In this chapter, base on the strong whispering-gallery (WG) mode dependence on cavity geometry we investigated in Chapter 4, we propose a photonic crystal (PhC) circular-shaped microcavity sustaining a high quality (Q) factor WG mode. The modal properties and lasing actions of sustained WG mode are investigated and obtained in numerical simulations and experiments, respectively. Due to the presence of WG mode, we initiatively investigate the uniform coupling property between the microcavity and the different inserted waveguides geometries both in simulations and experiments. Besides, in numerical simulations, we propose an idea of applying this PhC microcavity design on a double-layered structure for serving as a highly sensitive optical stress micro-sensor. In the end of this chapter, we propose a nanocavity design sustain the lowest order WG mode with ultra-small mode volume based on square PhC lattice.
5-1 Photonic Crystal CD2 Microcavity
So far, we have investigated and demonstrated various WG modes lasing actions in microcavities formed by various quasi-PhC (QPhC) lattices in Chapter 3 and 4. In measurements, all of them show the properties of high Q factors and low thresholds. However, when considering the integration in photonic integrated circuits (PICs), this kind of QPhC microcavity is difficult to integrate with the present PhC-based system due to its spatially non-periodic lattice structure. Although it is possible to realize PICs by QPhC waveguide
system [55, 101], many further investigations are required. To overcome this difficult situation, one realistic approach is to enhance or well-localize WG modes in PhC microcavity by some modifications. And the strong WG mode dependence on the cavity geometry (positions of twelve nearest air holes) in 12-fold QPhC D2 microcavity we investigate in Chapter 4 provides us a design direction to enhance WG mode in PhC D2 microcavity.
5-1-1 Microcavity Design & Simulated Modal Prperties
According to the WG mode dependence on cavity geometry, it is possible to sustain a WG mode in PhC D2 microcavity by modifying the twelve nearest air holes positions as the way in 12-fold QPhC D2 microcavity. The scheme of our design is shown in Fig. 5-1. The PhCs are formed by air holes on a thin dielectric slab and the original PhC D2 microcavity is formed by removing seven air holes. The positions of twelve nearest air holes are rearranged to be the same with those of 12-fold QPhC D2 microcavity. Thus, six of them are shifted inward and the others are shifted outward. The microcavity is re-named as PhC circular-D2 (CD2) microcavity due to its cavity shape, as shown in Fig. 5-1.
Fig. 5-1: Scheme and cavity design of PhC D2 microcavity. The microcavity is modified from PhC D2 to CD2 microcavity by shifting the twelve nearest air holes inward or outward to make the spacing between air holes equal to one lattice constant.
To confirm this design, we apply three-dimensional (3D) finite-difference time-domain (FDTD) method to simulate the WG modal characteristics in PhC CD2 microcavity. The designed lattice constant (a) and the air-hole radius (r) over a (r/a) ratio are 500 nm and 0.34.
The refractive index of the dielectric material (InGaAsP) is set to be 3.4. The simulated electric- and magnetic-field distributions of sustained WG6, 1 mode whose lobes match with the gears formed by the twelve nearest air holes are shown in Fig. 5-2(a)-(c). From the mode distribution in the x-z plane, one can observe a significant zero-distribution region at the center of the microcavity. It is also observed from the electric field in the x-y plane that only a very small fraction of energy radiates into vertical directions due to the modal cancellation [102] of WG mode, which contributes to its high Q factor. This can also be seen from the electric-field distribution in the wave-vector (k) space by Fourier transformation, as shown in Fig. 5-2(d). Only very few leaky components are inside the light cone. The simulated Q factor and effective mode volume are 36,000 and 1.6(λ/n)3, respectively. We also calculate other resonance modes in PhC CD2
Fig. 5-2: 3D FDTD simulations of WG6, 1 mode in PhC D2 microcavity. Electric-field distribution in (a) x-z and (b) x-y planes. (c) Magnetic-field distribution in the x-z plane. (d) WG6, 1 mode electric-field distribution in k-space by Fourier transformation.
