Low-power thermo-optic silicon modulator for large-scale photonic integrated systems

SungWon Chung; Makoto Nakai; Hossein Hashemi

doi:10.1364/OE.27.013430

1. Introduction

Silicon photonics has entered the commercial market in the context of high-speed optical transceivers for data centers [1]. The same silicon photonics platform enables emerging applications such as large-scale optical phased arrays for LiDAR and imaging [2–4], large-scale optical quantum systems [5, 6], large-scale optical machine learning processors [7, 8], and large-scale optical switches [9, 10]. These new applications have vastly different design requirements on their components compared to those of the more traditional high-speed optical transceiver application. A common feature across these applications is the large number of optical components that must be integrated. Large-scale integration necessitates low-power, small footprint, and low loss for every component. For instance, an integrated platform that requires 10,000 optical modulators, each consuming 10-mW of power, will require 100 W of power which is prohibitively large for many applications especially those meant for hand-held battery-operated devices. Fortunately, in most of these emerging applications with large-scale programmable photonic integrated systems, the required modulation speed need not be in the GHz range [11]. For instance, in an optical phased arrays used for automotive LiDAR, a required optical beam scanning rate of 10-frames/second sets the modulation speed to around 12-kHz for a 0.1° resolution across 120° field-of-view.

The most energy-efficient optical modulators in silicon to date are based on optical resonance [12–16], multi-mode structures [17–19], and slow light propagation in photonic crystal waveguides [20–22]. The optical bandwidth of these modulators is limited so that they cannot be used in certain applications or require a calibration process that needs a significant power consumption. For instance, in a one-dimensional optical phased array, the optical beam can be steered in two dimensions by sweeping the wavelength over hundreds of nanometers [23, 24]. The thermo-optic modulators based on multi-mode structures [17] are unable to support such a wide optical bandwidth and also suffer from a large optical loss compared to single-mode waveguide modulators. Non-resonant optical modulators based on carrier-depletion mechanism in silicon [25–30] have an ultralow power consumption below 10 μW and support a broad optical bandwidth, but they are not suitable for very large-scale programmable photonic integrated systems due to their large silicon footprint and relatively high optical loss.

Silicon thermo-optic waveguide modulators [2,3,12,17,18,22,31–55] have a potential to meet such unique design requirements for large-scale integration where optical modulation speed is not demanding. Non-resonant thermo-optic waveguide modulators can be designed with less than 1-dB optical loss and less than 0.001-mm² silicon footprint while providing optical bandwidth larger than several hundred nanometers in the near-infrared spectrum. Therefore, large-scale programmable photonic processors for optical quantum information processing [6] and optical machine learning [7], as well as large-scale low-loss optical switches [10], have been implemented with thermo-optic modulators. The energy efficiency of thermo-optic modulators, which limits the scalability of a system, depends on the efficiency of transferring the heat to and confining it in the silicon waveguide. The power consumption of thermo-optic modulators can be lowered by enhancing the heat isolation between the silicon modulator and surrounding cladding materials. A common approach is to create vertical air-gap trenches or selective silicon under-etching surrounding the silicon waveguide modulator to reduce the unwanted heat dissipation through the buried oxide to the underlying silicon substrate [31,37,38,43,44]. This heat isolation technique significantly reduces the power consumption (Fig. 1) but comes with drawbacks. Densely placed trench isolation structure or silicon substrate removal over a large area limits the scalability of integration [13] and also reduces the reliability due to the accumulated mechanical fatigue from temperature stress [56]. Furthermore, since the air-gap trenches and silicon undercuts reduce the effective thermal conductivity of the modulators [31, 44, 45], the increased heat isolation adversely affects the modulation bandwidth of suspended thermo-optic modulators (Fig. 1), to the extent that they are unattractive for many emerging applications.

Fig. 1 Performance trade-offs of thermo-optic silicon waveguide modulators at near-infrared without inherent optical bandwidth limitation. Data points from the published experimental results show that the geometrical design optimization demonstrated in this work improves the thermo-optic modulator performance trade-offs among optical loss, modulation bandwidth, and power consumption for 180° (π) phase shift. See Appendix for a feasible design space around the measured performance of this work, which can be explored with geometrical optimization.

Download Full Size | PDF

In this article, we report an experimental demonstration of geometrical design optimization for improving the efficiency of a low-loss silicon thermo-optic waveguide modulator in a standard silicon photonics technology. The modulator achieves the lowest power consumption among thermo-optic modulators with more than 100-nm optical bandwidth but without air-gap trench or silicon undercut for heat isolation. We exploit geometrical design optimization, which improves the trade-off between energy efficiency and modulation bandwidth without changing heat flow. The foundation of our geometrical design optimization is a dense dissimilar waveguide routing technique [31,55] whose efficiency improvement limit has not yet been fully investigated. This work further elaborates on the geometrical design optimization, so that a high efficiency can be achieved by using a commercial foundry silicon photonics process without air-gap trench [55] or silicon undercut [31]. The thermo-optic modulation efficiency with the geometrical design optimization is limited by optical cross-talk among adjacent waveguides [55] and thermal cross-talk among adjacent modulators. We consider not only the optical cross-talk but also the thermal cross-talk, which depends on the modulator pitch in a large-scale photonic integrated system, and accordingly determine the modulator geometry by compromising the efficiency. As a result, the record efficiency with the experimental prototype is achieved with a conservative design with redundant reliability on electromigration limits. Computational thermal characterization and theoretical analysis predict further efficiency improvement and size reduction are achievable with an optimized geometrical design.

2. Thermo-optic silicon modulator with geometric design optimization

2.1. Fabricated device structure

The physical structure of the fabricated thermo-optic phase modulator is illustrated in Fig. 2. The modulator is implemented in a commercial foundry silicon photonics process that features a silicon-on-insulator (SOI) wafer with 2-μm thickness buried oxide (BOX) and 220-nm thickness silicon waveguide. The modulator consists of 14 straight waveguides connected by non-circular Clothoid bends [57,58], constructing a 1.22-mm path length for light propagation. The 1550-nm wavelength optical signal couples in and out of the chip using grating couplers. Figure 2(d) and the inset SEM microphotographs in Fig. 2(c) show the fabricated modulator on a test chip with 275-μm thickness. The spacing between the straight waveguides should be small to result in a compact form factor and high efficiency (explained later). To reduce the unwanted optical crosstalk between adjacent waveguides, the waveguide widths are alternated [31]. To limit the crosstalk below −40 dB, the minimum pitch between two waveguides with identical 500-nm widths is 2550-nm, while it reduces to 1000-nm if the waveguide widths are 400-nm and 500-nm, respectively. Each bend consists of two pairs of symmetric Clothoid curves with a waveguide taper in between (formal specification on the bend structure is described in Appendix A-1 and A-2). The bend geometry is designed and optimized so that it occupies small silicon footprint (4-μm bend height) with less than 0.1 dB optical loss. Details on the design methodology are discussed in Appendix A-3. The TiN heater has 60-μm length, 3-μm width, and 110-nm thickness with 900-nm separation to the silicon waveguide layer. Aluminum metal with 6-μm width conveys electrical current to the TiN heater. The TiN heater length is determined by the current handling capability of the TiN heater for a desired power delivery. When using a pulsed modulator driver [47], which increases metal current handling capability, the heater length and correspondingly the modulator length can be reduced by more than a factor of 2. Optical phase shift at a wavelength λ near 1550 nm in a silicon waveguide with a length of L for a temperature rise ΔT is given by Δϕ = 2π(L/λ)(dn/dT)ΔT where dn/dT is 1.86 × 10⁻⁴K⁻¹ at temperatures near 300 K [54, 59]. Note that the optical mode overlapping with densely folded waveguides may contribute to an additional optical phase shift. In a conventional low-power thermo-optic waveguide modulator, with air-gap trench isolation and silicon undercut for increased heat isolation, all optical waveguides are placed beneath the metal heater to be exposed to a higher temperature rise [Fig. 2(b)]. On the contrary, the modulator in Fig. 2(c) places only 15% of the 1.22-mm waveguide path length directly under the TiN heater while the rest 85% of the total modulator waveguide length is not directly under the TiN heater. Perhaps counter-intuitive at first, this design enables more efficient utilization of the total thermal energy that is generated by the heater towards optical modulation. Figure 2(a) shows a typical cross-section of these peripheral waveguides, which recollect the wasted heat energy and improve the modulator efficiency without air-gap trench isolation and silicon undercut. Figure 2(e) shows the simulated vertical cross-section temperature profile of the modulator with 10-mW heater power dissipation and constant 300-K thermal boundary condition at the chip bottom surface. The horizontal temperature profile at the plane parallel to the silicon waveguide center is shown in Fig. 2(f). Although the temperatures of waveguide sections that are directly placed under the TiN heater increase the most, the temperatures of all other waveguide sections also increase. Hence, all waveguide sections lead to optical modulation.

Fig. 2 Fabricated structure of a silicon thermo-optic modulator with geometrical design optimization for efficiency improvement. (a) Typical cross-section of the proposed silicon thermo-optic phase modulator with additional peripheral waveguides. (b) Typical cross-section of a conventional ultralow power thermo-optic phase modulator with air-gap trench and silicon undercut for increased heat isolation [31]. (c) Physical layout of the fabricated thermo-optic phase modulator on a silicon-on-insulator (SOI) wafer, which leverages geometrical design optimization for improving the energy efficiency of thermo-optic modulation for large-scale integration, with SEM photographs on multi-section Clothoid bend structures and waveguide array with alternating widths. (d) Microphotograph of a fabricated modulator test chip, which characterizes the modulator in a Mach-Zhender interferometer. (e) Vertical cross-section of the fabricated modulator on a SOI wafer (2 μm buried oxide (BOX) thickness) with simulated temperature profile. (f) Horizontal cross-section of the fabricated modulator with simulated temperature profile at the center of the silicon waveguides (the complete modulator structure is superimposed on the temperature profile as a guide).

Download Full Size | PDF

2.2. Computational characteristics

Figure 3 summarizes computational characterization on the impact of geometrical design optimization on thermo-optic modulation efficiency, modulation speed, and thermal crosstalk. These simulation results show that geometrical design optimization may improve the thermo-optic modulation efficiency by more than an order of magnitude. From simulation, we observe that the contribution of bend waveguides to the modulation efficiency, speed, and thermal crosstalk is less than 5% due to relatively small temperature rise at the bend waveguides.

Fig. 3 Computational characterization. (a) Temperature profile in the middle of the fabricated thermo-optic modulator at the plane parallel to the waveguide center. (b) Power consumption necessary for 180° (π) phase modulation depending on modulator size, showing a relation between the thermo-optic modulation efficiency and the number of peripheral waveguides for blackgeometrical design optimization. Experimental result is plotted as an upward triangle point, simulated result is plotted as other discrete points, and analytical result is plotted as a dashed line (See Appendix B-1 for the first-order analytical mode to predict the modulator power consumption from waveguide temperature profile). (c) Transient step response of the fabricated thermo-optic modulator with 10-mW heater input, showing a relation between the modulation speed and the number of peripheral waveguides for geometrical design optimization. (d) Thermal crosstalk induced by an adjacent modulator depending on the number of peripheral waveguides for geometrical design optimization.

Download Full Size | PDF

Figure 3(a) shows the waveguide center-plane temperature profile in the middle of the fabricated thermo-optic modulator with 2-μm BOX layer for 10-mW heater power dissipation. Due to the low thermal conductivity of silicon dioxide compared to the silicon substrate, a thicker BOX layer leads to a higher waveguide temperature, resulting in a higher thermo-optic modulation efficiency. A thicker BOX layer, however, reduces the thermo-optic modulation speed. This temperature profile can be approximated as a first-order model for a one-dimensional cylinder, which is represented as T(r) = c₁ ln (r) + c₂ where the negative constant c₁ and the positive c₂ are model fit parameters and r is the distance from the heater to a waveguide center assuming that the heater and the waveguides are in the same plane (see Appendix B-1). The first-order approximation holds for a limited region near the heater (r < 10 μm) where the impact of geometrical design optimization is significant.

