1 Introduction

The on-chip interconnection network or Network-on-Chip (NoC) is exploiting photonic solution as opposed to its electrical counterparts, especially in the multicore scenario where, in order to provide energy efficient performance with low latency and high throughput [1,2,3,4,5], one of the most promising technology is Silicon-On-Insulator (SOI) platform due to its compatibility with CMOS technology and high refractive index contrast [5,6,7,8].

An important element of integrated optical device is microring resonator which can be used as filter, modulator etc. Microring can also be used to create a NoC as proposed in [1] avoiding any waveguide crossings thus reducing crosstalk. Many methods have been proposed to improve the performances of crossings based on multimode interference [9], mode expansion [10, 11], polymer waveguide bridges [12], Bloch waves [13] etc. [14]. Although the amount of crosstalk of single waveguide crossing can be reduced to as low as − 20 dB [15], it can be substantially large in an optical NoC where multiple waveguide intersections are needed, and thus seriously degrade the signal quality. The use of Multi Micro Ring (MMR) structure in the all-optical NoC reported in [1] is a feasible solution to this problem which avoids the waveguide crossings completely.

With the increasing complexity of the device designs and increasing density of the functions integrated onto a single device, the characterization of the building blocks of all-optical Network-on-Chip (NoC) are becoming important to highlight the discrepancies between the fabrication outcome and the intended design [16]. The importance of accurate characterization lies in the fact that it allows to understand the tolerances of the fabrication process and the results found during the characterization stage can be used to mitigate the effect of fabrication inaccuracies by optimizing the design afterwards [17].

This paper presents an unique add-drop filter model to characterize the microring coupling efficiency by way of curve fitting. The novelty of this model lies in the fact that, unlike other models, it measures the effective coupling efficiency very precisely considering both the waveguide propagation loss and backscattering effect due to sidewall roughness.

Comparing the experimental curve with the theoretical fit obtained from the model the effective coupling efficiency for different coupling angles from five differently positioned chips of two wafers are measured to verify the reliability of the chosen coupling angle for the required effective coupling efficiency in Multi Micro Ring (MMR) NoC reported in [1, 5]. Then the measured values are compared with the results obtained in the design phase using Finite Difference Time Domain (FDTD) simulation, to estimate the fabrication inaccuracies.

Little et al. developed an add-drop filter model in [18] considering the backscattering effect, but the authors didn’t consider the waveguide propagation loss, whereas this model considers both the loss and backscattering effect due to sidewall roughness.

2 Microring coupling efficiency characterization

2.1 Add-Drop filter model

The all-pass filter model described in [19] is extended here for add-drop filter configuration shown in Fig. 1. To account for the backscattering, this model considers bidirectional propagation both in the bus and the ring as opposed to the usual model of [20, 21] where only unidirectional mode of the resonator is considered.

Such a system is described by an \(8\times 8\) scattering matrix:

$$\begin{aligned} \begin{bmatrix} A_1^- \\ A_2^- \\ A_3^- \\ A_4^- \\ A_5^- \\ A_6^- \\ A_7^- \\ A_8^- \end{bmatrix} = \begin{bmatrix} 0&0&t_1&i \kappa _1&0&0&0&0 \\ 0&0&i \kappa _1&t_1&0&0&0&0 \\ t_1&i \kappa _1&0&0&0&0&0&0 \\ i \kappa _1&t_1&0&0&0&0&0&0 \\ 0&0&0&0&0&0&t_2&i \kappa _2 \\ 0&0&0&0&0&0&i \kappa _2&t_2 \\ 0&0&0&0&t_2&i \kappa _2&0&0 \\ 0&0&0&0&i \kappa _2&t_2&0&0 \end{bmatrix} \begin{bmatrix} A_1^+ \\ A_2^+ \\ A_3^+ \\ A_4^+ \\ A_5^+ \\ A_6^+ \\ A_7^+ \\ A_8^+ \end{bmatrix}. \end{aligned}$$
(1)
Fig. 1
figure 1

