Coherent interaction between two orthogonal travelling-wave modes in a microdonut resonator for filtering and buffering applications Qingzhong Huang1,* and Jinzhong Yu1,2 1
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China 2 Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100083, China *
[email protected]
Abstract: We theoretically investigate the coherent interaction between two orthogonal travelling-wave modes in a microdonut resonator symmetrically coupled to two bus waveguides. An analytical model has been developed to describe this structure using transfer matrix method. The simulation reveals that the two-mode microdonut can exhibit either a flat-top response or allpass transmission, governed by the resonance spacing. Then, we implement analytical simulations to characterize the device and analyze the influence of coupling efficiencies and propagation losses of two resonant modes on behavior. Consequently, finite difference time domain simulations have been performed. The numerical results validate our theoretical analysis, and optical buffering effect is demonstrated in a pulse propagation simulation, when the two resonances are aligned. In addition, we show that the device function can be switched between flat-top filtering and all-pass filtering by tuning the local refractive index in a microdonut. Hence, this structure is promising for on-chip optical filtering and buffering applications. ©2014 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (130.7408) Wavelength filtering devices; (230.3990) Micro-optical devices; (230.5750) Resonators; (030.4070) Modes.
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1. Introduction Coherent interaction between photonic resonators has attracted substantial interests in recent years. Great efforts are made to map the coherent interference between excitation pathways in atomic systems [1, 2] to resonances in optical resonator systems [3–16]. Optical analogues of electromagnetic induced transparency (EIT) and Fano effect in atomic systems have been studied based on coupled photonic crystal slabs [3, 4], planar photonic crystal cavities [5–9], and whispering-gallery resonators [10–15]. As claimed in [16], coherent interference between two resonant modes in a cavity generates intriguing characteristics. It can exhibit an EIT-like or Fano-like response for two non-orthogonal modes, while behaving as either a flat-top filter or an all-pass filter for two orthogonal modes. All-pass filters can provide lossless and strong group delay, since they produce strong on-resonance phase variation while maintaining unity transmission both on and off resonance. Unlike EIT-like and Fano-like effect, flat-top and allpass responses do not appear in an atomic system, indicating that the coherent interaction in a multimode cavity is more versatile for practical applications. By far, the reported two-mode cavities are constituted by one or two photonic crystal slabs having two resonant modes of opposite symmetry for normally incident light [17, 18], or two adjacent photonic crystal cavities side-coupled to a bus waveguide on a chip [19]. Here, we employ a planar microdonut resonator [20, 21] coupled to two bus waveguides that can support two orthogonal modes and has many merits with respect to filtering and buffering applications. Firstly, the travelling-wave microdonut resonator behaves as an add-drop filter, having the capability of dropping the forbidden wavelengths to another channel, instead of reflecting them to the input port [22]. Secondly, this structure is suitable for on-chip photonic integration, and the design is also flexible. Thirdly, the device function is switchable between
#222056 - $15.00 USD Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014 (C) 2014 OSA 20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25172
flat-top filtering and all-pass filtering via tuning the local refractive index in a microdonut resonator by heating [15] or free-carrier injection/extraction [23]. In this paper, the coherent interaction of two orthogonal modes in a two-bus waveguides coupled microdonut resonator is investigated analytically and numerically. An analytical model is developed to describe this structure using transfer matrix method. It can function as either a flat-top filter as the two resonances are spectrally detuned, or an all-pass filter as the two resonances are aligned. Analytical simulations are implemented to characterize the device and study the influences of coupling efficiencies and propagation losses of two whisperinggallery modes (WGMs) on performance. Consequently, we perform finite difference time domain (FDTD) simulations, and the numerical results validate our theoretical analysis. As the two resonances are aligned, optical buffering effect is demonstrated in a pulse propagation simulation. Furthermore, it is shown that the function of device is switchable by tuning the local refractive index in a microdonut resonator. 2. Structure, theory and principles Figure 1(a) shows the schematic of a microdonut resonator symmetrically coupled to two bus waveguides. The resonator can support two transverse modes, while the bus waveguide is single mode. Here, we use the two lowest-order transverse modes in a microdonut resonator, namely the first-order radial WGM (WGM1) and second-order radial WGM (WGM2). WGM1/WGM2 and bus waveguide mode are evanescently coupled, while WGM1 and WGM2 are indirectly coupled in the bent-straight waveguide coupling region which can be regarded as a directional 3 × 3 coupler. Hence, both WGM1 and WGM2 are possibly triggered by the input optical field in the bus waveguide, and the two-mode interference will become significant when their resonances are nearby. In fact, due to the perturbation in the coupling region, the operating wavelength may shift slightly from the resonant wavelength [24]. For simplicity, the influence of phase shifts in transmitted fields in the coupling region are not considered in our model, as the field coupling is relatively weak.