Figure 3(b) shows the heater power consumption to achieve 180° (π) thermo-optic phase modulation with a varying modulator size, which is determined by the number of peripheral waveguides for geometrical design optimization. For a single waveguide (N = 0) without any peripheral waveguides, the heater needs 24.4 mW power dissipation for 180° thermo-optic phase modulation. With 26 peripheral waveguides (N = 26) with 800-nm pitch, the heater power consumption necessary for 180° phase modulation can be as low as 1.97 mW, which is 12.4 times reduction compared to the single waveguide modulator. The fabricated modulator having 14 peripheral waveguides with 1000-nm pitch achieved 180° phase modulation with 2.56 mW heater power consumption, representing 9.53 times efficiency improvement compared to the single waveguide modulator.

Figure 3(c) shows the simulated transient evolution of the waveguide center-plane temperature profile of the fabricated modulator with 10-mW step input given to the heater at time zero, with which we investigate the impact of geometrical design optimization on the thermo-optic modulation bandwidth. The x-axis position of the 14 peripheral waveguides with 1-μm pitch in the modulator extends from −7 μm to 7 μm. As the transient temperature profile is symmetric around the origin, only the positive x-axis part is plotted. The inset graphs show that the geometrical design optimization slows down the thermo-optic modulation speed. The 10–90% rise time of 27.6 μs for a single waveguide modulator increases only slightly to 34.8 μs with the fabricated modulator having 14 peripheral waveguides. These results indicate that the geometrical design optimization provides a better trade-off between efficiency and modulation bandwidth, compared to that of the heat isolation process (surveyed in Fig. 1) using air-gap trench and silicon undercut (See Appendix B-2 for the transient analytical model of the thermo-optic modulator with the geometrical design optimization, and Appendix B-3 for the explanation on the cause of the trade-off enhancement).

Figure 3(d) shows the simulated thermal crosstalk induced by an adjacent modulator. For a given modulator pitch, as the number of the peripheral waveguides increases, the thermal crosstalk increases. With the fabricated modulator (N = 14), the crosstalk increases by 3.2 dB compared to a single waveguide modulator (N = 0). To suppress thermal crosstalk below −30 dB, the minimum pitch with the fabricated modulator is 47 μm, which is 7 μm larger than that of the single waveguide modulator. When the number of peripheral waveguides increases from 14 to 28, comparing the increased thermal crosstalk [Fig. 3(d)] with the decreased power consumption [Fig. 3(b)] shows that the thermal crosstalk continues to increase after the modulation efficiency improvement saturates.

2.3. Experimental evaluation

Figure 4 shows the experimentally measured characteristics of the thermo-optic modulator using an on-chip Mach-Zhender interferometer (MZI) as illustrated in Fig. 4(a). Optical loss and optical bandwidth are characterized by measuring the MZI extinction ratio by sweeping a laser source from 1520 nm to 1600 nm, which couples into the chip through a grating coupler. Modulator power consumption is characterized by measuring the free spectral range (FSR) of the MZI test structure while sweeping the heater driving power from zero to 10 mW. Thermo-optic modulation bandwidth is characterized by measuring the rise time and the fall time of the MZI modulated by a 1-kHz square-wave voltage input to the heater, and verified by measuring the MZI frequency response with a sinusoidal voltage input to the heater. For the modulation bandwidth characterization, a trans-impedance amplifier (TIA) is used instead of an optical power meter. See Appendix C for further details on the experimental setup.

Fig. 4 Experimental results. (a) Mach-Zhender interferometer (MZI) for the experimental characterization of the thermo-optic modulator. (b) Extinction ratio of −23.03 dB on average with three test chips for optical loss characterization, corresponding to the optical loss of 1.23 dB. (c) Power consumption of 2.56 mW for 180° thermo-optic phase modulation. (d) Normalized optical transmission, which is extracted from the measured MZI output power, for optical bandwidth characterization. (e) Thermo-optic modulation bandwidth of 10.1 kHz, which is measured in time domain by observing 10–90% rise and fall time. (f) Thermo-optic modulation bandwidth, which is measured in frequency domain by using a sinusoidal modulation signal.

Download Full Size | PDF

The average extinction ratio from the MZI measurement is −23.03 dB at 1550 nm (Fig. 4b), which corresponds to the optical loss of 1.23 dB. See Appendix A-4 for design techniques to further reduce the optical loss from 1.23 dB to sub-1 dB by using Clothoid bend waveguides with a continuously varying width. Three modulators in the same wafer but at a different sample die are measured and results in less than ±0.1 dB variation in the loss. The modulator dissipates 2.56 mW for 180° phase shift [Fig. 4(c)]. The 1-dB optical bandwidth of the modulator is much larger than 80 nm from 1520 nm to 1600 nm [Fig. 4(d)]. This optical bandwidth measurement range is limited by the laser source and the grating coupler. With an improved experimental setup, a 1-dB optical bandwidth of larger than 100 nm will be observed.

Concerning thermo-optic modulation bandwidth characterization, the measured rise and fall time are 34.8 μs and 34.4 μs, respectively [Fig. 4(e)]. The 1550.2-nm wavelength aligns with the MZI null point, where the MZI output power becomes the minimum, to the zero heater power. With this zero calibration applied, the peak voltage of the 1-kHz square-wave pulse train is set to 808 mV such that the heater dissipates 2.56-mW power. This two-step calibration balances the rise time with the fall time. The minimum and maximum sinusoid input voltage to the heater is matched with that of the square-wave pulse train. The measured 3-dB frequency of the MZI frequency response [Fig. 4(f)] is 10.1 kHz, which is close to the first-order approximation 10.06 kHz from the rise time as well as the first-order approximation 10.17 kHz from the fall time. The modulator bandwidth can be further increased by applying digital pre-emphasis techniques [60, 61] with a moderate increase in power consumption. As the heater power dissipation increases from zero to 2.56 mW for 180° phase modulation, the TiN heater resistance increases from 243.7 Ω to 249.5 Ω, showing a temperature dependency of 0.61% with the TiN heater. Summary of the experimental results is listed in Table 1 in comparison with previously reported silicon thermo-optic modulators.

Table 1. Comparison of Experimental Results with Previously Reported Near-Infrared Silicon Thermo-Optic Modulators.

View Table | View all tables in this article

3. Discussion

Energy-efficient, compact, low-loss optical modulators are key components in many large-scale silicon photonic integrated circuit applications. Past research has focused on the realization of high-speed energy-efficient optical modulators for optical communication transceivers. Such carrier-injection and carrier-depletion electro-optic modulators are not suitable to large-scale integration due to high optical loss and large physical size. Optical resonance based modulators, while energy-efficient and compact, do not support a wide optical bandwidth that is needed in many emerging applications. Thermo-optic waveguide modulators have a low optical loss, and can be energy efficient if heat energy can be delivered to and confined in the optical waveguide efficiently. In the past, air-gap trench isolation and silicon undercut post-processing have been used to increase the energy efficiency of thermo-optic modulators. However, these approaches are not reliable for large-scale integration and result in a lower modulation speed typically below 0.5 kHz.

Here, we experimentally demonstrate a geometrically-optimized compact structure that leads to a record-low power consumption among thermo-optic modulators without air-gap trench and silicon undercut while maintaining a low loss on the order of 1 dB and a large optical bandwidth beyond 100 nm. The geometrical design optimization for efficiency and modulation bandwidth improvement, which is demonstrated in this work with a TiN heater and silicon waveguides beneath the heater, is not specific to a particular implementation but can be applied to other structures and devices.

The geometrical design optimization may realize a large-scale integration of 1,000 low-loss optical modulators on a chip with a less than 5 mm² area and an average power consumption below 1 W by exploring a feasible design space with geometrical optimization. This level of integration will provide optical phased array performance beyond the present state-of-the-art with a reduced system cost while enabling a number of other emerging applications with large-scale programmable photonic integrated circuits.

Appendix A: Multi-Clothoid 180° bend waveguide

Low-loss 180° bend waveguide structure is critical to the fabricated low-power thermo-optic modulator, which consists of 15 parallel waveguides with 1 μm-pitch and alternating widths (400 nm and 500 nm). Since these waveguides are connected by 14 U-turn (180°) bend waveguides, if a conventional circular bend waveguide with 0.5 μm-radius connects these parallel waveguides, the overall optical loss of the thermo-optic modulator becomes larger than 50 dB.

In order to reduce such a huge loss, we propose a multi-Clothoid 180° bend waveguide structure, which can achieve less than 0.08 dB optical loss to connect two parallel waveguides with different widths (400 nm and 500 nm) and thus we can reduce the modulator optical loss from 50 dB to approximately 1 dB. Clothoid curve, which is also called as Euler curve [62], has a property that its curvature at any point is proportional to the distance along the waveguide center line, measured from the origin. Cherchi et al. [58] and Nakai et al. [57] showed that bend waveguides satisfying this property significantly reduce the bend waveguide loss due to the excitation of high-order modes as well as the mode mismatch between straight and bend waveguides. Note that the curvature κ at a point P on a curve is defined as the inverse of the radius R of the tangent circle that approaches the curve the most tightly at P. In other words, κ = 1/R, and such a tangent circle, which best approximates a curve at a point on the curve, is referred as an osculating circle of the curve on the given point [63].

In the following for sub-sections in Appendix A, after briefly reviewing the geometry of a Clothoid curve, we describe 1) the specification on the geometrical structure of a multi-Clothoid 180° bend waveguide, 2) design equations to create a multi-Clothoid bend waveguide structure for a given geometrical specification, 3) a systematic design procedure, which allows the minimization of the optical loss for a given physical size constraint on the bend structure, 4) and techniques to further reduce the optical loss of the multi-Clothoid 180° bend waveguide.

A-1. Geometrical structure

Figure 5(a) shows a multi-Clothoid 180° bend waveguide, which consists of six segments, in the fabricated thermo-optic modulator. Nakai et al. [57] showed that the minimum optical loss of a Clothoid bend waveguide is with a bend angle larger than 90°, and thus we design the segment 1 and its symmetrically transposed shape (segment 2) to provide a combined bend angle larger than 180°. Note that the segment 3 and its symmetrically transposed shape (segment 4) have a small bending angle, such that the segment 1 and 2 dominate the optical loss while allowing the overall bend structure to provide 180° bending. Since the adjacent waveguides that are connected through the multi-Clothoid 180° bend structure have different widths (to reduce the coupling between the waveguides), a taper waveguide (segment 5) provides necessary width transition. The segment 6 is a straight waveguide, aligning the two end points.

Fig. 5 Multi-Clothoid 180° bend waveguide structure in the fabricated thermo-optic modulator. 15 parallel straight waveguides with alternating widths in the modulator are connected by two types of 14 U-turn bend structures, each of which is constructed by four Clothoid bend waveguides (Segment 1–4), a tilted taper waveguide (Segment 5), and a straight strip waveguide (Segment 6). (a) bend waveguide structure, which is analytically drawn by using geometrical design equations derived in Appendix A-2, in the fabricated modulator. (b) Layout of a Clothoid bend waveguide segment 1. (c) Curvature change of the fabricated multi-Clothoid bend structure as a function of path length L starting from the segment 1. Note that the curvature of each Clothoid curve segment is zero at the point where the curve interfaces a straight waveguide or a taper waveguide. The curvature reaches the maximum at the point where the waveguide bend angle ϕ has the maximum value of θ.

