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Abstract
A shortcut to adiabatic light transfer in mode-evolution coupler by axis bending is proposed. Phase mismatch due to waveguide width difference can be cancelled by a proper bending profile, realizing shortcut to adiabatic light transfer in the coupler. The bent coupler is more compact than the conventional straight coupler without compromising the device performance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Mode-evolution couplers have been widely studied in integrated optics due to their large operating bandwidths and high fabrication tolerances [1–3]. In addition to applications as optical couplers and splitters, mode-evolution couplers have been used to realize polarization splitters [4], polarization rotators [5], and polarization splitter-rotators [6]. In mode-evolution couplers, light evolution follows the local eigenmodes adiabatically, and therefore, the device performance is less sensitive to operating wavelength and device geometry. However, the major drawback of mode-evolution couplers is that, in order to avoid coupling between the eigenmodes, device geometry needs to be smoothly varied, which often results in long devices. Various approaches have been devised to optimize the device geometries in order to reduce their lengths. These approaches include limiting unwanted mode-coupling [7,8], shape function optimization [9,10], polynomial approach [11], and multi-dimensional optimization [12,13], among others.
One particular class of optimization borrows the shortcut to adiabaticity (STA) concepts in quantum theory [14], which are originally developed to accelerate slow adiabatic processes in quantum systems. Owing to quantum-optical analogies in optical waveguides [15], STA protocols have found successful applications and lead to compact and robust waveguide devices [16,17]. One approach is based on the counter-diabatic (CD) or transitionless tracking method [18,19], where the non-adiabatic transitions are countered by the addition of a counter-diabatic term so that the system evolution follows the eigenmodes of the uncorrected system exactly. An equivalent approach is based on the Lewis-Riesenfeld invariant in which the device design is inverse-engineered to accelerate adiabatic process [20]. The fast quasi-adiabatic dynamics (FAQUAD) [21] and adiabaticity engineering (AE) [22] approaches redistribute system adiabaticity to achieve shortcuts to adiabatic light evolution. In these STAs, the waveguide spacing and widths are engineered to obtain the required coupling coefficient and phase mismatch corresponding to the design protocol. An often overlooked degree of freedom in device optimization is the axis of the coupler, which has not been considered until recently. Longhi first showed that Landau-Zener dynamics can be realized in directional couplers with cubically bent axis [23]. Bent directional couplers have been shown to be robust against parameter variations [24,25]. Multi-dimensional optimization including geometry tilt has showed great promise in realizing very compact couplers [12,13]. Axis bending in directional couplers with identical width also leads to compact and robust couplers [26,27].
In this paper, we demonstrate a shortcut to adiabaticity in mode-evolution couplers by axis bending. Unlike the CD protocol [27] where waveguide spacing is engineered along the device length, the current shortcut uses a fixed waveguide spacing, which is advantageous in terms of fabrication. Axis bending is used to cancel the phase mismatch between dissimilar waveguides in a conventional mode-evolution coupler, thus realizing an effective resonant coupling scheme in the bent waveguide reference frame. The resonant coupling scheme accelerates light coupling and realizes an effective shortcut to adiabatic light coupling in the lab reference frame. The resulting bent mode-evolution coupler is shorter then the conventional straight coupler and has the same robustness against wavelength and dimension variations. In this work, we demonstrate the proposed shortcut to adiabatic light transfer by showing complete light transfer between two waveguides in a bent mode-evolution coupler. The same design protocol can be applied to power splitters with intended splitting ratios by adjusting waveguide widths at the output [28]. These 2$\times$2 power couplers are one of the most fundamental components in integrated optics with applications such as power splitting, light switching [29], and wavelength multiplexing [30].
2. Theoretical analysis
In Fig. 1, we consider a bent mode-evolution coupler of length $L$ consisting of two evanescently coupled waveguides with a fixed spacing $d$ (center to center). The axis of the coupler is bent along $z$ with a bending profile described by $x_0(z)$. The waveguide widths at the input $w_1(-L/2)=W_1$ and $w_2(-L/2)=W_2$ are linearly tapered to $W_2$ and $W_1$ at the output. We note that $W_1$ and $W_2$ at the output can be adjusted to obtain various splitting ratios [28]. Here, we show complete light transfer between the waveguides to demonstrate the shortcut.
