer function,

Here $\frac{\partial {\tilde{\tau }}_j}{\partial {\varphi }_j}=-i{{\tilde{\tau }}_{j-1}{\tilde{\kappa }}_j}_{}$ and $\frac{\partial {\tilde{\tau }}_j}{\partial {\tilde{\tau }}_{j-1}}=-{{\tilde{\kappa }}_j}_{}$ can be obtained from Eq. (2) readily,

where

Hence τ͂₀ = 1,ϕ͂₀ ^(eff) = 0 , we get

It is noticed that the maximum effect is achieved at resonant points; where ϕ_j,ϕ_j-1,…,ϕ₁ mod 2π = 0, it leads that τ͂_j and κ͂_j are real numbers at resonances. Hence ϕ͂^(eff) _j ≡ arg(τ͂_j) = 0, therefore

Additionally, dτ͂_j is an imaginary number at resonances, as Eq. (10) indicated, and ∣dτ͂_j∣«∣τ͂_j∣, so that the Sagnac phase shift at resonances can be written as:

For the two-identical-rings configuration analogous to the degenerated three-level system, it can be written as

With a₁ = 1 and ${\mathit{d\varphi }}_2=-{\mathit{d\varphi }}_1=d\varphi =\frac{4\mathit{\omega A}}{c^2}\mathrm{\Omega },$ we get the simplified explicit form of the Sagnac phase shift

Equation (14) gives the explicit expression of the effective Sagnac phase shift for a basic CRIT configuration. It shows that the phase shift is mostly relevant with the reflection coefficients of the two couples. As Smith et al. explained [15], the CRIT appears in the weak-coupling limit as r ₁ → 1, and as $\mathrm{\Delta \omega }=\frac{\mathrm{\Delta \varphi }}{{\tau }_R}=\frac{2\sqrt{2\left(1-r_1\right)}}{{\tau }_R}$, the two split modes become close together. Then the interference occurs accompanied with strong dispersion. Besides, r₂ is related to the single-mode linewidth, which is γ = 2(1-r₂a₂)/√ητ_R. The decrease of r₂ broadens the linewidth and compresses the relative mode split; it also influences the CRIT dispersion property.

 

Fig. 5. Sagnac-effect-induced normalized phase shift at resonance as a function of the reflection coefficients for the two-rings configuration: a ₁ = 0.9999 , a ₂ = 0.88 . The numerical result from transfer function and theoretic result from explicit expression Eq. (14).

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The relation between the normalized effective Sagnac phase shift at resonance and the reflection coefficient is illustrated in Fig. 5, on which numerical calculation from transfer function and theoretic result from Eq. (14) are both presented. According to this result, we are convinced that Eq. (14) gives an appropriate prediction for the structure’s behavior. Either the increase of r₁ or the decrease of r₂ effects the increase of the phase shift. Moreover when r₁ → 1, the phase shift increases rapidly in an exponential-like manner. It shows r ₁ → 1 is a critical condition for the phase shift enhancement, what with the factor 1/1- r ₁ appears in Eq. (14).

Since we have discussed the basic configuration of CRIT, the EIT-like effect can also be achieved by using the two rings with unequal size or larger numbers of rings with equal size, as Smith et al. suggested [15]. The analysis for these situations can directly follow the approach discussed above, and the Sagnac effect is also notably enhanced in these configurations.

4. Gyroscope based on the CRIT structure

As the analysis of the CRIT structure in previous sections indicated, the Sagnac effect is notably enhanced in coupled-resonator structures with an EIT-like property, as the result of the high dispersion of the structures. Naturally, one is led to ask whether this effect can be utilized as a gyroscope device. In this section, we discuss some implementation considerations for the CRIT-structure-based gyroscope.

First, fabrication possibilities: The premise of the CRIT property in coupled resonators is the highly intrinsic quality factor, which leads to a ₁ → 1, and the weak coupling limit, which leads to r ₁ → 1. Unlike the absorption in atomic systems, the loss in micro-resonators is typically dominated by surface scattering, whereas intrinsic absorption is negligible. With the recently improved micro-fabrication technique, the quality factor of the resonator can exceed 10⁸. However, the size of the resonator may still be constrained by the quality factor, as the high-quality-factor resonator with larger size brings more technical challenges for fabrication processes. On the other hand, the coupling reflectivity is determined by the separated distance of resonators. Because the evanescent field attenuates exponentially of the distance, it can be carefully adjusted and approach unity by increasing the separation. Actually, an experiment realization of an on-chip CRIT structure was reported by Qianfan Xu et al. recently, which is constructed by micrometer-scale micro-rings [18].

