# Double critical coupled ring resonator-based add–drop filters

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## Abstract

In this paper, we introduced double-critical coupling condition (DCCC) for lossy mode couplers of the add–drop resonator to achieve the desirable and tunable signals required for optical communication and photonics sensors applications. The performance of add–drop resonator under double-critical condition is simulated and analyzed. Some equations were derived for optical parameters including full width at half maximum (FWHM), the out-of-band rejection ratio (OBRR), quality factor and the crosstalk of the add–drop resonator, and the output signals were examined as a function of DCCC. Based on double-critical condition, the OBRR values larger than 40 dB, the crosstalk larger than 50 dB, highest quality factor of 9000 and the FWHM as small as 0.17 nm were realized in silicon add–drop resonator, which are quite compatible with the reported experimental data. Double-critical condition shows that the lossless coupling is not enough condition to acquire high-quality filtered signal with a large crosstalk and in presence of coupling losses, the DCCC can determine the optimum relation between the strength of coupling coefficients and the coupling losses.

## Keywords

Add–drop filter Double-critical coupling condition Microring resonator Calibration signals Tuning FWHM Crosstalk## Introduction

Optical communication has been receiving considerable attention due to the growing demand for high-speed internet, multimedia communication and data-dense applications [1]. The capacity limitation of electro-optical networks gives rise to the emergence of all optical networks, which are capable to support various optical add–drop multiplexers with a vast bandwidth. Integrated ring resonators have found a number of usages in integrated optics due to their compact size, low cost of fabrication and their ability to integrate with on chip circuits. Today, micro-size resonators are practical devices for switching [2, 3, 4], multiplexing, demultiplexing [5] and wavelength filtering [6, 7], modulation applications [8], wide band antenna [9, 10], modulator and logic gate in optical signal processing [11, 12], and sensitive sensors and biosensors [13, 14, 15]. One of the primary applications of ring resonators is their application as an add–drop notch type filter in optical networks. Based on the coupler position and type, different layouts of add–drop resonators have been reported for filtering signals, which include the symmetrically coupled [16] and asymmetrically coupled add–drop resonator [17] and one-stage parallel-coupled add–drop resonators [18]. Applying multiple stage resonators can also increase the quality of the output signals and improve the crosstalk between channels [19]. Generally, the crosstalk in optical filters emerges due to the miss match in channel wavelengths [20]. One of the approaches in reducing the mismatch in the channel wavelengths is considering the resonance condition in the ring resonator and its effect on the free spectral range. Indeed, the resonant mode number, the perimeter of the ring resonator, the group refractive index of ring’s waveguide and the resonance wavelength are determining factors [21]. These parameters together with critical condition need to be considered in the fabrication and modeling part. It is reported that the crosstalk can be controlled by applying vertical coupling in add–drop resonator [22], and it can be reduced via the scheme of a pair of the parallel coupled of critical resonators [23]. However, a mathematical relation is required to tune the crosstalk. Totally, to use ring resonator as an optical filter or an optical sensing device, the output signals need to be tunable. Since traveling of signals via different couplers can combine the fields and cause the interference, adjusting the parameters associated with couplers contribute to achieve high quality signals.

In this work, we introduced a simultaneous two-lossy-mode critical coupling condition to achieve the optimum transmission from add–drop resonator. Based on this condition, the through and drop port signals from add–drop resonator were tuned to achieve the desirable signals. The quality of signals was examined in terms of the full width at half maximum (FWHM), quality factor, out-of-band rejection ratio and the crosstalk and compared with data from fabricated silicon add–drop filter. The introduced double-critical condition is applicable in optical filtering, photonics sensors, terrestrial microwave, Wi-Fi devices and WDM technologies.