Fig. 1. (a) Schematic of a two-mode microdonut resonator symmetrically coupled with two bus waveguides. (b) General schematic of the interference between two coupling-out fields in the drop channel as the resonances of WGM1 and WGM2 are aligned or appropriately detuned.
Using transfer matrix method, the input-output relations for a 3 × 3 coupler are given by b0 t0 b = − jk 1 1 b2 − jk2
− jk1 t1 − kc
− jk2 a0 −kc a1 t2 a2
(1)
where a0,1,2 and b0,1,2 are the optical fields (the subscripts 0, 1, and 2 represent the waveguide mode, WGM1, and WGM2, respectively) at the input and output ports in the coupling-in #222056 - $15.00 USD Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014 (C) 2014 OSA 20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25173
region, respectively; -jk1,2 and -kc are the field coupling coefficients between these modes; t0,1,2 is the field transmission coefficient. The symbols c1,2 and d0,1,2 in Fig. 1(a) denote the optical fields at the input and output ports in the coupling-out region, respectively. It is seen that the output field in the drop channel is the interference result of the couplingout fields from two WGMs. The phase difference of the two coupling-out fields, associated with the azimuthal order difference of two WGMs, plays an important role. The azimuthal order of WGM1,2 is denoted by m1,2 = 2πRneff1,2/λ1,2, where R, neff1,2 and λ1,2 are the microdonut outer radius, effective index and resonant wavelength of WGM1,2, respectively. When (m1-m2) is even, the phase difference is 2pπ (p is an integer) and the interference is constructive in the vicinity of the aligned resonances of non-orthogonal WGM1 and WGM2, as studied in [25]. On the other hand, when (m1-m2) is odd, the phase difference is (2p + 1)π and the interference is destructive around the aligned two orthogonal modes. The conception of “non-orthogonal” and “orthogonal” modes in a cavity coupled by multiple ports is explained in [16]. Moreover, as the two resonances of WGM1 and WGM2 are spectrally detuned, the phase difference changes, and the inter-mode interference may become constructive for odd (m1-m2). In this paper, the subject of interest is the interference effect of two WGM resonances and its applications as (m1-m2) is odd. To describe the two-mode interference effect in our structure graphically, we consider an operation wavelength at (λ1 + λ2)/2 and assume two coupling-out fields of equal strength. Figure 1(b) depicts the field coupling of WGMs and waveguide mode and the interference of two coupling-out fields from the aligned or detuned two WGMs. When the two resonances are rightly aligned, the two fields coupled out from on-resonant WGMs combine destructively and cancel each other out, whereas when the resonances are detuned, the two fields coupled out from off-resonant WGMs combine constructively and form a stronger field in waveguide. The phenomenon suggests that this structure can possibly behave as either a flat-top filter or an all-pass filter, which will be further explained in the following text. We use θ = 2π2R(neff1 + neff2)/λ to describe the phase shift of two-bus waveguides coupled two-mode microdonut, where λ denotes the vacuum wavelength. The optical fields of WGMs are phase shifted and cross-coupled in the resonator, and the relations are described as for (2n0.5) π
