II. PROPOSED WAVEGUIDE STRUCTURE

To design a waveguide structure that can support both microwave and optical signals, we investigated conventional microwave planar waveguides such as embedded symmetric striplines, microstrip, slotlines, and CPWs. Considering fabrication limitations, our initial simulations using HFSS software ensured the feasibility of the CPW structure with a characteristic impedance of 50 Ω [Reference Banan, Hai, Berini and Liboiron-Ladouceur11]. In this regards, the signal trace must be made thicker than the skin depth to minimize the microwave loss. Furthermore, the ground traces supporting the optical mode propagation must be thin to lower the optical propagation loss over the metal structures. The schematic cross-section of the structure is depicted in Fig. 1(a). The waveguide structure comprises three metal strips of thicknesses t ₁ and t ₂ with widths w _g and w _s , respectively. A distance s separates the metal strips. Metal strips with permittivities of ε₁ for the thin metal strips and ε₂ for the thick signal strip are surrounded by a finite homogeneous dielectric of thickness t ₃ with permittivity ε₃. The bottom dielectric has a thickness t ₄ with permittivity ε₄. This structure forms an embedded FGCPW capable of supporting quasi transverse electromagnetic (TEM) waves at microwave frequencies [Reference Wen13]. Each ground strip is designed to be thin enough (a few tens of nanometer) to support long range surface plasmon polaritons (LRSPPs) at optical frequencies [Reference Berini14].

Fig. 1. (a) Schematic cross-section of the proposed structure excluding the ground pads with connections as well as optical tapers. (b) Microscope image of a fabricated structure at one end of the transmission line.

The microwave mode is excited and captured via ground–signal–ground (GSG) radio frequency (RF) probes from the top. To provide contact between the probes and the FGCPW transmission line, two ground pads with the same thickness as the signal strip has (t ₁ + t ₂) are designed (Fig. 1(b)). The two ground pads are connected to the corresponding ground strips via five parallel connections of 5 µm width and t ₁ thickness. The use of these narrower connections instead of a single wide one is motivated by a need to allow propagation of the LRSPP along the ground strips without minimum perturbation. The length of these pad connections is adjusted such that the pitch is compatible to the GSG RF probes (150 µm). The LRSPP modes are excited and captured through edge-coupling technique via single mode fibers. To increase the optical coupling efficiency between the fibers and the ground strips, optical tapers are designed. A 2 µm gap between optical tapers and ground strips is inserted to provide electrical isolation of the ground strips.

According to the experimental optical performance of the waveguide structure detailed in [Reference Banan, Hai, Berini and Liboiron-Ladouceur11], an optical attenuation of approximately 1.87 dB/mm and coupling loss of 3.9 dB/facet were measured for a 25 nm thick and 30 µm wide waveguide. Additionally, the optical attenuation might be further decreased mainly by reducing the thickness of the propagating waveguides, however it comes at the cost of larger optical mode profile. The coupling loss can be improved by redesigning and optimizing the taper section.

This work focuses on the investigation of the microwave performance of the proposed waveguide structure. We provide more insight on how the proposed FGCPW dimensions affect transmission and reflection responses. The discussion presented in this research work allows an optimization of the structure for enhanced RF performance beyond the results obtained in [Reference Banan, Hai, Berini and Liboiron-Ladouceur11] of 40 GHz bandwidth for a 3 mm long waveguide.

Assuming that the propagation mode in the CPW is quasi-static (i.e., a pure TEM mode), and using conformal mapping technique, the FGCPW can be characterized to derive the analytical expressions of the line capacitance per unit length [Reference Chen and Chou15, Reference Ghione and Naldi16]. Indeed, conformal mapping transforms the geometry of the CPW into a parallel plate capacitor whose capacitance is a linear function of dielectric constant. In this analysis, the metal structure is considered to be a PEC (perfect electrical conductor, ε₁ = ε₂ = ∞) embedded in Cytop polymer, i.e., a fluoroscope-polymer, with a dielectric constant of ε₃ = 2 [17], which is spin coated on a silicon wafer with a dielectric constant of ε₄ = 11.7. Using this technique, the total line capacitance of the CPW can be obtained whereby the characteristic impedance Z _c can also be determined by

(1) $$Z_c = \displaystyle{1 \over {C_{tot} v_p}}, $$

where v _p is the phase velocity given by $v_p = \left( {c/(\sqrt {{\rm \varepsilon} _{eff}} )} \right)$ . Here, c is the speed of light, and ε _eff is the effective dielectric constant which can be obtained from ε _eff  = (C _tot (ε _r )/C _tot (ε _r  = 1)) where ε _r denotes the dielectric constant of the medium between the parallel plates of the capacitor.

