LIU Chun-juan，MU Zhou，WU Xiao-suo，ZHENG Li-jun

(School of Electronic Information and Engineering, Lanzhou Jiaotong University, Lanzhou 730070，China)

Abstract：The transmission equation of microdisk resonator is obtained by the transfer matrix method.The physical model is built and the electric field distribution, output spectrum and phase of the microdisk resonator are simulated by three-dimensional finite element software.The influence of the structural parameters on transmission characteristics and the temperature sensing property of the microdisk resonator are studied deeply.The results show that the output spectrum will change significantly with the distance between the microdisk and the straight waveguide within a certain range but there is no apparent change in the phase of the output port.The extinction ratio and maxima sensitivity of the device will reach 30 dB and 45 pm/℃, respectively.Microdisk has higher integration, higher quality factor and wider free spectral range compared with common microring resonator.

Key words：microdisk; whispering gallery mode; transfer matrix method; temperature sensing

0 Introduction

As an important photonic device, optical microcavites has received extensive attention in basic research and application[1].Optical microcavities can be divided into Fabry-Perot microcavity, photonic crystal microcavity and whispering-gallery-mode(WGM)microcavity[2-3]according to the different limiting mechanism of cavity to light field.The optical signal undergoes multiple reflections in the cavity for achieving phase coherence enhancement to produce a stable wave mode.Thus most of the energy is confined within the cavity, and high-quality factor can be obtained[4].Some whispering-wall mode resonators with high-quality factor include micro-rings, microdisks, etc., which can enhance significantly light interaction have a wide range of applications in integrated optics such as microring resonator-based sensors, logic modules, and microdisk based lasers proposed inRefs.[5-7].Driven by the demand for high-sensitivity, real-time portable detector systems especially micro-nano-scale optical sensors have been studied extensively in recent years.Generally, silicon-based photonic devices are often used for environmental and biological sensing[8].Compared with microring resonator, the microdisk resonator has smaller device size and higher quality factor except for good compatibility with the CMOS technology, which make it applicable as a potential optical sensor[9-10].Therefore it is significant to investigate microdisk resonator for the improvement of the performance of the sensor device under relatively simple technology processes.

In our work, the transmission characteristics of the microdisk resonator are analyzed theoretically by transmission matrix method, and the transmission equation is obtained.The influence of structural parameters on transmission and performance are simulated numerically.The output spectrum and temperature sensing property of microdisk resonator based on silicon-on-insulator(SOI)waveguide is investigated deeply.The extinction ratio(ER)can reach as high as 30 dB, which means the device has many potential applications in the field of optical devices.

1 Theoretical analysis

The structure of the microdisk resonator with a straight waveguide and a microdisk is shown in Fig.1.Assuming that the first-order radial whispering gallery mode(WGM1)and the second-order radial whispering gallery mode(WGM2)of the structure are excited simultaneously, thereby indirectly forming a 3×3 coupler.

Fig.1 Structural diagram of microdisk resonator

According to transfer matrix method, the relationship between the direct waveguide and the microdisk coupling can be given by[11]

(1)

wherea0,1,2andb0,1,2are different modes of light field of the input and output ports, respectively(subscripts 0, 1 and 2 indicate different modes of the waveguide, respectively);-jk1,2and-kcare coupling coefficients between different modes, andt0,1,2respresents the transmission coefficients of different modes.

According to coupling mechanism, the phase shiftθcaused by the coupling of the straight waveguide and the microdisk can be defined as

θ=2π2Rneff/λ,

(2)

whereR,neffandλrepresent the radius of the disk, the effective refractive index of the whispering gallery mode, and the wavelength in vacuum, respectively.When(2m-0.5)π<θ<(2m+0.5)π(mis a positive integer), the relationship between the field amplitudes at which resonance occurs between the straight waveguide and the microdisk can be written as

(3)

and when(2m-0.5)π<θ<(2m+0.5)π or(2m+0.5)π<θ<(2m+1)π, there is

(4)

whereα1andα2are the losses of the microdisk, andφ1andφ2are the phase shifts of WGM1 and WGM2, respectively.The phase shiftφ1,2=(4π2Rneff1,2)/λ.

