The Sagnac effect in Coupled-Resonator Slow-Light Waveguide Structures

We study the effect of rotation on the propagation of electromagnetic waves in slow-light waveguide structures consisting of coupled micro-ring resonators. We show that such configurations exhibit new a type of the Sagnac effect which can be used for the realization of highly-compact integrated rotation sensors and gyroscopes. PACS Numbers: 42.60.Da, 42.70Qs When an electromagnetic wave propagates in a moving medium it accumulates additional phase shift, compared to a wave propagating in a stationary medium, which depends on the scalar product between the wave propagation direction and the velocity vector of the medium [1, 2]. A particularly interesting configuration is that of a wave propagating along a circular path in a rotating medium. In such scenario, the additional phase accumulated by the wave depends on the relation between the propagation directions of the medium and the wave (co-directional or counter-directional). This phase difference is often referred to as the Sagnac effect and in addition to its scientific importance, it has numerous practical application such as detection and high-precision measurement of rotation. In the past few years, much attention was devoted to slowing down the propagation speed of light and to coherently stop and store pulses of light [3-6]. There are two major approaches to achieve significant reduction of the group velocity of light, which employ either electronic or optic resonances. Because of the inherent constraints associated with the conversion of the optical signals to coherent electronic states, the electronic resonance approach is less attractive for practical implementations of slow-light devices. Consequently, significant efforts were focused on controlling the speed of light using photonic structure incorporating microcavities and photonic crystals. Substantial delays and storage of light pulses were predicted in various coupled-cavities structures such as coupled resonator optical waveguides (CROWs) [7] and side-coupled integrated spaced sequence of resonators (SCISSORs) [8]. Recently, Leonhardt et al. pointed out the advantages of using the Sagnac effect is slowlight medium generated by electromagnetically induced transparency (EIT) for the realization of an ultra-sensitive optical gyroscope [9]. Subsequently, Steinberg studied the effect of rotation in coupled photonic crystal defect cavities [10] and Matsko et al. proposed to utilize the dispersive characteristics of slow-light propagation in a closed loop SCISSOR-like configuration to realize a high-sensitivity miniaturized optical gyroscope [11]. In that study, however, the SCISSOR was modeled as a highly-dispersive conventional waveguide where the slow group velocity of the light in the SCISSOR stems from the average interaction of the light with the high-Q resonators. In this letter, we study the properties of the Sagnac effect in a CROW which is wrapped around itself, with application for a highly compact rotation sensor or an optical gyroscope. Figure 1 illustrates the geometrical configuration: light is launched into the input waveguide and equally divided between the two channels of the 3dB coupler. The signal in each arm is coupled to a different end of the circular CROW consisting of directly coupled ring resonators. Finally, the counter propagating signals (marked by the black and white arrows) are combined by the 3dB coupler where the output signal in each arm of the coupler depends on the relative phase difference between the signals: ( ) ( ) φ φ ∆ = ∆ = 2 1 2 2 2 2 1 2 2 2 sin ; cos in r in r E B E A (1) where Ein and ∆φ are correspondingly the input amplitude and the phase difference between the counter-rotating fields. When the device is stationary, the overall phases accumulated by both signals are identical i.e., ∆φ = 0, resulting in complete cancellation of Br. On the other hand, when the device is rotating, the phases accumulated by the signals differ, resulting in a non-vanishing intensity Br. To evaluate the phase difference ∆φ in a CROW it is convenient to divide the structure into sections as illustrated in Fig. 1: An input section which consists of the input coupler and part of the first micro-ring (this section is marked by the dashed white line “I”); A recurring section consisting of two halves of a micro-ring coupled to a complete ring, constituting the main body of the CROW (this section is defined by two successive dashed white lines: I→S1, S1→S2, etc.); And an output section which is similar to the input section (from the line marked by “O” to the output coupler). Because of the recurring section, it is convenient to represent each section by a transfer matrix linking between the input and output ports of the section. The overall transfer matrix of the structure is then found simply by multiplying these matrices in the correct order. The phase accumulated by a wave propagating in non-stationary waveguide depends primarily on the scalar product of the waveguide velocity and the wave-vector k. In the configuration studied here, the contribution of each segment r d in each micro-ring is different because the center of rotation does not necessarily coincide with the center of any of the micro-rings. Therefore, in order to construct the transfer matrix of each section we have to evaluate the phase accumulated by a wave propagating along a curved waveguide segment which is rotating around an arbitrary point. Figure 2 illustrate the geometry of this problem: a wave propagating in micro-ring resonator with radius R while the center of this ring is rotating with angular velocity Ω around a fixed point. The distance between the center of the micro-ring and the center of rotation is 0 ~ R . The phase accumulated by the wave as it propagates along a segment dr stem from two contributions: The conventional phase due to the propagation d(∆φprop)=ω/cn|dr| and the rotation related phase shift which is given by [1]: ( ) r d V c n d r r ⋅ − = ∆ α ω φ 1 ) ( 2 2