Download Full Size | PDF

The geometry of a Clothoid curve can be uniquely determined by two parameters as illustrated in Fig. 5(b) — a bend angle θ and a constant A. Once these two parameters are given, Cartesian coordinate pairs (x, y), which define a corresponding Clothoid curve, can be obtained as follows. For one point P(x, y) on the bend waveguide center line connecting two end points O and E, the bend angle of the waveguide front at the point P is denoted by ϕ. For another point P′(x′, y′) on the center line, which is adjacent to the point P with a small distance dl, an infinitesimal dϕ represents an increase in the bend angle of waveguide front at the point P towards the adjacent point P′. Let R(ϕ) be the radius of the osculating circle at the point P on the Clothoid curve. Then the following relations hold.

d l = R (ϕ) d ϕ, d x = d l cos ϕ, d y = d l sin ϕ .

The constant A constrains the Clothoid curve such that

R (ϕ) L (ϕ) = A^{2}

where L(ϕ) is the path length from the point O to the point P′, which is obtained from Eq. (1) as

L (ϕ) = \int_{0}^{ϕ} R (ψ) d ψ .

Eq. (1) and Eq. (2) lead to

\begin{matrix} d l = \frac{A^{2}}{L (ϕ)} d ϕ \\ \int_{0}^{L (ϕ)} l d l = \int_{0}^{ϕ} A^{2} d ψ \\ \frac{L {(ϕ)}^{2}}{2} = A^{2} ϕ, \end{matrix}

which can be rewritten as

L (ϕ) = A \sqrt{2 ϕ}

Substituting (2) and (5) into (1) gives

d x = (\frac{A}{\sqrt{2 ϕ}} cos ϕ) d ϕ, d y = (\frac{A}{\sqrt{2 ϕ}} sin ϕ) d ϕ .

Using a series expansion, for 0 < ϕ < θ, Euler [62] showed that the coordinate P(ϕ) = [x(ϕ) y(ϕ)]^T is obtained from (6) as

\begin{array}{l} x (ϕ) & = & \frac{A}{\sqrt{2}} \int_{0}^{ϕ} \frac{cos ψ}{\sqrt{ψ}} d ψ \\ = & \frac{A}{\sqrt{2}} \int_{0}^{ϕ} \frac{1}{\sqrt{ψ}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)!} ψ^{2 k} d ψ \\ = & \frac{A}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{k}}{(2 k)!} \int_{0}^{ϕ} ψ^{2 k - \frac{1}{2}} d ψ] \\ = & \frac{A}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{k}}{(2 k + \frac{1}{2}) \cdot (2 k)!} ϕ^{2 k + \frac{1}{2}}] \end{array}

and

\begin{array}{l} y (ϕ) & = & \frac{A}{\sqrt{2}} \int_{0}^{ϕ} \frac{sin ψ}{\sqrt{ψ}} d ψ \\ = & \frac{A}{\sqrt{2}} \int_{0}^{ϕ} \frac{1}{\sqrt{ψ}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} ψ^{2 k + 1} d ψ \\ = & \frac{A}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{k}}{(2 k + 1)!} \int_{0}^{ϕ} ψ^{2 k + \frac{1}{2}} d ψ] \\ = & \frac{A}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{k}}{(2 k + \frac{3}{2}) \cdot (2 k + 1)!} ϕ^{2 k + \frac{3}{2}}] . \end{array}

The inner and the outer edge coordinates of the Clothoid bend waveguide are determined with a waveguide width W for the center line coordinates P(ϕ):

P^{i} = [\begin{matrix} x (ϕ) - \frac{W}{2} sin ϕ \\ y (ϕ) + \frac{W}{2} cos ϕ \end{matrix}], P^{o} = [\begin{matrix} x (ϕ) + \frac{W}{2} sin ϕ \\ y (ϕ) - \frac{W}{2} cos ϕ . \end{matrix}]

Therefore, the complete geometry of the proposed multi-Clothoid 180° bend structure, which consists of 6 segments, can be specified by the six parameters listed in Table 2. Note that the two parameters, θ_b and A_b, for the segment 3, 4 can be derived from other parameters, as will be shown in Appendix A-2 and A-3.

Table 2. Geometrical Parameters of a Multi-Clothoid 180° Bend Waveguide.

View Table | View all tables in this article

A-2. Geometrical design equations

We derive a set of design equations to create a multi-Clothoid 180° bend waveguide from a geometrical specification listed in Table 2. These design equations are also used to optimize the bend waveguide such that the optical loss is minimized, as will be discussed in Appendix A-3.

The center line geometry of the segment 1 (Clothoid bend) in Cartesian coordinate P₁(x₁, y₁) = [x₁(ϕ) y₁(ϕ)]^T is determined by Eq. (7) and Eq. (8) with 0 ≤ ϕ ≤ θ_a from the two parameters in Table 2 — θ_a and A_a. The inner and the outer coordinates of the segment 1 are determined by Eq. (9) from the parameter W_a, which specifies the width.

The center line geometry of the segment 2 (symmetrically transposed shape of the segment 1) in Cartesian coordinate P₂(x₂, y₂) = [x₂(ϕ) y₂(ϕ)]^T is obtained by applying three geometrical transformations on the segment 1. First, the segment 1 is transposed by switching its x and y coordinates:

P_{1}^{S} (ϕ) = M [\begin{matrix} x_{1} (ϕ) \\ y_{1} (ϕ) \end{matrix}] = [\begin{matrix} x_{1} (ϕ) \\ y_{1} (ϕ) \end{matrix}] = [\begin{matrix} y_{1} (ϕ) \\ x_{1} (ϕ) \end{matrix}]

where M is a 2×2 permutation matrix, which interchanges two rows. Then, the origin of the segment 2 is adjusted, such that the segment 2 adjoins the segment 1 at the point E(x₁(θ_a), y₁(θ_a)):

\begin{array}{l} P_{1}^{A} (ϕ) & = & P_{1}^{S} (ϕ) + P_{1} (θ_{a}) - P_{1}^{T} (θ_{a}) \\ = & [\begin{array}{l} y_{1} (ϕ) + x_{1} (θ_{a}) - y_{1} (θ_{a}) \\ x_{1} (ϕ) + y_{1} (θ_{a}) - x_{1} (θ_{a}) \end{array}] . \end{array}

The third transformation is to rotate the adjusted second segment by the angle δ around the point E, which gives us the coordinates of the segment 2 as

\begin{array}{l} P_{2} (ϕ) & = & {[x_{2} (ϕ) y_{2} (ϕ)]}^{T} = J (P_{1}^{A} (ϕ - E) + E \\ = & [\begin{array}{l} - x_{1} (ϕ) sin δ_{a} + y_{1} (ϕ) cos δ_{a} + x_{1} (θ_{a}) [1 + sin δ_{a}] - y_{1} (θ_{a}) cos δ_{a} \\ x_{1} (ϕ) cos δ_{a} + y_{1} (ϕ) sin δ_{a} - x_{1} (θ_{a}) cos δ_{a} + y_{1} (θ_{a}) [1 - sin δ_{a}] \end{array}] . \end{array}

In Eq. (12), J is a rotation matrix given by

J = [\begin{array}{l} cos δ_{a} & - sin δ_{a} \\ sin δ_{a} & cos δ_{a} \end{array}] .

In order to connect the segment 2 with the segment 1 such that the osculating circle of the segment 1 at the interfacing point E coincides with that of the segment 2 at the point E, it is geometrically obvious that the necessary rotation angle δ_a is given by the following relation:

δ_{a} = 2 θ_{a} - \frac{3 π}{2} .

The center line geometry of the segment 3 and the segment 4 are determined in the same way as the segment 1 and the segment 2, but with a different origin position. Assuming that the origin of the segment is (0, 0), let the Cartesian coordinates of the segment 3 and the segment 4 be P′₃(ϕ) = [x′₃(ϕ) y′₃(ϕ)]^T and P′₄(ϕ) = [x′₄(ϕ) y′₄(ϕ)]^T , respectively. Since δ_b = 2θ_b + π/2, we obtain P′₃ and P′₄ for 0 < ϕ < θ_b from Eqs. (7)–(8) and (12) as

\begin{array}{l} P_{3}^{'} (ϕ) & = & [\begin{matrix} \frac{A_{b}}{\sqrt{2}} \sum_{k = 0}^{\infty} (\frac{{(- 1)}^{k}}{(2 k + \frac{1}{2}) \cdot (2 k)!} ϕ^{2 k + \frac{1}{2}}) \\ \frac{A_{b}}{\sqrt{2}} \sum_{k = 0}^{\infty} (\frac{{(- 1)}^{k}}{(2 k + \frac{3}{2}) \cdot (2 k + 1)!} ϕ^{2 k + \frac{3}{2}}) \end{matrix}], \\ P_{4}^{'} (ϕ) & = & [\begin{array}{l} - x_{3}^{'} (ϕ) sin δ_{b} + y_{3}^{'} (ϕ) cos δ_{b} + x_{3}^{'} (θ_{b}) [1 + sin δ_{b}] - y_{3}^{'} (θ_{b}) cos δ_{b} \\ x_{3}^{'} (ϕ) cos δ_{b} + y_{3}^{'} (ϕ) sin δ_{b} - x_{3}^{'} (θ_{b}) cos δ_{b} + y_{3}^{'} (θ_{b}) [1 - sin δ_{b}] \end{array}] . \end{array}

Note that the bend angle θ_b for the segment 3 and the segment 4 is related to θ_a such that the overall structure has 180° bending:

θ_{b} = θ_{a} - \frac{π}{2} .

We can show that the necessary origin O_b for the segment 3 and the segment 4 is

O_{b} = [\begin{matrix} O_{b x} \\ O_{b y} \end{matrix}] = [\begin{matrix} - x_{4}^{'} (0) + x_{2} (0) - L_{t} cos (2 θ_{b}) \\ g \end{matrix}]

where g is the pitch of the two straight waveguides that are connected by the multi-Clothoid 180° bend structure and L_t is the length of the taper waveguide (segment 5). Therefore, the center line Cartesian coordinates of the segment 3, P₃(ϕ) = [x₃(ϕ) y₃(ϕ)]^T , and the segment 4, P₄(ϕ) = [x₄(ϕ) y₄(ϕ)]^T , with a proper origin location are given by

P_{3} (ϕ) = P_{3}^{'} (ϕ) + O_{b} = [\begin{matrix} x_{3}^{'} (ϕ) + O_{b x} \\ y_{3}^{'} (ϕ) + g \end{matrix}],

P_{4} (ϕ) = P_{4}^{'} (ϕ) + O_{b} = [\begin{matrix} x_{4}^{'} (ϕ) + O_{b x} \\ y_{4}^{'} (ϕ) + g \end{matrix}],

where

\begin{array}{l} O_{b x} & = & - [x_{3} (θ_{b}) (1 + sin δ_{b}) - y_{3} (θ_{b}) cos δ_{b}] \\ + [x_{1} (θ_{a}) (1 + sin δ_{a}) - y_{1} (θ_{a}) cos δ_{a}] - L_{t} cos (2 θ_{b}) \\ = & - x_{3} (θ_{a} - π / 2) (1 - cos (2 θ_{a})) + y_{3} (θ_{a} - π / 2) sin (2 θ_{a}) \\ + x_{1} (θ_{a}) (1 + cos (2 θ_{a})) + y_{1} (θ_{a}) sin (2 θ_{a}) \\ + L_{t} cos (2 θ_{a}) . \end{array}

The center line geometry of the segment 5 (tilted taper waveguide) in Cartesian coordinate P₅(x₅, y₅) = [x₅(l) y₅(l)]^T for 0 ≤ l ≤ L_t is determined by the parameter L_t and θ_b. Since the tile angle of the segment 5 is 2θ_b and the origin coordinate is given by O_t = [x₄(0) y₄(0)]^T , the center line coordinate of the segment 5 is obtained as

P_{5} (l) = [\begin{matrix} x_{5} (l) \\ y_{5} (l) \end{matrix}] = [\begin{matrix} x_{4} (0) + l sin (2 θ_{b}) \\ y_{4} (0) + l cos (2 θ_{b}) \end{matrix}] .

