A coarse wavelength division multiplexing (CWDM) system requires a wavelength demultiplexing function as well as an optical signal detection function to implement a receiver. Considering that the CWDM system is usually a more economic solution than a dense WDM system, a more economic implementation of its receiver is desirable. As for the wavelength demultiplexing function, many solutions have been studied, which include an arrayed waveguide grating (AWG) [1], a DBR-filter [2], a microring resonator [3], and a grating-assisted coupler [4-8]. Among these, the grating-assisted asymmetric waveguide directional coupler [9] is useful in that the structure is simple compared to other filter structures and the coupling efficiency can be effectively controlled depending on a grating length. In particular, a contra-direction coupler is more advantageous than a co-directional coupler because the former is more fabrication process tolerant: its performance is less sensitive to the grating length variation since there is no re-coupling [10]. As an effort to combine the signal detection function with the wavelength-multiplexing function for a simplified receiver, wavelength-selective photodetectors have been suggested in previous studies, which include a grating-assisted co-directional coupler with a III-V absorption layer [11] and an anti-resonant reflective optical waveguide (ARROW) structure with a quantum well (QW) absorption layer [12]. These structures have rather long coupling lengths (~1 mm), require a complicated fabrication process, and more importantly, are not compatible with Si-photonic devices.

In this paper, we propose a wavelength-selective photodetector based on a grating-assisted contra-directional coupler with a graphene absorption layer. The proposed device consists of a side-polished fiber and a silicon waveguide with a uniform grating and monolayer graphene. In general, an introduction of an absorbing layer in wavelength filters changes the central wavelength and the bandwidth of the filters, and degrades the filter efficiency. Whereas, graphene with monoatomic thickness is a good candidate for absorbing material in waveguide-type filter devices because it has little effect on the properties of the filter in spite of high absorption. Graphene has been widely studied for fast broadband photodetectors because of its wide absorption bandwidth from the visible to the far infrared region and high carrier mobility [13-16]. Recently, there also have been a lot of studies on other optical devices such as polarizers [17, 18], modulators [19, 20], and exotic sensors [21] based on the tunable plasmonic effect in graphene. We have studied theoretically the feasibility of the proposed device and have shown that an array of the proposed devices of different grating periods can be used for receivers in a CWMD system which have combined functions of wavelength demultiplexing and signal detection. According to our investigation, a properly designed device of a 500 µm long grating region has a sufficiently high absorption efficiency (95.1%). We believe that the proposed device can be a compact and economic solution to the CWDM receiver implementation because of the small device size and its simple Si-photonics compatible structure.

Figure 1 shows the proposed wavelength-selective photodetector structure, which consists of a side-polished optical fiber and a silicon waveguide with a uniform grating and monolayer graphene. Figure 1(a) presents an array of photodetectors with various resonant wavelengths for wavelength channel demultiplexing and optical signal detection. Figure 1(b) shows the cross-sectional view of a single device structure. If the silicon waveguide supports only a single guided-mode, two supermodes are formed by the coupling between the modes of the fiber and the silicon waveguide. Due to the large index difference between the fiber’s core and the silicon, those two supermodes usually have very asymmetric field distributions in the vertical direction: one has a field profile that is concentrated in the fiber’s core and the other has a field profile that is concentrated in the silicon. They also show a large difference in their effective indices, which results in huge phase mismatch and thus, hinders direct coupling between them without a grating structure. Therefore, the grating on the silicon waveguide with an appropriate period enables wavelength-selective phase matching and coupling of those two supermodes. That is, the light input to the fiber is coupled to the silicon waveguide at the wavelength where the phase matching condition by the grating is satisfied, and is absorbed by the graphene layer located below the silicon waveguide. In this work, a TE wave, which has a horizontal (z direction) electric field, was considered as the incident light. Since the graphene layer is located just beneath the silicon waveguide, the supermode concentrated in the silicon experiences much higher loss than the one concentrated in the core of the fiber. According to our calculation, which be shown later, the propagation length of the supermode concentrated in the silicon has ~100,000 times shorter than that of the other supermode. This implies that light passing through the core of the fiber is hardly absorbed, and only at the wavelength of the coupling to the silicon waveguide, light is absorbed into the graphene layer. Electron-hole pairs generated by the light absorbed in monolayer graphene make electric currents by a field-effect transistor formed by two horizontally arranged electrodes (source and drain) as shown in Fig. 1(a).