Considering that the dielectric–air boundaries (magnetic walls) are along the electric field lines, C _tot can be expressed as the sum of all partial capacitances without affecting the field [Reference Chen and Chou15, Reference Ghione and Naldi16, Reference Hoffmann18]. Therefore, C _tot  = C ₀ + C ₁ + C ₂ + C ₃, where C ₀ represents the air-capacitance in the absence of any dielectric. C ₁ is the capacitance of the top cladding (Cytop polymer) with a thickness of h ₁ = t ₃/2 and an equivalent dielectric constant of (ε₃ − 1) which is the difference between the cladding and the air dielectric constants. Similarly, C ₂ expresses the capacitance of the bottom cladding layer (Cytop polymer) that is beneath the signal and ground traces with a thickness of h ₂ = t ₃/2 and an equivalent dielectric constant of (ε₃ − ε₄). Finally, C ₃ denotes the capacitance of the bottom substrate (silicon) layer with a thickness of h ₃ = t ₄ and an equivalent dielectric constant of (ε₄ − 1) [Reference Chen and Chou15, Reference Ghione and Naldi16].

These partial capacitances can be determined by using the conformal mapping technique which tailors a separation into two parallel plates by using complete elliptical integrals of the first kind, K(k) [Reference Chen and Chou15]. Thus, C _tot can be expressed as

(2) $$\eqalign{C_{tot} & = 4{\rm \varepsilon} _0 \displaystyle{{K(k_0^{\prime} )} \over {K(k_0 )}} + 2{\rm \varepsilon} _0 ({\rm \varepsilon} _3 - 1)\displaystyle{{K(k_1^{\prime} )} \over {K(k_1 )}} \cr &\quad+ 2{\rm \varepsilon} _0 ({\rm \varepsilon} _3 - {\rm \varepsilon} _4 )\displaystyle{{K(k_2^{\prime} )} \over {K(k_2 )}} + 2{\rm \varepsilon} _0 ({\rm \varepsilon} _4 - 1)\displaystyle{{K(k_3^{\prime} )} \over {K(k_3 )}},} $$

where the four terms on the right side of equation (2) represent C ₀, C ₁, C ₂, and C ₃, which can be determined using the complete elliptical integrals of the first kind as the following

(3) $$\eqalign{k_0 = & \displaystyle{{x_3} \over {x_2}} \sqrt {\displaystyle{{(x_2^2 - x_1^2 )} \over {(x_3^2 - x_1^2 )}}}, \cr k_i = & \displaystyle{{\sinh \left( {\displaystyle{{\pi x_3} \over {2h_i}}} \right)} \over {\sinh \left( {\displaystyle{{\pi x_2} \over {2h_i}}} \right)}}\sqrt {\displaystyle{{\mathop {\sinh} \nolimits^2 \left( {\displaystyle{{\pi x_2} \over {2h_i}}} \right) - \mathop {\sinh} \nolimits^2 \left( {\displaystyle{{\pi x_1} \over {2h_i}}} \right)} \over {\mathop {\sinh} \nolimits^2 \left( {\displaystyle{{\pi x_3} \over {2h_i}}} \right) - \mathop {\sinh} \nolimits^2 \left( {\displaystyle{{\pi x_1} \over {2h_i}}} \right)}}},} $$