With the fabricated modulator, since the width difference between the two ends of the taper is 0.1 μm, a linear taper with 2 μm length is used, which has less than 0.01 dB optical loss. The inner and the outer coordinates of the linear taper are obtained from the center line coordinate P₅(l) as

P_{5}^{i} (l) = P_{5} (l) - W_{e} (l), P_{5}^{o} (l) = P_{5} (l) + W_{e} (l)

where the taper edge offset coordinate W_e(l) is given by

W_{e} (l) = [\begin{matrix} (\frac{W_{b} + (W_{a} - W_{b}) / L_{t}}{2}) cos ϕ \\ - (\frac{W_{b} + (W_{a} - W_{b}) / L_{t}}{2}) sin ϕ \end{matrix}]

with the segment 1–2 width W_a and the segment 3–4 width W_b.

The center line geometry of the segment 6 (straight waveguide) is defined by its length, which is given by |x₃(0)| = |O_bx|.

Table 3 summarizes the complete geometrical design equations for a given parameter set listed in Table 2. For simplicity, design equations for the center line geometry of six segments are listed. The edge coordinates of each segment can be obtained by using Eq. (9) for the Clothoid curve segments and Eq. (22) for the taper segment. Note that modification on the calculated coordinates using these design equation are necessary when generating final mask design data for silicon photonics foundry. This additional step is to satisfy foundry mask design rule requirements.

Table 3. Geometrical Design Equations to Create a Multi-Clothoid 180° Bend Waveguide (see Eq. (20) for O_bx) and Eqs. (27)–(30) for A_b).

View Table | View all tables in this article

A-3. Multi-Clothoid bend design method

To the fabricated low-power thermo-optic modulator, a simple design methodology is applied, which finds the geometry of a multi-Clothoid 180° bend such that its overall optical loss is minimized when the height of the bend H is constrained. Considering thermal crosstalk between adjacent modulators as well as optical crosstalk between adjacent waveguides in a modulator, three parameters (waveguide widths W_a, W_b, and waveguide pitch g) in Table 2 and the height of the bend H are determined. With the predetermined H, the other parameters in Table 2 can be calculated for a valid range of θ_a. Note that θ_a less than 90° leads to a physically very large structure while θ_a larger than 110° makes the segment 2 to undesirably intersect the segment 1. Therefore, Clothoid bend structures are created for 90° < θ_a < 110°. The overall optical loss for each bend structure is obtained by 3D FDTD simulations, so that a design with the minimum loss can be selected. A pseudo code description for this design methodology can be stated in Algorithm 1.

Algorithm 1. Multi-Clothoid 180° bend waveguide design procedure under a physical size constraint.

View Table | View all tables in this article

In what follows, to apply this design methodology, we first derive two equations, which correspond to the function f_a and f_b in the pseudo code description. Then we showcase a multi-Clothoid bend waveguide design with the fabricated thermo-optic modulator.

For finding A_a corresponding to a given θ_a and H, we derive an equation that uniquely relates these variables. The height of the multi-Clothoid bend H, which we define as the distance between two waveguide centerlines as shown in Fig. 6(a), is equal to y₂(ϕ′) where the slope of the segment 2 is zero with the value of ϕ = ϕ′. When this zero-slope condition holds,

\frac{d y_{2} (ϕ)}{d x_{2} (ϕ)} = \frac{d y_{2} (ϕ)}{d ϕ} / \frac{d x_{2} (ϕ)}{d ϕ} = 0,

using Eq. (12), we obtain

\begin{array}{l} \frac{d y_{2} (ϕ)}{d ϕ} & = & cos δ_{a} \frac{d x_{1} (ϕ)}{d ϕ} + sin δ_{a} \frac{d y_{1} (ϕ)}{d ϕ} \\ = & cos δ_{a} (\frac{A}{\sqrt{2 ϕ}} cos ϕ) + sin δ_{a} (\frac{A}{\sqrt{2 ϕ}} sin ϕ) \\ = & \frac{A}{\sqrt{2 ϕ}} cos (δ_{a} - ϕ) = 0 . \end{array}

Thus we find

ϕ^{'} = δ_{a} + \frac{π}{2} = 2 θ_{a} - π,

and the 180° bend height is obtained by using Eq. (12) as

\begin{array}{l} H & = & y_{2} (ϕ^{'}) = [y_{2} (ϕ^{'}) - y_{2} (θ_{a})] + y_{1} (θ_{a}) \\ = & [x_{1} (2 θ_{a} - π) - x_{1} (θ_{a})] cos δ_{a} + [y_{1} (2 θ_{a} - π) - y_{1} (θ_{a})] sin δ_{a} + y_{1} (θ_{a}) \\ = & \frac{A_{a}}{\sqrt{2}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} ({(2 θ_{a} - π)}^{2 k + \frac{1}{2}} - θ_{a}^{2 k + \frac{1}{2}})}{(2 k + \frac{1}{2}) (2 k + 1)!} cos δ_{a} \\ + \frac{A_{a}}{\sqrt{2}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} [({(2 θ_{a} - π)}^{2 k + \frac{3}{2}} - θ_{a}^{2 k + \frac{3}{2}}) sin δ_{a} + θ_{a}^{2 k + \frac{3}{2}}]}{(2 k + \frac{3}{2}) (2 k + 1)!} \end{array}

where δ_a = 2θ_a − 3π/2. Therefore, A_a can be determined from H and θ_a by using Eq. (27).

Fig. 6 Multi-Clothoid 180° bend waveguide design. (a) Analytically created bend structures with H = 4 μm, g = 1 μm, W_a = 500 nm, and W_b = 400 nm for a varying Clothoid bend angle θ_a. Four parameters (H_t, g, W_a, W_b) are predetermined by considering thermal and optical crosstalk while θ_a provides a degree of freedom to minimize the optical loss of the bend structure. (b) Experimentally measured optical loss of multi-Clothoid bend waveguides with θ_a = 105° (0.077 dB in average with 0.014 dB standard deviation from 5 samples) in comparison with simulation results. The inset SEM photographs show the two types of the bend structure (the measured optical loss is an average on the two types). (c) Comparison of the type-A multi-Clothoid 180° bend structure with conventional 180° bend structures based on standard circular bend waveguides. The improved type-A multi-Clothoid bend structure is described in Appendix A-4.

Download Full Size | PDF

Likewise, for finding A_b corresponding to a given θ_a, H, and g, we first find an equation on the combined height of the segment 3 and the segment 4:

\begin{array}{l} h_{2} & = & H - g - L_{t} sin (2 θ_{b}) - [y_{2} (ϕ^{'}) - y_{2} (0)] \\ = & [- x_{1} (θ_{a}) cos δ_{a} + y_{1} (θ_{a}) (1 - sin δ_{a})] - g + L_{t} sin (2 θ_{a}), \end{array}

which is computed as

\begin{array}{l} h_{2} & = & - g + L_{t} sin (2 θ_{a}) + \frac{A_{a}}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{2} sin (2 θ_{a})}{(2 k + \frac{1}{2}) \cdot (2 k)!} θ_{a}^{2 k + \frac{1}{2}}] \\ + \frac{A_{a}}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{k} (1 - cos 2 θ_{a})}{(2 k + \frac{3}{2}) \cdot (2 k + 1)!} θ_{a}^{2 k + \frac{3}{2}}] \end{array}

where θ_b is given by Eq. (16) as θ_b = θ_a − π/2. Then, since the combined height h₂ can be also represented as y′₄(0), using Eq. (15) and a relation δ_b = 2θ_b + π/2 = 2θ_a − π/2, we obtain an alternative expression on h₂ in terms of A_b by

\begin{array}{l} h_{2} & = & y_{4}^{'} (0) = - x_{3}^{'} (θ_{b}) cos (δ_{b}) + y_{3}^{'} (θ_{b}) (1 - sin δ_{b}) \\ = & \frac{A_{b}}{\sqrt{2}} \sum_{k = 0}^{\infty} [- \frac{{(- 1)}^{k} sin (2 θ_{a})}{(2 k + \frac{1}{2}) \cdot (2 k)!} {(θ_{a} - \frac{π}{2})}^{2 k + \frac{1}{2}}] \\ + \frac{A_{b}}{\sqrt{2}} \sum_{k = 0}^{\infty} [\frac{{(- 1)}^{k} (1 + cos 2 θ_{a})}{(2 k + \frac{3}{2}) \cdot (2 k + 1)!} {(θ_{a} - \frac{π}{2})}^{2 k + \frac{3}{2}}] . \end{array}

Therefore, A_b can be determined from H and θ_a by matching Eq. (29) with Eq. (30).

The overall center line length of the multi-Clothoid bend structure, which contributes to the overall bend loss, can be obtained from Eq. (5) as

L_{u} = 2 (A_{a} \sqrt{2 θ_{a}} + A_{b} \sqrt{2 θ_{b}} + L_{t} + O_{b x})

where O_bx is given by Eq. (20).

With the fabricated prototype low-power thermo-optic modulator with 15 parallel waveguides with 1 μm pitch, to suppress thermal crosstalk below −30 dB with 47 μm modulator pitch, the width of the modulator waveguide array needs to be less than 15 μm. This design requirement can be met by designing two types of the multi-Clothoid 180° bend waveguide structure (type-A and type-B) both with H = 4 μm and g = 1 μm. The Type-A bend structure is designed with W_a = 400 nm and W_b = 500 nm while the Type-B bend structure is designed with W_a = 500 nm and W_b = 400 nm. The length L_t of the segment 5 (linear taper waveguide) is set to 2 μm, which introduces less than 0.01 dB loss.

Figure 6(b) shows the optical loss of multi-Clothoid 180° bend waveguides with the four predetermined parameters (H, W_a, W_b, g) for 90° < θ_a < 110°. Twenty bend structures are created each with a different value of θ_a by using Eqs. (27)–(30) with design equations in Table 3. Then, the optical loss of each bend structure is evaluated by 3D FDTD simulation. Waveguide bending loss dominates for a small θ_a due to a large curvature while optical propagation loss dominates for a large θ_a due to a long optical path length. Therefore, a value of θ_a exists, which minimizes the overall optical loss of the multi-Clothoid bend. The optical loss of the designed bend structure is experimentally measured as an average of eight Type-A structures and six Type-B structures that are cascaded in the fabricated modulator. The measured average bend loss is 0.077 dB with a standard deviation of 0.014 dB, which is in-between the simulated bend loss of bend type A and bend type B, showing good agreement between measurement and simulation. Figure 6(c) compares the simulated optical loss of the multi-Clothoid bend with standard circular bends, showing that the multi-Clothoid bend has a smaller size and a lower optical loss at the same time.

Table 4 summarizes the geometric design parameters of the multi-Clothoid 180° bend waveguide structures in the fabricated thermo-optic modulator with θ_a = 105°, H = 4 μm, and g = 1 μm, which resulted in 0.077 dB measured optical loss.

Table 4. Geometrical Design Parameters of the Multi-Clothoid 180° Bend Waveguide Structures in the Fabricated Thermo-Optic Modulator.

View Table | View all tables in this article

A-4. Design technique for sub-1dB optical loss

Since the optical loss from 14 bend structures, which is 1.08 dB on average, dominates the overall optical loss of the fabricated modulator, we are exploring techniques to further reduce the optical loss. Figure 6(b) shows that the optical loss of the type-A (segment 1 and segment 2 have 400 nm width) multi-Clothoid bend structures with θ_a = 105° has 0.099 dB loss while the type-B bend structure with the same θ_a has only 0.033 dB loss. Based on this observation, we have investigated techniques to reduce the optical loss of the type-A bend.

