Performance Analysis of Three-Wavelength Multi-Channel Photonic Crystal Filters of Different Sizes

Abstract

Multi-wavelength and multi-channel photonic crystal filters are designed with different sizes considered by using a two-dimensional quadric lattice photonic crystal structure to solve the problems of a multi-channel filter with structure complexity, single-wavelength download, and channel interference. The designed filter consists of a waveguide, reflection wall, multimode microcavity, and output port. Each port can download three different wavelengths. In the communication band from 1.500 to 1.600 μm, the transmittance of each channel is greater than 90%, and the filtering efficiency is high. The size of the non-simplified filter is only 27 μm × 17 μm. On the premise of ensuring low loss transmittance (that is, the transmittance of each port is changed by no more than 10% at the wavelength from 1.5–1.6 μm), the size of the filter can reach 15 μm × 7 μm. This design will greatly reduce the overall structure size of the filter and is suitable for multiplexing and demultiplexing in WDM systems.

1. Introduction

Photonic crystal (PC) was independently proposed by John S. [1] and Yablonovitch E. [2] in 1987. PC is an artificial microstructure formed by periodic arrangement of media with different refractive indexes. The complete photonic crystal structure can generate a photonic band gap, and the light wave whose frequency is in the range of the photonic band gap is forbidden from propagating. When the defect is introduced into the photonic crystal, the defect mode can be introduced into the photonic band gap, so that the defect mode is localized at the defect position. The photonic crystal waveguide can be formed by introducing a line defect, and the line defect mode can propagate lossless at the defect position. If a point defect is introduced, the photonic crystal micro-resonator can be formed, and the point defect mode is strongly bound to the point defect position. Because photonic crystals have a photonic band gap and photonic localization, light propagation in media can be controlled. The optical filter, as the basic component unit of a wavelength division multiplexing system (WDM), is used to realize the up-and-down function of an optical signal. Photonic crystal with a micro/nano size can easily realize integration of the optical system, so the photonic crystal filter has always been the focus of many research teams. Fan et al. [3,4] of Stanford University first designed a two-dimensional photonic crystal filter with a micron size, which consists of two straight waveguides and a resonant cavity to form a four-port system. On the basis of Fan’s theory, Min et al. designed a flat down-path filter based on two-dimensional triangular lattice photonic crystal with an air hole arrangement, and the down-path efficiency reached 65% [5]. In the light of the same theoretical model, Qiu et al. designed another down-path filter based on triangular lattice photonic crystal plates. This system consists of two photonic crystal waveguides and a microcavity with two degenerate states [6]. Kim et al. [7] designed a reflecting wall at the end of the main waveguide to form a three-port filtering system in order to achieve efficient down filtering. Noda et al. of Kyoto University in Japan [8,9] also designed a series of flat down-path filters based on two-dimensional photonic crystal. Nkonde et al. presented a silicon photonic-crystal waveguide composed of three different microcavities, and the three output ends for waveband de-multiplexing with the peak transmission at the three optical windows could reach 86.6%, 63.5%, and 97.7% [10]. The proposed structure by Seifouri M. et al. can filter the central wavelength of 1548 nm with a transmission coefficient of over 95% [11]. A structure was made by Moloudian G. et al. of air holes in a dielectric background [12]. F. Brik proposed the design of a tunable add drop filter (ADF) based on photonic crystal [13].

At present, the study of upper and lower filters based on photonic crystal is a very active hot direction in the field of photonic crystal, and many researchers have done a lot of research work on the theoretical and experimental aspects [9,14]. Photonic crystal filters can be fabricated by using photonic crystal waveguides and point defects near the waveguides. When an optical signal in the waveguide has the same resonant frequency as the microcavity defect mode, the photons with resonant frequency are captured by the defect and emitted from the defect region into free space. Since the frequency of the defect mode is determined by the geometry size of the defect, it is possible to download multiplexed optical signals of different frequencies from the defects of different sizes by setting several defects of different geometry sizes in the photonic crystal plate near the waveguide.

The quality factor of the photonic crystal transmission spectrum and the normalized transmittance of the photonic crystal filter are the main parameters to measure the performance of photonic crystal filters. In order to distinguish between the filter waveform quality factor and micro cavity quality factor Q, the filter waveform quality factor is expressed by q, q = λ/Δλ, where λ is the center wavelength and Δλ is the full width half maximum (FWHM). Fan and others studied the impact of the medium cavity center column radius on the micro cavity quality factor, and found that when the dielectric cylinder radius is less than 0.15 a, the microcavity supports only a single mode while when the radius is 0.25~0.4 a, the microcavity supports the secondary mode. With the increase of the radius, the microcavity can also present a variety of patterns.

