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

The Z-transform has been employed for analyzing the discrete time systems and manipulating the discrete data sequences extensively in the fields of digital filtering and signal processing [24, 25], applied mathematics [26], economics [27] and control theory [28]. The signal flow graph (SFG) method takes advantage of Mason’s rule [29, 30] and the Z-transform [31, 32] to calculate optical transfer function of photonic devices. The SFG method can be interpreted as a transformation of either the method of successive substitution of simultaneous equations or the transfer matrix method to a topological approach. Based on the Mason’s rule, the transfer function from input node Ein to output node Eout in a SFG diagram is given by

$${\text{OTF}} = \frac{{E_{\text{out}} }}{{E_{\text{in}} }} = \frac{{\sum\nolimits_{m = 1}^{N} {T_{m} \Delta_{m} } }}{{1 - \sum\nolimits_{i} {L_{i} + \cdots } }}$$
(1)

where OTF is the input to output signals’ ratio, Tm displays the gain of the mth frontward path from input to output port, N shows the total number of frontward tracks from input to output ports, and Li 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 k1 and k2 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:

$${\text{OTF}}_{\text{drp}} = \frac{{E_{\text{drp}} }}{{E_{\text{in}} }} = \frac{{ - \sqrt {(1 - \gamma_{1} )k_{1} } \sqrt {(1 - \gamma_{2} )k_{2} } e^{{ - (\alpha L/4 + i\pi n_{g} L/\lambda )}} }}{{1 - \sqrt {(1 - \gamma_{1} )(1 - k_{1} )} \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{{ - (\alpha L/2 + i2\pi n_{g} L/\lambda )}} }}$$
(2)

where \(k_{m}\) (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

$$I_{\text{drp}} = \frac{{(1 - \gamma_{1} )(1 - \gamma_{2} )k_{1} k_{2} e^{ - \alpha L} }}{{\begin{array}{*{20}l} {\left[ {1 - \sqrt {(1 - \gamma_{1} )(1 - k_{1} )} \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{ - 0.5\alpha L} } \right]^{2} } \hfill \\ { + \,4\sqrt {(1 - \gamma_{1} )(1 - k_{1} )} \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{ - 0.5\alpha L} \sin^{2} \left( {\frac{\varphi }{2}} \right)} \hfill \\ \end{array} }}$$
(3)

here \(\varphi\) represents the phase constant that emerged in each round-trip traveling of light via the ADR system. The optical transfer function for through port of ADR for lossy mode couplers is

$${\text{OTF}}_{\text{thr}} = \frac{{E_{\text{thr}} }}{{E_{\text{in}} }} = \frac{{\sqrt {(1 - \gamma_{1} )(1 - k_{1} )} - \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{{ - (\alpha L/2 + i2\pi n_{g} L/\lambda )}} }}{{1 - \sqrt {(1 - \gamma_{1} )(1 - k_{1} )} \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{{ - (\alpha L/2 + i2\pi n_{g} L/\lambda )}} }}$$
(4)
Fig. 1
figure 1

Schematic diagram of single add–drop resonator

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The normalized intensity of through port of ADR is

$$I_{\text{thr}} = 1 - \frac{{(1 - \gamma_{1} )k_{1} [1 - e^{ - \alpha L} (1 - \gamma_{2} )(1 - k_{2} )]}}{{\begin{array}{*{20}l} {\left[ {1 - \sqrt {(1 - \gamma_{1} )(1 - k_{1} )} \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{ - 0.5\alpha L} } \right]^{2} } \hfill \\ { + 4\sqrt {(1 - \gamma_{1} )(1 - k_{1} )} \sqrt {(1 - \gamma_{2} )(1 - k_{2} )} e^{ - 0.5\alpha L} \sin^{2} \left( {\frac{\varphi }{2}} \right)} \hfill \\ \end{array} }}$$
(5)

Another significant parameter in characterizing the output signal quality from optical devices is the FWHM. According to Eqs. (3) and (5), the width of output signals at half of the maximum amplitude can be changed by the phase constant of \(\varphi = 4\pi^{2} fn_{g} R/c\). The maximum intensity at the drop port of the ADR corresponds to \(\varphi = 2m\pi\) (m = 0, 1, 2,…); thus by considering the drop port intensity equal to half of its maximum value, the phase constant becomes

$$\varphi = \,\cos^{ - 1} \left\{ {\begin{array}{*{20}c} {[4\sqrt {(1 - \gamma_{1} )(1 - \gamma_{2} )(1 - k_{1} )(1 - k_{2} )} e^{ - \alpha L} + } \\ {\frac{{ - (1 - \gamma_{1} )(1 - \gamma_{2} )(1 - k_{1} )(1 - k_{2} )e^{ - \alpha L} - 1]}}{{2\sqrt {(1 - \gamma_{1} )(1 - \gamma_{2} )(1 - k_{1} )(1 - k_{2} )} e^{ - \alpha L} }}} \\ \end{array} } \right\} = \cos^{ - 1} (\psi )$$
(6)

Suppose that the difference in the round-trip phases occurs between the point of the maximum intensity and the point at half of the maximum intensity, thus the difference in the round-trip phases can be written as [38]

$$\Delta \varphi = \varphi_{1} - \varphi_{2} = 2\cos^{ - 1} (\psi )$$
(7)

