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Spectrum control by anisotropy in a cylindrical microcavity made of electric anisotropic medium was studied. A finite-difference time domain method for electric anisotropic medium and Volume-average Effective Permittivity approximation are applied to calculate the resonant frequencies and quality factors of Whispering-gallery modes. The resonant frequency for different whispering-gallery modes has a similar shift in direct proportion to the relative difference of two principal refractive indices. The quality factors decay exponentially due to directional emission when the difference of two principal refractive indices increases. This novel tuning characteristic of anisotropic cylindrical microcavity will play an important role in many areas, such as light source with tunable wavelength, tunable filter and sensor.
©2009 Optical Society of America
1. Introduction
In the recent years, many kinds of optical microcavities with dimension of wavelength, including dielectric cylinder [1,2], disk [3–6], ring [7], and sphere [8,9], have attracted an increasing attention due to the Whispering-gallery modes (WGMs). The WGMs are the natural oscillation modes in these axial symmetry cavities and can be deduced directly from the Maxwell’s equations in cylindrical coordinates by using method of separation of variables. They are formed by continuous total internal reflection at the interface of microcavity. Therefore the special features of WGMs are high quality factor (Q factor) and tight confinement of their inner field near the cavity boundary which result in a small mode-volume. This is why these axial symmetry microcavities have potential application in low threshold laser [7,10], filter [7] and sensitive sensor [11–13]. Additionally photonic crystal cavity is also studied recently and thoroughly owing to its Whispering-gallery-like modes.
Resonant wavelength tunable microcavity is very desirable because of its potential in many applications such as wavelength division multiplexing, optical communication and optical interconnect. This wavelength tunability is achieved by modulating either the refractive index of cavity or the cavity length. The microcavity mode frequencies can be tuned by thermoelastic expansion of the microcavity material and thermo-induced change of the refractive index, i.e. a variation of the effective optical path [14–16]. However, the thermal tuning mechanism is limited by the low tuning speed, the narrow tuning range and the difficulty to control the temperature [14–16]. Micromachined tunable cavities [17–20] are introduced and do not suffer from all the drawbacks mentioned above. Some deformable membranes are employed in both coupled-cavity and vertical cavity to modulate the effective cavity lengths and realize the wavelength tuning. However, these configurations increase the complexities in fabrication or occupy a big volume due to the external cavity. Additionally, the tuning speed is still limited by the inherent mechanical characteristic of membrane materials.
In this paper we propose a two-dimensional (2D) cylindrical microcavity made of electric anisotropic medium. In the next section we introduce the numerical simulation method, including a 2D finite-difference time domain (FDTD) method for electric anisotropic medium [21,22] and Volume-average Effective Permittivity (V-EP) approximation [23,24] to deal with the microcavity boundary, successively demonstrate the validity of this numerical method in solving the microcavity problem by comparing the analytical solution and the numerical solution for isotropic cylindrical microcavity, and finally investigate the influence of medium electric anisotropy on the cavity resonant frequencies and Q factors.
2. 2D anisotropic FDTD method and V-EP approximation
The FDTD method for electric anisotropic medium have been studied in [21,22].
For electric anisotropic medium, the relative permittivity ε is a tensor expressed by a matrix with size of 3 × 3. If the directions parallel to the three principal axes are set to be the axes of Cartesian coordinate system, the matrix ε will be diagonal,
For TE polarization (with the electric field lying within x-y plane) in a lossless and nonmagnetic medium, by applying the regular Yee’s cell and central-difference approximations, the iterative formula can be obtained as follows:
where ε 0 and μ 0 is permittivity and permeability of free space, Δx and Δy represents spatial step size in the x, y direction, respectively, Δt is the temporal step size, and i, j is the position label, n is the time label.V-EP approximation [23,24] is more effective than staircase approximation to deal with the interface between two mediums. The location of every field component in V-EP approximation is the same as that in regular Yee’s cell, whereas every electric field component has its own private cells centered with it to calculate its permittivity. The private cells associated to the different electric field components partially overlap. A concept of an effective permittivity ε eff for certain electric field calculated by taking the weighted volume average of the two dielectrics is introduced and can be expressed as
where V (i, j, k) is the fill factor of the medium with permittivity ε 1 (εx for Ex and εy for Ey) of the private cell (i, j, k) for this electric field.The perfectly matched layer (PML) [25] is introduced as the absorbing boundary condition to absorb outgoing waves from the computation domain.