Prior to the detailed design of the device, modal analysis was performed to find out a proper range of the silicon waveguide thickness. We investigated the characteristics of the supermode caused by the coupling between the two guided modes of the fiber and the silicon waveguide. For the mode calculation, a uniform silicon waveguide without a grating was assumed, and a Finite element method (FEM) based commercial tool (COMSOL) was used. A standard single mode fiber was assumed: a core diameter (t_SMF) of 8.2 µm and 0.36% index difference between the core and the cladding were assumed. The gap distance (t_gap) between the fiber’s core and the silicon waveguide was set to be 500 nm. The complex refractive index of monolayer graphene, which was used as an absorbing layer, was calculated by , where C₁ = 5.446 µm⁻¹ and λ was a wavelength in µm [22]. This equation was the extracted from experimental data. The refractive indices of silicon and SiO₂ were 3.445 and 1.444, respectively.

Figure 2(a) shows the effective index of the supermodes with the change of silicon waveguide thickness at the wavelength of 1550 nm. The supermodes were numbered according to the number of null points in their electric field distributions: mode₀ has no null point and mode₁ has one. When the silicon waveguide is thin (~15 nm), the effective index of the silicon waveguide is very close to that of the fiber, so that two fundamental modes of individual waveguides interact strongly and the effective indices of the supermodes differ from those of the fundamental modes as shown in the inset of Fig. 2(a), an enlarged view of the region where the silicon thickness is ~15 nm. The strong interaction also results in rather severe deformation of the field profiles of the supermodes as shown in Fig. 2(b). In this case, if light is launched from the single-mode fiber to the device, both of the supermodes will be excited and the part of the incident light will be directly coupled to the silicon waveguide. This is undesirable for the wavelength-selective device operation as aforementioned. Moreover, the mode profile mismatch between the fiber mode and mode₁ will result in a large insertion loss. On the other hand, as the silicon waveguide becomes thick, the interaction between the individual waveguides becomes weak because of their increased effective index difference, so that the effective indices of the supermodes are very close to those of the individual fundamental modes. Figure 2(c) shows the electric field distributions of the supermodes when the silicon waveguide is 130 nm thick. The field distributions of mode₀ and mode₁ are quite similar to those of the silicon waveguide and the fiber, respectively. Therefore, the incident light launched from the single-mode fiber will mainly excite mode₁ and the insertion loss will be relatively small. More importantly, the direct coupling of the incident light to the silicon waveguide will be prevented as desired. So, a rather thick silicon waveguide is needed. However, if the silicon waveguide is too thick, higher-order guided-modes will appear. Figure 2(d) shows the electric field distributions of the supermodes when the silicon waveguide is 255 nm thick. While the characteristics of mode₀ remains the same, the field distribution of mode1 is deformed in the region of the silicon waveguide, and more importantly, a new supermode (mode₂), which results from the interaction between the fundamental mode of the fiber and the second order guide-mode of the silicon waveguide, appears, whose field distribution has considerably large overlap with that of the fundamental fiber mode. This is undesirable as aforementioned. So, too thick silicon waveguide should also be avoided. Figure 3(a) shows the cutoff wavelengths of mode₁ and mode₂ as a function of the silicon waveguide thickness. Since the existence of mode₂ will cause unwanted direct coupling of the incident light to the silicon waveguide, its cutoff wavelength should lie below the wavelength range of our interest, i.e., 1470 nm < λ < 1610 nm (corresponding to 8 CWDM channels), which is represented as a shaded region in Fig. 3(a). This limits the allowed silicon waveguide thickness to a range 5 nm to 219 nm. Figure 3(b) shows the insertion loss as a function of the silicon waveguide thickness, which was calculated from the mode overlap integral between the mode of the standard single-mode fiber and mode₁. A low insertion loss of approximately 0.01 dB or less is obtained when the silicon waveguide thickness lies in a range from 85 nm to 241 nm. Therefore, considering the higher-order mode avoidance and the acceptably low insertion loss, a proper range of the silicon waveguide thickness appears to be from ~85 nm to ~219 nm.