When the straight waveguide and the microdisk are coupled, Eqs.(3)and(4)can be reduced to

(5)

Under these two conditions, the pass-through transmission functionStis

(6)

2 Experiment and analysis

To illustrate the transmission characteristics and sensing performance, the physical model of the microdisk was established by COMSOL software.The device is constructed on silicon-on-insulator(SOI)wafer with 2 μm buried SiO2layer and 220 nm top silicon layer.The radius of the microdisk is 3 μm.The length and width of the straight waveguide are 10 μm and 0.4 μm, respectively.The spacing between the waveguide and the microdisk is 70 nm.The electric field distribution, output spectrum and phase change of the device were analyzed.The effect of structural parameters on transmission characteristics and temperature sensing was studied.

2.1 Microdisk transmission characteristics

Figs.2 and 3 are a two-dimensional and a three-dimensional electric field distribution for different wavelengths when the microdisk is coupled to a straight waveguide, respectively.

Fig.2 Two-dimensional electric field distribution of microdisk

Fig.3 Three-dimensional electric field in microdisk

It can be seen that different modes of optical signals are excited inside the microdisk under different resonant wavelength conditions.And mode coupling plays an important role in the research of silicon-based optical components.

The transmission characteristic of the microdisk resonator is demonstrated in Fig.4.It is observed that the extinction ratio reaches-30 dB and-29.8 dB at the wavelengths of 1 536.4 nm and 1 564.2 nm, respectively.The larger the extinction ratio, the better the performance and the higher the sensitivity of the device.

Fig.4 Transmission spectrum of output port

Fig.5 shows the phase change of the microdisk resonator.We can see that the input-output spectrum produces a phase difference of about 2π due to the mutual interference between different modes, and the optical isolation increases when the spacing of adjacent resonant peaks increases.The change in phase means that the effective refractive index of the device changes, and the group velocity of the device also changes.

Fig.5 Phase change of output port

2.2 Effect of structural parameters on transmission characteristics

The dependence of transmission characteristics on device structural parameters is shown in Figs.6 and 7.It can be seen from Fig.6 that the output spectrum changes as the gap between the microdisk and the straight waveguide changes from 28 nm to 40 nm.The maximum output intensety is-29.8 dB at the resonant wavelength of 1 564.2 nm when the gap is 70 nm, and the output intensity is only-16 dB when the gap is 50 nm.Therefore, the transmission characteristics of the device can be optimized by adjusting the gap between microdisk and straight waveguide.

Fig.6 Dependence of transmission spectrum on the gap between microdisk and direct waveguide

There is little change in the phase of the output port when the gap between the microdisk and the straight waveguide increases from 50 nm to 70 nm as shown in Fig.7.The influence of the gap on the phase of the output port is negligible relative to on the output spectrum.Normally, the change in phase means a change in the effective refractive index of the device, therefore the small variation of the gap the between straight waveguide and the microdisk does not obviously affect the effective refractive index of the device.That is to say, the effect on the optical signal transmission may be negligible from the above analysis.

Fig.7 Dependence of output port phase on the gap between microdisk and direct waveguide

2.3 Temperature sensing

Using above optimal parameters, the characteristics of the temperature sensing is investigated by monitoring the shift of resonant wavelength induced by the surrounding environment.In the simulation, the temperature variation is changed from 0 to 80 K.The transmission spectra of the device under different temperatures are demonstrated in Fig.8.

Fig.8 Transmission characteristics at different temperature values

The sensitivitySis an interesting index for sensing applications.The sensitivity can be expressed as[12]

(7)

where Δλis the change of resonance wavelength corresponding to a temperature change ΔT.The high sensitivity corresponds to the maximal sharpness of resonance peak and wide resonant wavelength shift.We can see from Fig.8 that the shifting wavelengths are 1 561, 1 561.8, 1 562.7, 1 563.3 and 1 564.2 nm, respectively when the temperature changes from 0 to 80 K.Therefore, the minimum sensitivity is 30 pm/℃ and the maximum value is 45 pm/℃ in the course according to Eq.(7), which means the device has great potential for sensor.

3 Conclusion

The basic structure of microdisk resonator is built and the transmission characteristics are analyzed by the transfer matrix method.Then the electric field distribution, output port transmission spectrum and phase change diagram of the structure are illustrated.The effect of the spacing between the microdisk and the straight waveguide on its transmission spectrum and phase change is studied by parameter scanning.It can be seen from the electric field stereogram that under different resonant wavelengths, the microdisk internally stimulates different modes.And the output spectrum is changed significantly as the spacing between the microdisk and the straight waveguide varies within a certain range.Therefore, the transmission characteristics of the device can be optimized by changing the spacing.Finally, we analyze and demonstrate output temperature sensing property of the microdisk resonator, the ER can reach 30 dB and the sensitivity can be up to 45 pm/℃, thus allowing the realization of high performance sensors.