Fig. 5-3: (a) Plot of normalized frequency versus PhC r/a ratio of the resonance modes in PhC CD2 microcavity by 3D FDTD simulations. The hollow circles, squares, and triangles denote the measured lasing actions from devices with lattice constants from 490 to 510 nm. (b) The measured resonance spectrum from well-fabricated device with lattice constant and r/a ratio of 500 nm and 0.33, respectively. The gain region of MQWs is indicated by the shadow region.
microcavity within our designed range. The plot of normalized frequency of the resonance modes versus PhC r/a ratio is shown in Fig. 5-3(a).
5-1-2 Measured Lasing Action & Mode Identification
The scanning electron microscope (SEM) pictures of well-fabricated PhC CD2
microcavity by the fabrication process are shown in Fig. 5-4. The fabricated microcavities are optically pulse-pumped at room temperature and WG6, 1 single mode lasing action is obtained.
The typical light-in light-out (L-L) curve of PhC CD2 microcavity with a = 500 nm and r/a ratio ~ 0.33 is shown in Fig. 5-5(a) and the threshold can be estimated as 0.24 mW. The typical lasing spectrum at wavelength 1536 nm is shown in Fig. 5-5(b). We also show the spectrum near 0.8 times threshold in Fig. 5-5(c). The line width ΔλFWHM is estimated as 0.15 nm by Lorentzian fitting, which corresponds to Q factor of 10,000 by λ / ΔλFWHM in experiments [103]. Besides, the side-mode suppression-ratio (SMSR) of 18 dB is estimated from the spectrum in Fig. 5-5(d). In Chapter 3, we have proposed a simple approach to
increase SMSR by inserting a central air hole in the microcavity to destroy other resonance modes without affecting the WG mode. However, this approach cannot be applied here because the main side mode observed in longer wavelength in Fig. 5-5(d) is WG5, 1 mode.
Nevertheless, the side mode reduction can be still achieved by other loss mechanism managements [104, 105].
In mode identification, to confirm the lasing mode is WG6, 1 mode, we increase the sensitivity of the optical spectrum analyzer to collect the weak radiations from other
Fig. 5-4: (a) Top- and (b) tilted-view SEM pictures of fabricated PhC CD2 microcavity lasers.
The fabricated lattice constant and r/a ratio are 500 nm and 0.33, respectively.
Fig. 5-5: (a) Typical L-L curve and (b) lasing spectrum at 1536 nm of PhC CD2 microcavity laser. The threshold can be estimated as 0.24 mW from the L-L curve. The measured Q factor can be estimated as 10,000 from the line width of 0.15 nm in (c) the spectrum near threshold.
The SMSR is also estimated as 18 dB from (d) the lasing spectrum in dB scale.
resonance modes of the microcavity. The measured lasing spectrum in decibel scale is shown in Fig. 5-3(b). Comparing it with the 3D FDTD simulation results in Fig. 5-3(a), we obtain a very good match and clearly identify the lasing mode as WG6, 1 mode. Almost all resonance modes are observed and identified except for WG7, 1 mode near the gain region edge of multi-quantum-wells (MQWs). Besides, statistical measured lasing actions with a = 490 - 510 nm and r/a ratio = 0.32 - 0.36 are also obtained and denoted by hollow circles, squares, and triangles (different shapes mean different lattice constants) in Fig. 5-3(a), which also match with the FDTD simulated results quite well. The slight normalized frequency differences between measured and simulated results are arisen from the fabrication imperfections and the estimation inaccuracy from SEM pictures.
5-1-3 Uniform Coupling Properties
The PhC-based microcavity-waveguide structure [106, 107] is an important basic building block for various applications in PICs, such as optical interconnectors, couplers, optical buffers, and so on. Moreover, it is also a critical approach to convert most present PhC microcavity lasers with vertical emissions to in-plane emissions in planar PICs. One of the key issues is the efficient coupling between the cavity and the waveguide. For efficient coupling, not only the mode frequencies but also the spatial mode distributions of cavity and waveguide should match with each other. Recently, A. Faraon et al. [108] and K. Nozaki et al.