Fig. 1. Schematic of the bent mode-evolution coupler. The waveguide widths $w_1(z)$ and $w_2(z)$ vary with $z$, tapering linearly from $W_1$ and $W_2$ at the input to $W_2$ and $W_1$ at the output. The waveguide spacing (center-to-center) is fixed at $d$. The black dashed line shows the bent axis described by $x_0(z)$. The device length is $L$
The waveguide core region has a refractive index of $n_a$ and is surrounded by the cladding region with refractive index $n_c$, with an index difference defined by $\Delta n=n_a-n_c$. In the bent waveguide reference frame $(x'=x-x_0(z), z'=z)$ and under the scalar and paraxial approximation, the amplitude of the guided mode in individual waveguides $|\Psi _c\rangle =[c_1(z'), c_2(z')]^{T}$ can be written using the coupled-mode equation as [23]
2.1 Straight mode-evolution coupler
For a conventional straight mode-evolution coupler, $x_0(z')=0$, and the lab reference frame $(x, z)$ is identical to the waveguide reference frame $(x', z')$, the Hamiltonian $\mathbf {H}$ reads
2.2 Shortcut using resonant coupling in a bent mode-evolution coupler
In the waveguide reference frame $(x', z')$, we can eliminate the mismatch by setting
so that the mismatch $\Delta =0$, and we also assume that the system has an updated coupling coefficient $\Omega _b$ that will be determined later. In this case, the Hamiltonian is The Hamiltonian in Eq. (8) describes resonant coupling between the individual waveguide modes $|1\rangle$ and $|2\rangle$ with the solution (consider excitation of waveguide 1 at the input, $|\Phi _c(-L/2)\rangle =|1\rangle$) given by where We can obtain a shortcut to perfect coupling from waveguide 1 at the input to waveguide 2 at the output by imposing the following conditions through a properly chosen $\Omega _b$ such that In other words, using resonant coupling scheme in the waveguide reference frame, we can realize an effective shortcut to adiabatic light coupling in the lab reference frame. Perfect light coupling is guaranteed by the boundary conditions in Eq. (11) with a properly chosen $\Omega _b$.3. Device design and simulation
In this section, we design a bent mode-evolution couplers in a conventional planar integrated optics platform to realize the shortcut and perform beam propagation method (BPM) simulations to verify the designs. The scalar 2D BPM code used in the simulations solves the scalar and paraxial wave equation using the finite difference scheme with the transparent boundary condition [31]. For device design and simulations, we consider the polymer waveguide platform which is a potential candidate for optical interconnects [32,33]. The design parameters are $n_c$=1.5, $\Delta n$=0.016, and $\lambda =1.55$ $\mu$m based on realistic numbers from previous experimental works [34]. In the design, the waveguide widths at the input are chosen as $W_1=4.2$ $\mu$m and $W_2=2.6$ $\mu$m, which are linearly tapered to $W_2$ and $W_1$ at the output. For the chosen waveguide system, the relation between mismatch $\Delta _s$ and width difference $\delta w$ can be approximated by a linear relation [10]. Because $\delta w$ is linearly varied in this design, we can conclude that $\Delta _s$ is a linear function of $z$ given by
where $\Delta _0$ is the mismatch at $z=-L/2$.3.1 Straight mode-evolution coupler
We first investigate the straight mode-evolution coupler with a fixed center-to-center spacing of $d=6$ $\mu$m. The light evolution dynamic is identical to the Landau-Zener dynamics of two-level quantum systems [23]. Considering input into waveguide 1, we plot the fractional beam power in waveguides 1 and 2 at the output as a function of device length $L$ in Fig. 2. We can observe the characteristic oscillations with $L$ in adiabatic devices with decreasing oscillation amplitude as the length increases, indicating that the device is more adiabatic. It can be seen that close to complete light coupling can be obtained for devices longer than 2.2 mm. On the other hand, the shortcut can achieve complete coupling at arbitrarily short device length by axis bending according to Eq. (7) and the boundary conditions in Eq. (11).
Fig. 2. Fractional beam power in waveguides 1 and 2 at the output as a function of device length for the straight mode-evolution coupler.
As an example, we show the device geometry of the $L=2.2$ mm straight coupler and the light evolution by exciting waveguide 1 at the input in Fig. 3(a). The corresponding fractional light power evolution in the individual waveguides is shown in Fig. 3(b). We can observe complete light transfer from waveguide 1 to waveguide 2, and the light evolution dynamics mimics the Landau-Zener dynamics [23].
Fig. 3. (a) Light evolution in the $L=2.2$ mm straight mode-evolution coupler showing Landau-Zener dynamics. White lines indicate the waveguide cores. (b) The corresponding fractional light power evolution in the individual waveguides.