In one of our recent experimental works, Nakai et al. [57] have successfully demonstrated that the optical loss of a Clothoid bend waveguide with θ_a = 105° and 400 nm width can be reduced from 0.099 dB to 0.048 dB by gradually changing the inner wall of the Clothoid bend waveguide as shown in Fig. 7. If we incorporate this modified Clothoid 105° bend waveguide into the segment 1 and 2 of the type-A multi-Clothoid 180° bend structure, our measurement results [57] predict that the overall thermo-optic modulator loss can decrease from 1.23 dB to 0.82 dB. The loss of the type-B bend can be reduced with the same technique, which will further reduce the overall thermo-optic modulator loss.

Fig. 7 SEM photograph of a low-loss Clothoid bend waveguide that we designed such that the inner wall width (distance between the center line and the inner edge) continuously varies from 400 nm to 500 nm by following a Heaviside function described in [57].

Download Full Size | PDF

Appendix B: Analytical models of thermo-optic modulator

To understand the fundamental design trade-off of thermo-optic modulators, we develop a simple analytical model, which relates heater power consumption to a corresponding optical phase shift. Heat conduction process and its analytical framework have been extensively studied with basic geometries and boundary conditions [64]. We build the analytical model by combining these well-known heat conduction solutions with a first-order thermo-optic modulation model. In the following sections, we present a steady-state model and a transient model in comparison to simulation and measurement results. The steady-state model allows us to predict an upper bound on the minimum power consumption achievable by adding peripheral waveguides. With the transient model, we discuss how geometrical design optimization affects the relation between energy efficiency and optical modulation bandwidth.

B-1. Steady-state model for estimating thermo-optic modulator power consumption

Figure 8(a) shows the physical model of a thermo-optic modulator with peripheral waveguides along with a simplified thermal analysis model, which approximates the physical model into a cylindrical shape. The outer surface of the cylinder model corresponds to the silicon substrate of the physical model. To simplify the model geometry, the cylinder model assumes that the heater and the waveguides are placed on the same plane. In addition, the heater itself is modeled as a smaller inner cylinder with a radius of r_i, which overlaps with the center waveguide.

Fig. 8 Steady-state analytical model of the thermo-optic modulator with peripheral waveguides for geometrical design optimization. (a) Physical modulator structure and simplified cylindrical thermal analysis model, which represents an array of 2k +1 waveguides and a heater. Note that the heater overlaps with the center waveguide in the analysis model (this assumption simplifies the mathematical representation of the model). (b) Comparison on the thermo-optic modulator power consumption between analytical model prediction, 3D numerical device simulation, and experimental measurement result. Note that N represents the number of peripheral waveguides.

Download Full Size | PDF

The symmetry of the cylinder model simplifies the general heat diffusion equation [64],

\nabla \cdot (k \nabla T) + \dot{q} = ρ c_{p} \frac{\partial T}{\partial t},

into

\frac{1}{r} \frac{\partial}{\partial r} (k r \frac{\partial T (r)}{\partial r}) = 0

where ρ is the density [Kg/m³], c_p is the specific heat [J/Kg·K], k is the thermal conductivity [W/m·K] for silicon-dioxide [SiO₂], and q̇ is the heat generation rate [W/m³] per unit volume. Note that, with the heater power consumption being considered as a thermal boundary condition, the volumetric heat generation rate q̇ can be represented as zero. Assuming that the heater interfaces with the inner wall of the cylinder, the heater imposes a Neumann (constant heat flux) boundary condition. In a thermal steady-state, we assume that the outer wall of the cylinder with a radius of r_o is exposed to a Dirichlet (constant temperature) boundary condition. Then, the temperature of the cylinder inner wall T(r_i) is kept constant by the heater while the outer wall temperature T(r_o) is kept constant by an external temperature controller:

T (r_{i}) = T_{i}, T (r_{o}) = T_{o} .

With the boundary conditions imposed by Eq. (34), the temperature profile of the cylinder T(r) is obtained as the solution of the differential heat conduction equation given in Eq. (33):

\begin{array}{l} T (r) & = & T_{i} - \frac{T_{i} - T_{o}}{ln (r_{o}) - ln (r_{i})} (ln (r) - ln (r_{i})) \\ = & c_{1} ln (r) + c_{2} \end{array}

where

c_{1} = - \frac{T_{i} - T_{o}}{ln (r_{o}) - ln (r_{i})}, c_{2} = \frac{T_{i} ln (r_{o}) - T_{o} ln (r_{i})}{ln (r_{o} / r_{i})} .

The heater power consumption q_h [W] can be obtained from Fourier’s law [65] and the temperature profile T(r) as,

q_{h} = - k A \frac{d T}{d r} = - k (2 π r L) \frac{d T}{d r},

where A is the area of the inner cylinder wall, L is the length of the cylinder. Note that the Fourier’s law [65] on heat conduction in solids was empirically developed, which states that the heat transfer rate per unit area [W/m²] is proportional to the temperature gradient [K/m] and the heat is transferred in the direction of decreasing temperature (q/A = −k∇T), and can be approximated from a statistical mechanics model [66,67]. By substituting the temperature profile T(r) given by Eq. (35) into Eq. (37), the heater power consumption q_h can be written as

q_{h} = \frac{2 π k L (T_{i} - T_{o})}{ln (r_{o}) - ln (r_{i})},

from which the heater temperature T_i as a function of the heater power consumption can be obtained as

T_{i} = T_{o} + \frac{ln (r_{o} / r_{i})}{2 π k L} q h .

For a single straight optical waveguide with a length L, an optical phase shift ϕ_c is given by

ϕ_{c} = (\frac{2 π L}{λ}) \frac{d n}{d T} Δ_{T} = (\frac{2 π L}{λ}) \frac{d n}{d t} (T_{i} - T_{o}) .

For simplicity, as we assumed that the center waveguide overlaps with the heater, the center waveguide temperature can be represented by the heater surface temperature T_i. By substituting Eq. (39) into Eq. (40), we obtain the relation between the optical phase shift ϕ_c and the heat power consumption q_h as follows:

ϕ_{c} = \frac{d n}{d T} (\frac{ln (r_{o} / r_{i})}{k λ}) q h .

The heater power consumption q_h,π to achieve 180° phase shift is obtained from Eq. (41) when ϕ_c = π:

q_{h, r} = (\frac{k λ}{ln (r_{o} / r_{i})}) \frac{ϕ_{c}}{\frac{d n}{d T}} = (\frac{k λ}{ln (r_{o} / r_{i})}) \frac{π}{\frac{d n}{d T}} .

With geometrical design optimization using the 2k peripheral optical waveguides with a pitch p, the thermo-optic modulator’s overall optical phase shift ϕ_m is expressed as

\begin{array}{l} ϕ_{m} & = & \sum_{m = - k}^{k} (\frac{2 π L}{λ}) \frac{d n}{d T} (T | m | p) - T_{o}) \\ = & ϕ_{c} + 2 \sum_{m = 1}^{k} (\frac{2 π L}{λ}) \frac{d n}{d T} (T (mp) - T_{o}) \\ = & ϕ_{c} + 2 \sum_{m = 1}^{k} (\frac{2 π L}{λ}) \frac{d n}{d T} (\frac{ln (r_{o}) - ln (mp)}{ln (r_{o}) - ln (r_{i})} (T_{i} - T_{o})) . \end{array}

By denoting the additional phase shift achieved by the geometrical design optimization as ϕ_e, we obtain

\begin{array}{l} ϕ_{e} & = & ϕ_{m} - ϕ_{c} \\ = & (\frac{4 π L}{λ}) \frac{d n}{d T} \frac{T_{i} - T_{o}}{ln (r_{o} / r_{i})} ln (\frac{{(r_{o} / p)}^{k}}{k!}) . \end{array}

When the overall phase shift ϕ_m is 180°, let the phase shift contributed by the center waveguide at location r = 0 be ϕ′_c = ϕ_m − ϕ_e = π − ϕ_e. Then, the necessary heater power consumption to achieve π phase modulation with geometrical design optimization q′_h,π can be determined as

q_{h, π}^{'} = (\frac{k λ}{ln (r_{o} / r_{i})}) \frac{ϕ_{c}^{'}}{\frac{d n}{d T}} = (\frac{k λ}{ln (r_{o} / r_{i})}) \frac{π - ϕ_{e}}{\frac{d n}{d T}} .

Thus the reduction of the heater power consumption Δq_h,π by the geometrical design optimization can be written as

Δ q_{h, r} = q_{h, r} - q_{h, r}^{'} = (4 π k L) \frac{T_{i} - T_{o}}{{ln}^{2} (r_{o} / r_{i})} ln (\frac{{(r_{o} / p)}^{k}}{k!}) .

Since r_o/p ≥ k, this results reveal that the geometrical design optimization can reduce an arbitrarily large amount of heater power consumption by increasing the width of the modulator waveguide array 2kp. Nevertheless, under thermal crosstalk requirement, which sets the modulator pitch and the maximum modulator width, the reduction of heater power consumption Δϕ_h,π has an upper bound.

Figure 8(b) shows the analytically obtained power consumption of the fabricated thermo-optic modulator as a dashed line for 2 ≤ N ≤ 14, showing good agreement with 3D numerical simulation as well as experimental measurement result. It should be also noted that, due to the geometrical approximation of the analytical model to a cylindrical shape, the analytical model is valid for a limited range (r < 10 μm with the fabricated design). Simulation results also predict, with the availability of poly-crystalline waveguides as an additional optical routing layer above the heater, that the efficiency of geometrical design optimization will increase roughly by a factor of 2 so that 180° phase shift will be achieved with one milliwatt power consumption.

B-2. Transient model for estimating thermo-optic modulation bandwidth

We derive a transient analytical model, which relates a thermal step input to the temperature profile of the first-order cylinder model, and then to the resulting transient optical phase shift. The resulting model explains that the performance trade-off between the thermo-optic modulator energy efficiency and modulation bandwidth is more favorable with the geometrical design optimization compared to the conventional heat isolation techniques using air-gap trench and silicon undercut to reduce thermal conductivity. The thermal step response from the transient analytical model is useful to the design of a thermal predistortion equalizer [61], which extends the thermo-modulation bandwidth by pre-emphasizing the high frequency contents of the heater control signal.

With the simplified cylinder model shown in Fig. 8(a), which represents a thermo-optic modulator with peripheral waveguides, a general heat diffusion equation for materials with homogeneous thermal conductivity can be written in a differential form [64] as

\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial T (r, t)}{\partial r}) = \frac{1}{α} \frac{\partial T (r, t)}{\partial t}

where the temperature profile T(r, t) is a function of spatial position r and time t, and α is a thermal diffusivity of silicon dioxide, which is related to the thermal conductivity k, the density ρ, and the specific heat c_p of silicon dioxide as α = k/ρc_p. To simplify the complementary solution of the partial derivative equation (PDE), which will be shown later, we first consider a cooling problem instead of a heating problem where a thermal step input is applied. After the PDE solution for the cooling problem is obtained, we will transform it to the desired solution for the heating problem by using a time property of a first-order system. In the cooling problem, the steady-state temperature profile given in Eq. (35) provides an initial condition to Eq. (47):

T (r, 0) = - \frac{T_{i} - T_{o}}{ln (r_{o} / r_{i})} ln (r) + \frac{T_{i} ln (r_{o}) - T_{o} ln (r_{i})}{ln (r_{o} / r_{i})} .