Add-drop filter model

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Here, \(A_i^+(i=1,2,\ldots ,8)\) and \(A_j^-(j=1,2,\ldots ,8)\) are the eight inputs and eight outputs, respectively. The real coefficients \(\kappa _1\), \(\kappa _2\) and \(t_1\), \(t_2\) denotes the field coupling from bus to ring and field transmission respectively. The relation between these coefficients are \(t_1^2+\kappa _1^2=1\) and \(t_2^2+\kappa _2^2=1\) for lossless coupling. We have assumed here that both the directional couplers are identical, so \(\kappa _1= \kappa _2\) and \(t_1=t_2\). The ring feedback equations are expressed as follows:

$$\begin{aligned} A_2^+=aA_8^- {\text {e}}^{(-i\delta )}+R_{\text {c}} A_2^-;\nonumber \\ A_4^+=aA_6^- {\text {e}}^{(-i\delta )}+R_{\text {c}} A_4^-; \nonumber \\ A_6^+=aA_4^- {\text {e}}^{(-i\delta )}+R_{\text {c}} A_6^-;\nonumber \\ A_8^+=aA_2^- {\text {e}}^{(-i\delta )}+R_{\text {c}} A_8^-; \end{aligned}$$
(2)

where \(\delta\) and a are the accumulation of phase and field loss over half round trip, respectively, and they are expressed as \(\delta = \text {optical length of half round} \times \frac{2 \pi }{\lambda }=\pi r n_{\text {eff}} \times \frac{2 \pi }{\lambda }\) and \(a={\text {e}}^{-\pi r \frac{\alpha }{2}}\), where r is the ring radius, \(\alpha\) is the loss in dB/cm, \(n_{\text {eff}}\) is the effective refractive index. The reflection coefficient \(R_{\text {c}}\) is expressed as \(R_{\text {c}}=R{\text {e}}^{i\gamma }\), where R is the reflectivity or backscattering and \(\gamma\) is the phase of \(R_{\text {c}}\). The detail about these parameters can be found in [18, 19]

The transfer matrix of the add-drop filter obtained by solving the scattering matrix of Eq. (1) and ring feedback equations of Eq. (2) for \(A_3^+=0\) and \(A_7^+=0\) is as follows:

$$\begin{aligned} \begin{bmatrix} A_3^- \\ A_7^- \end{bmatrix} = \begin{bmatrix} t_1- \frac{\kappa _1^2 F_8}{1- F_8 t_1}&\frac{i \kappa _1 F_7}{1- F_8 t_1} \\ \frac{i \kappa _2 F_7^\prime }{1- F_8^\prime t_2}&t_2- \frac{\kappa _2^2 F_8^\prime }{1- F_8^\prime t_2} \end{bmatrix} \begin{bmatrix} A_1^+ \\ A_5^+ \end{bmatrix}. \end{aligned}$$
(3)

In this equation,

$$\begin{aligned} F_7=F_4+\frac{F_5 F_2}{F_3 }; F_8= \frac{F_5 F_1}{F_3}+F_6; \nonumber \\ F_7^\prime =F_4^\prime +\frac{F_5^\prime F_2^\prime }{F_3^\prime }; F_8^\prime = \frac{F_5^\prime F_1^\prime }{F_3^\prime }+F_6^\prime . \end{aligned}$$
(4)

where

$$\begin{aligned} \begin{array}{lc} F_1=a{\text {e}}^{-i\delta }+\frac{aR_{\text {c}}^2t_1t_2{\text {e}}^{-i\delta }}{F};\\ F_2=R_{\text {c}}^2 t_2i\kappa _2+\frac{a^2 R_{\text {c}}^2 t_1 t_2^2 {\text {e}}^{-2i\delta }i\kappa _2}{F};\\ F_3=1-R_{\text {c}}^2 t_2^2-\frac{a^2 R_{\text {c}}^2 t_1 t_2^3 {\text {e}}^{-2i\delta }}{F};\\ F_4=a{\text {e}}^{-i\delta } i\kappa _2+\frac{a R_{\text {c}}^2 t_1 t_2 {\text {e}}^{-i\delta } i\kappa _2}{F};\\ F_5= a{\text {e}}^{-i\delta } t_2+ \frac{aR_{\text {c}}^2 t_1 t_2^2 {\text {e}}^{-i\delta }}{F};\\ F_6=\frac{R_{\text {c}}^2 t_1}{F};\\ F=1-a^2t_1t_2 {\text {e}}^{-2i\delta }. \end{array} \end{aligned}$$