where $k^{\prime} = \sqrt {1 - k^2}, {\kern 1pt} \;x_1 = (w_s /2),{\kern 1pt} \;x_2 = x_1 + s,{\kern 1pt} \;x_3 = x_2 + w_g, $ and i = 1, 2, and 3. Using the obtained transmission line capacitance per unit length (C _tot ), the effective dielectric constant can be calculated by ε _eff  = 1 + Q ₁ (ε₃ − 1) + Q ₂ (ε₃ − ε₄), where, Q ₁ and Q ₂ can be given by $Q_1 = \displaystyle{1 \over 2}\displaystyle{{K(k_0 )} \over {K(k_0^{\prime} )}}\left( {\displaystyle{{K(k_3^{\prime} )} \over {K(k_3 )}} + \displaystyle{{K(k_1^{\prime} )} \over {K(k_1 )}}} \right),\;Q_2 = \displaystyle{1 \over 2}\displaystyle{{K(k_0 )} \over {K(k_0^{\prime} )}}\displaystyle{{K(k_2^{\prime} )} \over {K(k_2 )}}$ . Therefore, the characteristic impedance of the CPW can be determined by

(4) $$Z_c = \displaystyle{{30\pi} \over {\sqrt {{\rm \varepsilon} _{eff}}}} \displaystyle{{K(k_0 )} \over {K(k_0^{\prime} )}}.$$

Based on the above analysis, the impact of the separation distance, and the width of the signal and ground traces on the FGCPW line characteristic impedance, Z _c , is investigated. As illustrated in Fig. 2(a), at 40 GHz for a fixed gap size of s = 15 μm, increasing the signal strips width w _s decreases the impedance, while increasing the ground strip width w _g has relatively small impact on the impedance Z _c . Furthermore, Fig. 2(b) shows that an increase in the signal separation s increases the characteristic impedance.

Fig. 2. Characteristic impedance of the FGCPW at 40 GHz as a function of signal strip width (w _s ) for (a) $s = 15{\kern 1pt} \;{\rm \mu m}$ and different ground strip widths, and (b) $w_g = 30\;{\kern 1pt} {\rm \mu m}$ and different gap sizes.

To characterize the proposed waveguide structure with its data transmission, high-speed test and measurement equipments are exploited and a transmission line impedance of 50 Ω is set as a design requirement. In such context, as shown in Fig. 2(b), the optimal signal strip dimension for a 50 Ω FGCPW line is w _s  = 50 μm. Additionally, according to Fig. 2(a), the requirement of 50 Ω for a signal strip width of $50{\kern 1pt} \;{\rm \mu m}$ is met if the values for the two other parameters w _g and s are 30 μm and 15 μm, respectively. The proposed structure is fabricated through a custom fabrication process. The complete fabrication process is detailed in [Reference Banan, Tait, Liboiron-Ladouceur and Berini19].

III. CHARACTERIZATION OF THE FGCPW

After the design and fabrication, the transmission response of the waveguide structure is characterized. Figure 3 illustrates the schematic of the electrical S-parameter measurement set-up using a 50 MHz–40 GHz vector network analyzer (VNA). An electrical input power of −10 dBm is provided by the VNA which is used to excite and capture the quasi-TEM mode of the proposed FGCPW. The waveguide is excited using an electrical GSG probe at its input. The transmitted signal is collected with another GSG probe at the output of the waveguide structure. Both probes are controlled with three-axis micropositioners. Prior to the measurements, the set-up is calibrated using a calibration substrate for short, open, load (50 Ω) as well as through, i.e., standard short, open, load, and through (SOLT) calibration, to normalize the impact of the 40 GHz RF probes and cables on the measurements.

Fig. 3. Schematic of the experimental set-up for the S-parameter measurements.

Figure 4 presents the S-parameter magnitude results for various 3 mm long waveguides with a 50 µm wide signal strip (w _s ), three different ground strip widths (i.e., w _g of 5, 20, and 30 µm) and seven different separation distances (i.e., s of 2, 5, 8, 10, 12, 15, and 18 µm). As shown in Fig. 4, all structures show a transmission (S ₂₁) drop and high reflections (S ₁₁) at low frequencies up to approximately 1 GHz. Afterwards, the transmission improves up to approximately 12 GHz, and then decreases as frequency increases. The transmission drop at low frequencies is largely due to reflection caused by impedance mismatch in the structure. This will be discussed in Section IV. The measurement shows that the wider the ground strips w _g and separation distance s the lower the transmission loss. The results confirmed that a wider ground strip leads to lower reflection due to improved impedance match (i.e., see Fig. 2), whereas, a larger separation distance leads to higher reflections.

