Investigation of feedback characteristics in single planar waveguide-coupled square microcavity by FDTD technique

Guoping Lina, Le Maa, Benrui Zhengb, Huiying Xub, Zhiping Cai*b

aDepartment of Physics, Xiamen University, Xiamen 361005, China; bDepartment of Electronic Engineering, Xiamen University, Xiamen 361005, China

ABSTRACT

Feedback characteristics in single planar waveguide-coupled square microcavity are investigated by using the two-dimensional finite-difference time-domain method. A square microcavity with refractive index of 3.5 and side length of 2.2µm plays as a wavelength selective feedback element, and a single planar waveguide with width of 0.2µm is used to input and output light in the simulation. Simulation results show that most of the on-resonance extinction in transmitted optical power through the planar waveguide can be reflected. The highest reflection observed is more than 92% of the incident power at the resonance wavelength. For better understanding the high reflection phenomenon, the steady field patterns of single waveguide coupled square microcavity has also been obtained. The influence of coupling gap between the cavity and waveguide, on the resonance wavelength, reflection efficiency and the linewidth of reflection peak has been studied. With increasing gap, the resonance position shifts to the longer wavelength, and the linewidth narrows. The results also show that an optimized gap exits for the highest optical power reflection.

Keywords: microcavity, FDTD method, feedback element, resonance

1. INTRODUCTION Recently, optical microcavities, such as the microspheres [1], microdisks [2], microrings [3], square microcavities [4,5] , have received remarkable attention for low-threshold microlasers [1,2], channel add-drop filters [3], high-sensitivity bio-sensors [6], cavity quantum electrodynamics [7] applications due to their compact sizes and high Q resonances. Light at particular resonance wavelengths in such microcavities can be strongly confined by repeated total internal reflection (TIR) at the boundary. In circular microcavities, such resonance modes are called whispering gallery modes (WGMs). While in other microcavities like square microcavities, whispering gallery (WG)-like [8] modes still exist. To couple light in and out of the microcavities effectively, it is necessary to utilize overlapping of the evanescent field of the whispering gallery modes with the phase-matched evanescent field of other optical components. Such coupling has been implemented by use of high-index prisms [9], tapered optical fibers [10], side-polished optical fibers [11] and planar waveguides [3]. Among these, the planar waveguide coupling techniques are particularly suited for high-density integrated photonics applications. The planar waveguide coupling can either be lateral or vertical. Lateral coupling technique has the advantages of monolithic structures and can be fabricated by relatively standard processes. Compared with conventional circular microcavities, square microcavities have longer interaction length between the cavity sidewall and the straight waveguides for lateral coupling.

The whispering-gallery like modes in planar waveguide-coupled square and rectangular microcavities have been investigated using coupled mode theory [12], mode expansion method[13] and finite-difference time-domain (FDTD) [4]. Previous works are primarily focus on the two planar waveguides coupled microcavities, normally for the add-drop filter applications. Recently, Khanh Kieu have proposed and demonstrated a new type of feedback mirror for fiber laser applications based on fiber coupled microsphere resonator [14]. This technique has taken full advantage of the feedback characteristic of the waveguide coupled WGM microresonators. The proposed scheme should also work with other types of microcavities in the integrated photonics circuits such as microdisks and square microcavities. It has implied many optical applications [14,15].

*zpcai@xmu.edu.cn; phone 86 592 2580141; fax 86 592 2580141

Optoelectronic Devices and Integration II, edited by Xuping Zhang, Hai Ming, Maggie Yihong Chen,Proc. of SPIE Vol. 6838, 68381D, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.752523

Proc. of SPIE Vol. 6838 68381D-12008 SPIE Digital Library -- Subscriber Archive Copy

Little work has been done to investigate the feedback characteristics in a single waveguide coupled microcavity. In this paper, we consider a single planar waveguide coupled square µ-cavity as a wavelength selective feedback element. We model and analyze the mode coupling between the square microcavity and the laterally coupled single planar waveguide using a two-dimensional (2D) FDTD method [16]. Both transmission and reflection spectrums in the waveguide are obtained. High reflection can be obviously observed due to the WG-like mode resonances in the microcavity at the resonance wavelengths. The mode field patterns when the light is highly reflected and highly pass are also presented for better understanding the feedback phenomenon in a single waveguide coupled microcavity. The responses of the resonance wavelength, reflection efficiency and line-width of the reflection peak to the gap between the planar waveguide and the square µ-cavity have been investigated. Simulated results show that the reflection peak position shifts to longer wavelength when the gap is increasing, and an optimal coupling gap exits for maximum light energy reflection.

