1 Introduction

Photonic crystal ring resonators (PhCRRs) are promising structures proposed for designing all optical devices suitable photonic integrated circuits (PICs). A PhCRR mostly consists of ring-shaped resonator located between two waveguides; these waveguides are called bus and drop waveguides. The working mechanism of the PhCRR is such that in a specific wavelength namely resonant wavelength, optical waves traveling in bus waveguide will drop to the drop waveguide through ring-shaped resonator. Therefore, PhCRRs can be used as wavelength selectors for designing a large variety of optical devices. The most significant advantage of PhCRRs upon other wavelength selectors like resonant cavities [1, 2] is their high transmission efficiencies, which mostly reaches to 100 %.

The most common devices realized using PhCRRs are channel drop filters (CDFs). One of the earliest implementation of PhCRRs has been proposed by Kumar et al. [3]. Robinson and Nekkeeran [4, 5] used circular structures as the core of the resonant ring. They obtained 114 and 94 of quality factors for circular and hexagonal rods, respectively. The sensitivity of their proposed structure resonant wavelength upon the variation of refractive index was \({\Delta }\lambda /{\Delta }\hbox {n}=1\,\hbox {nm}\,/\,0.01\). It means that for every 0.01 unit increase in the refractive index of the structure the resonant wavelength will shift 1 nm toward upper wavelengths. Mahmoud et al. [6–8] used X-shaped structures as the ring resonator for realizing optical CDFs. The quality factor and refractive index sensitivity of their proposed structures were 196 and \({\Delta }\lambda /{\Delta }\hbox {n}=1.4\,\hbox {nm}\,/\,0.01\). By using \(45^{\circ }\) PhCRR based on square lattice silicon rods, Bai et al. obtained quality factor and transmission efficiency equal to 830 and 90 % [9]. Tavousi and Mansouri-Birjandi [10] designed an optical CDF with 1050of quality factor. Most recently, an H-shaped resonator was used for realizing optical CDF, whose quality factor and refractive index sensitivity were 224 and \({\Delta }\lambda /{\Delta }\hbox {n}=1\,\hbox {nm}\,/\,0.01\) [11]. A quasi-square PhCRR was proposed by Rakhshani and Mansouri-Birjandi [12]; they obtained 842 and \({\Delta }\lambda /{\Delta }\hbox {n}=1\,\hbox {nm}\,/\,0.01\) for quality factor and refractive index sensitivity. By cascading three resonators with different refractive indices, they designed a four-channel optical demultiplexer. 12-fold quasi-crystal was another structure employed as the core of ring resonator, which resulted in 344 and \({\Delta }\lambda /{\Delta }\hbox {n}=1\,\hbox {nm}\,/\,0.01\) for quality factor and refractive index sensitivity [13]. Elliptical rings produced 647 of quality factor. Also it has been shown that by cascading three elliptical rings between bus and drop waveguides one can improve the quality factor to 1559 at the expense of reducing the transmission efficiency [14].

Optical demultiplexers [15, 16], switches [17] logic gates [18] and sensors [19] are the other examples of optical devices proposed using PhCRRs. By integrating multiple PhCRRs with different resonant wavelengths, one can design optical demultiplexer [16]. In optical demultipexers reducing the channel spacing and crosstalk are very crucial, for this purpose the exploited ring resonators should have very low bandwidth and high quality factor. As far as we know, the working mechanism of optical switches, logic gates and sensors are based on the shift of the resonant mode which is mostly induced by changing the refractive index of the resonator. Therefore, having high refractive index sensitivity and also high quality factor is very important in designing high-speed optical switches and logic gates and high-accuracy optical sensors.

Considering the above-mentioned discussion, one can conclude that in designing PhCRRs maximizing the quality factor and refractive index sensitivity is very crucial. Therefore, in this work we are going to propose and design a high quality factor and high refractive index sensitive PhCRR.

The rest of the paper is structured as follow: in Sect. 2, we discuss the design procedure of the structure. In Sect. 3, we propose the simulation results; in this section, we obtained the output spectrum of the filter and then investigate the refractive index sensitivity of the PhCRR, and in Sect. 4, we conclude from our work.