Note that here we consider the steady-state condition as T(r, 0) rather than T(r, ∞). In the transformed solution for the heating problem, the steady-state condition will be reached as time t approaches infinity.

Using the method of the separation of variables, the solution of Eq. (47) can be assumed as T(r, t) = Γ(r)Λ(t). With this assumed solution, the transient heat equation is written as

\frac{1}{r Γ (r)} \frac{d}{d r} (r \frac{d Γ (r)}{d r}) = \frac{1}{α Λ (t)} \frac{d Λ (t)}{d t} .

It is well known that this heat equation can have physically meaningful solutions when both sides are equal to a negative constant, leading to two independent ordinary differential equations (ODEs):

\frac{1}{r} \frac{d}{d r} (r \frac{d Γ (r)}{d r}) = - ψ^{2}

and

\frac{1}{α} \frac{d Λ (t)}{d t} = - ψ^{2}

where −ψ² is a negative constant. By denoting x = ψr, the first ODE, Eq. (50), can be rewritten as

x^{2} \frac{d^{2} Γ}{d x^{2}} + x \frac{d Γ}{d r} + x^{2} Γ = 0,

which is a zero-order Bessel differential equation. The two boundary conditions of the model, T(r_o, t) = 0 and ∂T(r, t)/∂r = 0, limit the solution of the Bessel differential equation only to the first kind of a zero-order Bessel function J₀,

Γ (r) = J_{0} (ψ r) = \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m}}{{(m!)}^{2}} {(\frac{ψ r}{2})}^{2 m} .

The second ODE, Eq. (51), can be rewritten as

\frac{d Λ}{d t} + α ψ^{2} Λ = 0

whose solution is obtained as

Λ (t) = C e^{- α ψ^{2} t}

where C is a constant to be determined with a boundary condition. The geometrical boundary condition T(r_o, t) = 0 mandates Γ(r_o) = 0, leading to the multiple values of ψ satisfying J₀(ψr) = 0. By denoting each of such eigenvalues as ψ_n, the general solution of T(r, t) is obtained as

\begin{array}{l} T (r, t) & = & \sum_{n = 1}^{\infty} Γ_{n} (r) Λ_{n} (r) \\ = & \sum_{n = 1}^{\infty} J_{0} (ψ_{n} r) C_{n} e^{- α ψ_{n}^{2} t} . \end{array}

We use the steady state condition T(r, 0) as a basis to determine the constant C_n.

T (r, 0) = \sum_{n = 1}^{\infty} C_{n} J_{0} (ψ_{n} r) .

This equation represents the Fourier-Bessel series expansion [68] of T(r, 0), which is a sum of Bessel functions J₀(ψ_nr) where each Bessel function is scaled by a coefficient C_n. It is known that the Fourier-Bessel series expansion coefficients C_n are calculated as

C_{n} = \frac{2 H (ψ_{n})}{r_{o}^{2} J_{1}^{2} (ψ_{n} r_{o})}

where J₁ is the first kind of a Bessel function with the order of one,

J_{1} (ψ_{n} r_{o}) = \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m}}{m! (m + 1)!} {(\frac{ψ_{n} r_{o}}{2})}^{2 m + 1},

and H(ψ_n) is the Hankel transform [69] of T(r, 0),

H (ψ_{n}) = \int_{0}^{\infty} r J_{0} (ψ_{n} r) T (r, 0) d r .

Thus, with the initial condition given by Eq. (48) and the boundary condition given by T(r, t) = 0 for r > r_o, the general solution for the transient heat equation given in Eq. (47) can be obtained as

T (r, t) = \frac{2}{r_{o}^{2}} \sum_{n = 1}^{\infty} [\frac{J_{0} (ψ_{n} r)}{J_{1}^{2} (ψ_{n} r_{o})} e^{- α ψ_{n}^{2} t} \int_{0}^{r_{o}} r J_{0} (ψ_{n} r) T (r, 0) d r] .

We transform the solution given by Eq. (61), which is obtained for the cooling problem, to the desirable solution T_h(r, t) for the heating problem:

\begin{array}{l} T_{h} (r, t) & = & T (r, 0) - T (r, t) \\ = & \sum_{n = 1}^{\infty} C_{n} J_{0} (ψ_{n} r) (1 - e^{- α ψ_{n}^{2} t}) . \end{array}

Therefore this result, with Eq. (43), allows us to find the transient optical phase response ϕ_m(t) to a given thermal step input as follows:

\begin{array}{l} ϕ_{m} (t) & = & \sum_{m = - k}^{k} (\frac{2 π L}{λ}) \frac{d n}{d T} T_{h} (| m | p, t) \\ = & (\frac{4 π L}{λ r_{o}^{2}}) \frac{d n}{d T} \sum_{m = - k}^{k} \sum_{n = 1}^{\infty} [\frac{J_{0} (ψ_{n} | m | p)}{J_{1}^{2} (ψ_{n} r_{o})} (1 - e^{- α ψ_{n}^{2} t}) \int_{0}^{r_{o}} | m | p J_{0} (ψ_{n} | m | p) T (r, 0) d r] \end{array}

where 2k is the number of peripheral waveguides for geometrical design optimization, p is the waveguide pitch, and L is the heater length, as defined in Appendix A-1 on the steady-state analytical model. Using Eq. (38) from the steady-state model, also with Eq. (62), the transient heater power consumption to generate the thermal step input is obtained as

\begin{array}{l} q_{h} (t) & = & \frac{2 π k_{c} L (T_{s} (0, \infty)}{ln (r_{o} / r_{i})} u (t) \\ = & \frac{2 π k_{c} L}{ln (r_{o} / r_{i})} [- T_{o} + \sum_{n = 1}^{\infty} C_{n}] u (t) \end{array}

where k_c is the effective thermal conductivity of the cladding material surrounding the modulator, and u(t) is a unit step function.

Figure 9(a) shows the transient temperature profile T_h(r, t) from the analytical model, Eq. (62), in comparison with numerical thermal simulation results. The transient optical phase response for a thermal step input from the analytical model, Eq. (63), is compared with experimental measurement and numerical simulation in Fig. 9(b). The experimental result from the fabricated thermo-optic modulator with 14 peripheral waveguides for geometrical design optimization (N = 14) agrees with the analytical model prediction on the transient optical phase response. With geometrical design optimization, the 10–90% rise time from analytical model and experimental measurement is 54.2 μs and 58.5 μs, respectively. Analytical model prediction shows that the geometrical design optimization slows down the 10–90% rise time by 26.6% from 42.8 μs to 54.2 μs. By using an integrated waveguide heater [34,49], the rise and fall time will get faster roughly by an order of magnitude while maintaining a significant reduction in power consumption from the geometrical design optimization. In addition to using integrated waveguide heaters, by applying digital pre-emphasis techniques [4,60], thermo-optic modulators with geometrical design optimization may achieve a modulation bandwidth larger than 1 MHz.

Fig. 9 Transient analytical model of the thermo-optic modulator with geometrical design optimization. (a) Transient analytical model prediction on the waveguide temperature profile in comparison with numerical simulation. (b) Transient analytical model prediction on the optical phase response in comparison with numerical simulation and experimental measurement.

Download Full Size | PDF

B-3. Relation between efficiency and modulation bandwidth

With the conventional low-power thermo-optic modulators that rely on heat isolation techniques, air-gap trench and silicon undercut reduce the effective thermal conductivity k_c of the cladding material surrounding the silicon waveguides, which in turn increase the time constant 1/αψ² of the transient thermal response given by Eq. (63) where α = k_c/ρc_p. For example, in order to reduce the power consumption by half, doubling the thermal conductivity will reduce the thermal diffusivity α by half, which doubles the time constant of the first-order transient phase response from 1/αψ² to 2/αψ².

Fig. 10 Experimental setup for characterizing steady-state and transient thermo-optic phase response.

Download Full Size | PDF

On the contrary, with the proposed thermo-optic modulators with geometrical design optimization, there is no material change so that the thermal diffusivity α does not change. Although geometrical design optimization does not change the time constant of each peripheral waveguide, the additional peripheral waveguides affect the overall transient optical phase response. Figure 8(b) shows that geometrical design optimization with the fabricated thermo-optic modulator reduces the power consumption approximately 10 times while Fig. 9(b) shows that the geometrical design optimization slows down the optical phase response less than 50% as discussed in Appendix B-2.

This indicates that the geometrical design optimization technique provides a better performance trade-off between energy efficiency and thermo-optic modulation bandwidth compared to the conventional heat isolation techniques that require process modification or post-processing steps.

Appendix C. Experimental characterization setup

C-1. Steady-state optical phase response measurement

For steady-state phase response characterization Fig. 10, the tunable laser source (Ando AQ4321D) is programmed to sweep the wavelength of its output signal around 1550 nm. A polarization controller is adjusted to maximize the light coupling into the on-chip grating coupler through TE mode. An optical power meter (Newport 2936R) measures the output power from the MZI-based on-chip modulator test structure. The steady-state optical phase response is extracted from the measured optical power [4,34,70]. A programmable DC supply (Agilent E3631) is controlled such that the heater power consumption of the thermo-optic modulator sweeps from 0 to 10 mW. For precise measurement, Keithley 2002 multimeter measures the output current and the voltage of the DC supply. Experimental characterization has been fully automated by a custom developed Python software through GP-IB interface to the measurement equipment.

C-2. Transient optical phase response measurement

For transient phase response characterization Fig. 10, the wavelength of the tunable laser source output is fixed and an arbitrary waveform generator (Tektronics AWG3000) drives the thermo-optic modulator with a square-wave signal while the DC supply and the multimeter are disconnected to avoid electrical loading effects. The transient optical response is measured by a photo-detector (Thorlabs D400FC) with 1-GHz bandwidth. A custom-designed amplifier provides 40-dB gain to the photo-detector output signal and drives the input channel of the oscilloscope with 2-GHz sampling rate (Agilent DSO1000). The oscilloscope acquisition is synchronized with the output trigger signal of the arbitrary waveform generator, such that the oscilloscope output resolution can be enhanced by synchronous averaging. The transient optical phase response is extracted in the same way as the steady-state response measurement.

Funding

Toyota Central R&D Labs., Inc.; TowerJazz, Inc.

Acknowledgments

The authors thank Edward Preisler, Oleg Martynov, Farnood Rezaie, Qamar Yasir, and Minoli Pathirane at TowerJazz for the chip fabrication and technical support.

