A compact thermo-optical multimode-interference silicon-based 1×4 nano-photonic switch

Haifeng Zhou; Junfeng Song; Edward K. S. Chee; Chao Li; Huijuan Zhang; Guoqiang Lo

doi:10.1364/OE.21.021403

1. Introduction

Optical packet switching (OPS) holds the promise of a bandwidth-efficient and highly-reconfigurable optical layer to scale the payload capacity and increase the bit-rates beyond the capability of electronics [1, 2]. A generic OPS node architecture generally consists of a set of multiplexers(MUX) and demultiplexers(DEMUX), an input interface, a space switch fabric with wavelength converters and optical buffers, an output interface, and a switch control unit. The space switch fabric receives the packets from the input interface that extracts the optical packet header and routes it with instructions from the switch control unit. The (DE)MUX is usually realized by arrayed waveguide gratings (AWGs), whose largest I/O channel was reported to reach 64 × 64 to date [3]. Each channel of the AWG must be interconnected by the space switch fabric after the input interface. Therefore, the largely scalable switch fabric is a key enabling technology to this end.

Nowadays, microelectromechanical system (MEMS)-based spatial switch consisting of movable mirrors has been demonstrated to be as large as 1100 × 1100 [4], dominating on the large-port-count optical switch market. However, its relatively slow switching speed at a scale of mili-second and indispensable movable motors tends to be problematic for future OPS and optical bursting switching applications. Therefore, much effort and attention have been devoted to studying integrated optical switches with large port counts. Most of them are demonstrated on III–V semiconductor substrates [5–7]. In the past decade, the silicon-on-insulator (SOI) platform has attracted much attention due to its matured fabrication technologies, high index contrast for low-loss waveguiding and bending properties and flexible modulation schemes for various operation speed requirements. Very compact silicon-based 1 × N optical switches were demonstrated by using serial cascading of many 1 × 2 interferometers [8].

In order to further simplify the device architecture, this work reports a 1 × 4 silicon-based optical switch by adopting the concept of generalized Mach-Zehnder interferometer. With the modulation arms paralleled, the overall footprint is potentially smaller than that of the serialized case. Following the device layout and introducing the fabrication process, we present the performance of the constituent sub-components. We propose a statistical method to conduct tolerance analysis on the critical parameters. Finally, we present the static and dynamic switching performance of the first demonstration in detail.

2. Architecture, fabrication and sub-components

As shown in Fig. 1(a), the 1×4 switch consists of one MMI section as 1×4 splitter and another MMI section as 4 × 4 power combiner, which are connected by 4 interferometric arms. By appropriately feeding the phases on the arms, the optical beam from the 1 × 4 optical splitter can be recombined through the 4 × 4 MMI coupler so that the light can be directed into any one of the output waveguides. All the heaters share a common ground pad to simplify the electrically driven scheme by connecting each one of their ends using aluminum wires. A set of isolation grooves are designed between heaters to minimize thermal interference. The overall device length is mainly determined by the heater lengths on the interferometric arms L_Arm. The lateral dimension is mainly determined by the waveguide spacings d_Arm. With fixed L_Arm and d_Arm, the footprint of this architecture is nearly one third smaller than that of a conventional 1× 4 optical switch, which normally consists of 2-stage cascaded 1× 2 optical switchs and has 6 interferometric arms in total.

Fig. 1 (a) Microscope image of the fabricated 1 × 4 MMI-based optical switch and (b) the z-cut cross-sectional view in the arm region with thermo-heater. In (a), the electrode pads can act as a scale bar, whose widths are 70 μm. The total length of the two MMI sections is just 21 μm. The active part of the thermo-heater is 185 μm long and 2.0 μm wide. d_Arm is set to be 30 μm.