3.2 Bent mode-evolution coupler
Next, we illustrate the process of obtaining the shortcut by axis bending. To achieve the shortcut, we first bend the coupler axis according to Eq. (7). Using Eqs. (7) and (13), the bending profile is obtained by integration
In this example, we implement the shortcut with an $L=1$ mm coupler, which for a straight coupler can only achieve 82.6% coupling efficiency as shown in Fig. 2. In this coupled waveguide system, the relation between $\Omega$ and $d$ is well fitted by the exponential relation $\Omega =\Omega _0\exp [-\gamma (d-d_0)]$ [35] with $\Omega _0=0.1745$ mm$^{-1}$ and $\gamma =0.7249$ $\mu$m$^{-1}$. Using the relation, we find that, at the same fixed center-to-center spacing of $d=6$ $\mu$m and its corresponding $\Omega _b$, the boundary conditions in Eq. (11) are satisfied. We apply the bending profile in Eq. (14) and $d=6$ $\mu$m to the mode-evolution coupler and the resulting waveguide geometry is shown in Fig. 4(a). In the same figure, we show the light evolution in the shortcut coupler by exciting waveguide 1 at the input. The corresponding fractional light power evolution in the individual waveguides is shown in Fig. 4(b). Clearly, complete light transfer from waveguide 1 to waveguide 2 is realized via the shortcut at $L=1$ mm. In Fig. 4(c), light evolution is shown in terms of the adiabatic basis $|\Phi _+\rangle$ and $|\Phi _-\rangle$ of the straight coupler by field projection onto the local eigenmodes. We can observe that light evolution indeed closely follows $|\Phi _-\rangle$, proving that shortcut to adiabaticity in the lab frame is indeed achieved via resonant coupling in the waveguide frame.Fig. 4. (a) Light evolution in the $L=1$ mm bent mode-evolution coupler showing the shortcut. White lines indicate the waveguide cores. (b) The corresponding fractional light power evolution in the individual waveguides. (c) Light evolution in terms of the adiabatic basis of the straight coupler.
3.3 Device robustness
Next, we examine the robustness of the shortcut coupler. For the same $L=1$ mm bent coupler, the coupling efficiency from waveguide 1 to waveguide 2 as a function of wavelength is shown in Fig. 5(a). For comparison, we also plot the spectrum of the $L=2.2$ mm straight mode-evolution coupler in the same figure. In Fig. 5(b), we show the coupling efficiencies for the $L=1$ mm bent coupler and $L=2.2$ mm straight coupler as a function of waveguide width error $\Delta w$ to account for errors in fabrication. We observe almost identical spectra and fabrication tolerance for the 1 mm shortcut and the 2.2 mm conventional coupler, indicating that the proposed shortcut not only can effectively shorten the device length but also can preserve the robustness of the original device. Other fabrication imperfections such as thickness variation and gap error can also lead to similar deteriorations in device performance. The key result from the above analysis is to illustrate that the $L=1$ mm bent shortcut coupler has the same performance as the $L=2.2$ mm straight coupler, showing the effectiveness of the shortcut protocol.
Fig. 5. Coupling efficiency of the $L=1$ mm bent coupler and the $L=2.2$ mm straight coupler as a function of (a) wavelength and (b) width variation.
4. Discussion and conclusion
Similar to the CD shortcut, both the original coupling coefficient and the phase mismatch need to be modified in the proposed shortcut. Rather than modifying the phase mismatch via waveguide widths, we engineer the bending profile to achieve the shortcut. Compared with CD and invariant-based protocols where both the coupling coefficient and the phase mismatch are utilized to achieve shortcuts, the current shortcut eliminates the phase mismatch in the Hamiltonian to achieve the shortcut, which is faster than other protocols due to its resonant nature. Unlike conventional resonant coupling devices such as the parallel directional coupler, in which the system evolution is described by an equal excitation of the symmetric and anti-symmetric eigenmodes, system evolution in the bent mode-evolution coupler follows one of the eigenmodes closely. The observed robustness can be attributed to the mode-evolution nature of light evolution in the bent coupler.
In conclusion, we propose a new shortcut to adiabatic mode-evolution by elimination of phase mismatch via axis bending. By engineering the bending profile to cancel the waveguide width mismatch, shortcut to adiabatic light transfer can be realized. The shortcut bent coupler is more compact than the conventional straight coupler and preserves the robustness of the conventional coupler. The result further demonstrates that axis bending is a useful degree of freedom in waveguide coupler design. In addition to engineering the waveguide material, spacing, and width, axis bending adds a useful degree of freedom in device design. The addition of this new degree of freedom could lead to new designs and applications.
Funding
Ministry of Science and Technology, Taiwan (111-2221-E-006-052-MY3).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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