[109] have investigated high coupling efficiency between PhC single-defect waveguides and high Q PhC L3 microcavity and point-shifted nanocavity, respectively. However, the coupling efficiencies are not uniform in different inserted waveguide geometries due to the specific resonance direction of the mode. This would be an un-neglected problem under some specific designs where multiple input/output ports are needed.
Fig. 5-6: FDTD simulated WG6, 1 mode profile reveals that the evanescent field of each lobe propagates along twelve different directions from the microcavity.
Based on PhC CD2 microcavity, we propose our initiative design to solve this problem. It can be found that the evanescent field of each lobe of WG6, 1 mode propagates along twelve different directions from the microcavity as shown in Fig. 5-6. Intuitively, because every lobe of WG6, 1 mode is identical to each other, uniform coupling efficiencies can be obtained between the microcavity and the inserted PhC waveguides in these geometries. To investigate this uniform coupling behavior, we first numerically study the transmission of waveguide-cavity-waveguide structure in 180° line-to-line geometry (labeled as A6 type) as shown in Fig. 5-7(a) by two-dimensional (2D) FDTD method with approximated index of 2.7.
The separation between the cavity and waveguide is properly chosen as two lattice periods for the purpose of high transmission. Under the parameters of a = 500 nm and r/a ratio = 0.36, the detected transmission from the output port is around 60 %. The propagating field distribution and transmission spectrum are shown in Fig. 5-7(b) and (c), respectively. The transmission is defined as the ratio of the detected powers five lattice period distance after and before the microcavity. The transmission is also optimized by varying r/a ratio from 0.28 to 0.36. The highest transmission is over 90 % when r/a ratio = 0.30 - 0.32, as shown in Fig.
5-7(d). The propagation loss caused by the PhC waveguide has been considered and normalized.
Fig. 5-7: (a) Scheme of waveguide-cavity-waveguide coupling system based on PhC CD2
microcavity with different waveguide geometries. (Different output ports, numbered as port 1 - 10) (b) Propagating field distribution and (c) transmission spectrum of A6 type coupler with r/a ratio of 0.36. (d) Optimization of transmission versus r/a ratio. The maximum transmission appears when r/a ratio = 0.30 - 0.32.
To initiatively confirm the uniform coupling characteristic in different propagating geometries corresponding to each WG modal lobe, we calculate the transmissions of geometries with the same input A but different output ports numbered 1 to 10 as denoted and shown in Fig. 5-7(a). The r/a ratio is set as 0.32 according to the optimization result. The simulated transmissions and wavelengths of different geometries are summarized in Table.
5-1. In Table. 5-1, the transmissions are found to be in the range of 91 – 93 %, which indicates the uniform coupling in different waveguide-cavity-waveguide geometries. We also calculate the transmission of an inserted output waveguide between ports 5 and 6 for comparison, and we find that for WG6, 1 mode, the transmission dramatically decreases to lower than 2 %. It is necessary to notice that we do not consider A11 type because this geometry will involve additional coupling effects for the wave propagation in the parallel waveguides [110] and make the analysis more complicated.
Table. 5-1: Transmissions and wavelengths of different waveguide-cavity-waveguide geometries named A1 to A10 type.
Type Transmission Wavelength Type Transmission Wavelength A1 91.9 % 1586.4nm A6 90.9 % 1586.2nm A2 91.5 % 1586.3nm A7 92.9 % 1586.3nm A3 91.9 % 1586.3nm A8 91.3 % 1586.3nm A4 92.5 % 1586.3nm A9 91.8 % 1586.3nm A5 93.0 % 1586.3nm A10 92.9 % 1586.3nm
Fig. 5-8: (a) Scheme of A4-8 coupler with power splitting function and (b) its propagating field distribution. The output powers of port 4 and port 8 are almost the same with 42 % transmission.
Based on the uniform coupling property, we also design a coupler with one input port A and two output ports 4 and 8 named A4-8 type as shown in Fig. 5-8(a). Almost the same transmission of 42 % in each output port is achieved and the propagating field distribution is shown in Fig. 5-8(b). This indicates that PhC CD2 microcavity with WG6, 1 mode combined with PhC waveguide is very suitable in designing PhC-based components that need multi-port functions.