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As the SEM image of the cross-sectional view is shown in Fig. 1(b), the device fabrication began with a commercially avialable SOI wafer with 220 nm-thick top silicon layer and 2.0 μm-thick buried dioxide. The standard 248 nm UV photo-lithography was accompanied by a bottom anti-reflective coating layer (BARC) to define the waveguide patterns with high resolution. With a 70 nm-thick PECVD-grown oxide as hard mask, the silicon waveguide was etched by the inductively coupled plasma (ICP) etcher. Subsequently, in order to relieve the stress issues, undoped silicate glass (USG) and low-temperature high-density-plasma (HDP) deposited oxide were grown alternatively over the silicon waveguide to reach a thickness of 1.2 μm for eliminating evanescent wave absorption. Later on, 150 nm-thick titanium nitride (TiN) heaters were deposited over the oxide layer with the aid of 5 nm Ti thin layer. The heaters was etched with 50 nm SiN as mask. To form the metal interconnects, 250 Å TaN/0.75 μm AlSiCu metal layers were deposited, patterned and etched consecutively and contacted with the heaters through pre-defined via holes. In order to effectively isolate heat between interferometric arms, the isolation grooves were etched to tens of microns, together with the deep trenches that fiber butt-coupling calls for [9].

2.1. 1×4 optical splitter/combiner

In general, light beam entering into a step-index multimode (MMI) waveguide from a single-mode waveguide excites multiple guided modes with propagation constants nearly quadratically dependent on the mode number [10]. In the center-excited case of a symmetric MMI waveguide, 4-folded images can be reproduced at a length of 3L_π/16, where the beating length L_π is approximately related to the waveguide effective width W_MMI, the effective refractive index n_eff and the vacuum wavelength λ by $L_{π} = n_{eff} W_{MMI}^{2} / λ$ . The four images are ideally with the same intensities and equally spaced by a distance of W_MMI/4. While taking such an MMI segment as a 1 × 4 splitter, its main difference from 2-stage cascaded 1 × 2 splitters is that the images have relative phase differences of ϕ^1×4 = [0, π/2, π/2, 0].

The compactness of an MMI splitter is mainly determined by the widths and the spacing of the access waveguides. Using finite difference frequency domain (FDFD) method, the coupling length of two parallel 450 nm-wide waveguides can be estimated by calculating the effective refractive indices of the two lowest supermodes. It exponentially increases from 16 μm to 270 μm when the gap width in between increases from 150 nm to 500 nm. Thus, in this experiment, the gap width of the access waveguides was chosen to be 350 nm to avoid undesired mode coupling after the MMI section. With this dimension, the effective MMI width is 3.2 μm and the physical MMI width is 50 nm narrower if considering the penetration depth of transverse electric (TE) mode. In addition, 2 μm-long adiabatic tapers were placed at both the input and output ends of the MMI section to prohibit high-order modes excitation that deteriorates the imaging performance.

We took the simulated imaging length L_MMI of 4.2 μm as a baseline and fabricated 7 different devices with the physical length of the MMI section varied. The insertion loss and the power imbalance for all the devices are examined in Figs. 2 (b) and 2(c), respectively. The insertion loss is calculated by referring to a straight waveguide. The power imbalance is defined as the power contrast between the highest and lowest transmissions through the four outputs. The large differences in loss and imbalance of different curves indicate that the physical length of MMI section is critical to the beam splitting performance. An error of 0.02L_MMI (≈80 nm) may degrade badly splitting performance. The best device for 1550 nm is with a length of 0.98L_MMI, whose insertion loss and power imbalance are less than 0.65 dB and 0.5 dB, respectively. Within the C band from 1530 nm to 1565 nm, this device also has relatively low insertion loss (basically < 1.0 dB) and power imbalance and thus shows good wavelength insensitivity.

Fig. 2 (a) The Scanning-electron-microscope (SEM) image of a typical 1×4 optical splitter. The dependence of (b) insertion loss and (c) power imbalance on the MMI section length L, whose ratios to the baseline L_MMI value varying with a step of 2.0% are shown in the legend.

