Silicon photonics (SiPh) transceivers are prevalent in data centers. The complexity of their photonic integrated circuits (PICs) is increasing, as the channel and wavelength multiplications continue to increase. The PICs are required to be low-noise as well as low-loss for secure links. While a major noise source is signal distortion arising from the limited bandwidths of the optical modulators and detectors, another notable source is optical reflection from the elemental devices in the circuits [1]. SiPh circuits often include edge couplers, grating couplers (GCs), and multi-mode interferometers in addition to waveguides (WGs), and excessively large reflections can occur particularly from these elemental devices [2,3]. Reflections in transceiver PICs can ripple their optical input/output signals. The output of a laser diode mounted on a chip can be destabilized by back reflections [4]. Accurately measuring the reflection as well as transmission of actual elemental devices is important for designing highly integrated SiPh circuits.

Optical frequency-domain reflectometry (OFDR) is useful for measuring backscatter along an optical circuit [5,6]. Its frequency-domain (FD) data can be converted into time-domain (TD) data equivalent to that of optical time-domain reflectometry (OTDR) by using an inverse fast Fourier transform (FFT), and vice versa by an FFT. OFDR can be used to obtain the group index n_g and propagation loss of a WG, as well as the reflection amplitudes and phase derivatives of the discontinuities along it. The devices under test (DUTs) for OFDR now include silicon oxide/nitride or glass WGs on chips [7–12] in addition to the latest optical fibers [13,14]. The reflections from their discontinuities such as connectors, bends, crosses, and strained sections, are small and thus multiple reflections are suppressed, which helps identify the remaining reflection peaks. There are few reports on OFDR on SiPh circuits [15–18], even semiconductor-WG circuits [19,20], despite their recent prevalence. In general, the elemental devices in a SiPh circuit exhibit large reflections. Hence, many multiple-reflection peaks appear in the TD data, which makes it difficult to locate corresponding points in the circuit, particularly when the circuit complexity is high. In this Letter, we demonstrate that the transmittance and reflectance of elemental devices in SiPh circuits can be derived from FD data of multiple-reflection peaks if the peaks are identified. We analyze the FD data of GC-WG-GC circuits, which are simple but useful for illustrating the key points of the analysis method. As well as n_g and the propagation loss of the WG, we determine the wavelength-dependent transmittance and reflectance of the GC and confirm their values in a numerical simulation.

Figure 1 illustrates a backscatter model of a WG circuit that takes account of time-equivalent multiple-reflection paths. The WG has two reflection points RP1 and RP2, which are apart by L from each other. The light propagating along the WG has wavenumber ${k_0} = 2\pi /\lambda $, where λ is the light wavelength in a vacuum and extinction coefficient ${n_{\textrm{ei}}}$. RP1 has transmission coefficients ${t_{1\textrm{f}}}$ and ${t_{1\textrm{b}}}$ for the forward and backward directions, respectively, and has reflection coefficients ${r_{1\textrm{f}}}$ and ${r_{1\textrm{b}}}$; RP2 has ${t_{2\textrm{f}}}$, ${t_{2\textrm{b}}}$, ${r_{2\textrm{f}}}$, and ${r_{2\textrm{b}}}$. For light input to the WG from the RP1 side, path 1 is the fastest of all the reflection paths, and its reflection is given by ${R_1} = {|{{r_{1\textrm{f}}}} |^2}$. The light along the second fastest path 2 is reflected as backscatter halfway in the WG with reflection coefficient ${r_{z\textrm{f}}}$ at position ${z_1}$. Because the light makes a round trip within the WG and passes RP1 twice, the reflected power is given by ${R_{1,2}} = t_{1\textrm{f}}^2\; t_{1\textrm{b}}^2\; {|{{r_{z\textrm{f}}}} |^2}\; \textrm{exp}({4{k_0}{n_{\textrm{ei}}}{z_1}} )$. The light along the third fastest path 3 is reflected at RP2 with the reflection ${R_2} = t_{1\textrm{f}}^2\; t_{1\textrm{b}}^2\; {|{{r_{2\textrm{f}}}} |^2}\; \textrm{exp}({4\; {k_0}{n_{\textrm{ei}}}L} )$. There are three reflection paths 4–6 that are the fourth fastest and equivalent in time, as shown in Fig. 1. The light beams along them are reflected halfway in the WG with reflection coefficient ${r_{z\textrm{f}}}$ at ${z_2}$ (paths 4 and 5) and with ${r_{z\textrm{b}}}$ at $L - {z_2}$ (path 6), and return to the photodetector of the OFDR at the same time. If the WG is uniform, ${r_{z\textrm{f}}}$ and ${r_{z\textrm{b}}}$ are constant throughout the WG and ${r_{z\textrm{f}}} ={-} {r_{z\textrm{b}}}$. Then, the reflections for the three paths sum to ${R_{2,3}} = t_{1\textrm{f}}^2\; t_{1\textrm{b}}^2\; {|{{r_{2\textrm{f}}}} |^2}\; {|{{r_{z\textrm{f}}}} |^2}\; {|{2{r_{1\textrm{b}}} - {r_{2\textrm{f}}}} |^2}\; \textrm{exp}[{4{k_0}{n_{\textrm{ei}}}({L + {z_2}} )} ]$. The fifth fastest path 7 makes two round trips, and its reflection is ${R_3} = t_{1\textrm{f}}^2\; t_{1\textrm{b}}^2\; {|{{r_{1\textrm{b}}}} |^2}\; {|{{r_{2\textrm{f}}}} |^4}\; \textrm{exp}({8{k_0}{n_{\textrm{ei}}}L} )$. Reflection formulas for even slower paths can be built in the same manner. Reflection paths with two or more halfway-reflections are ignored as being too small.