## Theoretical background

*E*

_{in}to output node

*E*

_{out}in a SFG diagram is given by

*T*

_{m}displays the gain of the

*m*th frontward path from input to output port,

*N*shows the total number of frontward tracks from input to output ports, and

*L*

_{i}is the transmittance gain of each loop [33]. Schematic design of a ring resonator with two bus waveguides and a couple of 2 × 2 optical couplers known as add–drop ring resonator (ADR) is shown in Fig. 1. The coupling power

*k*

_{1}and

*k*

_{2}specified by the gap and the length that the waveguides were coupled. Using the Mason rule for ADR layout in Fig. 1, the optical transfer function for the drop port of add–drop resonator becomes:

*m*= 1,2) is the coupling strength for each coupler, the amount of loss in each coupling region is given by \(\gamma_{m}\) (

*m*= 1,2), the waveguide loss denoted by \(\alpha\), L is the circumference of the ring resonator, \(n_{g}\) denotes the group refractive index of the ring’s waveguide, and \(\lambda\) shows the wavelength of the input light. The exponential term in OTF \(\xi = e^{{ - (\alpha L/2 + i2\pi n_{g} L/\lambda )}}\) is known as loss-unit delay parameter [34, 35, 36, 37]. The normalized intensity relation for the output-to-input port of resonating system can be obtained from the scalar product of OTF by its complex conjugates for each port. The normalized intensity of the drop port of ADR is calculated as

*m*= 0, 1, 2,…); thus by considering the drop port intensity equal to half of its maximum value, the phase constant becomes

*λ*

_{0}to the FWHM [39]. Based on Eq. (8), the quality factor for ADR can be determined as

*OBRR*, as

## Result and discussion

*R*= 1.5 μm and the group refractive index of

*n*

_{g}= 4.2 [40] are used for simulation. As simulated in Fig. 2, the variation of coupling coefficients between bus waveguide and the ring waveguide in ADR can affect the intensities at the drop and through ports, the OBRR and the FWHM of output signals. Simulated results in Fig. 2a, b show that the intensity in the drop port of ADR increased by increasing the values of coupling coefficients; however, an inverse reaction can be realized for the output signals from the through port. A trade-off behavior can be observed between simultaneous responses of the through and drop ports. Based on the result in Fig. 2c, the out-of-band rejection ratio larger than 20 dB can be achieved by power coupling values < 0.3. The OBRR values larger than 40 dB can be realized by conducting 10% of input signals into the ring’s waveguide. Figure 2d shows that the sharp and narrow signals with desired FWHM < 2 nm can also be measured for the coupling coefficients < 0.1. As demonstrated in Fig. 2d, for the specific relation in coupling coefficients, the FWHM is reached to a plateau with a constant value as large as 60.85 nm, which is equal to the FSR of the applied ADR. It means the coupling coefficients < 0.1 can again fulfill our expectations to achieve the resonance peaks with narrow FWHM and high OBRR, which are desirable for sensing applications.

### Lossy mode critical coupling

*γ*

_{1}and

*γ*

_{2}. For simplicity, the first term of Taylor expansion is considered for the exponential terms in OTF \(\xi = e^{{ - (\alpha L/2 + i2\pi n_{g} L/\lambda )}} \approx 1\). Based on Eqs. (12) and (13), both critical coupling conditions are considered and the critical coupling values were plotted for add–drop resonator with lossy mode couplers. It is supposed that both couplers have equal coupling losses and the amount of light coupled into the ring waveguide can change from 0 to 100 percent. Figure 3a, b shows the critical coupling values for zero through port output Eq. (12) and all-pass light via the drop port Eq. (13), respectively. Both critical conditions are demonstrated in Fig. 3c. Both critical conditions lead to the same line for lossless couplers (

*γ*

_{1}=

*γ*

_{2}= 0), which is the intercept line of two surfaces. The view of Fig. 3c from the

*z*direction is given in Fig. 3d. In order to find the most effective over coupling areas on the light transmission of add–drop resonator, 12 points are selected from both critical surfaces as addressed from A to L in Fig. 3d. The area pinned by B, E, H and L is located on the common intercept line, which is fulfilled over DCCC. The optical transmission of the add–drop resonator against wavelength was plotted for selected points (A–L) as shown in Fig. 4. The coupling coefficients and coupling losses of these selected points are given in Table 1. The points on the intercept line at Fig. 3c, d (B, E, H and L) represent the lossless coupling that fulfills the double-critical coupling condition. However, increase in coupling losses (

*γ*

_{1}and

*γ*

_{2}) contributes to vanishing crosstalk, which can be seen for the A, D, G, K and L points in Fig. 4.

