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Abstract
The fast quasi-adiabatic dynamics (FAQUAD) protocol has proven to be an effective approach to provide shortcuts to adiabatic light evolution in optical waveguides, resulting in short and robust devices. However, the FAQUAD approach of homogeneously distributing device adiabaticity only works for a single mode (polarization, wavelength, or spatial mode group) system. We propose an adiabaticity engineering approach to redistribute the adiabaticity of optical waveguides in multi-mode systems. By engineering the adiabaticity distribution using a single control parameter, we obtain shortcuts to adiabaticity in optical waveguides for multi-mode systems. The concept is applied to the design of a compact polarization-independent adiabatic 3-dB coupler on silicon.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The concept of adiabaticity, ever since its introduction by Born and Fock in 1928 [1], has found extensive applications in various fields of physics. For time-evolution in a quantum system, adiabaticity means that the system follows an eigenstate trajectory closely if it is initially prepared in one of the eigenstates. In guided-wave optics, a large class of devices operate on the same principle, where light evolution in space follows the evolution of an eigenmode closely if one of the eigenmodes is excited initially. These adiabatic devices (also referred to as mode-evolution-based devices) rely on gradual change of the device geometry to control the eigenmodes, and as long as the adiabaticity criterion is satisfied, light evolution can be accomplished with large bandwidth and high fabrication tolerances, which are their major advantages. The advantages have led to the development of many adiabatic optical waveguide devices [2–17]. Like their quantum mechanical counterparts that require long evolution times to achieve adiabaticity, adiabatic optical waveguide devices generally require long device lengths in order to avoid coupling to unwanted modes. There have been numerous approaches to optimize the geometry of adiabatic devices in order to reduce device lengths [4,5,8,18–21]. One approach is to limit the fraction of power scattered into unwanted modes below a constant value [8,18]. Another approach is based on the equalization of taper loss along each propagation step [19–21]. Various shape functions were also considered to minimize coupling between eigenmodes [4,5]. However, these approaches are only focused on a single polarization, i.e., either for the transverse electric (TE) mode or for the transverse magnetic (TM) mode.
Recently, a new class of protocols in quantum control for speeding up slow adiabatic evolutions called shortcuts to adiabaticity (STA) [22,23] has found success in optical waveguides [24–26], resulting in short and robust functional devices. One of the STA protocols, fast quasi-adiabatic dynamics (FAQUAD) [27], achieves shortcuts in optical waveguides by homogenizing the adiabaticity using only one control parameter [13,16,28–31]. Despite its success in Y-junctions [28,29], polarization splitters [30,31], and beam splitters [13,16], the simple FAQUAD strategy of making the evolution as fast as possible by making it as adiabatic as possible anywhere during evolution fails when more than one mode (polarization, wavelength, or spatial mode group) needs to be taken into consideration. In this paper, we introduce the concept of adiabaticity engineering in optical waveguides. Using this method, we can achieve shortcuts to adiabaticity in multi-mode systems. When applied to a mode-evolution-based 3-dB coupler, we show that the device adiabaticity distribution can be optimized for both the TE and TM polarizations simultaneously, resulting in a compact and polarization-independent device with 69% length reduction over the conventional design.