To investigate this uniform coupling property in experiments, we fabricate PhC CD2
microcavities with three different geometries. The SEM pictures and lasing spectra near threshold are shown in Fig. 5-9(a)-(c). The measured Q-factors degrade to around 6,900, 6,700, and 7,000 from the Lorentzian fit line width of 0.220, 0.225, and 0.215 nm near 0.8
Fig. 5-9: (a)-(c) SEM pictures and measured lasing spectra near threshold of PhC CD2
microcavities with three inserted waveguide geometries. From left to right, the Lorentzian fit line widths degrade to 0.220, 0.225, and 0.215 nm, respectively.
times threshold for the three cases. This uniform degradation also indicates the uniform coupling behavior in different cavity-waveguide geometries. We can conclude that this microcavity provides us more flexibility and freedom in designing various waveguide-cavity-waveguide geometries for applications in PICs due to its uniform coupling property.
5-2 Photonic Crystal CD2 Microcavity for Stress Sensor Applications
Due to the high Q factor [9, 10] and small mode volume [11-13] in PhC micro- and nano-cavities, they are potential in achieving advantages of highly sensitive, portable, condensed, and so on, for serving as optical sensors. Very recently, highly sensitive optical index and particle sensors have been investigated and reported by several groups [19-23, 111-113] based on PhC micro- and nanocavities with ultra-small mode volume and high Q factors, which are potential in chemical sensing and biological labeling [114, 115]. On the other hand, optical stress sensor is another important component in mechanical and semiconductor applications, especially in micro-electromechanical systems (MEMS).
Although some interesting optical stress sensor designs based on PhCs or PhC waveguides have been reported [116-120], the designs based on high Q micro- or nano-cavities are still
difficult to find in literatures and the sensitivity of present designs can be further improved.
5-2-1 Principle of Optical Stress Sensor
In optical stress sensors, according to the measured optical property change due to the structural variation, one can estimate the applied stress that leads to the corresponding structural variation. Among present reports of PhC optical stress sensors, we can roughly classify them into two categories by the detected optical property variations, optical spectrum and intensity. In the former one, researchers can measure the optical transmission spectrum shift caused by the displacement between PhC membranes [116] or by the elongated cavity length in microcavity-waveguide system on cantilever and suspended PhC membranes [117, 118] to estimate the applied stress on these structures. In the latter one, researchers can measure the transmitted optical intensity degradation due to the waveguides misalignment [119, 120] caused by the applied stress. However, in real case, there will be larger inaccuracy in measuring optical intensity variation than optical spectrum shift. Thus, estimating applied stress by optical spectrum shift would be a better and promising approach.
Generally, at first, researchers can find the relationship between the applied stress and the corresponding structural variation of their designed structure, for example, membrane displacement in ref. [116] and elongation of microcavity in ref. [117]. And then the relationship between the structural variation and the corresponding optical spectrum shift will be addressed. According to these two relationships, one can estimate the applied stress by the measured optical spectrum shift and decide how small the detectable stress variation per wavelength unit is. And the minimum detectable stress variation δF can be defined in newton unit instead of that per wavelength unit by considering the minimum spectral resolution of the measured light wave, which is decided by the measured optical line width in spectrum, or equivalently, Q factor. Thus, we can define a simple equation to illustrate this relationship:
where ΔF / Δd represents the applied stress needed to cause specific structural variation and Δd / Δλ represents the structural variation needed to cause specific optical spectrum shift, respectively. And the notations of λ, and λ / Q represent the wavelength and optical line-width in spectrum, respectively. Therefore, except for small ΔF / Δd and Δd / Δλ, high Q factor of the measured light wave is also necessary to achieve small δF. For simplicity, in the following investigations and discussions, we define factors of S = Δd / ΔF and W = Δλ / Δd to represent the structural variation rate due to the applied stress and wavelength shift rate due to the structural variation, respectively. And the δF can be expressed as λ / SWQ as in equation (5-1).