The selected points from the critical coupling surfaces

Selected points | |
| |
---|---|---|---|

A | 0.0082 | 0.2751 | 0.9111 |

B | 0.0090 | 0.0000 | 0.0090 |

C | 0.2088 | 0.1084 | 0.4669 |

D | 0.3815 | 0.2129 | 0.9955 |

E | 0.3815 | 0.0000 | 0.3815 |

F | 0.5221 | 0.3072 | 0.0042 |

G | 0.7068 | 0.1205 | 0.9996 |

H | 0.7108 | 0.0000 | 0.7108 |

I | 0.7349 | 0.4799 | 0.0200 |

J | 0.7631 | 0.2691 | 0.5565 |

K | 0.9598 | 0.4940 | 0.8432 |

L | 0.9960 | 0.0000 | 0.9960 |

*Q*-factor, the OBRR and the crosstalk for selected points. As shown in Fig. 5a, the minimum FWHM can be measured by selecting a small and lossless coupling coefficient. The optimum performance of bandpass filter can be shown by measuring the higher OBRR [41]. Among the selected points on the critical surfaces, the lossless point B possess the smallest FWHM of 0.17 nm, the highest quality factor of 9000 and an OBRR as high as 47 dB, which is desirable for filtering and sensing applications. Based on the definition of

*Q*-factor in Eq. (9), the value of FWHM is a decisive gauge for the quality factor. As shown in Fig. 5a, almost all selected points have the FWHM as large as 10 nm, but the FWHM for point B reaches to 0.17 nm, which is almost 10 times less than that of from other points. It obviously brings about a

*Q*-factor almost 10 times larger other points.

Large *Q*-factor is associated with low-loss resonator. These results for point B are quite compatible with the reported results in [40], which shows compatibility of DCCC with experimental data. For the bandpass filters (here drop port of ADF), the out-of-band rejection represents the amount of applied suppression to any of the signal outside of the chosen wavelength band. The out-of-band rejection specifies a rejection value centered on resonance wavelength, which is defined in relation to a peak efficiency of a filter. In order to increase the measurement accuracy in measuring instruments, the wavelength span should be larger than out-of-band limits. According to Eq. (11), the crosstalk was defined as the difference between the maximum intensity at the drop port and the minimum intensity at the through port. As shown in Fig. 4, for some selected points located on the critical surfaces, there is no overlap between the through and drop signals. It leads to emergence of the negative values for the crosstalk. The crosstalk larger than 20 dB is desirable for filtering and optical communications [20, 22]. As shown in Fig. 5d, all of the selected points fulfill the desirable crosstalk condition (> 20 dB), but the A, D, G, K and L points.

The intercept line of both critical conditions can provide the best quality for output signals. Moreover, a narrow FWHM and high OBRR can be achieved by coupling coefficient values < 0.1 according to the results of Fig. 2. Here, point G fulfills only one of the critical conditions and its coupling coefficients are larger than 0.1. As shown in Fig. 4c, there exists no overlap between the drop and the through ports of point G. It leads to a negative crosstalk for this point as shown in Fig. 5d. The lossless points of E and H have the highest crosstalk; however, point L reveals that the lossless coupling is not enough condition to acquire large crosstalk and the strength of coupling coefficients is also effective parameter. These resonance features can be achieved by any add–drop resonator, which fulfills the introduced double-critical coupling condition region.

## Conclusion

Double-critical coupling condition (DCCC) is introduced for lossy mode couplers. Obtaining the high-quality and tunable filtered signals from add–drop resonator was the main aim of interest. The effect of coupling loss and coupling strength on the FWHM, OBRR, *Q*-factor and crosstalk was examined. The OBRR larger than 40 dB, the crosstalk larger than 50 dB, highest quality factor of 9000 and the FWHM as small as 0.17 nm can be realized under the DCCC. The performance of add–drop resonator under DCCC reveals that the coupling strength of couplers and the amount of coupler loss should simultaneously be considered to achieve high-quality output signals in practice. The introduced DCCC can determine the optimum relation between the strength of coupling coefficients and the coupling losses.

## Notes

### Acknowledgements

IS Amiri would like to acknowledge for the research facilities of Ton Duc Thang University, Vietnam.

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