In the mode calculation, the propagation losses of the supermodes due to the graphene layer beneath the silicon waveguide were also investigated, which are not shown here. When the silicon waveguide thickness lies in the range from 50 nm to 200 nm, the loss of mode₀ is about 10⁵ times higher than that of mode₁. For example, when the silicon waveguide is 100 nm thick, attenuation coefficients (α) of mode₀ and mode₁ are, respectively, ~3.06 × 10⁻² µm⁻¹ and ~1.39 × 10⁻⁷ µm⁻¹, corresponding to propagation lengths of 32.6 µm and 7.21 m. So, the loss of mode₁ does not play an important role, and the relatively short propagation length of mode₀ guarantees almost complete absorption of the light coupled to the silicon waveguide.

In the modal analysis above, we assumed a semi-infinite SiO₂ substrate. In practice, the SiO₂ substrate thickness should be finite. Moreover, an SiO₂ layer of a finite thickness deposited or grown on a silicon substrate may be adopted to fully exploit the silicon photonics compatibility of the proposed structure for economic fabrication. So, we also investigated the effect of the finite SiO₂ layer thickness in both cases. In the SiO₂ substrate case, the guided super modes become stable when the substrate is thicker than 1.0 µm. In the silicon substrate case, an SiO₂ layer thicker than 1.0 µm is also enough to not affect the guided supermodes. Therefore, in all the calculations shown later in this paper, a 2.0 µm thick SiO₂ layer is assumed for safe elimination of the finite substrate thickness effect. If other materials whose indices are higher than that of SiO₂ are used, the required thickness of the lower clad will be larger in the silicon substrate case.

For more efficient device design base on numerical calculations, the proposed structure is also modeled by the coupled-mode theory (CMT) with an assumption that the direct coupling between the supermodes is prevented. The phase matching condition for compensation of the phase difference between two supermodes with a grating given by

where β_i represent the propagation constant of the mode_i, Λ is a period of the grating, and the sign in the parentheses indicates the propagation direction: + is for the forward direction and – is for the reverse direction. In other words, if – (+) is chosen in the parentheses, contra (co)-directional coupling of two supermodes occurs at the wavelengths satisfying Eq. (1). One can see that the required grating period for the contra-directional coupling is always smaller than that of the co-directional one. A smaller grating period makes the whole device length shorter for the same number of gratings. In addition, since the smaller grating period causes less unwanted scattering at a given operating wavelength, so that the characteristics of the wavelength-selective directional coupler are maintained even for the deeper grating. As will be explained later, the deep grating enhances the coupling coefficient (κ₀₁), which reduces the device length for a given absorption efficiency. Therefore, in this work, we considered only the design of the contra-directional coupler based integrated photodetector. The Bragg reflection condition of a grating in which each excited supermode is reflected back to its own mode of the counter propagation direction is given by

However, in the proposed device scheme, this Bragg reflection should be avoided because it creates unnecessary energy loss and noise. Figure 4 shows the phase matching conditions graphically when the silicon waveguide is 100 nm thick and Λ = 426 nm. The mode₀-mode₁ coupling occurs at λ_couple = 1550 nm, while the Bragg reflections of mode₀ and mode₁ occur at λ_mod0 = 1275 nm and λ_mod1 = 1770 nm, respectively.

Since the Bragg reflection wavelengths are sufficiently far apart from the wavelength range of our interest, we can ignore the self-reflection terms in the CMT modeling. Therefore, coupled mode equations [23, 24] of two waveguides can be simplified as follows:

where A and B are the amplitudes of the supermodes, the superscripts (+, −) indicate propagation directions (forward or backward), and Δγ_i = Δβ_i ± jα_i is positive in the forward direction and negative in the backward direction). The coupling coefficient (κ₀₁) is calculated as

where Δε(x) is the dielectric perturbation in a grating, ω is an angular frequency, and E_{mode i} is the electric field distribution of the supermode. The solution of the coupled mode equation is given by

where , and A₀ and L are a modal amplitude of an incident wave and a grating length, respectively. The throughput transmission in the fiber core and the drop efficiency in the silicon waveguide are represented by η_through = |A⁺(L)|²/A⁺(0)|² and η_drop = |B⁻(0)|²/|A⁺(0)|², respectively. Due to the substantial loss of mode₀ mainly guided in the silicon waveguide, the wave propagating through the drop port is also absorbed into the graphene layer within a short length. Thus, the final graphene absorption efficiency can be simply calculated η_absorption = 1 − η_through.