 

Fig. 1. Backscatter model of fast reflection paths 1–7 of optical circuit with L-long waveguide (WG) between reflection points RP1 and RP2. Detected by optical frequency-domain reflectometry (OFDR) are reflections R₁, R₂, and R₃ from RP1 and RP2 only, and R_1,2 and R_2,3 from RP1, RP2, positions z₁, z₂, and $L - {z_2}$. They can be expressed with transmission and reflection coefficients t_1f, r_1f, t_1b, r_1b, r_2f, r_zf, wavenumber k₀, and extinction coefficient n_ei.

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While the values of L, ${k_0}$, and z are known, there are eleven unknown parameter values. If they are fewer than the independent features of TD data of the PIC, they can be determined from the data. Here, we impose three conditions on the PIC:

 

Fig. 2. Schematic of (a) grating coupler (GC)–WG–GC circuit with probing single-mode fiber (SMF) and (b) OFDR measuring system for frequency-domain (FD) and time-domain (TD) analyses.

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As ${z_1}$ and ${z_2}$ tend to 0 or L, Eqs. (2) and (4) further break down into

These equations can be solved in terms of the individual parameters. Noting the round trip of light, we have the WG propagation loss:

Long and short GC–WG–GC circuits were fabricated on a 775-µm-thick silicon-on-insulator (SOI) wafer with a 200-nm-thick top Si layer on a 2-µm-thick buried oxide (BOX) layer. The Si cores of the circuits were defined in the top Si layer by ArF immersion lithography with half-tone photo masks for high uniformity. A 2-µm-thick SiO₂ was deposited on the Si cores as an upper cladding. The lower cladding was the BOX layer. The Si cores of the WGs had dimensions of 350 nm (H) × 200 nm (V). The WG lengths were 40.609 and 2.449 mm. All the GCs were of the same design, in which the grating pitch was 509 nm, and the width and depth of the grooves were 254 and 70 nm, respectively. A Si chip with the circuits was diced out of the wafer. One GC of each circuit was coupled to a probing SMF, as shown in Fig. 2(a). The probing SMF had an 8°-angled flat end with an antireflection coating (ARC), exhibiting a reflection of less than –84 dB for $\lambda = $1280–1340 nm. Figure 2(b) is a schematic of the OFDR measuring system used.

The DUT chip was kept at 25 °C with a Peltier device. The probing SMF was 0.658-m-long and was connected to a 1-m-long SMF patch cord with FC/APC connectors, which was further connected to an optical reflectometer (OBR 4613 from Luna Innovations Inc.) with FC/APC connectors via an in-line polarization controller. The time resolution of the reflectometer was 0.1 ps, and the sensitivity of the photodetector was –130 dB. The light input to the GCs was set to be of transverse-electric (TE) polarization. The probing SMF and GCs were actively aligned at a target $\lambda $ of 1310 nm. The measured FD data were converted to TD data and displayed on a monitor. The optical loss of the probe and cord SMFs was measured to be 0.4 dB from the backscatter level immediately before the probe tip [21]. This loss was not incorporated in the circuit modeling here, for simplicity. Instead, we will correct an analysis result with it later.