\(F_1^\prime\) to \(F_6^\prime\) can be found from \(F_1\) to \(F_6\) by replacing \(t_1\) by \(t_2\) and \(t_2\) by \(t_1\) and by replacing \(\kappa _1\) by \(\kappa _2\) and \(\kappa _2\) by \(\kappa _1\).

Through fitting the drop port simulated spectrum (obtained from the above model) with the drop port experimental spectrum the coupling efficiency between the bus and ring waveguide of the microring can be measured.

2.2 Position of the characterized chips

Five differently positioned chips are selected from two different wafers for characterization. One center chip and four corner chips are chosen as shown in Fig. 2. The measurements and the analysis of all these differently positioned chips will allow to find out the wafer scale uniformity of ring coupling efficiency across two wafers. In a multiproject wafer full-active run the chips were manufactured by IME through CMC microsystem.

Fig. 2
figure 2

The position of the characterized chips on the wafer

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2.3 Device architecture and design

Fig. 3
figure 3

An add-drop filter having bent bus waveguides

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Figure 3 shows an add-drop filter where the coupling region is extended following the ring shape i.e. unlike usual straight bus waveguide, here the bus waveguide has a bending. This bending creates an arc that subtends an angle \(\theta\) at the center of the ring. The more the arc length increases i.e. the angle increases, the more the coupling region will be extended which will result in a higher coupling efficiency. The angle \(\theta\) is called as ‘coupling angle’. To test the model, multiple add-drop rings with different coupling angles have been realized and characterized.

Fig. 4
figure 4

Mask image of seven add-drop filters having bent bus waveguides of different angles

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Fig. 5
figure 5

Mask image of an add-drop filter showing the position of the ports

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The fabricated PIC that contains the Multi Micro Ming (MMR) architecture based Network-on-Chip (NoC) has seven add-drop filters of different coupling angles in order to characterize the effective coupling efficiency after the fabrication. The coupling angles of these seven add-drop filters are 0\(^{\circ }\), 4\(^{\circ }\), 8\(^{\circ }\), 16\(^{\circ }\), 24\(^{\circ }\), 32\(^{\circ }\), 40\(^{\circ }\) from right to left respectively as shown in the mask image of Fig. 4. Figure 5 shows one add-drop filter from the mask separately where the position of the ports of the add-drop filter are specified.

The cross-section of the bus and ring in the coupling region are schematically shown in Fig. 6. The cross-section of the bus consists of a single-mode channel waveguide (with embedded core) having a width of 460 nm and a height of 220 nm, whereas the cross-section of the ring consists of a single-mode half-rib waveguide having a width of 480 nm and height of 220 nm. A 90 nm thick internal slab of doped silicon is placed 1 \(\upmu\)m away from the microring waveguide to tune the rings independently by exploiting the Joule effect through electrical current injection in this doped region.

Fig. 6
figure 6

Device cross-section in the coupling region

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Fig. 7
figure 7

Experimental setup

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2.4 Experimental measurement

The schematic diagram of the experimental setup is shown in Fig. 7. Spectral scans are taken using an external cavity Tunable Laser (TL) by generating a continuous wave signal having 0 dBm power in the range of 1500–1600 nm. A fiber Polarization Controller (PC) maximizes the coupling efficiency between the optical fiber and the input grating coupler. The output spectrum is collected with a spectral resolution of 1 picometer from the ‘drop’ port grating coupler using a synchronized power meter. A temperature controller is used to maintain a constant temperature of 25 \(^{\circ }\)C.

2.5 Characterization of microring coupling efficiency by curve fitting using the add-drop filter model

In this section, how to step by step fit the simulated curve obtained from the described add-drop filter model with the experimental curve from the drop port will be shown to measure the actual coupling efficiency. To illustrate this an add-drop filter having coupling angle of 32\(^{\circ }\) is chosen from of the top corner chip of wafer 2.