2 Design procedure

For designing our proposed PhC-based ring resonator, we use a \(40^{*}30\) square lattice of dielectric rods immersed in air. The effective refractive index of dielectric rods is 3.46. And the radius of dielectric rods is \({R}=0.25\hbox {a}\), where a is the lattice constant of the PhC structure. Before proceeding the design procedure of the proposed PhCRR, we should calculate the band structure and extract the potential PBG region of the fundamental structure. For this purpose, we used plane wave expansion (PWE) method [20]. The obtained band structure diagram is shown in Fig. 1. As we see there are three PBGs in the band structure diagram, two PBG in TM mode (blue-colored areas) and one in TE mode (red-colored area). The TM PBGs are in \(0.25<\hbox {a}/\lambda <0.35\) and \(0.47<\hbox {a}/\lambda <0.54\) range, and the TE PBG is in \(0.69<\hbox {a}/\lambda < 0.71\) range. By choosing \(\hbox {a}=786\,\hbox {nm}\) for the lattice constant of the structure, the second PBG in TM mode will be in \(1455\,\hbox {nm}<\lambda <1672\,\hbox {nm}\) wavelength range which completely covers the third wavelength window for optical communication. Therefore, all the simulations will be done in TM mode.

Fig. 1
figure 1

Band structure diagram of the fundamental PhC

Full size image

After investigating the PBG region of the structure, it is the time to implement the proposed PhCRR structure by creating required defects in appropriate locations inside the fundamental PhC structure. First of all, we created two parallel waveguides namely bus and drop waveguides. There are 13 rows of dielectric rods between these waveguides. Then for creating the resonant ring, we removed an \(11^{*}11\) array of dielectric rods and obtained an empty square at the middle of the PhC structure between bus and drop waveguides. In the next step, we put an octagon-shaped structure at the center of the empty square as the core part of the resonant ring. The radius of the core rods is \({R}=0.25\hbox {A}\), where \({A}=0.85\hbox {a}\) is the lattice constant of the core structure. The refractive index of core structure is 3.46 too. The proposed PhCRR structure has four ports: input port (A), forward transmission port (B), backward drop port (C) and forward drop port (D). In order to improve the transmission efficiency and the performance of the resonant ring, we introduced four scattering rods at the corners of the square; these scattering rods will cancel the backward reflections from the corners of the square which can reduce the transmission efficiency due to destructive interference of forward and backward traveling optical waves. The radius and the refractive index of these scattering rods are the same as the initial structure. The final sketch of the proposed PhC-based ring resonator is shown in Fig. 2.

Fig. 2
figure 2

Final sketch of the proposed PhCRR

Full size image

3 Simulation and results

Fullwave toolbox of RSoft software was used to implement FDTD for simulating the proposed structure. Accurate modeling of PhC-based structures requires 3D simulation which is very time-consuming and requires very power full computers. Therefore, we used effective refractive index approximation for reducing the 3D simulations to 2D simulations [22]. Also we used perfectly matched layers (PML) boundary condition [21] surrounding our structure; the width of the PML is 500 nm. The output spectrum of the proposed PhCRR is shown in Fig. 3. The normalized transmission of the structure at port B, C and D is depicted with blue, red and green curves. Figure 3 shows that optical waves in all the wavelengths will go toward port B except at \(\lambda =1551\,\hbox {nm}\) in which optical waves will drop to the drop waveguide and travel toward port C we have no output wave at port D. The drop efficiency of the structure is 99.6 % at \(\lambda =1550\,\hbox {nm}\), and the \(-\)3 dB bandwidth is 0.3 nm, so the quality factor (\({Q}=\lambda _0/{\Delta }\lambda \)) will be 5170. The distribution of the optical wave inside the structure for two different wavelengths is shown in Fig. 4. Figure 4a shows that at \(\lambda =1551\,\hbox {nm}\) the optical waves due to dropping function of the resonant ring will drop to the drop waveguide and travel toward port C; however, at \(\lambda =1553\,\hbox {nm}\) optical waves will not drop to the drop waveguide and only will travel toward port B because their central wavelengths do not coincide with the drop wavelength of the structure. At the following, we are going to investigate the refractive index sensitivity of the proposed PhCRR. For investigating the effect of the refractive index of every section of the structure on the resonant wavelength of the ring resonator, we simulated the proposed PhCRR for different refractive indices. In every simulation, we varied the refractive index of one section and considered the other refractive indices to be constant.