In designing the proposed device, design parameters include a silicon waveguide thickness (t_Si), a grating depth (t_grat), a grating period (Λ), and a grating length (L). The main goal of our design is to minimize the device (or grating) length for a sufficiently high absorption efficiency (> 95%). In most of the design process, the clad thickness (t_gap) between the fiber core and the silicon waveguide was set to be 500 nm considering the fabrication tolerance in side-polishing of the fiber. The effect of t_gap variation was also examined. All the calculations in this work were conducted by using the commercial FEM tool (COMSOL), which were confirmed by comparing with the CMT analysis described in Section II. The same material constants that were used in the mode analysis in Section II were used in all the calculations.

First, the influence of the grating depth (t_grat) was investigated for silicon thicknesses of t_Si = 100 nm, 130 nm and 160 nm. Figure 5(a) shows the coupling coefficient (κ₀₁) as a function of the grating depth. The coupling coefficient is determined by the electric field distribution of the two supermodes in the grating area as shown in Eq. (4). However, Eq. (4) is valid for small perturbation of the dielectric constant of the grating. For a deep grating of high index contrast, it is difficult to calculate the coupling coefficient accurately by an analytical method. In this work, we estimated the coupling coefficient by fitting the CMT result derived in section II with the absorption and transmission spectra calculated with the FEM at a specific grating length (L = 100 µm). The validity of the estimation was confirmed by comparing the FEM calculated spectra for other grating lengths with the CMT results based on the estimated coupling coefficient. As will be shown later, those two calculations agree well up to L = 1000 µm, which is long enough to obtain a sufficiently high absorption efficiency (>95%). In Fig. 5(a), one can see that the coupling coefficient monotonically increases with the grating depth when t_Si = 100 nm and 130 nm. Whereas, when t_Si = 160 nm, the coupling coefficient decreases to zero at a grating depth of ~37% and increases again. This can be understood from the electric field distribution of mode₁. Figure 5(b) shows the electric field distributions of mode₁ for various t_Si. The inset is an enlarged view near the boundary between the silicon waveguide and the clad of the fiber which is represented by the red dotted line. As the thickness of the silicon increases, the null point at which the sign of the electric field changes shifts into the silicon waveguide. On the other hand, the field distribution of mode₀ remains almost the same for the considered values of t_Si. So, when the null point of mode1 lies in the silicon waveguide (for example, when t_Si = 160 nm), the overlap integral between mode₀ and mode₁ can be zero for a certain grating depth. The mode overlap integral also explains why the thinner silicon waveguide has the larger coupling coefficient for the same relative grating depth. According to our investigation, when t_Si < 100 nm the, coupling coefficient increases further for a small relative grating depth, but the maximum allowed grating depth becomes smaller. As a result, the maximum achievable coupling coefficient remains about the same as the case of t_Si = 100 nm. Therefore, considering either insertion loss and coupling coefficient (or device length), t_Si = 100 nm appeared to be a proper choice. When the relative grating depth is above 80%, the coupling coefficients remain the same for all t_Si. We surmise that this is because the increased scattering of the deep grating partially destroys the guiding of the silicon layer. Coupling effect is restricted. So, as for the relative grating depth, 80% appears to be a proper choice.

Figure 6 shows peak absorption efficiencies and transmissions at the through port as a function of grating length (L) when t_Si = 100 nm with t_grat = 60 nm and 80 nm. The grating periods were chosen to have the peak absorption at a wavelength of 1550 nm: Λ = 474 nm for t_grat = 60 nm and Λ = 499 nm for t_grat = 80 nm. The solid lines show the FEM calculations and the dotted lines show the CMT results with the estimated coupling and the attenuation coefficients. Note that the attenuation coefficient (α) is estimated simultaneously with the coupling coefficient in the aforementioned fitting process. In general, the FEM calculations agree well with the CMT results. As mentioned earlier, the coefficients estimated for L = 100 µm were used for all the other grating lengths, which confirms the validity of the CMT modeling of our device. We surmise that the ripple in the FEM results from Fabry-Perot resonance due to the reflections at the ends of the grating region. For an absorption efficiency of 95%, a grating length of about 500 µm (~1.4 mm) is required when the grating depth is 80 nm (60 nm). This can be understood from the coupling coefficients: for t_grat = 60 nm and 80 nm, κ₀₁ = 4349 m⁻¹ and for t_grat = 80 nm, κ₀₁ = 7340 m⁻¹, respectively, as shown in Fig. 5(a). In the proposed device, the loss of mode₀ is very high, so that the attenuation coefficient is much larger than the coupling coefficient. This implies that one the incident wave carried by mode₁ is coupled to mode₀, it is quickly absorbed by the graphene layer. Therefore, the absorption efficiency is mainly determined by the magnitude of the coupling coefficient for a fixed grating length.