References

1. A. Rickman, “The commercialization of silicon photonics,” Nat. Photonics 8, 579–582 (2014). [CrossRef]

2. J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013). [CrossRef] [PubMed]

3. H. Abediasl and H. Hashemi, “Monolithic optical phased-array transceiver in a standard SOI CMOS process,” Opt. Express 23, 6509–6519 (2015). [CrossRef] [PubMed]

4. S. Chung, H. Abediasl, and H. Hashemi, “A monolithically integrated large-scale optical phased array in silicon-on-insulator CMOS,” IEEE J. Solid-State Circuits 53, 275–296 (2018). [CrossRef]

5. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]

6. N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017). [CrossRef]

7. Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017). [CrossRef]

8. K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 3541 (2014). [CrossRef] [PubMed]

9. Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2, 242–246 (2008). [CrossRef]

10. P. Dumais, D. J. Goodwill, D. Celo, J. Jiang, C. Zhang, F. Zhao, X. Tu, C. Zhang, S. Yan, J. He, M. Li, W. Liu, Y. Wei, D. Geng, H. Mehrvar, and E. Bernier, “Silicon photonic switch subsystem with 900 monolithically integrated calibration photodiodes and 64-fiber package,” J. Light. Technol. 36, 233–238 (2018). [CrossRef]

11. N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alising, and D. Englund, “Linear programmable nanophotonic processors,” Optica 12, 1623–1631 (2018). [CrossRef]

12. P. Dong, W. Qian, H. Liang, R. Shafiiha, D. Feng, G. Li, J. E. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “Thermally tunable silicon racetrack resonators with ultralow tuning power,” Opt. Express 18, 20298–20304 (2010). [CrossRef] [PubMed]

13. C. Sun, M. Wade, M. Georgas, S. Lin, L. Alloatti, B. Moss, R. Kumar, A. H. Atabaki, F. Pavanello, J. M. Shainline, J. S. Orcutt, R. J. Ram, M. Popović, and V. Stojanović, “A 45 nm CMOS-SOI monolithic photonics platform with bit-statistics-based resonant microring thermal tuning,” IEEE J. Solid-State Circuits 51, 893–907 (2016). [CrossRef]

14. M. Bahadori, A. Gazman, N. Janosik, S. Rumley, Z. Zhu, R. Polster, Q. Cheng, and K. Bergman, “Thermal rectification of integrated microheaters for microring resonators in silicon photonics platform,” J. Light. Technol. 36, 773–788 (2018). [CrossRef]

15. P. Dong, S. Liao, D. Feng, H. Liang, D. Zheng, R. Shafiiha, C.-C. Kung, W. Qian, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low V_pp, ultralow-energy, compact, high-speed silicon electro-optic modulator,” Opt. Express 17, 22484–22490 (2009). [CrossRef]

16. S. Manipatruni, K. Preston, L. Chen, and M. Lipson, “Ultra-low voltage, ultra-small mode volume silicon microring modulator,” Opt. Express 18, 18235–18242 (2010). [CrossRef] [PubMed]

17. Y.-C. Chang, S. P. Roberts, B. Stern, and M. Lipson, “Resonance-free light recycling,” Prepr. at https://arxiv.org/abs/1710.02891 (2017).

18. S. A. Miller, C. T. Phare, Y.-C. Chang, X. Ji, O. A. J. Gordillo, A. Mohanty, S. P. roberts, M. C. Shin, B. Stern, M. Zadka, and M. Lipson, “512-element actively steered silicon phased array for low-power LIDAR,” in Conf. on Lasers and Electro-Optics (CLEO) Technical Digest, (OSA, 2018). [CrossRef]

19. S.-H. Kim, J.-B. You, H.-W. Rhee, D. E. Yoo, D.-W. Lee, K. Yu, and H.-H. Park, “High-performance silicon MMI switch based on thermo-optic control of interference modes,” IEEE Photonics Technol. Lett. 30, 1427–1430 (2018). [CrossRef]

20. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). [CrossRef] [PubMed]

21. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campenhout, P. Bienstman, and D. V. Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Light. Technol. 23, 401–411 (2005). [CrossRef]

22. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

23. J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Opt. Express 19, 21595–21604 (2011). [CrossRef] [PubMed]

24. C. V. Poulton, A. Yaacobi, D. B. Cole, M. J. Byrd, M. Raval, D. Vermeulen, and M. R. Watts, “Coherent solid-state LIDAR with silicon photonic optical phased arrays,” Opt. Lett. 42, 4091–4094 (2017). [CrossRef] [PubMed]

25. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004). [CrossRef] [PubMed]

26. C. V. Poulton, M. J. Byrd, E. Timurdogan, P. Russo, D. Vermeulen, and M. R. Watts, “Optical phased arrays for integrated beam steering,” in IEEE Int’l Conf. Group IV Photonics (GFP) Technical Digest, (2018).

27. J. Ding, H. Chen, L. Yang, L. Zhang, R. Ji, Y. Tian, W. Zhu, Y. Lu, P. Zhou, R. Min, and M. Yu, “Ultra-low-power carrier-depletion Mach-Zehnder silicon optical modulator,” Opt. Express 20, 7081–7087 (2012). [CrossRef] [PubMed]

28. N. Dupuis, B. G. Lee, A. V. Rylyakov, D. M. Kuchta, C. W. Baks, J. S. Orcutt, D. M. Gill, W. M. J. Green, and C. L. Schow, “Design and fabrication of low-insertion-loss and low-crosstalk broadband 2×2 Mach-Zehnder silicon photonic switches,” J. Light. Technol. 33, 3597–3606 (2015). [CrossRef]

29. G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3, 229–245 (2014). [CrossRef]

30. D. Marris-Morini, L. Vivien, J. M. Fédéli, E. Cassan, P. Lyan, and S. Laval, “Low loss and high speed silicon optical modulator based on a lateral carrier depletion structure,” Opt. Express 16, 334–339 (2008). [CrossRef] [PubMed]

31. Z. Lu, K. Murray, H. Jayatilleka, and L. Chrostowski, “Michelson interferometer thermo-optic switch on SOI with a 50-μw power consumption,” IEEE Photonics Technol. Lett. 22, 2319–2322 (2015).

32. M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond submilliwatt silicon-on-insulator thermooptic switch,” IEEE Photonics Technol. Lett. 16, 2514–2516 (2004). [CrossRef]

33. A. Densmore, S. Janz, R. Ma, J. H. Schmid, D.-X. Xu, A. Delâge, J. Lapointe, M. Vachon, and P. Cheben, “Compact and low power thermo-optic switch using folded silicon waveguides,” Opt. Express 17, 10457–10465 (2009). [CrossRef] [PubMed]

34. N. C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, and C. Galland, “Efficient, compact and low loss thermo-optic phase shifter in silicon,” Opt. Express 22, 10487–10493 (2014). [CrossRef] [PubMed]

35. R. L. Espinola, M. C. Tsai, J. T. Yardley, and R. M. Osgood, “Fast and low-power thermooptic switch on thin silicon-on-insulator,” IEEE Photonics Technol. Lett. 15, 1366–1368 (2003). [CrossRef]

36. J. Xia, J. Yu, Z. Fan, Z. Wang, and S. Chen, “Thermo-optic variable optical attenuator with low power consumption fabricated on silicon-oninsulator by anisotropic chemical etching,” Opt. Eng. 43, 789–790 (2004). [CrossRef]

37. R. Kasahara, K. Watanabe, M. Itoh, Y. Inoue, and A. Kaneko, “Extremely low power consumption thermooptic switch (0.6 mw) with suspended ridge and silicon-silica hybrid waveguide structures,” in Eur. Conf. Opt. Commun. (ECOC) Technical Digest, (2008).

38. Y. Hashizume, S. Katayose, T. Tsuchizawa, T. Watanabe, and M. Itoh, “Low-power silicon thermo-optic switch with folded waveguide arms and suspended ridge structures,” Electron. Lett. 48, 1234–1235 (2012). [CrossRef]

39. Q. Fang, J. F. Song, T.-Y. Liow, H. Cai, M. B. Yu, G. Q. Lo, and D.-L. Kwong, “Ultralow power silicon photonics thermo-optic switch with suspended phase arms,” IEEE Photon. Technol. Lett. 23, 525–527 (2011). [CrossRef]

40. M. R. Watts, J. Sun, C. DeRose, D. C. Trotter, R. W. Young, and G. N. Nielson, “Adiabatic thermo-optic Mach-Zehnder switch,” Opt. Lett. 38, 733–735 (2013). [CrossRef] [PubMed]

41. L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photonics Technol. Lett. 19, 342–344 (2007). [CrossRef]

42. J. Sun, E. Timurdogan, A. Yaacobi, Z. Su, E. S. Hosseini, D. B. Cole, and M. R. Watts, “Large-scale silicon photonic circuits for optical phased arrays,” IEEE J. Sel. Top. on Quantum Electron. 20, 264–278 (2014). [CrossRef]

43. P. Sun and R. M. Reano, “Submilliwatt thermo-optic switches using freestanding silicon-on-insulator strip waveguides,” Opt. Express 18, 8406–8411 (2010). [CrossRef] [PubMed]

44. D. Celo, D. J. Goodwill, J. Jiang, P. Dumais, M. Li, and E. Bernier, “Thermo-optic silicon photonics with low power and extreme resilence to over-drive,” in Optical Interconnect Conf. (OIC) Technical Digest, (IEEE, 2016), pp. 26–27.

45. A. Masood, M. Pantouvaki, G. Lepage, P. Verheyen, J. V. Campenhout, P. Absil, D. V. Thourhout, and W. Bogaerts, “Comparison of heater architectures for thermal control of silicon photonic circuits,” in IEEE Int’l Conf. Group IV Photonics (GFP) Technical Digest, (2013), pp. 83–84.

46. H. Jayatilleka, H. Shoman, R. Boeck, N. A. F. Jaeger, L. Chrostowski, and S. Shekhar, “Automatic configuration and wavelength locking of coupled silicon ring resonators,” J. Light. Technol. 36, 210–218 (2018). [CrossRef]

47. R. Fatemi, A. Khachaturian, and A. Hajimiri, “A nonuniform sparse 2-D large-FOV optical phased array with a low-power PWM drive,” IEEE J. Solid-State Circuits (in press 2019). [CrossRef]

48. T. Kim, P. Bhargava, C. V. Poulton, J. Notaros, A. Yaacobi, E. Timurdogan, C. Baiocco, N. Fahrenkopf, S. Kruger, T. Ngai, Y. Timalsina, M. R. Watts, and V. Stojanovic, “A single-chip optical phased array in a 3D-integrated silicon photonics/65nm CMOS technology,” in IEEE Int’l Solid-State Circuits Conference (ISSCC) Digest of Technical Papers, (IEEE, 2019).

49. H. Jayatilleka, H. Shoman, L. Chrostowski, and S. Shekhar, “Photoconductive heaters enable control of large-scale silicon photonic ring resonator circuits,” Optica 6, 84–91 (2019). [CrossRef]

50. M. W. Pruessner, T. H. Stievater, M. S. Ferraro, and W. S. Rabinovich, “Thermo-optic tuning and switching in SOI waveguide fabry-perot microcavities,” Opt. Express 15, 7557–7563 (2007). [CrossRef] [PubMed]

51. J. Song, Q. Fang, S. H. Tao, T. Y. Liow, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Fast and low power michelson interferometer thermo-optical switch on SOI,” Opt. Express 16, 15304–15311 (2008). [CrossRef] [PubMed]

52. Y. Shoji, K. Kintaka, S. Suda, H. Kawashima, T. Hasama, and H. Ishikawa, “Low-crosstalk 2×2 thermo-optic switch with silicon wire waveguides,” Opt. Express 18, 9071–9075 (2010). [CrossRef] [PubMed]

53. M. Nedeljkovic, S. Stanković, C. J. Mitchel, A. Z. Khokhar, D. J. T. Scott, A. Reynolds, F. Y. Gardes, C. G. Littlejohns, G. T. Reed, and G. Z. Mashanovich, “Mid-infrared thermo-optic modulators in SOI,” IEEE Photon. Technol. Lett. 26, 1352–1355 (2014). [CrossRef]

54. G. Cocorullo and I. Rendina, “Thermo-optical modulation at 1.5μm in silicon etalon,” Electron. Lett. 48, 83–85 (1992). [CrossRef]

55. K. Murray, Z. Lu, H. Jayatilleka, and L. Chrostowski, “Dense dissimilar waveguide routing for highly efficient thermo-optic switches on silicon,” Opt. Express 23, 19575–19585 (2015). [CrossRef] [PubMed]

56. S.-K. Ryu, J. Im, P. S. Ho, and R. Huang, “A kinetic decomposition process for air-gap interconnects and induced deformation instability of a low-k dielectric cap layer,” J. Mech. Sci. Technol. 28, 255–261 (2014). [CrossRef]

57. M. Nakai, T. Nomura, S. Chung, and H. Hashemi, “Geometric loss reduction in tight bent waveguides for silicon photonics,” in Conf. on Lasers and Electro-Optics (CLEO) Technical Digest, (OSA, 2018). [CrossRef]

58. M. Cherchi, S. Ylinen, M. Harjanne, M. Kapulainen, and T. Aalto, “Dramatic size reduction of waveguide bends on a micron-scale silicon photonic platform,” Opt. Express 21, 17814–17823 (2013). [CrossRef] [PubMed]

59. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980). [CrossRef]

60. H. Matsuura, S. Suda, K. Tanizawa, K. Suzuki, K. Ikeda, H. Kawashima, and S. Namiki, “Accelerating switching speed of thermo-optic MZI silicon-photonic switches with “turbo pulse” in PWM control,” in Optical Fiber Communication Conf. (OFC) Technical Digest, (OSA, 2017). [CrossRef]

61. S. Chung and H. Hashemi, “Extending modulation bandwidth of SOI thermo-optic phase shifters through digital pre-emphasis,” in Conf. on Lasers and Electro-Optics (CLEO) Technical Digest, (OSA, 2018). [CrossRef]

62. L. Euler, A Method for Finding Curved Lines Enjoying Properties of Maximum or Minimum, or Solution of Isoperimetric Problems in the Broadest Accepted Sense (Lausanne, 1744).