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2.2. 4×4 optical coupler

To compose a 1×4 optical switch, one needs another 4×4 MMI coupler to connect the four split beams and four switching output ports. For an N × N MMI coupler with the the shortest imaging length of $\frac{3}{N} L_{π}$ , the phase retardation between the nth input port and the lth output port is expressed by [11]

ϕ_{nl} = ϕ_{0} - \frac{π}{2} {(- 1)}^{n + l + N} - \frac{π}{4 N} \times {[{(- 1)}^{n} (n - \frac{1}{2}) - {(- 1)}^{N + l} (l - \frac{1}{2})]}^{2},

where ϕ₀ is a constant phase of −3β₀L_π/N − π/N +3π/4. According to this, the phase relations between the I/O ports of a 4 × 4 MMI coupler can be written as

ψ^{4 \times 4} = (\begin{matrix} 0 & \frac{3}{4} & - \frac{1}{4} & 0 \\ \frac{3}{4} & 0 & 0 & - \frac{1}{4} \\ - \frac{1}{4} & 0 & 0 & \frac{3}{4} \\ 0 & - \frac{1}{4} & \frac{3}{4} & 0 \end{matrix}) π .

The elements of this matrix also indicate the necessary relative phases for different switching states in front of the 4 × 4 MMI coupler. Theoretically, light power from any of the four inputs is uniformly split into the four output ports. Due to the further magnification of the estimation error of MMI length, very large discrepancy was observed in our experiment. In order to better understand the influential factors of device performance, the transmissions of all the I/O combinations of the 1 × 4 MMI coupler are shown in Fig. 3. In Fig. 3(a), it is found that the loss is around 0.7 dB for each channel when λ = 1550 nm, which is comparable to that of a 1 × 4 splitter. However, the power imbalance between the output ports depicted in Fig. 3(b) is very serious, which is as high as 3.0 dB. In addition, we should also expect that the phase relations between the I/O combinations also derivate from the matrix ψ^4×4 of the ideal case. Therefore, the 4 × 4 MMI coupler is the most vulnerable sub-component of the switch.

Fig. 3 The wavelength dependence of (a) the insertion loss and (b) power imbalance of the 4 × 4 MMI coupler. The cases with different input port excited are distinguished in color. The loss and imbalance are calculated with different input port excited. The unexpected oscillations around 1560nm were actually found throughout the C band in the raw transmission data, which might be attributed to the reflection wave interference within the MMI section.

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With the known transfer matrix of 1 × 4 MMI splitter and 4 × 4 MMI coupler, the complex field at the lth output port of the 1 × 4 switch can be described as following

r_{l} exp (i ϕ_{l}) = \sum_{n = 1}^{4} T_{nl}^{4 \times 4} r_{n} exp [j (ϕ_{n}^{mod} + ϕ_{n}^{const} + ϕ_{n}^{1 \times 4})],

where

T_{nl}^{4 \times 4} = \frac{1}{2} exp (- j ψ_{i, j}^{4 \times 4})

for the ideal case;

ϕ_{n}^{mod}

is in charge of the external phase modulation;

ϕ_{n}^{const}

is the constant phase interferometric arms which includes the phase delay from waveguide bends and fabrication errors, and those from designed delay lines and static biasing modulation.

2.3. Estimation of power consumption

It is well known that the magnitude of phase modulation is proportional to the static power consumption. Let us briefly comment the efficiency of different modulation schemes and make a comparison with conventional 1 × 4 optical switch. In Table 1, in the ideal case of $ϕ_{n}^{const} = 0$ , we enumerated the phase modulations $ϕ_{n}^{mod}$ required by all the switching states. The values in (a–c) are equivalent in phase but within different intervals to account for different physical effects. For example, carrier injection effect changes the local refractive index negatively and thus produces positive phase change. Therefore, the smallest phase change through current enhancement that meets the switching conditions are within the interval of [0, 2π]. This is the case (a). Likewise, Si thermo-optical effect and carrier depletion effect correspond to case (b). The case (c) has an example that both carrier injection and depletion effects are available for each of the interferometric arms [12]. If different switching states are assumed to occur evenly in practical switching fabric, the averaged magnitude of phase modulation $\bar{ϕ^{mod}}$ in time can be calculated by