2. Adiabaticity engineering in optical waveguides
The adiabaticity parameter for vectorial field propagating in the $z$ direction in an optical waveguide can be defined as [29]
2.1 Fast quasi-adiabatic dynamics (FAQUAD)
The FAQUAD protocol [13,16,27–31] works by imposing a constant adiabaticity parameter $c(z)=\varepsilon$ along the length of the device $L$. Integrating Eq. (3) with $c(z)=\varepsilon$, we obtain $\varepsilon =(1/L)\int F(D)dD$. Also from Eq. (3), we obtain the rate of change of $z$ with respect to $D$ for FAQUAD as
In subsequent analysis, $F(D)$ is obtained from Eq. (3) by assuming a linearly varying $D$ such that $R(D)$ is a constant. Equation (4) can then be integrated to obtain the FAQUAD device geometry $z_{FAQUAD}(D)$. With this geometry, the device achieves constant adiabaticity, which has been found to be an optimal approach in adiabatic passage [32]. However, the FAQUAD strategy only optimizes device geometry for one mode. Next, we propose an adiabaticity engineering approach to optimally distribute adiabaticity in multi-mode systems. Note that for multi-mode systems here, we refer to multiple polarization modes, wavelengths, or spatial mode groups.2.2 Adiabaticity engineering in multi-mode systems
The goal of adiabaticity engineering is to find $R_{AE}(D)=d z_{AE}/d D$ such that the adiabaticity for each considered mode is as evenly distributed as possible along the evolution. Considering $M$ modes and denoting each mode by $i$, we can first obtain the FAQUAD design [Eq. (4)] for each mode:
We first let $R_{AE}(D)=A\times r_{AE}(D)$ ($A$ is a constant to be determined later), then impose the condition where $w_i$ is the weight assigned to each mode and $\sum _1^{M} w_i=1$. The weight determines the contribution of each mode in the engineered adiabaticity. From Eq. (3), we know that for each mode, the adiabaticity is With the criterion in Eq. (6), we can ensure that local adiabaticity is at the smallest possible value for all modes involved. Physically, $R_{AE}$ is the rate of change of device length with respect to $D$, and the criterion in Eq. (6) selects the maximum value among the modes for all $D$, meaning that the change in $D$ with $z$ is the slowest (most adiabatic). Finally, we can use the relation where $L$ is the device length, to find the constant $A=L/\int r_{AE}(D)dD$ and ensure that the resulting $z_{AE}(D)$ from the integration of $R_{AE}(D)$ satisfies the boundary conditions.We will now apply adiabaticity engineering to a two polarization system to obtain a compact polarization-independent 3-dB coupler.
3. Adiabaticity engineering in an adiabatic (mode-evolution-based) 3-dB coupler
We consider an adiabatic 3-dB coupler based on SOI strip waveguides consisting of 2 $\mu$m thick buried oxide layer, 220 nm silicon strip, and covered with silica cladding. The top view of the coupler is shown schematically in Fig. 1. The device is divided into three regions as shown in the figure. In the 30 $\mu$m long Region 1, two waveguides with $W_1$=300 nm and $W_2$=500 nm are brought together using an S-bend to reduce the gap from 1.65 $\mu$m to 200 nm. The key element of this design is the mode evolution region (Region 2) of length $L$, while a constant gap of 200 nm is maintained, a taper function $D(z)$ is applied to both waveguides so their widths becomes $W_1+D(z)$ and $W_2-D(z)$. With the boundary conditions, $D_i\equiv D(0)=0,~~D_f\equiv D(L)=100\textrm { nm},$ the dissimilar waveguides are converted to two identical waveguides of 400 nm width at the end of Region 2. In the 12 $\mu$m long Region 3, two s-bends separate the waveguide gap from 200 nm to 1.6 $\mu$m.
Fig. 1. Top-view schematic of the adiabatic (mode-evolution-based) 3-dB coupler. The insets show the first four eigenmodes of the coupler at the device input and at the input and output of Region 2.
Using a finite-difference three-dimensional vectorial mode (FDM) solver, we solve for the first four eigenmodes of the coupler at different positions as shown in the insets of Fig. 1. Depending on the input mode type, TE or TM, and the input port, port 1 or port 2, only one of the four local eigenmodes is excited initially in Region 2. As the excited mode adiabatically propagates along the coupler, very little or no power is coupled into the other eigenmode of the same polarization. At the output, the excited mode evolves into the even/odd mode of the two identical waveguides, thus naturally splits the power evenly in each waveguide. The key in the design is thus to ensure that the evolution in Region 2 is done slowly (adiabatically) enough to minimize unwanted coupling into the other eigenmode. If the adiabaticity criterion is satisfied for both TE and TM polarization modes, the device functions as a polarization-independent 3-dB coupler. The existing polarization-independent designs [12,33] are based on linearly-varying taper functions to gradually bring the dissimilar waveguides to be identical ones. However, as we will shown in subsequent analysis, the device adiabaticities for both polarization modes are highly unevenly distributed during modal evolution with a linear taper function, thus requiring long device lengths to keep the overall adiabaticity parameter below an acceptable level for both polarizations.