63. V. Blåsjö, Transcendental Curves in the Leibnizian Calculus (Academic, Cambridge, MA, 2007).

64. F. P. Incropera, T. L. Bergman, A. S. Lavine, and D. P. DeWitt, Fundamentals of heat and mass transfer (Wiley, 2011).

65. A. Freeman, The Analytical Theory of Heat by Joseph Fourier, Translated, with Notes (University, London, 1878).

66. P. Gaspard and T. Gilbert, “On the derivation of Fourier’s law in stochastic energy exchange systems,” J. Stat. Mech. Theory Exp. 11, 1–21 (2008).

67. A. Kupiainen, “On the derivation of Fourier’s law,” in New Trends in Mathematical Physics, V. Sidoravicius, ed. (Springer, New York, 2009), pp. 421–431. [CrossRef]

68. J. Schroeder, “Signal processing via Fourier-Bessel series expansion,” Digit. Signal Process. 3, 112–124 (1993). [CrossRef]

69. R. Piessens, “The Hankel transform,” The Transforms and applications handbook: Second Edition, A. D. Poularikas, ed. (CRC, 2000), pp. 128–150.

70. A. M. Gutierrez, A. Brimont, M. Aamer, and P. Sanchis, “Method for measuring waveguide propagation loss by means of a Mach-Zehnder interferometer structure,” Opt. Commun. 285, 1144–1147 (2012). [CrossRef]

Year	Inst.	Power (mW)^†	Loss Mod. (dB)	BW (kHz)	Under-cut	Opt. BW limit	Wavelen. (nm)	Si thick. (nm)	Size (μm²)	ΔT (K)^a	Waveguide type
2005 [22]	IBM	2	2	3400	Yes	20–30 nm	1630	220	20 × 20	-	Photonic crystal
2008 [37]	NTT	0.6	3.1	0.1	Yes	No	1550	300	500 × 130	-	Single-mode
2011 [39]	OSU	0.49	1.2	2.4	Yes	No	1550	220	1000 × 30	4.5	Single-mode
2012 [38]	NTT	0.40	0.55	0.2	Yes	No	1550	200	500 × 20	-	Single-mode
2015 [31]	UBC	0.05	3.3	0.4	Yes	No	1550	220	300 × 32	1	Single-mode
2016 [44]	Huawei	0.5	0.58	0.26	Yes	No	1550	220	100 × 24	-	Single-mode

2003 [35]	Columbia	50	3.1	100	No	No	1550	260	1200 × 14	8	Single-mode
2007 [41]	UT Austin	80	5	18	No	20–30 nm	1550	260	80 × 13	9	Photonic crystal
2009 [33]	NRCC	6.5	0.5	25	No	No	1550	260	130 × 130	0.7	Single-mode
2013 [40]	MIT	12.7	1.7	146	No	No	1550	220	6 × 6	480	Single-mode
2015 [55]	UBC	3.8	1.2	7.8	No^b	No	1550	220	800 × 180	-	Single-mode
2016 [44]	Huawei	10	0.58	5	No	No	1550	220	100 × 24	-	Single-mode
2017 [17]	Columbia	1.7	4.6–6.0	53	No	100 nm	1601	250	850 × 25	-	Multi-mode
2019	this work	2.56	1.23	10.1	No	No	1550	220	109 × 21	7.4	Single-mode

Parameter	Description	Associated Segments
g	Pitch of the two waveguides to be connected by the bend	Segment 3, 4, 6

θ_a	Bend angle (larger than π/2) of segment 1, 2	Segment 1, 2, 3, 4, 5, 6
A_a	Geometrical size constraint of segment 1, 2	Segment 1, 2, 3, 4, 6
W_a	Waveguide width of segment 1, 2	Segment 1, 2, 5, 6

W_b	Waveguide width of segment 3, 4	Segment 3, 4, 5

L_t	Taper length	Segment 3, 4, 5, 6

Seg.	Waveguide center line x-coordinate	Waveguide center line y-coordinate	Variables
1	$x_{1} (ϕ) = \frac{A_{a}}{\sqrt{2}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} ϕ^{2 k + \frac{1}{2}}}{(2 k + \frac{1}{2}) \cdot (2 k)!}$ .	$y_{1} (ϕ) = \frac{A_{a}}{\sqrt{2}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} ϕ^{2 k + \frac{3}{2}}}{(2 k + \frac{3}{2}) \cdot (2 k + 1)!}$ .	0 ≤ϕ ≤ θ_a
2	x₂(ϕ) = −x₁(ϕ) sin δ_a + y₁(ϕ) cos δ_a + x₁(θ_a)[1 + sin δ_a] − y₁(θ_a) cos δ_a.	y₂(ϕ) = x₁(ϕ) cos δ_a + y₁(ϕ) sin δ_a − x₁(θ_a) cos δ_a + y₁(θ_a)[1 − sin δ_a].	0 ≤ϕ ≤ θ_a, δ_a = 2θ_a − 3π/2
3	$x_{3}^{'} (ϕ) = \frac{A_{b}}{\sqrt{2}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} ϕ^{2 k + \frac{1}{2}}}{(2 k + \frac{1}{2}) \cdot (2 k)!}$ , x₃(ϕ) = O_bx + x′₃(ϕ).	$y_{3}^{'} (ϕ) = \frac{A_{b}}{\sqrt{2}} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} ϕ^{2 k + \frac{3}{2}}}{(2 k + \frac{3}{2}) \cdot (2 k + 1)!}$ , y₃(ϕ) = g + x′₃(ϕ).	0 ≤ϕ ≤ θ_b, θ_b = θ_a − π/2
4	x₄(ϕ) = O_bx − x′₃(ϕ) sinδ_b + y′₃(ϕ) cos δ_b + x′₃(θ_b)[1 + sinδ_b] − y′₃(θ_b) cosδ_b	y₄(ϕ) = g + x′₃(ϕ) cos δ_b + y′₃(ϕ) sin δ_b − x′₃(θ_b) cosδ_b + y′₃(θ_b) [1 + sinδ_b]	0 ≤ϕ ≤ θ_b, δ_b = 2θ_a − π/2
5	x₅(l) = x₄(0) + l sin(2θ_b).	y₅(l) = y₄(0) + l cos(2θ_b).	0 ≤ l ≤ L_t
6	x₆(x) = −x.	y₆(x) = 0.	0 ≤ x ≤ \|O_bx\|

U-turn type	Segment number	Waveguide type	Width (μm)	θ(°)	$A = \sqrt{RL}$ (μm)	R(θ) (μm)	L(θ) (μm)	δ (°)	O(x, y) (nm)
A	6	Straight Strip	400	-	-	-	6.268	−180	(0.000, 0.000)
A	1	Clothoid	400	105	2.914	1.522	5.579	0	(0.000, 0.000)
A	2	Trans. Clothoid	400	105	2.914	1.522	5.579	−60	(−1.302, 1.302)
A	5	Taper	400 → 500	-	-	-	2.000	30	(−2.536, 2.000)
A	4	Trans. Clothoid	500	15	2.719	1.968	3.758	120	(−5.485, 5.485)
A	3	Clothoid	500	15	2.719	1.968	3.758	0	(−6.268, 1.000)

B	6	Straight Strip	500	-	-	-	6.268	−180	(0.000, 0.000)
B	1	Clothoid	500	105	2.914	1.522	5.579	0	(0.000, 0.000)
B	2	Trans. Clothoid	500	105	2.914	1.522	5.579	−60	(−1.302, 1.302)
B	5	Taper	500 ! 400	-	-	-	2.000	30	(−2.536, 2.000)
B	4	Trans. Clothoid	400	15	2.719	1.968	3.758	120	(−5.485, 5.485)
B	3	Clothoid	400	15	2.719	1.968	3.758	0	(−6.268, 1.000)

Year	Inst.	Power (mW)^†	Loss Mod. (dB)	BW (kHz)	Under-cut	Opt. BW limit	Wavelen. (nm)	Si thick. (nm)	Size (μm²)	ΔT (K)^a	Waveguide type
2005 [22]	IBM	2	2	3400	Yes	20–30 nm	1630	220	20 × 20	-	Photonic crystal
2008 [37]	NTT	0.6	3.1	0.1	Yes	No	1550	300	500 × 130	-	Single-mode
2011 [39]	OSU	0.49	1.2	2.4	Yes	No	1550	220	1000 × 30	4.5	Single-mode
2012 [38]	NTT	0.40	0.55	0.2	Yes	No	1550	200	500 × 20	-	Single-mode
2015 [31]	UBC	0.05	3.3	0.4	Yes	No	1550	220	300 × 32	1	Single-mode
2016 [44]	Huawei	0.5	0.58	0.26	Yes	No	1550	220	100 × 24	-	Single-mode

2003 [35]	Columbia	50	3.1	100	No	No	1550	260	1200 × 14	8	Single-mode
2007 [41]	UT Austin	80	5	18	No	20–30 nm	1550	260	80 × 13	9	Photonic crystal
2009 [33]	NRCC	6.5	0.5	25	No	No	1550	260	130 × 130	0.7	Single-mode
2013 [40]	MIT	12.7	1.7	146	No	No	1550	220	6 × 6	480	Single-mode
2015 [55]	UBC	3.8	1.2	7.8	No^b	No	1550	220	800 × 180	-	Single-mode
2016 [44]	Huawei	10	0.58	5	No	No	1550	220	100 × 24	-	Single-mode
2017 [17]	Columbia	1.7	4.6–6.0	53	No	100 nm	1601	250	850 × 25	-	Multi-mode
2019	this work	2.56	1.23	10.1	No	No	1550	220	109 × 21	7.4	Single-mode

Low-power thermo-optic silicon modulator for large-scale photonic integrated systems

Abstract

1. Introduction

2. Thermo-optic silicon modulator with geometric design optimization

2.1. Fabricated device structure

2.2. Computational characteristics

2.3. Experimental evaluation

3. Discussion

Appendix A: Multi-Clothoid 180° bend waveguide

A-1. Geometrical structure

A-2. Geometrical design equations

A-3. Multi-Clothoid bend design method

A-4. Design technique for sub-1dB optical loss

Appendix B: Analytical models of thermo-optic modulator

B-1. Steady-state model for estimating thermo-optic modulator power consumption

B-2. Transient model for estimating thermo-optic modulation bandwidth

B-3. Relation between efficiency and modulation bandwidth

Appendix C. Experimental characterization setup

C-1. Steady-state optical phase response measurement

C-2. Transient optical phase response measurement

Funding

Acknowledgments

References

Cited By

Figures (10)

Tables (5)

Equations (64)

Optics Express