In terms of the frequency, the round-trip phases can be written by \(\Delta \varphi = 4\pi^{2} n_{g} R\Delta f/c\). This equation can be expressed in terms of the wavelength by \(\Delta \varphi = 4\pi^{2} n_{g} R\Delta \lambda /\lambda^{2}\). By considering Eq. (7), the FWHM of add–drop resonator, \(\Delta \lambda_{\text{FWHM}}\), is calculated as

$$\Delta \lambda_{\text{FWHM}} = \frac{{\lambda^{2} \cos^{ - 1} (\psi )}}{{2\pi^{2} n_{g} R}}$$
(8)

The other quantity for characterization of the quality of the output signals is the quality factor, which shows the stored energy to the energy loss per cycle. In the context of resonators, the ability of the waveguide to confine the field is described by the quality factor, which shows a measure of the sharpness of the resonance peaks. It is defined as the ratio of the resonance wavelength λ0 to the FWHM [39]. Based on Eq. (8), the quality factor for ADR can be determined as

$$Q\,{\text{factor}} = \frac{{\lambda_{o} }}{{\Delta \lambda_{\text{FWHM}} }} = \frac{{\pi n_{g} \,L}}{{\lambda \cos^{ - 1} (\psi )}}$$
(9)

Determining minimum and maximum intensities contributes to specify the out-of-band rejection ratio, OBRR, as

$${\text{OBRR}} = 10\log_{10} \left( {\frac{{I_{\text{drop}}^{ \text{max} } }}{{I_{\text{drop}}^{ \text{min} } }}} \right)$$
(10)

The difference in channels wavelengths causes a crosstalk between optical channels in optical filters. The overlapping areas between relative through/input port transmission and relative drop/input port transmission known as crosstalk areas can be calculated by

$${\text{Crosstalk}} = 10\log_{10} \left( {\frac{{I_{\text{drop}}^{\text{max} } }}{{I_{\text{thr}}^{\text{min} } }}} \right)$$
(11)

Result and discussion

In order to achieve the desirable resonance peaks for practical use in optical communication and photonics sensor applications, it is required to achieve the sharpest resonance peaks with narrowest FWHM and the maximum OBRR. To this end, some ring’s parameters like coupling coefficients, coupling losses and waveguide loss are determining factors, which are required to be optimized. In this work, the experimental data from the silicon microring resonator with radius as small as R = 1.5 μm and the group refractive index of ng = 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.

Fig. 2
figure 2

The effect of changing coupling coefficients of the add–drop resonators on the a drop port intensity. b Through port intensity. c Out-of-band rejection ratio (OBRR). d Full width at half maximum (FWHM)

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Lossy mode critical coupling

The relative phases of the signals through the recombination process determine whether they interfere destructively or constructively. Sharp and narrow resonance peaks are required for optical communications and sensor applications. To obtain a mathematical relation between couplers, it is supposed that light transmission via the through port of ADR reaches to its minimum value, \({\text{OTF}}_{\text{Thr}} (\phi = 2m\pi ) = 0\); in the other words, the maximum transmission can be realized at the drop port of ADR, \({\text{OTF}}_{\text{Drp}} (\phi = 2m\pi ) = 1\). Two critical coupling coefficients are calculated from \({\text{OTF}}_{\text{Thr}} (\phi = 2m\pi ) = 0\) and \({\text{OTF}}_{\text{Drp}} (\phi = 2m\pi ) = 1\) as follows:

$$k_{Ct}^{{H{\text{th}}}} = 1 - \frac{{1 - k_{1} }}{{(1 - \gamma_{1} )(1 - \gamma_{2} )}}$$
(12)

and

$$k_{Ct}^{{H{\text{drp}}}} = \frac{\begin{aligned} \left\{ { \pm 4\sqrt {[(1 - \gamma_{1} )^{3} (1 - \gamma_{2} )^{3} - (1 - \gamma_{1} )^{2} (1 - \gamma_{2} )^{2} ](k_{1} - k_{1}^{2} )} } \right. \hfill \\ - \left. {(1 - \gamma_{1} )(1 - \gamma_{2} )[2(1 - \gamma_{1} )(1 - \gamma_{2} )(k_{1} - 1) - 4k_{1} + 2]} \right\} \hfill \\ \end{aligned} }{{2(1 - \gamma_{1} )^{2} (1 - \gamma_{2} )^{2} }}$$
(13)

where \(k_{1}\) shows a power coupling coefficient for one of the couplers in add–drop resonator system and \(k_{Ct}\) represents the value of the second coupling confident. The coupling losses in each coupling region of ADR are shown by γ1 and γ2. 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 (γ1 = γ2 = 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 (γ1 and γ2) contributes to vanishing crosstalk, which can be seen for the A, D, G, K and L points in Fig. 4.

Fig. 3
figure 3

All-inclusive critical coupling values for lossy mode couplers of ADR. Critical coupling surface for a\({\text{OTF}}_{\text{Thr}} (\phi = 2m\pi ) = 0\). b\({\text{OTF}}_{\text{Drp}} (\phi = 2m\pi ) = 1\). c Double-critical coupling condition \({\text{OTF}}_{\text{Thr}} = 0\;\& \;{\text{OTF}}_{\text{drp}} = 1\) and d double-critical coupling condition from z direction view (2D)

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Fig. 4
figure 4

Optical transmission of the add–drop resonator against wavelength for 12 selected points (A–L) from double-critical coupling condition surfaces a through port, b drop port, c through and drop ports

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Table 1 The selected points from the critical coupling surfaces
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The selected points represent different transmission characteristics. The output transmission from ADR was analyzed for every selected point at resonant peak (1.523 µm) by considering the bar chart of the FWHM, the 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.

Fig. 5
figure 5

The calculated optical parameters for each selected point. a The FWHM at resonant peak (1.52. µm). b The Q-factor, c The OBRR and d the crosstalk

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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.