Fan [14] et al. studied in detail the influence of the structural parameters of microcavities on quality factors. Fan [15] and others also studied the effects of the dielectric layer between the waveguide and the microcavity on the filter quality factor q, and put forward that along with the increase of the dielectric layer, the quality of the filter factor improved. However, the absorption and scattering loss would increase because of the increased layers, leading to a lower efficiency of the filter. Thus, in the design of the filter, the filtering efficiency and the quality factor must be considered overall. The effect of the distance between the reference surface and the reflection wall on the quality factor of the microcavity was studied.

In conclusion, large studies and designs of photonic crystal filters have found that each channel can download a resonant wavelength. If multi-channel filtering is needed, multiple microcavities and channels will be introduced. However, the increase of channels will inevitably lead to the expansion of the overall structure. Moreover, the microcavity near the back will be affected by the microcavity near the front, and the filtering efficiency will be reduced. To solve this problem, Fan [11] et al. designed a dual-wavelength photonic crystal filter. Based on the above theoretical analysis, multi-wavelength and multi-channel photonic crystal filters are considered for the design.

2. Design of the Multimode Resonator

The design is based on two-dimensional quadric lattice photonic crystals. In the structure, the medium column is arranged in the air periodically. Silicon (Si) material is selected as the material with a refractive index of n = 3.4, the radius of the medium column is 0.2 a, and a = 550 nm is the lattice constant. The finite difference time domain (FDTD) method is adopted, and the perfect matching layer (PML) is used as the absorption boundary condition. The domain min: x = −7.5, z = −2.5; domain max: x = 7, z = 4; grid size: Δx = a/20, Δz = a/20; PML width is a. The thickness is considered infinity.

The design of the multi-mode resonator cavity is shown in Figure 1. The central medium column of the micro-cavity is first set as 0.08 a. When the central medium column is less than 0.15 a, it can be seen from the literature [9] that the four vertices are moving outward and the movement of the four media columns with defects in the micro-cavity can change the cavity mold from the unipolar mode to the multistage mode, so this design chooses to remove these four media columns.

The structural parameters of the four medium columns marked in red and the central medium column of the microcavity were used to regulate the microcavity mode, and their radii are R1 and R2, respectively.

The energy band structure of the microcavity was calculated by plane wave expansion (PWE). In order to ensure the accuracy of energy band calculation, 7 × 7 supercells were used for calculation. The position of the four red columns is not changed. When R1 = 0.2 a and R2 = 0.08 a, the calculation results are shown in Figure 1b. Four defect modes appear in the band gap, and their corresponding normalized frequencies are 0.396, 0.379, 0.36, and 0.345, respectively.

The corresponding frequency of these modes is far away from the band edge, and there is a relatively large frequency interval between each mode, which provides the basic conditions for the three-wavelength multichannel filter designed by our target.

By changing the size of R1 and R2, the defect pattern distribution in the microcavity is changed, and the corresponding frequency will move. When R1 = 0.18 a and R2 = 0.08 a, the calculation results are shown in Figure 1c. The number of modes remains unchanged and the change in the radius R1 causes the frequency corresponding to each mode to move towards high frequency. The main reason for this phenomenon is that when the radius of the dielectric column decreases, the equivalent refractive index in the microcavity decreases, and the frequency corresponding to the microcavity mode will move towards the air band.

The comparison of Figure 1b shows that there is no overlap of each frequency after the movement, which further confirms the feasibility of this type of multi-channel filter. When R1 = 0.168 a and R2 = 0.07 a, the calculation results are shown in Figure 1d, and three defect modes appear. As R1 changes play a dominant role, the frequency still moves in the direction of high frequency, accompanied by the disappearance of the low frequency mode.

In addition to the above problems, Fan et al. [3,4] studied the photonic crystal microcavity in 1996 and found the following rule: when the radius is 0–0.15 a, the microcavity only supports one monopole mode; as the radius continues to increase, multiple patterns emerge. For example, when the radius is 0.6 a, there are nondegenerate quadrupole modes, second-order unipole modes, and degenerate hexpole modes. The main reason of the low frequency mode disappearing is as follows. The corresponding model of the energy band structure in Figure 1b in frequency 0.345 is the single polarity mode. The four pillars of the red medium with a dielectric microcavity center column also construct a small cavity. The unipolar mode is the small microcavity resonant mode. When the red medium column radius is reduced, the small microcavity resonant ability is abate, leading to the disappearance of the corresponding mode.

3. Structural Design and Numerical Simulation of the Multi-Channel Filter

The microcavities in the previous section are arranged linearly from right to left, and the cavity is separated from the main waveguide by 1 a. A line defect is introduced 1 a below the resonant cavity to form the waveguide, and a reflection wall is arranged at the end of the main waveguide.