\bar{ϕ^{mod}} = \frac{1}{4} \sum_{state} \sum_{n = 1}^{4} | ϕ_{n}^{mod} |_{ϕ_{n}^{const} = 0} - ϕ_{n}^{const} - ϕ_{state}^{ref} - m_{n} 2 π |,

where

ϕ_{state}^{ref}

is a variable reference phase for phase translation; m_n is an proper integer to translate the phase in the bracket into the targeted intervals of (a–c). The

\bar{ϕ^{mod}}

can be optimized to achieve the lowest power consumption by searching for suitable

ϕ_{state}^{ref}

for all the switching states. When

ϕ_{n}^{const} = 0

,

\bar{ϕ^{mod}}

of the cases (a–c) are 1.5π, 1.5π and π, respectively. However, in most cases,

ϕ_{n}^{const} \neq 0

and the

\bar{ϕ^{mod}}

becomes higher. The value range of

\bar{ϕ^{mod}}

can be statistically estimated by randomly generating the 1 × 4 optical switches with arbitrary

ϕ_{n}^{const}

of 2π [rand(), rand(), rand(), rand()]. The expression “rand()” stands for a random number within [0, 1] generated by computer program. For the cases (a–c), the

\bar{ϕ^{mod}}

can be optimized by varying

ϕ_{state}^{ref}

, whose optimal values are within the intervals [1.5π, 2π], [1.5π, 2π] and [π, 1.5π], respectively. In our experiment, only three heaters are activated and the fourth arm is free of modulation so that

ϕ_{state}^{ref}

is fixed and can not be used for optimization. Under such extra condition, the

\bar{ϕ^{mod}}

becomes even higher. Equation 4 shows that, in this case,

\bar{ϕ^{mod}}

are within [2π, 4π], [2π, 4π] and 1.5π, respectively. As for the conventional 1 × 4 switch realized by 2-stage 1 × 2 optical switches [8], the heaters of the three 1 × 2 switches are independent and only one switch is working on average in time. Hence,

\bar{ϕ^{mod}}

is within [π, 2π], [π, 2π], and π for the cases (a–c), respectively. In summary, although the averaged magnitude of modulation can be reduced to a limit of π if the scheme of case (c) is used, the architecture has no improvement from the aspect of power consumption compared with the conventional design.

Table 1. The phase modulations on the interferometric arms required by the four switching states. The cases (a–c) are equivalent in phase but distinct from each other in magnitude.

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2.4. Influence of the quality of 4 × 4 optical coupler

Since the quality of a 4 × 4 MMI coupler is most challenging to assure in experiment or simulation, let us establish a simple model to examine its influence to the crosstalk of the 1 × 4 switch. We neglect the insertion loss of the sub-components because they only contribute to the overall device insertion loss. We also fix the power imbalance of the 1 × 4 MMI power splitter as 0.5 dB. The variable factor is the power imbalance of 4 × 4 MMI coupler, which results in incomplete constructive or destructive interference at the output ports and is regarded as the most critical sub-component for design. Let us employ random number generator to account for fabrication uncertainty and use a variable σ_M to characterize the largest power imbalance of the 4 × 4 MMI coupler. To be specific, on each arm before the 4 × 4 MMI coupler, the field amplitude is described as $\frac{1}{2} [1 - σ_{M}^{1 \times 4} rand ()]$ , where the amplitude error $σ_{M}^{1 \times 4}$ is always fixed to 6% by considering the experimental power imbalance of nearly 0.5 dB. The phase derivation induced by the 1 × 4 power splitter is not of great importance to the switching crosstalk, as it can always be compensated by dynamic modulation. To simulate the 4 × 4 MMI coupler, we consider the derivation of transfer function in both amplitude and phase by writing

T_{n, l}^{4 \times 4} = \frac{1}{2} [1 - σ_{M}^{4 \times 4} rand ()] exp {- j [ψ_{n, l}^{4 \times 4} + σ_{M}^{4 \times 4} (rand () - \frac{1}{2}) π]} .