Figure 7(a) shows absorption and transmission (at the through port) spectra for t_grat = 80 nm and 100 nm with t_Si = 100 nm, and L = 500 µm. The grating period was chosen such that a peak absorption occurs at the wavelength of 1550 nm in each case: Λ = 499 nm for t_grat = 80 nm and Λ = 523 nm for t_grat = 100 nm. Since the coupling coefficient remains almost the same for t_grat > 80% as shown in Fig. 5(a), the absorption spectra for t_grat = 80 nm and 100 nm are similar in general. The maximum absorption efficiency of 95.1% and the 3 dB bandwidth of 7.5 nm were obtained. Owing to the stronger scattering, the case of t_grat = 100 nm shows ripples at the short wavelengths, where absorption is less than 10%. This scattering effect causes crosstalk of ~10 dB to the adjacent wavelength channels. In order to avoid such a scattering effect, the grating depth of 80 nm is more desirable. The black dotted curve represents the CMT result with κ₀₁ = 7340 m⁻¹, which agrees well with the FEM calculation. The spectrum of the transmission at the through port for t_grat = 80 nm is also plotted in Fig. 7(a), which demonstrates that the unwanted scattering of the grating is negligible, so that all the unabsorbed wave appears as the transmission at the through port with an insertion loss of 0.2 dB. Figures 7(b) and 7(c) show the electric field distributions (cross-section view at the center) when the TE wave is incident on the core of the fiber. At λ = 1550 nm (Fig. 7(b)), most of the incident wave is coupled to the silicon waveguide and absorbed into graphene. On the contrary, at λ = 1530 nm (Fig. 7(c)), most of the incident wave propagates through the fiber core with no coupling to the silicon waveguide.

We also investigated the influence of the thickness of the clad between the fiber core and the silicon waveguide (t_gap). Figure 8 shows the coupling coefficient as a function of t_gap for t_Si = 100 nm, t_grat = 80 nm, and Λ = 499 nm. As the gap increases, the coupling coefficient shows a linear decrease because the overlap integral between the two supermodes becomes smaller. So, for a smaller t_gap, the higher coupling coefficient and thus, a shorter device length are obtained. However, considering fabrication tolerance during the optical fiber polishing, t_gap = 500 nm seems to be a reasonable choice.

Figure 9(a) shows the center wavelength of the absorption spectrum as a function of the grating period for t_Si = 100 nm, t_grat = 80 nm, t_gap = 500 nm. Since the supermodes of the designed structure show linear-like dispersion in the wavelength range of our interest as shown in Fig. 4, the central wavelength varies linearly with the grating period. Figure 9(b) shows the FEM calculated absorption spectra for Λ = 492 nm, 499 nm, and 506 nm designed for the center wavelengths of 1530 nm, 1550 nm, and 1570 nm, which correspond to three central wavelength channels of the CWDM. The maximum absorption efficiencies were 96.0%, 95.1%, and 95.5%, respectively. The 3 dB bandwidths were 8 nm, 7.5 nm, and 7.6 nm, respectively, which causes −14 dB crosstalk with neighboring channels. As the center wavelength shifts with a period change, spectral characteristics remain almost the same, which is a desirable feature in fabrication as well as design.

In this work, we used an approximate formula for the index of graphene [22]. In Fig. 10, the approximate formula is compared to Kubo formula [25, 26] that is believed to be more accurate. The approximation seems to be acceptably close to the Kubo formula in the wavelength range of our interest. In order to investigate the effect of the approximation on the device performance, the absorption spectra were calculated with the graphene index extracted from the Kubo formula and plotted in Fig. 9(b) (dotted lines). One can see that the approximation of the graphene index makes negligible difference in the device performance.