Another key limitation to the linear resonant systems comes from the tradeoff between dispersion and linewidth, according to which the group dispersion is in scale with the available spectral width. This relation exists in both EIT and CRIT [15]. In a sense, the mode split Δω gives an estimation for the linewidth, but the more precise criterion is the full width at half maximum (FWHM) linewidth. It can be obtained from Eq. (4) with à ₂(δ) = à ^(env) ₂/2 and δ → 0, and it leads to

 

Fig. 6. CRIT linewidth versus the reflection coefficients r ₁and r ₂, for the two-rings configuration: a ₁ = 0.9999, a ₂ = 0.88.

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As shown in Fig. 6, either the increase of r ₂ or the decrease of r ₁ causes the increase of the linewidth, which is just the reverse of the trend of the effective Sagnac phase shift. The phase shift is enhanced as much as the light slows, but the linewidth is narrowed correspondingly. In addition, the linewidth in the optical spectra is also inversed proportional to round-trip time, τ_R, which is determined by the size of the resonator. As a result, the optic linewidth can be tailored by adjusting both the reflection coefficients and the resonator’s size to match a specific light source.

Further, the gyroscope based on a CRIT structure also suffers from the shot noise limitation as do other passive optical systems. Assuming the gyroscope works at π/2 phase offset, the shot-noise-limited sensitivity can be written as [4]

where P is the output light power, ħω is a single photon’s energy, t_M is the measurement time, η is the quantum efficiency of the photo-detector, and σ_Δϕ and σ_P are standard deviations of the phase shift and output light power. Combining Eqs. (5),(14),(16), the shot-noise-limited minimal detectable rotation rate is

The sensitivity is also enhanced with the factor as much as light slows. This factor is determined by the optical parameters we discussed before, and it may be as high as from 10²–10⁴. Therefore, a miniature optic gyroscope becomes possible, considering the sensitivity enhancement the CRIT structure achieved. Furthermore, we notice that the CRIT structure displays transparent property at resonances. It is beneficial for suppressing shot noise as the maximum output power is achieved in Eq. (17).

Since the EIT medium is composed of myriad atoms, the aggregation of CRIT structures transforms the “microscopic” effect to “macroscopic”. With the micro-fabrication technique, the micrometer-scale CRIT structure such as the “photonic atom,” can be integrated onto a single chip with high density in order to construct a sensitive gyroscope of compact size. As Smith et al. explained [15], the high dispersion can also be obtained by N rings configuration. But this approach compresses the available linewidth too, so cascading a series of basic CRIT structures should be a better choice for accumulating the effective Sagnac phase shift without narrowing the linewidth.

Based on the discussions above, we propose a rough design for a miniature optical gyroscope with the CRIT structure. Assuming the structure is fabricated on a SiO2 substrate whose refractive index is 1.50, and the diameter of the single-ring resonator is 10μm with a₁ = 0.9999, a ₂ = 0.88; r₁ = 0.99999, r₂ = 0.8, for a 1550 nm light source, the available linewidth is 114.8 MHz and the normalized effective Sagnac phase shift is about 2.2×10⁴. If the photonic chip is centimeter scale and is tiled with CRIT micro-resonators, the effective area is about several square meters. This is equivalent to an optic-fiber gyroscope with several hundred meters of fiber coil. But different from the optic-fiber gyroscope, the CRIT-based gyroscope is constructed with all-solid configuration and compact size, and it should be more stable against nonreciprocal perturbations.

5. Conclusion

In this letter, we propose to utilize a highly dispersive coupled-resonator structure with an EIT-like property for the purpose of rotation detection. Based on the analogy between optical parameters in the CRIT structure and atomic parameters in the EIT medium proposed by Smith et al. [15], we analyze the Sagnac effect in the CRIT structure and point out that the Sagnac phase shift contributed by the whole structure is notably enhanced. An explicit expression of the phase shift is derived and discussed. Furthermore, some issues for the implementation of the CRIT-structure-based gyroscope device are considered, such as the fabrication possibility, linewidth, shot-noise-limit sensitivity, and integration onto a single photonic chip.

Our analysis confirms that high dispersion is a prerequisite for a Sagnac effect sensitive structure, and the tradeoff between the linewidth and dispersion should be considered as an intrinsic constraint for both an EIT medium and a CRIT structure. These properties can be tailored by adjusting some optical parameters, such as the reflection coefficients and resonator size, in order to match the requirements of specific applications.

With an improved micro-fabrication technique, a miniature gyroscope constructed by high-density integrated CRIT structures is possible. This gyroscope should have all-solid configuration, compact size, and sensitivity comparable to the common optic-fiber gyroscope. It could be easily integrated to all-optical application and in constructing a high-performance rotation sensor.