By matching the phases for different switching states before the 4 × 4 MMI coupler, light transmission from all the four output ports can be calculated by using Eq. (3). The power ratio of the strongest transmission to the secondary strongest transmission is recorded, the lowest of which at the four different switching states is considered as the device crosstalk. The above process was repeated to consider 1000 randomly generated devices. The averaged value of the crosstalk is shown in Fig. 4. The result indicates that when the power imbalance reaches an extent of 3 dB, the switching crosstalk is expected to be around 10.44 dB. If a crosstalk of 20 dB is required, the power imbalance of the 4 × 4 optical coupler must be lower than 1 dB at least.

Fig. 4 Tolerance analysis on the influence of the quality of a 4 × 4 MMI coupler to the switching crosstalk of the 1 × 4 MMI-based optical switch.

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3. The 1×4 optical switch and the operation states

3.1. The operation states

The switch is symmetric in the horizontal direction, but for the purpose of heat isolation, its two outside arms are designed longer by introducing arc-shape bent waveguides to keep large enough arm spacing. The physical length differences contribute to the constant phase retardation $ϕ_{n}^{const}$ . Fabrication errors may also incur unwanted non-uniformity on the interferometric arms. Hence, the initial operation state is no longer of a uniform splitting as initially predicted with identical interferometric arms. Therefore, it is necessary to find out the four operation states in the beginning of device testing.

Since only the relative phase between interferometric arms is of importance, one can keep an arm free of modulation (here Arm 4 is chosen) and just apply voltage to the other three arms so that it requires less complex algorithm to coordinate the voltage driving. The heaters are driven by semiconductor device analyzer (Agilent B1500A) that can output 3-channel voltage signals. By simply expanding the matrix multiplication in Eq. (3), it can be easily seen that the complex field at any output port is a superposition of the phase-modulated complex fields at the input ports. For example, the complex field at port ”O2” can be expressed as

E_{2} = T_{32} r_{3} exp [i (ϕ_{3}^{mod} + ϕ_{3}^{const} + ϕ_{3}^{1 \times 4})] + \sum_{n \neq 3} T_{nl} r_{n} exp [i (ψ_{n})] .

The amplitude of the complex field |E₂| at the port “O2” is plotted in Fig. 5(a) as a function of the applied voltage on the 3rd arm. The maximum and minimum transmissions appear alternatively, where the intervals between adjacent peaks or valleys stand for one phase period.

Fig. 5 (a) The field amplitude at the port O2 varies with voltage that was only applied onto the 3rd arm. (b) The complex field at port O2 at iterations that sweep voltage alternatively to maximize the transmission. The three numbers in the legend are the voltages applied to the three interferometric arms. (a) is correspondent to the first iteration in (b).

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The electric resistance of the heaters are 1 kΩ by observing the V–I characteristics and thus the switching power for π phase shift is calculated to be 14 mW.

More intuitively, the complex field can also be plotted in a polar coordinate system, as shown in Fig. 5(b). In order to find the required phase modulation for the highest |E₂|, we fit the curve in Fig. 5(a) by using Eq. (6), single out the voltage correspondent to the first maximum, fix it and optimize the other arms, alternatively. Using this method, the phase combination converges to a stationary group for the highest transmission at the targeted output ports. The voltage combinations and the transmissions are listed in Table 2 for all the four switching states. At the designed wavelength of 1550 nm, the switching crosstalk is lower than −11.38 dB and the extinction ratio is higher than 11.48 dB. The averaged insertion loss is around 1.7 dB and the transmission non-uniformity of all the “ON” states is less than 1.05 dB. If considering the power necessary for all the four switching states, the average power consumption of this switch is calculated to be 31.7 mW. This value is within the range of [2π, 4π] mentioned in Sec. 2.3.