5-2-2 Structure Design, Simulated Modal Behaviors, and Sensing Resolution
We propose a double-layered (DL) PhC membrane microcavity design with air-gap distance d, as shown in Fig. 5-10, based on the PhC CD2 microcavity with WG6, 1 mode proposed in Chapter 5-1. The PhC CD2 microcavity design and WG6, 1 mode profile in electric-field are both shown in the insets A and B of Fig. 5-10. At first, the modal properties of DL PhC membrane microcavity are investigated by 3D FDTD method. The simulated domain is 24a × 24a × 12a with a/16 computed grid size. And the PhC r/a ratio, lattice constant, and d are set to be 0.32, 420, and 440 nm, respectively. The simulated bonding and anti-bonding modes profiles in electrical field in x-z plane based on WG6, 1 mode are shown in Figs. 5-11(a) and (b), which can be analog to the electronic bonding and anti-bonding states in chemical molecules. This “photonic molecule” design has been widely considered and investigated as the key component to construct logical PICs in different micro-structures as
we have mentioned in Chapter 4-1-3. This DL structure can be realized by MEMS technologies [120] and is potential for integrating in PICs.
As we mentioned before, at first, we need to know the wavelength shift rate W = Δλ / Δd arisen from the structural variation first. In our design, the air-gap distance d between the membranes is the structural variation parameter. In 3D FDTD simulations, d is varied from 165 to 660 nm. The relationship between d and the simulated resonance wavelength are shown in Fig. 5-11(c), which can be directly analog to the relationship between energy states and distance between atoms in chemical molecules. In Fig. 5-11(c), due to the weakened
Fig. 5-10: Scheme of DL PhC CD2 microcavity design. The microcavity design and the simulated WG6, 1 mode profile in electric-field are shown in the right inset-A and -B.
Fig. 5-11: The simulated mode profiles in electric-field in x-z plane of (a) bonding and (b) anti-bonding modes. (c) The simulated wavelengths of bonding and anti-bonding modes versus the air-gap distance d.
waveguide evanescence coupling when these two membranes become far apart, the optical mode will tend to act like the original WG6, 1 mode in single membrane and the wavelength difference between the bonding and anti-bonding modes becomes smaller when d increases.
Thus, there will be different wavelength shift rate W under different d and the W factor will decrease when d increases. The calculated W factor at different d is shown in Fig. 5-12 for the bonding mode. For example, the W factors are found to be 0.235 and 0.007 nm optical spectral shifts of bonding mode for 1 nm air-gap d decreasing when the initial d = 165 and 550 nm, respectively. These values are higher than previous reports [116-118], which also indicate the strong optical spectral response of this DL structure in serving as an optical stress sensor. As a result, we can say that high sensitivity (small δF) can be achieved in our design by choosing small d (large W factor). However, as we mentioned in equation (5-1), in terms of optical properties, the δF is not only determined by the W factor but also by the Q factor.
Thus, we calculate the Q factors of bonding and anti-bonding modes when d = 440 nm.
We obtain high Q factors of 75,200 and 22,700 from bonding (Qbonding) and anti-bonding modes, respectively. The former one is even higher than that of WG6, 1 mode (~ 36,000) in original PhC CD2 microcavity. Again, according to equation (5-1), high Q factor is benefit to obtain small δF. Therefore, in the following investigations, we will focus on the bonding mode with high Q factor. The simulated Qbonding with d varied from 165 to 660 nm is shown in Fig. 5-12. There are two main effects on Qbonding variation, waveguide evanescence coupling
We obtain high Q factors of 75,200 and 22,700 from bonding (Qbonding) and anti-bonding modes, respectively. The former one is even higher than that of WG6, 1 mode (~ 36,000) in original PhC CD2 microcavity. Again, according to equation (5-1), high Q factor is benefit to obtain small δF. Therefore, in the following investigations, we will focus on the bonding mode with high Q factor. The simulated Qbonding with d varied from 165 to 660 nm is shown in Fig. 5-12. There are two main effects on Qbonding variation, waveguide evanescence coupling