Figures 3(a) and 3(d) show the TD reflections of the long and short circuits, respectively, around the first three major peaks while Figs. 3(b), 3(c), and 3(e)–3(g) are magnified graphs of the major peaks. The reflection amplitudes were normalized by the optical power immediately before the output connector of the reflectometer, and, to sharpen reflection peaks, a Hann window as a frequency-domain window (FDW) was applied before inverse FFT. The horizontal axis is in time, and, for convenience, the origin is set at the first major peaks LP1 and SP1, which were exactly at the same time, although the time origin of the raw data was at the optical connector of the reflectometer. In Fig. 3(a), the major peaks LP1 and LP2 correspond to ${R_1}$ and ${R_2}$, respectively. A peak corresponding to ${R_3}$ was too weak to be observed, but its expected position is labeled “(LP3)” in Fig. 3(a). LPs1–LPs4 in Figs. 3(a)–3(c) are spurious sideband peaks of the major peaks created by the Doppler effect from undesirable vibration of the light path in the interferometer of the reflectometer [22]. Those peaks correspond to noise that distorts the backscatter graphs of the WG. LPback after LP1 in Fig. 3(b) is a reflection from the back surface of the DUT Si-chip, while no such reflection was observed near LP2, as shown in Fig. 3(c). For the short WG circuit, the first three major peaks SP1, SP2, and SP3 corresponding to ${R_1}$, ${R_2}$, and ${R_3}$ were distinctly observed, as shown in Fig. 3(d). SPs1–SPs3 are spurious sideband peaks. As shown in Figs. 3(e)–3(g), the signal-to-noise ratios (SNRs) of the peaks were 26, 21, and 15 dB, respectively.

 

Fig. 3. (a)–(c) TD reflections of long GC–WG–GC circuit around (a) major peaks (LP1 and LP2) and non-observable but expected peak (LP3), and magnified graphs around (b) LP1 and (c) LP2. (d)–(g) TD reflections of short GC–WG–GC circuit around (d) major peaks (SP1–SP3) and magnified graphs around (e) SP1, (f) SP2, and (g) SP3. Blue lines are in 0.1-ps resolution and black line in panel (a) is 10-ps average. LPs1–LPs4, SPs1–SPs3 are spurious sideband peaks, and LPback is reflection from the back surface of measured chip.

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The TD data of the long GC–WG–GC circuit was used to determine the group index n_g and $Los{s_{\textrm{WG}}}$ of the Si wire WG. Here, n_g is given by ${n_\textrm{g}} = c\; \Delta {T_{1,2}}/({2\; {L_{1,2}}} )$, where c is the speed of light in a vacuum, $\Delta {T_{1,2}}$ is the round-trip time difference between LP1 and LP2, and ${L_{1,2}}$ is the length of the WG between GC1 and GC2. From $\Delta {T_{1,2}} = $1.2344 ns and ${L_{1,2}} = 40.609\; \textrm{mm}$, ${n_\textrm{g}} = 4.5563$ is taken as an average over the 1280–1340 nm wavelength range. The time T measured from LP1 or the SP1 peak can be converted into a corresponding position z by using the relation $z = c\; T/({2\; {n_\textrm{g}}} )$, where z is the one-way length and T is the round-trip time. Also, the two-point spatial resolution corresponding to the 0.1-ps time resolution is found to be 3.3 µm. Low-noise data on ${ {{R_{1,2}}} |_{z \to 0 + 0}}$ and ${ {{R_{1,2}}} |_{z \to L - 0}}$ are required for the propagation loss extraction. As can be seen in Fig. 3(b), however, the WG backscatter fluctuated in a 20-dB range and there were peaks LPback near LP1, and LPs3 between the LP1 and LPback. Avoiding these noise sources, we selected a 0.2-ns (6.580 mm) span centered at ${T_{0 + 0}} = 0.1246\; \textrm{ns}$ (${z_{0 + 0}} = 4.098\; \textrm{mm}$) after LPback for ${ {{R_{1,2}}} |_{z \to 0 + 0}}$. Similarly, for ${ {{R_{1,2}}} |_{z \to L - 0}}$, we selected a 0.2-ns span centered at ${T_{L - 0}} = 1.1216\; \textrm{ns}$ (${z_{L - 0}} = 36.898\; \textrm{mm}$) before LP2, avoiding anticipated contamination by a counter peak of LPs4. These selected spans were transformed by FFT and further into wavelength-domain (WD) data. Figures 4(a) and 4(b) show the WD data of ${ {{R_{1,2}}} |_{z \to 0 + 0}}$ and ${ {{R_{1,2}}} |_{z \to L - 0}}$ before FDW application, and lines fitted to them, which are cubic polynomials of $\lambda $. The $Los{s_{\textrm{WG}}}$ was determined from the fit lines by using Eq. (10) and the corrected L of 32.800 mm $({ = {z_{L - 0}} - {z_{0 + 0}}} )$, as shown in Fig. 4(c). That was a little larger than 1.62 dB/cm previously measured by transmission [23], probably because of the different foundries used. The $Los{s_{\textrm{WG}}}$ can be converted into ${n_{\textrm{ei}}}$ by ${n_{\textrm{ei}}} ={-} 1.832 \times {10^{ - 2}}\; \lambda \; Los{s_{\textrm{WG}}}$.