Based on the above numerical analysis, a three-channel photonic crystal filter is designed, with the structure shown in Figure 2

From right to left are ports A, B, and C, and the radius of the microcavity’s central media column (R1) is 0.2a, 0.18a, and 0.168a, respectively. The radii (R2) of the four media columns around the central media column are 0.08a, 0.08a, and 0.07a, respectively. The distance between the central reference surface of the microcavity A and the interface of the reflecting wall is 3a. The size of the non-simplified filter is only 27 μm × 17 μm Firstly, there is a certain distance to avoid interference from the reflecting wall, and on the other hand, the phase condition to achieve 100% down-path efficiency is also satisfied [9].

The two-dimension finite-difference time-domain method was used to simulate and calculate the transmission spectrum of the filter, as shown in Figure 3. The light used is a Gaussian beam. Pulse light is used when calculating the transmission spectrum, and continuous light is used when calculating the distribution of the light field. The electric field intensity distribution and response time point can be calculated according to the calculation results as shown in Figure 4, where λ = 1.5938 μm.

The results show that the filter can download three wavelengths respectively at three ports: for port A, the downloadable wavelength is 1.5938, 1.4504, and 1.3901 μm; for port B, the downloadable wavelength is 1.5663, 1.4362, and 1.3827 μm; and for port C, the downloadable wavelength is 1.5125, 1.4271, and 1.3758 μm. For the three ports, in the communication band from 1.5 to 1.6 μm, the transmittance of each channel is greater than 90%, and the download efficiency of each channel to different wavelengths is shown in

Table 1. Download efficiency, crosstalk, and sensitivity of each channel of the filter for different wavelengths.

Wavelength (um)	Transmission (%)			Sensitivity (fs)
	Part A	Part B	Part C
1.5938	96.29	9.2	3.36	33.3
1.5663	3.39	98.18	1.36	33.3
1.5125	0.547	0.49	94.75	33.3
1.4504	77.69	2.45	0.95	40
1.4362	2.28	45.95	2.13	40
1.4271	1.98	1.09	79.13	40
1.3901	57.58	18.99	11.43	20
1.3827	8.29	61.91	17.23	20
1.3758	2.79	10.83	55.02	20

. Compared with [10] (three-port transmission of 86.6%, 63.5%, 97.7%) and [13] (maximum transmission of 98%), the three-port transmission can be up to 92.29%, 98.18%, and 94.75%. It can also be observed from Figure 3 that there are no other peaks between each transmission spectrum, indicating that there is little crosstalk between the ports. The field distribution E of photonic crystal is used to describe the specific electromagnetic mode or distribution state of the electromagnetic wave in the microstructure. Power generally refers to power/light intensity, abs (E) squared. From the results of Figure 3 and Figure 4, the transmittance, crosstalk rate, and response time can be calculated as shown in Table 1. The blue ones are crosstalk. Because cT = 1000 μm and the energy is almost 90% of the stable energy, then this moment is the response time point. As c = 3 × 10⁸ m/s, T = cT/C = 33.3 (fs).

In order to distinguish it from the quality factor Q of the micro-cavity, the quality factor of the filter waveform is expressed by q, q = λ/Δλ, where λ is the central wavelength and Δλ is the half-width. I will answer the question about pattern quality factors along with the next question. The FWHM and Q values of each channel of the filter for different wavelengths are shown in

Table 2. Download efficiency, FWHM, and q values of each channel of the filter for different wavelengths.

Port	Center Wavelength (um)	Transmittance (%)	FWHM (nm)	q Value
A	1.5938	92.29	3	531
	1.4504	77.69	1.5	967
	1.3901	57.58	5	278
B	1.5663	98.18	3	522
	1.4362	45.95	2.5	574
	1.3827	61.91	2.6	532
C	1.5125	94.75	6	252
	1.4271	79.13	3.6	396
	1.3758	55.02	8	172

. Figure 5 shows the distribution of the optical power at different wavelengths.

4. Influence of Size

For the fabrication of the photonic crystal integrated structure, the smaller the size, the easier the integration and the lower the production cost. Considering the production, we tried to reduce the size to investigate it. The number of layers in each direction is reduced respectively. For this filter, the position of the four edges is defined as shown in the figure, with the top side defined as Up, the left side as Left, the right side as Bight, and the bottom as Bottom. Take, for example, the rule for reducing the top size. The medium column is reduced from the top layer by layer. For each layer reduced, the size of the overall filter can be reduced by 27 μm × 0.55 μm, as shown in Figure 6. The dielectric column in the orange dotted line box was taken as the starting row, and the number of layers was gradually reduced down along the direction of the orange arrow. For each layer reduced, the two-dimensional FDTD method was used to simulate the filter and calculate its corresponding transmission spectrum, as shown in Figure 7.