Fig. 4. Measured transmission (S ₂₁) and reflection (S ₁₁) of the 3 mm long waveguides with w _s  = 50 μm for various separation distances s for w _g  = 30 μm (in (a), and (d)), w _g  = 20 μm (in (b), and (e)), and w _g  = 5 μm (in (c), and (f)), over the frequency range of 50 MHz to 40 GHz.

Additionally, Fig. 5 shows the measured transmission (S ₂₁) phase profile of the 3 mm long waveguides with (w _s ) = 50 μm (w _g ) = 30 μm and various separation distances s (in Fig. 5(a)), and for different lengths while s = 15 μm (in Fig. 5(b)). As illustrated, the phase response of the line is linear suggesting that the line is not dispersive.

Fig. 5. Measured transmission (S ₂₁) phase profile of the waveguides with w _s  = 50 μm, w _g  = 30 μm; (a) for the length of 3 mm and various separation distances s, and (b) for various lengths and a fixed separation distance s = 15 μm, over the frequency range of 50 MHz to 40 GHz.

IV. CIRCUIT MODEL AND TRANSMISSION PARAMETERS EXTRACTION

Transmission line models are necessary to predict interconnect properties such as impedance and propagation constant. In this work, the transmission line is modeled on the basis of its lumped circuit elements which requires careful consideration due to its operation frequency range from direct current (DC) to a few tens of GHz. Additionally such a circuit model should also reflect the effect of the parasitic elements originating from the pads and their connections to model the structure as precise as possible. The lumped circuit parameters of this accurate equivalent circuit model for the fabricated FGCPW are obtained using Advanced Design System (ADS) commercial software. In order to improve the accuracy of the model, the parasitic impact of the two electrical pads at both ends and their connections is modeled and compared with its conventional circuit model without these imposed effects, i.e., see Fig. 1(b). Indeed, the pads and their connections pose additional RLC parasitic elements to the FGCPW transmission line which degrade the performance of the device in terms of transmission (S ₂₁), particularly at low frequencies (i.e., below 10 GHz). This issue can be mainly associated with the sheet resistance, inductance, and capacitance of the pads and their connections.

Figure 6 demonstrates the accurate equivalent circuit model with the parasitic elements of the pads. To extract the values of the elements in the circuit model shown in Fig. 6, the measured S ₂₁ data are imported into ADS and the difference between the transmissions of the circuit model and the measured S ₂₁ is minimized. This process is done by applying the “Random” and “Gradient” optimization methods in the OPTIM engine of ADS. Table 1 summarizes the results for different waveguide lengths with a signal width (w _s ) of 50 µm, ground strips (w _g ) of 30 µm width, and a 15 µm gap (s).

Fig. 6. Equivalent lumped circuit model of the FGCPW transmission line in which the effects of the pads and their connections are considered.

Table 1. Extracted lumped circuit element values from the measured S-parameters based on the distributed circuit model presented in Fig. 6 for different lengths of the transmission line with w _s  = 50 μm, w _g  = 30 μm, and s = 15 μm.

As expected, increasing the length results in an increase in the transmission line's resistance R and inductance L. It also reveals that the line's conductance G (or dielectric loss) is negligible. Additionally, the line capacitance C of the proposed FGCPW is approximately two orders of magnitude less than conventional CPW structures on silicon substrate. It should be underlined that the small divergence in the parasitic parameters of the TX and RX pads and their connections for different lengths is mainly associated with the optimization process of the ADS software to find the minimum difference between the transmission of the circuit model and the measured S ₂₁.

Figure 7 shows the measured transmission S ₂₁ (solid line) for the four devices length investigated in Table 1. Overlaid are the calculated transmission responses obtained from the extracted circuit element values based on both the conventional circuit model from the Telegrapher's equations [Reference Eisenstadt and Eo20] (black dash line) where the pads are not considered, and the customized accurate model (red dash-dotted line) which includes the pads parasitic circuit elements. The circuit model results for four different lengths are plotted in two figures for proper scaling of the magnitude.

Fig. 7. Measured S ₂₁ parameter (solid line) along with the modeled transmission responses without and with considering the parasitic elements of the pads and connections, for the conventional circuit model (dashed line) and the accurate circuit models (dash-dot), respectively. The waveguides are 3, 5, 8, and 13 mm long with 50 µm signal strip width, 30 µm ground strip width, and 15 µm separation distance over the frequency range of 50 MHz to 40 GHz.