3.1 Adiabaticity analysis of the linearly-varying taper function
The adiabaticity parameter along any point of the mode evolution region can be obtained using Eq. (1) with mode data calculated using the FDM solver. In conventional designs [12,33], the waveguides in region 2 are tapered according to a linear taper function $D(z)=(D_f/L)z$ as shown by the solid line in Fig. 2(a). Using (1), we obtain the adiabaticity parameters of the conventional linear taper structure for both TE and TM polarizations as shown in Fig. 2(b). The adiabaticity parameters for both polarizations are unevenly distributed over the length of the device. For the TE mode, it is small at the beginning of the taper and steadily increases towards the end of the taper. For the TM mode, it increases almost linearly from the beginning towards the end of the taper. In order to satisfy the adiabatic criterion ($c\ll 1$) anywhere during evolution, the length $L$ would need to be increased to keep the overall adiabaticity parameter below an acceptable level. That is why conventional linear adiabatic 3-dB couplers have long device lengths in general. We also note that the adiabaticity of the TE mode is larger than that of the TM mode, indicating larger coupling for the TE mode.
Fig. 2. (a) Plot of different taper functions $D(z)$. (b) Calculated adiabaticity parameters of the TE ($c_{TE}$) and TM ($c_{TM}$) polarizations for the linear taper function.
We use a commercial software (FIMMPROP, Photon Design) employing a full vectorial eigenmode expansion method (EME) to simulate light propagation in the 3-dB couplers. To determine the required mode evolution region length $L$ for polarization-independent 3-dB coupling, we simulate the transmission at output ports 3 and 4 with port 2 input at $\lambda$=1.55 $\mu$m for both polarizations as a function of $L$ as shown in Fig. 3. At the length $L=91.4$ $\mu$m (dashed lines in the figures), the two transmission curves for both polarizations cross in the figures, meaning that the device now functions as a 3-dB coupler for both polarizations as shown in the insets. However, the current strategy of uniformly increasing $D$ with $z$ as in the linear taper function is not an efficient strategy to satisfy the adiabatic criterion. Due to the large adiabaticity parameter of the TE mode near the end of region 2, we observe large oscillations of the TE transmission curves in Fig. 3, meaning that long device length is needed to suppress unwanted coupling. Intuitively, the taper function $D(z)$ should instead vary faster when the adiabaticity is small, and vice versa. This is the main idea behind the FAQUAD approach.
Fig. 3. Simulated transmissions through the coupler with a linear taper function with port 2 input as a function of the device length $L$ at $\lambda$=1.55 $\mu$m for both polarizations. Dashed lines indicate the polarization-independent device length $L=91.4$ $\mu$m. Insets: EME simulated light distribution in the 3-dB coupler splitter with an $L=91.4$ $\mu$m.