According to the transmission spectrum, the transmittance of the three channels is compared, as shown in Figure 8. It should be explained here that only points in the dot plot have the actual meaning of representing the data, and the lines are only for the convenience of distinguishing different filters. In the figure, U0 represents the original value of the filter with no reduction at the top, U1 represents the value of the filter with one layer reduction at the top, U2, U3, and so on.

Figure 8a shows the transmittance of port A at the peak value of the transmission spectrum; Figure 8b shows the transmittance of port B at the peak value of the transmission spectrum; Figure 8c shows the transmittance of port C at the peak value of the transmission spectrum.

When the top of the filter is reduced by four layers, it can be clearly seen that the transmittance of the three channels is greatly reduced. One or two layers are reduced, and each channel still maintains a high and approximate transmittance. Considering the production complexity, the upper layer can be reduced by two layers. The bottom, left, and right can be simplified in the same way.

Based on the above results, under the condition of ensuring transmittance, the maximum size reduction scheme is as follows: for this filter, the position of the four edges is defined as shown in the figure, with the top side defined as up, the left side as left, the right side as right, and the bottom side as bottom. The bottom is reduced by five layers, the left by two layers, the right by three layers, and the top by two layers. The transmission spectrum of the filter is obtained by simulating the filter with the two-dimensional finite-difference time-domain method, as shown in Figure 9.

The results show that the filter can download three wavelengths at three ports (Figure 10): for port A, the downloadable wavelength is 1.5938, 1.4504, and 1.3901 µm. For port B, the downloadable wavelength is 1.5663, 1.4362, and 1.3828 µm. For port C, the downloadable wavelength is 1.5125, 1.4271, and 1.3759 µm. From the results of Figure 9, the transmittance, crosstalk rate, and response time can be calculated as

Table 3. Download efficiency, crosstalk, and sensitivity of each channel of the filter for different wavelengths.

Wavelength (um)	Transmission (%)			Sensitivity (fs)
	Part A	Part B	Part C
1.5938	96.13	9.19	3.35	33.3
1.5663	3.39	98.04	1.35	33.3
1.5125	0.54	0.49	94.7	33.3
1.4504	77.5	2.44	0.98	40
1.4362	2.29	45.82	2.13	40
1.4271	1.99	1.09	78.96	40
1.3901	57.36	18.96	11.4	20
1.3828	8.28	61.66	17.2	20
1.3759	2.78	10.83	54.8	20

. The blue ones are crosstalk. By comparison, the downloadable wavelength is almost constant. For the three ports, in the communication band from 1.500 to 1.600 µm, each channel also has high transmittance.

Table 4. Download efficiency of each channel of the filter to different wavelengths and the increase and decrease rates compared with those before optimization.

Port	Wavelength (um)	Efficiency (%)	The Increase and Decrease Rates (%)
A	1.5918	97.8	+5.3
	1.4508	56.09	−27.8
	1.3901	48.65	−15.5
B	1.5643	96.64	−1.5
	1.4362	47.26	+2.8
	1.3824	66.16	+6.9
C	1.5116	95.08	+0.3
	1.4276	57.62	−27.2
	1.3759	60.15	+9.9

shows the download efficiency of each channel to different wavelengths. The comparison of the transmittance of each channel of the filter to different wavelengths before optimization is shown in Table 4. It shows that transmissivity increases and decreases, and the size can be changed regularly as required. Figure 9 shows the stable electric field distribution at each wavelength of the photonic crystal filter.

5. Conclusions

Firstly, a three-wavelength multi-channel photonic crystal filter was designed by using a two-dimensional quadric lattice photonic crystal structure. The filter mainly has three down ports, and each port can download three different wavelengths. In the communication band from 1.5 to 1.6 µm, the transmittance of each channel is greater than 90% and the filtering efficiency is relatively high. The filter size is only 27 µm × 17 µm. Then, based on the filter, the size was investigated, and the minimum size can reach 15µm × 7 µm under the condition of ensuring transmittance (that is, the transmittance of each port is changed by no more than 10% at the wavelength from 1.5–1.6 µm). Compared with the single-wavelength filter, this design will greatly reduce the overall structure size of the filter, and is suitable for multiplexing and demultiplexing in WDM systems. In this paper, the size is discussed, but it is not the optimization of the size. It only provides a reference, and can choose different structure manufacturing schemes according to different requirements. It has certain guiding significance for the manufacturing of devices.