2. THEORETICAL MODEL AND METHOD A square microcavity coupled with a single-mode planar waveguide as shown in figure 1 is considered. A small air-gap distance g separates the microcavity from the waveguide. The square microcavity of side length a is laterally coupled with a straight single-mode waveguide of width w. In the simulations, an index of refraction of 3.5 is assumed for both the single-mode waveguide and the square microcavity while an index of refraction of 1 is assumed for the background, corresponding to the refractive index contrast between silicon and air.

Fig. 1. Schematic of a planar waveguide-coupled square microcavity

The EM field in the microcavity and waveguide coupling structure shown in figure 1 is described by time-dependent Maxwell’s equations as shown in Eq. (1), where E and H are the electric and magnetic field vectors, D and B are the displacement field and magnetic field, ε and µ are the dielectric constant and the magnetic permeability. The FDTD technique treats the Maxwell equations as a set of finite difference equations in both time and space, it has been proven to be one of the most powerful numerical techniques that has been usefully applied to a wide range of optical devices, especially for wavelength-scale microstructures. In this paper, a 2D-FDTD method is employed to simulate the mode coupling in planar waveguide coupled square microcavity (depicted in Fig. 1).

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3. FDTD SIMULATED SPECTRUM AND MODE FIELD To simulate the transmission and reflection characteristics in the waveguide coupled square microcavity. A short Gaussian pulse centered at vacuum wavelength 1.50 µm covering a wide band about 20nm was launched into the input port (as show in Figure 1). Both the transmitted power in the through port and the reflected power in the input port were calculated. Through dividing the power by the incident power at each wavelength, the transmission and reflection spectrum were obtained. In the 2D FDTD simulation, the spatial cell size is 15nm, and the time step �t is Courant limit. The spectral resolution was about 0.10 nm. A perfectly matched layer (PML) absorbing boundary with a thickness of 1.0µm in the x and y direction was used to absorb all incident light waves on the simulation boundaries.

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Figure 2 shows the FDTD simulated transmission and reflection spectra of a single planar waveguide coupled square microcavity, we consider the side length of the square microcavity a = 2.2µm, the width of the single mode waveguide w = 0.2µm, and the gap g = 200 nm. Recent advanced in the technology of nanofabrication have made it feasible to fabricate such structures of this size using standard photolithography. The polarization of incident light is set to TM (dominant component of the electric field perpendicular to the substrate plan). After running enough time steps, the normalized transmission and reflection spectrums are obtained in Figure 2. Multimode resonances can be clearly discerned from the spectrums. Due to different phase matching condition of the modes in the square microcavity with the waveguide mode, the coupling efficiencies are different at the resonance wavelengths. From figure 2, it is seen that the resonance at λ=1447.5 nm has the highest coupling efficiency ≈ 98% and the narrowest line width 0.47nm (responding to a cavity Q factor about 3080). At this resonance wavelength, only 2% of the incident light passes through the waveguide, and backreflection as large as 88% is observed. In Section 4, by tuning the coupling gap to 150nm, the reflection achieve 92.5%, suggesting the use of the waveguide coupled microcavity as wavelength selective reflector in integrated optics. This high reflection phenomenon in waveguide coupled microcavity is also observed in fiber taper coupled microsphere in experiment [17]. From Fig. 2, we can see that the transmission energy at non-resonance wavelength is more than 99% of the incident optical power.

For better understanding the high reflection phenomenon, the mode field patterns of single waveguide coupled square microcavity device were obtained by launching a continuous-wave (CW) at the resonance and non-resonance wavelengths separately. Figure 3 (a) shows the mode field pattern of the device when a continuous-wave at a non-resonance wavelength λ= 1443.0nm is launched. We can see that most of the incident light passes through the waveguide with a little light coupling to the microcavity. It agrees very well with the transmission spectrum in figure 2. Figure 3 (b) shows the mode field pattern of the device when launching a continuous-wave at a resonance wavelength λ= 1447.5nm. It is observed that most of the incident light at the wavelength can not pass through the waveguide. They are coupled to the wavelength scale square microcavity, form a strong resonance in the inner cavity, and then coupled back to waveguide in the backward direction. We also find that the electric field in the left part of the waveguide has been

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enhanced compared to the mode field pattern in figure 3 (a). Figure 4 shows the electric field amplitudes in the middle of the waveguide along x dimension. From it, we can see that the resonance light wave in the square microcavity has been coupled backward in to the waveguide and impacted the electric field patterns visibly. This feedback phenomenon has demonstrated that this device can be used as a wavelength selective reflector in integrated optics.