3. Comparison of analytical solution and numerical solution for isotropic cylindrical microcavity
The WGMs problem of isotropic and homogeneous cylindrical cavity can be solved analytically and rigorously [26]. The characteristic equation is given as follows
where n 1 and n 2 is the refractive index of cavity medium and background medium, respectively, k is the vacuum wavenumber, Jm is the first kind Bessel function, Hm (2) is the second kind Hankel function, the superscript sign(') denotes the first derivative of full variable for the corresponding function, m = 1,2…is the azimuthal mode number, η = n 1/n 2 for TM, and η = n 2/n 1 for TE modes. With a given m, this equation gives a series of complex solutions km,l, where l = 1,2… is the so-called radial mode number to denote these values in an increasing order. Thus a WGM with azimuthal mode number m and radial mode number l can be expressed by the symbol WGMm,l. The resonant frequency of WGMm,l iswhere c is the light speed in free space, and its Q factor isAs shown in Fig. 1 , the structure of microcavity to be studied is a cylindrical dielectric rod in air, i.e. n 2 = 1. The refractive index n 1 and the radius R of the dielectric rod are taken to be 3.2 and 1μm, respectively. The direction parallel to the rod is set to be axis z, and the two orthogonal directions lying within the cross section are set to be axis x and axis y to form a right-handed set with axis z. The rod is assumed infinite in the z direction. TM polarization has only one electric field component with direction parallel to rod axis. This means that it is not an electrical anisotropic problem any more. Therefore only TE polarization with the electric field is in the cross-sectional plane is considered in this paper. We set a magnetic line current source, a sine signal modulated in time by Gaussian pulse with a central wavelength λ s = 1400nm, at point S to excite the WGMs in the microcavity and a detection point D to detect the electromagnetic field, from which the resonant frequencies and Q factor of cavity will be calculated by Padé approximation with Baker’s algorithm [27]. Both the two points locate at axis y (to avoid exciting degenerate mode which will be discussed in another paper) and have a distance 15R/16 away from the cavity centre.
Fig. 1 Sketch of cylindrical cavity. The structure of anisotropic microcavity to be studied is a cylindrical dielectric rod with dielectric constant ε 1 and radius R = 1μm in background medium with dielectric constant ε 2 = 1. The direction along the rod is parallel to one of the three principal axes and is set to be axis z, and the two orthogonal directions lying within the cross section are parallel to the other two principal axes and are set to be axis x and axis y to form a right-handed set with axis z. The rod is assumed infinite in the z direction. Both source point S and detection point D locate at axis y (to avoid exciting degenerate mode which will be discussed in another paper) and have a distance 15R/16 away from the cavity centre.
For isotropic medium, the dielectric constant in Eq. (2) is εx = εy = n 1 2 = 3.22 = 11.56. We set spatial step size Δx = Δy = 10nm, temporal step size Δt = Δx/(2c), where c is light speed in free space. The size of the spatial computational domain, the total time iterations and the layers of the PML are set to 6μm × 6μm, 160 000 and 20, respectively.
Both the analytical solution and numerical solution for this isotropic cylindrical microcavity are given in Table 1 . This table shows that the resonant frequencies and the Q factors obtained by numerical solution accord with the analytical solution very well. This fact demonstrates the validity of the numerical solution in solving the microcavity problem, and then we will employ it to study the anisotropic microcavity.
Table 1. Comparison of analytical solution and numerical solution for isotropic cylindrical microcavity with refractive index n2 = 3.2 and radius R = 1μm.
4. Resonant frequency shift and Q factor stability of electric anisotropic cylindrical microcavity
In order to investigate the influence of medium electric anisotropy on the cavity resonant frequencies, change the y direction principal refractive index ny while maintaining that other parameters are the same as that in Fig. 1. The red real curve, green dashed curve and blue dotted curve in Fig. 2 is the spectrum of cylindrical microcavities with relative difference of two principal refractive indices (nx - ny)/nx equal to 0, 0.01 and 0.02, respectively. These spectrum ranges are chosen to be between 175 and 225 THz at which commutation wavelength 1.55μm (corresponding frequency 193.55THz) locates. It is obvious that there are resonant frequency shifts for WGMs of microcavities with different electric anisotropy.
Fig. 2 Influence of medium electric anisotropy on the cavity resonant frequencies. The red real curve, green dashed curve and blue dotted curve is the spectrum of cylindrical microcavities with relative difference of two principal refractive indices (nx - ny)/nx equal to 0, 0.01 and 0.02, respectively. Other parameters are the same as that in Fig. 1.