Since our model use both the waveguide propagation loss and backscattering values for accurately characterizing the coupling efficiency, we need to measure these values beforehand. The different ways to measure the waveguide loss are compared in [22] and the method based on the all-pass filter model [19] is shown to be the most simple and accurate, because this method is independent of input coupling losses and estimation of facet reflectivity is not required here. So, using the model of [19], the waveguide propagation loss and backscattering values from all the differently positioned chips of both the wafers are measured and reported in [23]. For the top corner chip of wafer 2, the measured value of loss \((\alpha ) = 2.55\) dB/cm and backscattering \((R) = 0.00425\).

Fig. 8
figure 8

A selected resonance near 1550 nm from the measured drop port experimental curve for 32\(^{\circ }\) coupling angle from the wafer 2 top corner chip

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Fig. 9
figure 9

The measured drop port experimental curve for 32\(^{\circ }\) coupling angle from wafer 2 top corner chip and the theoretical fit obtained from the add-drop model using loss \((\alpha ) = 2.55\) dB/cm, backscattering \((R) = 0.00425\) and power coupling efficiency \((K) = 8.38\%\)

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The step by step fitting is as follows:

  1. Step 1:

    A resonance near 1550 nm is selected from the experimental spectrum of the drop port which is shown in Fig. 8.

  2. Step 2:

    At first, both the loss \((\alpha )\) and backscattering (R) are considered as zero and by varying power coupling efficiency K \((=\kappa ^2)\) the simulated curve from the add-drop model is fitted with experimental spectrum.

  3. Step 3:

    After that the actual loss value for this chip i.e. loss \((\alpha ) = 2.55\) dB/cm is considered keeping the backscattering (R) value to be still zero. Loss will decrease the peak of the simulated curve. If we normalize it to compare both the curves, bandwidth of the simulated curve will change. So, to fit with the experimental curve we need to decrease the value of K.

  4. Step 4:

    Finally the actual value of both the loss and backscattering for this chip i.e. loss \((\alpha ) = 2.55\) dB/cm and backscattering \((R) = 0.00425\) are considered. Backscattering will again decrease the peak of the simulated curve. After the normalization the bandwidth of the simulated curve will change and we need to decrease the value of K to fit with the experimental curve. For \((K) = 8.38\%\) both the curves match properly as shown in Fig. 9.

So, in wafer 2 top corner chip the value of coupling efficiency for 32\(^{\circ }\) coupling angle is found to be \(8.38\%\).

Fig. 10
figure 10

The measured drop port experimental curve for 24\(^{\circ }\) coupling angle from wafer 2 center chip and the theoretical fit obtained from the add-drop model using loss \((\alpha ) = 3.1\) dB/cm, backscattering \((R) = 0.00505\) and power coupling efficiency \((K) = 4.84\%\)

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Figure 10 shows another example of fitting for 24\(^{\circ }\) coupling angle from wafer 2 center chip. For this chip the loss and backscattering were found to be 3.1 dB/cm and 0.00505 respectively as reported in [23]. From Fig. 10 it can be seen that for coupling efficiency of \(4.84\%\) the simulated and experimental curves match properly.

Fig. 11
figure 11

Variation of the coupling efficiency values across two wafers measured from five differently positioned chips for coupling angle of 0\(^{\circ }\), 4\(^{\circ }\), 8\(^{\circ }\), 16\(^{\circ }\), 24\(^{\circ }\), 32\(^{\circ }\), 40\(^{\circ }\). For each dataset the upper line denotes the maximum, lower line denotes the minimum, and middle dotted line denotes the average coupling efficiency

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In both the examples, the backscattering values are small (less than 0.005) and can be negligible. However, if the value is higher it can reduce the effective coupling efficiency significantly. For example, in the first case i.e. for 24\(^{\circ }\) coupling angle of wafer 2 top corner chip, if the backscattering value would be 0.0101 instead of 0.00425, the effective coupling efficiency would reduce to 7.802% from 8.38\(\%\).