We also investigated the effect of multilayer graphene on the device performance. Figure 11 shows peak absorption efficiencies as a function of grating length(L) with bilayer and trilayer graphene used when t_Si = 100 nm, t_grat = 80 nm, and Λ = 499 nm. The index of multilayer graphene was modeled by assuming the conductivity of n-layer graphene is n times of that of monolayer graphene [27, 28]. For comparison, the absorption spectrum for the monolayer graphene case presented in Fig. 6(a) was also plotted together. As the number of graphene layers increases, the peak absorption efficiency decreases for all grating lengths. This can be explained by the CMT modeling. When z = L, at the wavelength of resonance (satisfying the phase matching condition, β₀ − β₁ = 2π/Λ), the Eq. (6) becomes

where α₁ ≪ α₀ is assumed as aforementioned. One can see that as the loss of graphene (α₀) increases, the throughput transmission (η_through = |A⁺(L)|²/|A⁺(0)|²) increases for a given L and thus, the absorption efficiency (η_absorption = 1 − η_through) ) should decrease. This would look counterintuitive at first glance. However, it can be understood from the phase matching condition Eq. (1): the loss imbalance between those two supermodes ruins the phase matching condition and thus, the m₁-to-m₀ wave coupling through the grating as well. This implies that for a larger α₀, a larger L is needed to get a certain η_through, which will make the device long. It should also be noted that even for a large α₀, we can get an arbitrarily small η_through by increasing L since Eq. (7) tends to zero as L → ∞. The dotted lines in Fig. 11 represent the CMT calculation results with the estimated coupling and the attenuation coefficients. The estimated coupling coefficient (κ₀₁) was 7340 m^-1 for all three cases. The estimated attenuation coefficients (α₀) were 13377 m⁻¹, 28801 m⁻¹, and 44083 m⁻¹ for the cases of monolayer, bilayer, and trilayer graphene, respectively.

From Eq. (7), an optimal attenuation coefficient to minimize a device length appears to be zero. This implies that the graphene layer under the grating region should be removed and only placed in the drop port region. In this case, however, there is a huge scattering loss at the interface between the grating and the drop port waveguide (z = 0) for a large grating depth, which results in absorption efficiency degradation. So, the graphene layer was extended to the grating region and the scattering loss became negligible as verified in Fig. 7(b).

In this work, we assumed a TE wave as an incident wave. The attenuations of the silicon waveguide with the graphene layer for both TE and TM waves are dependent on the waveguide geometry, i.e., especially the waveguide thickness. According to our investigation, for the waveguide thickness up to ~190 nm, the attenuation of TM wave is lower than that of TE wave, and it has been found that the TM wave attenuation is too low to prevent the scattering at the interface between the grating and the drop port waveguide for a 100 nm thick waveguide. So, we focused on the TE wave case in this work.

We have proposed an integrated wavelength-selective graphene photodetector based on a grating-assisted contra-directional coupler, an array of which can be used for simultaneous wavelength demultiplexing and signal detection of CWDM channels. The proposed structure consists of a side-polished fiber and a silicon waveguide with uniform grating and monolayer graphene. The operation principle of the proposed device was studied with the guided-mode analysis and the coupled mode theory modeling. We also investigated proper ranges of geometric parameters considering their effect on the device performance. Considering an insertion loss and a device length, an optimal device structure was designed: when a silicon waveguide thickness, a grating depth, and a grating length are t_Si = 100 nm, t_grat = 80 nm, and L = 500 µm, respectively, a peak absorption efficiency of 95.1% and a 3 dB bandwidth of 7.5 nm were obtained at a center wavelength of 1550 nm with a grating period of Λ = 499 nm. We also demonstrated tuning of the center wavelength by changing the grating period with the device performance remained almost the same. For a CWDM application, an array of the designed wavelength-selective photodetectors will have a −14 dB crosstalk to adjacent wavelength channels. We believe that the proposed wavelength-selective photodetector array may be a compact and economic solution to the wavelength demultiplexing and signal detection in a CWDM communication system.