3.2 FAQUAD designs for single polarization mode
The FAQUAD approach was shown to be effective in reducing the device length of a single polarization mode (TE) implementation of the 3-dB coupler [13]. Using the adiabaticity parameters calculated using the linear taper function in Fig. 2(b) and Eq. (3) (where $d D/d z=D_f/L$ for the linear design), we first obtain $F(D)$ for both polarizations. We can then obtain the FAQUAD rate of change of $z$ with respect to $D$ for both polarizations $R_{TE}$ and $R_{TM}$ using Eq. (5) as shown in Fig. 4(a). Integrating $R_{TE}$ and $R_{TM}$, we have the FAQUAD taper functions for the TE (dashed) and TM (dash-dotted) polarizations as shown in Fig. 2(a). We also show the adiabaticity parameters for both polarizations corresponding to these two FAQUAD designs in Figs. 4(b) and (c). It is clear that the FAQUAD design can only homogenize the adiabaticity parameter for the intended polarization, and the adiabaticity parameter for the other polarization could even reach a larger value than the linear design. For example, the adiabaticity parameter for the TM polarization in the FAQUAD-TE design [dashed curve in Fig. 4(b)] is vary large at the beginning of the mode evolution region due to a fast changing $D$. In Fig. 5, we show the simulated transmission at output ports 3 and 4 as a function of $L$ for both polarizations using both the FAQUAD-TE and FAQUAD-TM designs. We can see that FAQUAD designs provide shortcuts to adiabaticity at shorter device lengths for their designed polarizations, achieving 3-dB power splitting at coupling lengths as short as 18.6 $\mu$m for the TE mode [Fig. 5(a)] and 8.8 $\mu$m for the TM mode [Fig. 5(d)]. However, due to unevenly distributed adiabaticity of the other polarization, even longer $L$’s are needed for polarization-independent operations. For example, the transmission for the TM mode input to the FAQUAD-TE design in Fig. 5(b) shows vary large oscillations indicating strong coupling between the eigenmodes. This is also evident from the large peak of adiabaticity parameter for the TM mode in the FAQUAD-TE.
Fig. 4. (a) The FAQUAD rate of change of $z$ with respect to $D$ for both polarizations. (b) Adiabaticity parameters corresponding to the FAQUAD-TE design in Fig. 2(a) for both polarizations. (c) Adiabaticity parameters corresponding to the FAQUAD-TM design in Fig. 2(a) for both polarizations.
Fig. 5. Simulated transmissions through the coupler with the FAQUAD-TE [(a)(b)] and FAQUAD-TM [(c)(d)] taper functions with port 2 input as a function of the device length $L$ at $\lambda$=1.55 $\mu$m for both polarizations.
3.3 Adiabaticity engineering
To optimally distribute device adiabaticity for both TE and TM polarization modes simultaneously, we use the condition in Eq. (6), that is, we require
where $R_{TE}$ and $R_{TM}$ are shown in Fig. 4(a), $w_{TE}$ and $w_{TM}$ are the weight functions for the TE and TM modes, and $w_{TE}+w_{TM}=1$. In Fig. 6, we show the $r_{AE}$’s for different weights ($w_{TE}=0.3,~0.5,~0.7,~0.34$). After normalizing for device length using Eq. (8) and integration, we obtain the engineered taper functions in Fig. 7. In Fig. 8, we show the calculated adiabaticity parameters corresponding to these taper functions with different weights for both polarizations. We can see that the device adiabaticity parameters for both polarizations are now redistributed with no large peaks. The $r_{AE}$’s in Fig. 6 are formed by the union of different sections of $w_{TE}R_{TE}$ and $w_{TM}R_{TM}$ curves, so we observe that the adiabaticity parameters for both polarizations have flat sections when $r_{AE}$’s correspond to their corresponding FAQUAD designs. The simulated transmissions at output ports 3 and 4 as a function of $L$ for both polarizations using these designs are shown in Fig. 9. Dashed lines indicate the polarization-independent device length, with lengths of $L=32.9$ $\mu$m, $L=50.9$ $\mu$m, $L=79.8$, and $L=28.1$ $\mu$m corresponding to $w_{TE}=0.3,~0.5,~0.7,~\textrm {and}~0.34$. We can see that polarization independent 3-dB coupling can be achieved at shorter lengths in these adiabaticity engineered devices than the conventional linear design with $L=91.4$ $\mu$m.Fig. 6. $r_{AE}$ (solid lines) obtained from different weights of the TE (dashed) and TM (dash-dotted) modes. (a) $w_{TE}=0.3$. (b) $w_{TE}=0.5$. (c) $w_{TE}=0.7$. (d) $w_{TE}=0.34$. $w_{TE}+w_{TM}=1$.
Fig. 7. Plot of different taper functions $D(z)$ obtained from adiabaticity engineering with different weights of the TE and TM polarizations.
Fig. 8. Adiabaticity parameters of the TE (solid) and TM (dashed) modes corresponding to the taper functions in Fig. 7. (a) $w_{TE}=0.3$. (b) $w_{TE}=0.5$. (c) $w_{TE}=0.7$. (d) $w_{TE}=0.34$.