By following the above mentioned approach the experimentally measured spectra from the drop ports of all the add-drop filters (having different coupling angles) of five differently positioned chips fabricated from two wafers are fitted with the simulated curve, and thus their effective coupling efficiencies are measured and shown in Fig. 11. This figure also shows the coupling efficiency variations across two wafers i.e. wafer 1 (W1) and wafer 2 (W2).

2.6 Comparison with the FDTD simulation results

Table 1 shows the comparison between the coupling efficiency values found using Finite-Difference Time-Domain (FDTD) simulation and from the add-drop filter model. For both the wafers the average values of coupling efficiencies from five different positioned chips for different coupling angles are shown. The FDTD simulations have been done for a directional coupler having the same cross section of the coupling region present on the fabricated PIC i.e. the ring width of 480 nm and bus width of 460 nm with a bus to ring gap of 200 nm.

Table 1 Comparison between the coupling efficiency values found using FDTD simulation and using add-drop model
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In Fig. 12 the plot of the discrepancy between these two results with the coupling angle for both the wafers are shown. From this figure, we can see that for both the wafers the discrepancies from the FDTD simulation results at different coupling angles are almost identical. So, we can say that both the wafers have uniform characteristics in terms of their effective coupling efficiencies due to the uniform fabrication outcomes. Also it can be noticed from the Fig. 12 that with the increasing coupling angle the discrepancy in both the wafer increases. This effect is related to the increment of the coupling length and longer couplers are more sensitive to the fabrication inaccuracy.

Fig. 12
figure 12

The plot of discrepancy (between the results of K found from the FDTD simulation and from the add-drop filter model) with the coupling angle in wafer 1 and wafer 2

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The required coupling efficiency for the proposed NoC reported in [1, 5] is 7.5% to guarantee the transmission BW for a 10 Gbs non return-to-zero on-off keyed signal. As can be seen from the FDTD simulation results (reported in Table 1), 24\(^{\circ }\) coupling angle could be exploited to achieve 7.5% coupling efficiency. But 32\(^{\circ }\) coupling angle was chosen instead to counteract the effect of the fabrication inaccuracies. We can see from Table 1 that the average effective coupling efficiency values in wafer 1 and wafer 2 for 32\(^{\circ }\) coupling angle are 7.61 and 7.65, respectively, instead of 10%. So, the result from the add-drop model verifies that 32\(^{\circ }\) coupling angle was appropriately chosen in [1, 5] to provide the required coupling efficiency for the MMR architecture based NoC.

The results of the effective coupling efficiency values for different coupling angles found from the developed add-drop model will serve as a very good design tool in future, as this model provide the effective coupling efficiency value very precisely considering not only the effect of waveguide propagation loss but also the backscattering effect.

On the other hand, the FDTD simulation, although being a valuable and flexible design tool, is very time and resource consuming. For this reason just the coupling region is typically simulated (not the entire add-drop microring). Also the fabrication process inaccuracies can not be considered.

The results from the developed model give an idea of the discrepancy range of the coupling efficiency values due to the fabrication inaccuracies which will help the designer to choose the appropriate coupling angle to achieve desired coupling efficiency.

3 Conclusion

An add-drop filter model in order to precisely characterize the effective coupling efficiency considering both the loss and backscattering effects is presented in this paper. The effective coupling efficiency values found for different coupling angles (from five differently positioned chips of two wafers) using the developed model will serve as a very good design tool in future to streamline the design of microring based structures on the same technological process. The results verified that 32\(^{\circ }\) coupling angle is appropriately chosen in order to get minimum 7.5\(\%\) effective coupling efficiency for the Multi Micro Ring (MMR) architecture based NoC reported in [1, 5]. Furthermore, the results are compared with the Finite-Difference Time-Domain (FDTD) simulation results which allows to measure the discrepancies from the design values due to the fabrication inaccuracies. It is expectedly found that the discrepancies increase with larger coupling angle, since longer couplers are more sensitive to the fabrication inaccuracies. It is also found that the discrepancies at different coupling angles are almost identical for both the wafers due to the uniform fabrication outcomes.