Fig. 3
figure 3

Output spectrum of the proposed PhCRR (Color figure online)

Full size image
Fig. 4
figure 4

Distribution of optical power at a \(\lambda =1551\,\hbox {nm}\) and b \(\lambda =1553\,\hbox {nm}\)

Full size image

The output spectra of the structure at port C for different refractive indices of adjacent rods (blue-colored rods) (\({n}_{1}\)) are shown in Fig. 5. For obtaining these spectra, we supposed the refractive indices of core rods and fundamental structure rods to be constant. As shown in Fig. 5 by increasing the refractive index of the dielectric rods, the output wavelengths shift toward higher wavelengths, such that for \({n}_{1}=3.46\), 3.465, 3.47, 3.475 and 3.48, the output wavelengths are \(\lambda =1551, 1551.1, 1551.2, 1551.3\) and \(1551.5\,\hbox {nm}\), respectively. The detailed specification of the output wavelengths for different values of the refractive index is listed in Table 1. The sensitivity of the resonant wavelength upon the refractive index of adjacent rods is \({\Delta }\lambda /{\Delta }\hbox {n}=0.2\,\hbox {nm}\,/\,0.01\).

Fig. 5
figure 5

Output spectra of the PhCRR for different values of \({n}_{1}\) (Color figure online)

Full size image
Table 1 Significant parameters of the proposed PhCRR for different values of \({n}_{1}\)
Full size table

The outermost rods of the core structure are shown with green color whose refractive index is labeled via \({N}_{1}\). In this part, we investigate the effect of \({N}_{1}\) on the resonant wavelength of the proposed PhCRR. The output spectra of the structure at port C for different values of \({N}_{1}\) are shown in Fig. 6. As shown in Fig. 6 by increasing \({N}_{1}\), the output wavelengths shift toward higher values, such that for \({N}_1=3.46\), 3.462, 3.465, 3.47 and 3.475, the output wavelengths are \(\lambda =1551\,\hbox {nm},\,1551.2\,\hbox {nm},\, 1551.5\,\hbox {nm},\,1552\,\hbox {nm}\) and \(1552.6\,\hbox {nm}\), respectively. The detailed specification of the output wavelengths for different values \({N}_{1}\) is listed in Table 2. The sensitivity of the resonant wavelength upon the refractive index of outermost core rods is \({\Delta }\lambda /{\Delta }\hbox {n}=1.06\,\hbox {nm}\,/\,0.01\).

Fig. 6
figure 6

Output spectra of the PhCRR for different values of \({N}_{1}\) (Color figure online)

Full size image
Table 2 Significant parameters of the proposed PhCRR for different values of \({N}_{1}\)
Full size table
Fig. 7
figure 7

Output spectra of the PhCRR for different values of \({N}_{2}\) (Color figure online)

Full size image

The refractive index of the second layer of core rods—yellow-colored rods–is labeled with \({N}_2\). The output spectra of the structure at port C for different values of \({N}_{2}\) are shown in Fig. 7. As shown in Fig. 7 by increasing \({N}_{2}\), the output wavelengths shift toward higher values, such that for \({N}_2=3.46\), 3.465, 3.47, 3.475 and 3.48, the output wavelengths are \(\lambda =1551, 1551.4, 1551.8, 1552.3\) and \(1552.7\,\hbox {nm}\), respectively. The detailed specification of the output wavelengths for different values \({N}_{2}\) is listed in Table 3. The sensitivity of the resonant wavelength upon the refractive index of second layer of core rods is \({\Delta }\lambda /{\Delta }\hbox {n}=0.8\,\hbox {nm}\,/\,0.01\).

Table 3 Significant parameters of the proposed PhCRR for different values of \({N}_{2}\)
Full size table

Now we investigate the effect of the refractive index of whole core structure. So we suppose the refractive index of the core rods including outermost, second layer and inner rods to be \({N}_{3}\). The output spectra of the structure at port C for different values of \({N}_{3}\) are shown in Fig. 8. As shown in Fig. 8 by increasing \({N}_{3}\), the output wavelengths shift toward higher values, such that for \({N}_3=3.46\), 3.463, 3.466, 3.469 and 3.472, the output wavelengths are \(\lambda =1551, 1551.8, 1552.8, 1553.7\) and \(1554.5\,\hbox {nm}\), respectively. The detailed specification of the output wavelengths for different values \({N}_{3}\) is listed in Table 4. The sensitivity of the resonant wavelength upon the refractive index of core rods is \({\Delta }\lambda /{\Delta }\hbox {n}=2.9\,\hbox {nm}\,/\,0.01\).