Fig. 3. FDTD simulated steady state electric field mode patterns of single waveguide coupled with square microcavity. a =

2.2 µm, w = 0.2 µm, g = 0.2 µm and TM-polarized. (a) non-resonant mode at λ= 1443.0nm (b) resonant mode at λ= 1447.5nm.

Fig. 4. Electric field amplitude in the middle of the waveguide versus the x position in the length. (a) no reflection at λ=

1443.0nm (b) high reflection at λ= 1447.5nm.

4. INFLUENCE OF THE COUPLING GAP ON THE FEEDBACK CHARACTERISTIC As described in Section 3, the light at the wavelength of 1447.5nm has a strong resonance in the square microcavity. Considering the near field coupling between waveguide and microcavity, the resonance mode should be sensitive to the near field coupling gap. In this section, we investigate the influence of changing in the coupling gap on the feedback characteristics. In the simulation, the parameters of the planar waveguide and the square microcavity are kept the same as

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depicted in Section 3. We change the coupling gap between the waveguide and microcavity and record the transmission and reflection spectrums around λ= 1447.5nm in different gaps.

Fig. 5. FDTD simulated transmission and reflection spectrums around the well resonant wavelength at different coupling gaps. (a) g =100nm, (b) g =150nm, (c) g =200nm, (d) g =250nm, (e) g =300nm, (f) g=350nm

Fig. 5 (a) and (b) show the spectrums in the device when the coupling gap g is 100nm and 150nm. Considering the small microcavity and the coupling gap, we change the spatial cell size to 10nm, and the time step △t is still Courant limit. The simulation runs for enough time to get the spectrums. It can be observed that the transmission at the through port of the waveguide drops obviously even at the non-resonance wavelength. It is because of the strong interaction between the cavity and the waveguide in the close gap. Fig. 5 (c)-(f) show the spectrums when the coupling gap g is 200nm, 250nm, 300nm and 350nm. From all the spectrums in Fig.5, we have observed obvious influence of the changing coupling gap on the feedback characteristics of the waveguide coupled square microcavity device.

Table. 1. Transmission, reflection, resonant wavelength and line width at different coupling gaps. (a) g =100nm, (b) g =150nm, (c) g =200nm, (d) g =250nm, (e) g =300nm, (f) g=350nm.

g (nm)

Transmission Reflection Wavelength (nm)

Line width (nm)

100 0.1% 88.1% 1442.00 3.00 150 0.6% 92.5% 1445.39 1.18 200 0.4% 88.2% 1447.57 0.47 250 2.4% 72.4% 1448.07 0.16 300 12.7% 42.7% 1448.25 0.08 350 41.6% 13.8% 1448.33 0.05

As shown in Table 1, when the coupling gap between waveguide and microcavity increases, the resonance wavelength shifts to longer wavelength, the minimum transmission power in the through port of waveguide becomes larger, and the linewidth of the reflection peak become smaller. It can be observed that the shift of the resonance wavelength and line width is much more obvious when the coupling gap is less than 250nm, while the minimum transmission is much more sensitive to the gap when gap is larger than 250nm. With increasing gap, the reflection power at the resonance wavelength fist increases and then decreases obviously. It can be proposed that there is an optimized gap between 100nm

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and 200nm for the maximum reflection. To prove this, we have made several simulations around 150nm air gap. The simulated results are summarized in Fig. 6. We found that at air gap g=150nm the reflection energy at resonance wavelength achieved the maximum, and the normalized transmission is kept below 0.1%.

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Fig. 6. Minimum transmission and maximum reflection at resonant wavelength versus gap dimension

5. CONCLUSION In summary, the optical feedback characteristics of the planar waveguide coupled square microcavity have been investigated by using the 2D-FDTD method. High reflection in the input port of waveguide due to the resonance of the square microcavity is observed. Through tuning the coupling gap, the responses of the resonance wavelength, reflection efficiency and the line-width of reflection peak have been studied. With increasing gap, the resonance position shifts to the longer wavelength, and the linewidth narrows. The results show that the highest light power reflection can be achieved when the gap is set to be 150nm, and the reflection is more than 92% of the incident power. Such phenomenon has indicated that waveguide coupled square microcavity can serve as a wavelength selective feedback element.

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