 

Fig. 4. Wavelength-dependent reflections of 0.2-ns spans (a) after LP1 and (b) before LP2 peaks for long GC–WG–GC circuit. Blue and black lines are measured and fitted, respectively. (c) Wavelength-dependent Si-wire-WG propagation loss extracted from the fit lines in panels (a) and (b).

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The TD data of SP1, SP2, and SP3 in Figs. 3(e)–3(g) were assigned to ${R_1}$, ${R_2}$, and ${R_3}$, respectively, whose time spans (length spans) were 7.00 ps (230 µm). Their corresponding WD data before FDW application were fitted with cubic polynomials of $\lambda $, as shown in Figs. 5(a)–5(c). The amplitude hikes near the 1280-nm wavelength in Figs. 5(b) and 5(c) were not included in the fits as they resulted from FFTs of the SP2 and SP3 TD-data having somewhat small SNRs. From the polynomial functions of ${R_1}$, ${R_2}$, and ${R_3}$, we extracted the $\lambda $ dependence of ${T_{\textrm{g1}}}$, ${R_{\textrm{g1,in}}}$, ${R_{\textrm{g1,out}}}$, and ${R_z}$ by using Eqs. (12)–(15), as shown in Figs. 6(a)–6(d). The ${T_{\textrm{g1}}}$ was increased by the 0.4-dB loss of the probing and patch cord SMFs as the planed correction. Also shown in Figs. 6(a)–6(c) are the ${T_{\textrm{g1}}}$, ${R_{\textrm{g1,in}}}$, and ${R_{\textrm{g1,out}}}$ predicted by a three-dimensional finite difference time-domain simulation, for which an SMF with a 9.2-µm mode diameter was directly joined to the upper cladding. The λ dependence of the measured ${T_{\textrm{g1}}}$, ${R_{\textrm{g1,in}}}$, and ${R_{\textrm{g1,out}}}$ graphs agreed with the predictions while the ${T_{\textrm{g1}}}$ peaks were –3.3 dB at a $\lambda $ of 1313 nm for the OFDR and –2.0 dB at 1300 nm for the simulation. The excessive loss and redshift of the measured peak can result from fabrication error and the slight difference in SMF–GC coupling structures and alignment.

 

Fig. 5. Wavelength-dependent reflections of 7.00-ps spans of (a) SP1, (b) SP2, and (c) SP3 peaks. Blue and black lines are measured and fitted, respectively.

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Fig. 6. Wavelength-dependent (a) transmittance ${T_{\textrm{g1}}}$, reflectance (b) ${R_{\textrm{g1,in}}}$ and (c) ${R_{\textrm{g1,out}}}$ of GC-to-SMF coupling, and (d) reflectance ${R_z}$ of Si wire WG. Red solid lines represent those extracted from OFDR measurement data, and blue dashed lines represent predictions from numerical simulation.

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All the OFDR data were analyzed in the linear regime. Because the Si wire WGs exhibit high nonlinearity, optical powers coupled to them should be small enough to sufficiently suppress nonlinearity-originated (NL) errors. However, too small powers deteriorate the SNRs of reflections because of the limited sensitivity of the photodetector. For minimal total errors, we chose 2.4 mW as the maximum output power of the reflectometer, observing no apparent NL loss, as shown in Fig. 3(a). The residual NL loss is the largest near the input ends of the circuits. If the NL loss in the WG between the ${R_1}$ and ${R_2}$ peaks of the short GC-WG-GC circuit is $\xi $ dB and that after the ${R_2}$ peak is negligible, the ${T_{\textrm{g1}}}$ from Eq. (12) is corrected to ${T_{\textrm{g}1,\textrm{c}}} \equiv {T_{\textrm{g1}}} + \xi /2$ in dB while Eqs. (13)–(15) can be used as they are. By first assuming a $\xi $ value ${\xi _\textrm{T}}$ for ${T_{\textrm{g}1,\textrm{c}}}$, we can derive the corresponding NL loss value ${\xi _{\textrm{NL}}}$ from a nonlinear propagation equation and the nonlinear parameters of the WG [24]. The assumed ${\xi _\textrm{T}}$ turns out to be correct if ${\xi _\textrm{T}} = {\xi _{\textrm{NR}}}$. In this way, we confirmed that even the largest $\xi $ at the ${T_{\textrm{g1}}}$ peak is as small as 0.02 dB and consequently the NL errors in all the analysis results are negligibly small.

In conclusion, we extracted the λ-dependent transmittance and reflectance of elemental devices of SiPh circuits from OFDR data. If necessary, simple PICs having elemental devices of the same design can be put near a target PIC on a chip or a wafer to separately measure the transmittance and reflectance of the elemental devices included in the target PIC. Thus, our analysis method can be extended to more complex SiPh circuits and modules.