Fig. 8
figure 8

Output spectra of the PhCRR for different values of \({N}_{3}\) (Color figure online)

Full size image
Table 4 Significant parameters of the proposed PhCRR for different values of \({N}_{3}\)
Full size table

Finally we investigate the effect of the refractive index of the whole structure (n) on the resonant wavelength of the proposed PhCRR. The output spectra of the structure at port C for different values of n are shown in Fig. 9. As shown in Fig. 8 by increasing n, the output wavelengths shift toward higher values, such that for \({n}=3.46\), 3.462, 3.464, 3.466 and 3.468, the output wavelengths are \(\lambda =1551, 1551.6, 1552.3, 1552.9\) and \(1553.5\,\hbox {nm}\), respectively. The detailed specification of the output wavelengths for different values \({N}_{3}\) is listed in Table 5. The sensitivity of the resonant wavelength upon the refractive index of total structure is \({\Delta }\lambda /{\Delta }\hbox {n}=3.1\,\hbox {nm}\,/\,0.01\).

Fig. 9
figure 9

Output spectra of the PhCRR for different values of n (Color figure online)

Full size image
Table 5 Significant parameters of the proposed PhCRR for different values of n
Full size table

The output spectra of the proposed structure for different radiuses of adjacent rods (blue-colored rods) (R) are shown in Fig. 10. As shown in Fig. 10 by increasing the radius of adjacent rods, the output wavelengths shift toward higher wavelengths, such that for \({R}=195\), 198, 201, 204 and 207 nm, the output wavelengths are \(\lambda =1550.6, 1551.3, 1551.9, 1552.4\) and \(1553\,\hbox {nm}\), respectively. The detailed specification of the output wavelengths for different values of R is listed in Table 6. The sensitivity of the resonant wavelength upon the radius of adjacent rods is \({\Delta }\lambda /{\Delta }{R}=0.2\,\hbox {nm}\,/\,1\). Also if we change the lattice constant of the structure into \(\hbox {a}_{1}=\hbox {k}{*}\,\hbox {a}\), the resonant wavelength of the structure will shift to \(\lambda _1=\hbox {k}{*}\lambda \).

Fig. 10
figure 10

Output spectra of the PhCRR for different values of R (Color figure online)

Full size image
Table 6 Significant parameters of the proposed PhCRR for different values of R
Full size table

Considering the obtained results shows that our proposed PhCRR has very high quality factor, which is much better than previously proposed structures [3–13]. Because in this structure we employed a relatively large resonant ring, which increases the reflection probability of optical waves from rods that causes destructive interference between the low amplitude resonant modes, which contributes for the lower mode cancelation mechanism, as a result the quality factor increases. The optical sensitivity to the refractive index of total structure and the resonant ring core is \({\Delta }\lambda /{\Delta }\hbox {n}=3.1\,\hbox {nm}\,/\,0.01\), and \({\Delta }\lambda /{\Delta }\hbox {n}=2.9\,\hbox {nm}\,/\,0.01\) which is much better than previously proposed structures [3–13]. Also because we employed a relatively large resonant ring, due to large footprint of the resonator the variation of effective refractive index of the resonator by changing the refractive index of rods is large, because the effective refractive index depends on the footprint and refractive index, and the resonant wavelength depends on the effective refractive index. Having such high quality factor values and high refractive index sensitivity ensures its suitability for realizing PICs such as optical filters, demultiplexers, sensors and optical switching devices. For designing optical filters and demultiplexers, high quality factor channels result in low channels spacing and low crosstalk values. In optical sensors and switching structures, high quality factor and high optical sensitivity result in high-accuracy, high-speed and low-sensing or switching threshold.

4 Conclusion

In this paper, we proposed PhCRR structure with a octagon-shaped core. The proposed structure has a resonant mode at 1551 nm with transmission efficiency equal to 99.6 %. The bandwidth and quality factor of the proposed structure are 0.3 nm and 5170. We also studied the refractive index sensitivity of the proposed structure. The optical sensitivity to the refractive index of total structure and the resonant ring core is \({\Delta }\lambda /{\Delta }\hbox {n}=3.1\,\hbox {nm}\,/\,0.01\) and \({\Delta }\lambda /{\Delta }\hbox {n}=2.9\,\hbox {nm}\,/\,0.01\).