Fig. 9. Simulated transmissions through the coupler with port 2 input as a function of the device length $L$ at $\lambda$=1.55 $\mu$m for both polarizations corresponding to the taper functions in Fig. 7. (a) $w_{TE}=0.3$. (b) $w_{TE}=0.5$. (c) $w_{TE}=0.7$. (d) $w_{TE}=0.34$. Dashed lines indicate the polarization-independent device length. (a) $L=32.9$ $\mu$m. (b) $L=50.9$ $\mu$m. (c) $L=79.8$ $\mu$m. (d) $L=28.1$ $\mu$m.
3.4 Optimization of adiabaticity-engineered devices
The strength of the adiabaticity parameter indicates unwanted coupling into the other modes, so the goal of adiabaticity engineering is to redistribute the adiabaticity parameters such that unwanted couplings are suppressed as much as possible during the evolution. The weights assigned to the modes in Eqs. (6) and (9) determine the contribution of the modes to the adiabaticity parameters. From the analysis above, we observe that the adiabaticity of the TE mode has a larger value than that of the TM mode, indicating larger coupling for the TE mode. This is also evident from the longer lengths required for the TE mode to reach adiabaticity shown in Fig. 3. As a result, the weight assigned to each mode needs to be further improved to obtain the optimal design.
To optimize the weights for the TE and TM modes, we plot the maxima of the adiabaticity parameters during evolution [maxima of the adiabaticity parameter curves in Fig. 8, indicative of the maximum coupling during evolution] for both polarizations ($c_{TE}$ and $c_{TM}$) and their sum ($c_{total}$) for various $w_{TE}$ weights from 0.1 to 0.9 in Fig. 10(a). From the minima of the sum, the optimal weight of $w_{TE}=0.34$ ($w_{TM}=0.66$) is found for this device. This is further evidenced by the shortest polarization-independent device lengths as a function of $w_{TE}$ shown in Fig. 10(b). At $w_{TE}=0.34$, we find the minimum device length for polarization-independent operation of $L=28.1$ $\mu$m, agreeing with the analysis in Fig. 10(a). For $w_{TE}=0.34$, the corresponding $r_{AE}$, taper function $D(z)$, adiabaticity parameters, and transmissions as a function of $L$ are shown in Figs. 6(d), 7, 8(d), and 9(d), respectively. For the $L$=28.1$\mu$m 3-dB coupler, we show the simulated light propagation in the device by exciting port 2 for both polarizations in Fig. 11. Clearly, the input light is evenly split into the two output ports for both the TE and the TM modes. The result shows that we have successfully used adiabaticity engineering to optimize the polarization-independent 3-dB coupler and shortened the device length from $L=91.4$ $\mu$m of the conventional linear design to $L=28.1$ $\mu$m.
Fig. 10. (a) Maxima of the adiabaticity parameters during evolution for both polarizations ($c_{TE}$ and $c_{TM}$) and their sum ($c_{total}$) for various $w_{TE}$. (b) The shortest polarization-independent device lengths as a function of $w_{TE}$.
Fig. 11. EME simulated light distribution in the polarization-independent 3-dB coupler with an $L=28.1$ $\mu$m for the (a) TE and (b) TM mode inputs.
In Fig. 12, we compare the wavelength dependence of the optimized $w_{TE}=0.34$ coupler with an $L$=28.1$\mu$m with that of the linear coupler at $L$=91.4$\mu$m (Fig. 3). For the optimized polarization-independent 3-dB coupler, the optimized coupler shows 3$\pm$0.5 dB splitting ratio for the full 200 nm range from 1.45 $\mu$m to 1.65 $\mu$m for both the TE [Fig. 12(a)] and the TM [Fig. 12(b)] modes. The smaller imbalance of the TM mode is due to its smaller adiabaticity parameter compared to the TE mode as shown in Fig. 8(d). For the linear polarization-independent 3-dB coupler [Figs. 12(c) and (d)], we also observe that the TM mode shows smaller imbalance due to smaller adiabaticity parameter as shown in Fig. 2(b). The 3$\pm$0.5 dB bandwidth for the TE mode [Fig. 12(c)] in the linear coupler is 110 nm (1.5 $\mu$m to 1.61 $\mu$m), less than that of the optimized coupler at 31% of its length. This is because the maximum value of the adiabaticity parameter of the linear coupler at $L$=91.4$\mu$m is still larger than that of the optimized coupler at $L$=28.1$\mu$m. From Fig. 12, we can see that the coupler optimized by adiabaticity engineering achieves performance better (TE mode) or similar (TM mode) to the conventional linear design with 69% length reduction.