As can be inferred from the figure, the conventional and custom circuit models for different lengths provide approximately the same results for frequencies above 10 GHz, while they differ at lower frequencies due to the effect of the parasitic elements originating from the pads and their connections, in which they act as parallel lossy LC oscillator (i.e., parallel RLC circuit). This can be confirmed by calculating the equivalent impedance (Z _in ) of the lumped circuit model as a function of frequency, in which the parallel RLC components cause high impedance mismatch at the input at low frequencies (below 10 GHz) resulting in high reflections. Therefore, the impact of the pads at lower frequency is significant and must be taken into account in order to have an optimized structure over the entire frequency range.

Figure 8 shows that the 13 mm long transmission line exhibits a resonance at 38 GHz which suggests the formation of a standing wave within the waveguide structure. Indeed, an impedance mismatch between the transmission line and the load will lead to reflections. If the incident signal is a continuous AC (alternating current) waveform, these reflections will mix with more of the oncoming incident waveform to produce stationary waveforms (i.e., standing waves) [Reference Connor21]. Measurement results depicted in this figure show that the resonance becomes weaker with smaller separation distances.

Fig. 8. Measured transmission of 13 mm long lines with w _s of 50 µm and w _g of 30 µm for different separation distances.

V. ENVISIONED APPLICATIONS

The hybrid structure presented in this work allows for data transmission in both the optical and electrical domains. One envisioned application is short interconnects in advanced packaging such as interposers [Reference Lee, Lee, Kim, Park and Sim22]. To implement such a hybrid transmission line, the cross-sectional dimensions of the structure can be designed (referring to Section II) to satisfy the impedance matching condition with any target electronic interface. Also, the electrical interconnects are implemented using fine pitch micro bumps (20 µm, [Reference Banijamali, Ramalingam, Liu and Kim23]). Thus, depending on the available fabrication technology, the ground pads as well as the signal connection (or pad) can be designed as small as possible to minimize their parasitic effects (referring to Section IV), and that they are compatible with micro bumps for the electrical interface. Additionally for efficiency, the laser source must be ideally edge coupled to the ground strips of the waveguide. As such, edge emitting semiconductor lasers are a good fit.

Bandwidth and reach of the discussed transmission line is comparable to recent works [Reference Mensink, Schinkel, Klumperink, van Tuijl and Nauta24, Reference Nazari and Emami-Neyestanak25]. The bandwidth density of the proposed waveguide structures is 1 Gb/s/μm [see reference Reference Banan, Hai, Berini and Liboiron-Ladouceur11] compared with 1.16 Gb/s/μm in [Reference Mensink, Schinkel, Klumperink, van Tuijl and Nauta24] and 1.5 Gb/s/μm in [Reference Nazari and Emami-Neyestanak25]. In the former case, signal and ground trace widths are 0.54 µm separated by 0.64 µm, supporting 2 Gb/s data rate over a 10 mm length. The bandwidth density of the proposed hybrid structure in this work is limited mainly by the fact that the line's impedance was designed to be 50 Ω for characterization purpose. By relaxing this limitation to a slightly larger impedance (see Fig. 2(a)), the lateral dimension can be reduced with a narrower signal width w _s for a larger bandwidth density.

The proposed structure has inherently unlimited bandwidth density due to the fact that it supports optical signal. Optical bandwidth limitations originate from electrical-to-optical (EO) and optical-to-electrical (OE) converters. An energy-efficient configuration for this structure is to provide on-chip and chip-to-chip interconnects by allocating high data rate channels to the optical section and sending lower data rates (e.g., control signals) over the electrical section. To reach this goal, assuming CMOS electrical transmitter and receiver designs with high impedance front-ends, the signal trace width can be as small as 0.5 µm and the ground traces width can be designed to be as small as 5 µm. The separation distance between the signal and ground traces should be such that the optical modes on the ground strips do not overlap with the signal strip. For the aforementioned ground width, a 10 µm separation must be considered for 20 nm thick ground strips. Structures with thicker strips (35 nm) can have 7 µm separation distances. Essentially, the geometry of the proposed structure can be optimized efficiently for a given characteristics of electrical transmitter and receiver.