Figure 3 demonstrates the relationship between the resonant frequencies of WGMs and relative difference of two principal refractive indices (nx - ny)/nx. The red circles, green asterisks and blue squares refer to numerical results for WGM8,1, WGM9,1 and WGM10,1, respectively. From this figure, we may naturally think that the resonant frequency shift for each WGM is in direct proportion to (nx - ny)/nx. Therefore, we fit these data points with a linear fitting function f = A (nx - ny)/nx + f 0. The red real line, green dashed line and blue dotted line are the corresponding fitting lines. Table 2 is the linear fit parameters of the three fitting lines. Now we can safely reach the conclusion that the resonant frequency shift is in direct proportion to (nx - ny)/nx just with different coefficients for each WGM. For a certain mode in an anisotropic cylindrical microcavity, the refractive index for different angular position is different because the direction of electrical field for different angular position is different. The electrical field near the axis y (or x) is almost parallel to y (or x) direction and so the corresponding refractive index is about ny (or nx). The change of principal refractive index ny in an anisotropic cylindrical microcavity corresponds to the change of the corresponding major axis ay in an isotropic elliptic microcavity [28,29] because both of them would cause a variation of optical path. When ny decreases, the optical path circulating around the perimeter decreases. As a result, there will be a blueshift for every WGM. Additionally the tuning range is wide enough because resonant frequency is very sensitive to relative difference of two principal refractive indices.
Fig. 3 The relationship between the resonant frequencies of WGMs and relative difference of two principal refractive indices (nx - ny)/nx. The red circles, green asterisks and blue squares refer to numerical results for WGM8,1, WGM9,1, and WGM10,1, respectively. The red real line, green dashed line and blue dotted line are the corresponding fitting lines.
Table 2. Linear fit parameters of the three fitting lines for WGM8,1, WGM9,1, and WGM10,1, respectively, in Fig. 3a.
Figure 4 illustrates the relationship between the Q factor of WGM8,1, WGM9,1 and WGM10,1 and relative difference of two principal refractive indices (nx - ny)/nx. The red circles, green asterisks and blue squares refer to numerical results for WGM8,1, WGM9,1 and WGM10,1, respectively. An exponential function Q = Q0 exp [B (nx - ny)/nx] is used to fit these data. The red real curve, green dashed curve and blue dotted curve are the corresponding fitting curves. Table 3 is the fit parameters of the three fitting curves. When (nx - ny)/nx increases, i.e. the ny decreases, the Q factor reduces exponentially due to directional output. In an anisotropic cylindrical microcavity, the axial symmetry as cylinder geometry configuration have destroyed by anisotropy and thus must result to a significant electromagnetic field directional output from the microcavity. This case is just like the decrease in Q factor of a cylindrical with quadrupolar deformation due to directional emission [29]. The areas near axis x (with highest refractive index nx) in an anisotropic cylindrical microcavity have an analogy to the areas near highest curvature surfaces in a cylindrical with quadrupolar deformation, where the highest directional emissions occur. In view of the length of paper, this problem will not be discussed further here and will be researched in another subsequently paper.
Fig. 4 The relationship between the Q factors of WGMs and relative difference of two principal refractive indices (nx - ny)/nx. The red circles, green asterisks and blue squares refer to numerical results for WGM8,1, WGM9,1, and WGM10,1, respectively. The red real curve, green dashed curve and blue dotted curve are the corresponding fitting curves.
Table 3. Parameters of the three fitting curves for WGM8,1, WGM9,1, and WGM10,1, respectively, in Fig. 4a.
5. Conclusion
We have studied the tuning of resonant frequency in 2D electrical anisotropic cylindrical microcavity in this paper. The resonant frequency shift is in direct proportion to relative difference of two principal refractive indices just with different coefficients for each WGM. The Q factors of WGMs decay exponentially when the difference of two principal refractive indices increases because that anisotropic cylindrical microcavity has a significant directional output.
With the electro-optic effect, the refractive index of cavity can be modulated with a very high speed at the lever of nanoseconds. If the anisotropy of cavity medium varies with pressure and temperature, the microcavity can be used as a sensor. Comparison with other tunable cavities, anisotropic cylindrical microcavity has many advantages such as wide tuning range, fast tuning speed, directional emission, high compactness and simple structure. Therefore it is the more attractive candidate for optical integrated circuits, including wavelength division multiplexing, optical communication and optical interconnect.
Acknowledgements
The authors thank the reviewers for their valuable comments very much and appreciate the editor’s kindness and work.
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