Fig. 12. Wavelength dependence of polarization-independent couplers with the optimized design ($L$=28.1 $\mu$m): (a) TE mode input and (b) TM mode input. Wavelength dependence of polarization-independent couplers with the linear design ($L$=91.4 $\mu$m): (c) TE mode input and (d) TM mode input.
3.5 Discussions
We can observe in Fig. 7 that the taper functions for $w_{TE}=0.3$ and $w_{TE}=0.34$ are very close to each other. Yet there is a large difference in polarization-independent device length (28.1 $\mu$m vs. 32.9 $\mu$m). This can be attributed to the larger TM mode adiabaticity parameter at the beginning of the evolution for $w_{TE}=0.3$ compared to $w_{TE}=0.34$ as shown in Figs. 8(a) and (d), while the difference for the TE modes is less pronounced. The larger adiabaticity parameter means larger coupling to the unwanted mode, as a result, the TM mode of $w_{TE}=0.3$ needs longer length then $w_{TE}=0.34$ one to reach shortcut to adiabaticity as can be seen in Figs. 9(a) and (d). We can see in Fig. 9(a) that when the TE mode reaches a shortcut to adiabaticity around $L$=28.1 $\mu$m, the TM mode still shows large oscillation. The result indicates that the device performance is sensitive to large local values of adiabaticity parameter and shows that our adiabaticity engineering approach of minimizing the total unwanted coupling does indeed optimize the taper function.
From Eq. (3), we know $c(z)dz=F(D)dD$. Integrating both sides, we obtain
The reason why these integrals are constant is because $F(D)$ is strictly dependent on the control parameter $D$ and not on the functional dependence between $z$ and $D$. As a result, the integrals in Eq. (10) always equal the same constant for the same mode. We can verify that the areas under the adiabaticity parameters $c(z)$ for the same polarization mode for different taper functions in Figs. 2(a) and 7 all equal $\varepsilon L=\int F(D)dD$ ($\varepsilon$ is the constant adiabaticity in FAQUAD protocol). In fact, we can regard adiabaticity engineering as the redistribution of $c(z)$ along $z$ under the constraint of a fixed area. Our adiabaticity engineering approach chooses the optimal distribution of adiabaticity parameters in multi-mode systems by redistributing them in such a way that the total unwanted coupling is minimized during evolution.4. Conclusions
We introduce the concept of engineering the adiabaticity parameters of optical waveguides using a single control parameter. The original fast quasi-adiabatic dynamics (FAQUAD) protocol homogenizes adiabaticity by engineering the rate of length change as a function of the control parameter. In this work, the concept of homogenizing device adiabaticity is extended to multi-mode systems by selecting the largest rate of length change (most adiabatic) among the modes along the propagation direction. We show that device adiabaticities for multi-mode systems can be optimized to obtain efficient shortcuts to adiabaticity. This approach is applied to the design of a polarization-independent 3-dB coupler on silicon, successfully reducing the mode-evolution region length from $L=91.4$ $\mu$m of the conventional linear design to $L=28.1$ $\mu$m, achieving a 69% length reduction. This protocol is applicable to the design of other multi-mode waveguide systems and could find applications in multi-level quantum control.
Funding
Ministry of Science and Technology, Taiwan (108-2218-E-992-302, 108-2221-E-006-204-MY3, 109-2224-E-992-001).
Disclosures
The authors declare no conflicts of interest.
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