Table 2. Light transmissions under the optimal phase modulations at 1550 nm

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Figure 6 presents the wavelength dependent transmissions at all the output ports while applying the different voltage combinations listed in Table 2. Because the switch is symmetric with respect to the central axis of propagation, the transmissions of O2 and O3 are similar and they are distinct from those of O1 and O4. The former two have more uniform transmissions than the latter two, which should be attributed to the imbalance of the 4 × 4 MMI coupler. Because the physical lengths of the two inner interferometric arms and the two outer ones are not the same, the transmissions at all the ports vary with wavelength periodically. This behavior is similar to the filtering function of conventional 1 × 2 optical Mach-Zehnder interferometers. By using FDFD method, the group refractive index n_g of a 220 nm-thick and 450 nm-wide waveguide is calculated to be 3.96. The length difference between the outer arms and the inner arms can be estimated by λ²/n_g/Δλ_FSR. The 8 nm λ_FSR in the spectrums are induced by an arm difference of 76 μm. The crosstalk increases with the decreased wavelength because the loss and imbalance performance of 4 × 4 coupler become poor in Fig. 3.

Fig. 6 The wavelength dependent transmission under different voltage combinations listed in Table 2. The transmissions at different output ports are distinguished in color.

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3.2. Thermo-optical heater and its dynamic response

To be compatible with the shared flowchart of our multiple processing wafer (MPW), we adopted the large thermo-optical (TO) effect of silicon to implement refractive index modulation. In optical packet switching system, the fast spatial speed is mainly determined by the length of the signal packet in the system. The speed of TO effect is usually at the scale of micro-second, which may find applications if multiple packets are assembled into containers (e.g. bursts or super-frames) and switched as an aggregate. In this study, using TO effect is also helpful to analyze the influence of imbalance of MMI couplers because of the negligible accompanying absorption. Without loss of generality, high speed carrier dispersion effect of silicon is more preferable to make fast switching and deliver short packets.

The ultimate characterization of the device dynamic response is to show all the four switching states by applying three or four synchronized multi-valued signals. This experiment requires multiple-channel arbitrary waveform generator, which is not available in our lab at present. Therefore, what we can do is to look for a scenario resembling the switching action at a specific output port. In Fig. 5(b), it is seen that the transmission at O2 almost touches the highest and the lowest value if one keeps the arm 2 and arm 3 biased and modulate the arm 1. Then, according to Table 2, we choose the biased voltages on arm 2 and arm 3 as 2.3 V and 0.3 V and dynamically modulate the arm 1 using non-return-to-zero (NRZ) signals. Figure 7 captured the dynamic response of the thermo-heater triggered by a 20 kHz square pulse, where the electrical and optical signals are plotted in the same panel in yellow and green, respectively. The power contrast of the optical signal between the “ON” and “OFF” state is around 8.5 dB, which is close to the extinction ratio of complete switching action. The rise time and the fall time of the optical signals are 5.9 μs and 6.7 μs, respectively.

Fig. 7 The dynamic response of the thermo-heater. A continuous electric square signal was applied to invoke the heaters for producing the highest transmission of O2. The markers ”start” and ”stop” in time denotes the time period of 1 bit non-return-to-zero signal. The time interval is 50 μs, corresponding to a frequency of 20 kHz. The rise time and fall time are the time intervals between 10% and 90% transmissions.

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4. Conclusion

An ultra-compact 1 × 4 optical switch was designed with self-imaging theory in multimode interference waveguide, and demonstrated on SOI platform with silicon thermo-optical effect. The performance of the sub-components, including the 1 × 4 power splitter, 4 × 4 power splitters, and thermo-optical heaters are also presented. The power loss and imbalance of the 1 × 4 optical splitter is comparable to that composed of cascaded 1 × 2 optical splitters. The absolute values of switching extinction ratio and crosstalk are below 10 dB. Using transfer matrix method and statistics, we also established an efficient theoretical model to investigate the in-fluences of 4 × 4 MMI coupler on the switching performance. Compared with the conventional 1×4 optical switch composed of 2-stage 1×2 optical switches, this architecture features small footprint but consumes more energy to realize signal transmission. This study may provide a reference for researchers who are interested in the applications of multi-channel multimode interference-based coupler to photonic circuits.

Acknowledgments

This study was supported by Singapore SERC/A^*STAR Grant 092-154-0098.

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States	[ $ϕ_{1}^{mod}$ , $ϕ_{2}^{mod}$ , $ϕ_{3}^{mod}$ , $ϕ_{4}^{mod}$ ] (π) when $ϕ_{n}^{const} = 0$			O1	O2	O3	O4
States	(a) 0 < $ϕ_{n}^{mod}$ < 2	(b) −2 < $ϕ_{n}^{mod}$ < 0	(c) −1 < $ϕ_{n}^{mod}$ < 1	O1	O2	O3	O4

1	[1/4, 1, 0, 1/4]	[0, −5/4, −1/4, 0]	[0, 3/4, −1/4, 0]	1	0	0	0
2	[0, 1/4, 1/4, 1]	[−1/4, 0, 0, −5/4]	[−1/4, 0, 0, 3/4]	0	1	0	0
3	[1, 1/4, 1/4, 0]	[−5/4, 0, 0, −1/4]	[3/4, 0, 0, −1/4]	0	0	1	0
4	[1/4, 0, 1, 1/4]	[0, −1/4, −5/4, 0]	[0, −1/4, 3/4, 0]	0	0	0	1

Cases	Voltage (V)	O1 (dB)	O2 (dB)	O3 (dB)	O4 (dB)
(a)	[2.6, 3.5, 3.6]	−1.4265	−16.9225	−12.8125	−22.0145
(b)	[3.7, 2.3, 0.3]	−24.9895	−1.6305	−13.9745	−22.0645
(c)	[1.9, 5.1, 1.7]	−21.4835	−27.7965	−1.3375	−19.8755
(d)	[4.9, 1.2, 4.2]	−19.3825	−18.4775	−18.4975	−2.3885

States	[ $ϕ_{1}^{mod}$ , $ϕ_{2}^{mod}$ , $ϕ_{3}^{mod}$ , $ϕ_{4}^{mod}$ ] (π) when $ϕ_{n}^{const} = 0$			O1	O2	O3	O4
States	(a) 0 < $ϕ_{n}^{mod}$ < 2	(b) −2 < $ϕ_{n}^{mod}$ < 0	(c) −1 < $ϕ_{n}^{mod}$ < 1	O1	O2	O3	O4

1	[1/4, 1, 0, 1/4]	[0, −5/4, −1/4, 0]	[0, 3/4, −1/4, 0]	1	0	0	0
2	[0, 1/4, 1/4, 1]	[−1/4, 0, 0, −5/4]	[−1/4, 0, 0, 3/4]	0	1	0	0
3	[1, 1/4, 1/4, 0]	[−5/4, 0, 0, −1/4]	[3/4, 0, 0, −1/4]	0	0	1	0
4	[1/4, 0, 1, 1/4]	[0, −1/4, −5/4, 0]	[0, −1/4, 3/4, 0]	0	0	0	1

Cases	Voltage (V)	O1 (dB)	O2 (dB)	O3 (dB)	O4 (dB)
(a)	[2.6, 3.5, 3.6]	−1.4265	−16.9225	−12.8125	−22.0145
(b)	[3.7, 2.3, 0.3]	−24.9895	−1.6305	−13.9745	−22.0645
(c)	[1.9, 5.1, 1.7]	−21.4835	−27.7965	−1.3375	−19.8755
(d)	[4.9, 1.2, 4.2]	−19.3825	−18.4775	−18.4975	−2.3885

A compact thermo-optical multimode-interference silicon-based 1×4 nano-photonic switch

Abstract

1. Introduction

2. Architecture, fabrication and sub-components

2.1. 1×4 optical splitter/combiner

2.2. 4×4 optical coupler

2.3. Estimation of power consumption

2.4. Influence of the quality of 4 × 4 optical coupler

3. The 1×4 optical switch and the operation states

3.1. The operation states

3.2. Thermo-optical heater and its dynamic response

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (2)

Equations (6)

Optics Express