Complete and robust light transfer in three-waveguide coupler by shortcut to adiabaticity Open Access

Waveguide coupler is a fundamental element device in integrated optics,¹ optical communication,² quantum information processing,³ and analogy of quantum optics.⁴ Due to these significant applications in both theoretical and experimental research studies, studying on the waveguide coupler is intriguing and vigorous realm in optical research. The directional coupler employs the parallel structure of two waveguides. However, the performance of a directional coupler is very sensitive to geometrical parameters of device, for example, device length and distance between two waveguides. This feature requires high precision of device fabrication. Thus, the key concept of introducing quantum control to design waveguide coupler is to increase the robustness of the waveguide coupler. It is well known that coupling equations between waveguides can be approximated to Schrodinger equation⁵ by employing coupled mode theory (CMT).^6–8 Therefore, we can exploit the quantum control technique to design robust waveguide couplers, which can improve the robustness contrasting with the traditional parallel waveguide coupler. For example, some analytical methods are applied to construct the specific functions of coupling strengths and detunings between the waveguides, including the Allen–Eberly scheme⁹ and phase mismatch model.¹⁰ Furthermore, adiabatic following in coherent quantum control is also widely used in manipulating the waveguide coupler, which can design robust and achromatic waveguide coupler, such as stimulated Raman adiabatic passage (STIRAP),^11,12 fractional STIRAP,¹³ and extension of STIRAP to multiple states^14,15 and adiabatic elimination.^16,17 STIRAP can also be used in various physical systems, such as many body physics,¹⁸ graphene,¹⁹ and THz²⁰ SPPs coupler. The biggest disadvantage of designing a waveguide coupler by using adiabatic following is that it requires much longer waveguide coupler due to the requirement of smooth coupling strength and detuning (well-known as the adiabatic condition).²¹ 

Recently, in order to overcome this detriment of adiabatic following, shortcut to adiabaticity (STA) had been proposed to decrease the quantum operation time.^22,23 Therefore, employing the quantum control technique of STA applies to design a waveguide coupler, which can increase the compactness of integrated optics.^24–26 Most recently, two remarkable papers propose the STA based on STIRAP in theoretical²⁸ and experimental²⁹ in a quantum atomic system. Thus, STA based on STIRAP intuitively can be employed to design a waveguide coupler to achieve much shorter device length and preserve the robustness of device. Therefore, this configuration of our design enhances the integration of the waveguide device. Furthermore, our device reduces the fabrication cost due to the maintenance of robustness against device geometry parameters by utilizing STA based on STIRAP.

In this paper, we propose a novel design of a three-waveguide coupler, which utilizes STA based on STIRAP to design a waveguide coupler, and our device achieves complete energy transfer with robustness against geometry parameters. According to the concept of STA based on STIRAP, our device contains a three waveguide coupler with a novel structure, which has some specific curved structures for input/output waveguides, and the middle waveguide is the straight structure (see Fig. 1).

Subsequently, we need to understand the relationship between coupling strength and distance based on the ion implantation technique. However, it is very hard to calculate in an analytical solution due to very complicated boundary conditions. Therefore, we simulate the two parallel waveguide configuration with the simulation of the ion implantation technique in CST software to obtain the relationship between coupling strength and distance [see Fig. 3(a)]. It is easy to obtain that the coupling strengths have an exponential relationship along with the distance between two adjacent waveguides. Therefore, we can engineer the distance functions between input/output and middle waveguides to get the coupling strengths given by STA based on STIRAP. Furthermore, we demonstrate that our device is much shorter compared to the device employing STIRAP (see Fig. 4), and we illustrate the robustness of our device, as shown in Fig. 5.

To sum up, the novelty of our paper can be concluded as follows: (i) we propose a novel design of a three waveguide coupler, which utilizes STA based on STIRAP to design a waveguide coupler for the first time. (ii) In contrast to the previous design of robust and complete achromatic three-waveguide couplers, our device has a much shorter device length due to the design of shortcut to adiabaticity (STA). (iii) We demonstrate that our design realizes highly efficient transfer with strong robustness against the perturbations of geometrical parameters.

In this paper, we consider the light energy transfer in a three-waveguide coupler, which corresponds to the three-level atomic Schrodinger equation^5,6 (with three differential equations), and these coupled equations can be written as

The components of the vector $C(z)=c1(z),c2(z),c3(z)T$ are the amplitudes of the fundamental modes of the corresponding waveguides, respectively, and $P1,2,3=c1,2,3(z)2$ are the corresponding normalized light intensities. The operator H(z) describes the coupling strengths and detunings between waveguides. Since the core of the three waveguides is made of the same material (with the same refraction index), there is no detuning between these waveguides (with the same propagation constant for each waveguide). Thus, operator H(z) can be obtained as

where Ω₁(z) [also Ω₂(z)] is the coupling strength between input and middle waveguides (also middle and output waveguides) and they can be calculated by CMT. According to CMT, the coupling strength has an exponential relationship with the distances between two adjacent waveguides. Therefore, we can design the function of distance between the input and middle waveguides (also middle and output waveguides) to schematize the function of coupling strength Ω₁(z) [also Ω₂(z)].

Based on these coupling equations, we can calculate adiabatic following of the three-waveguide coupler by employing diagonalization of the operator H(z). Subsequently, we can derive the eigenvalues Φ₀ = 0, $Φ±=±Ω12+Ω22$⁠, and their corresponding eigenstates,

where the mixing angle Θ is given by $tanΘ=Ω1Ω2$⁠. The adiabatic Hamiltonian H_a can be described as

where $Θ̇$ is the derivative of Θ. In the normal adiabatic following (STIRAP), if the adiabatic condition (⁠$Ω12+Ω22≫Θ̇$⁠) is satisfied, we can ignore all the off-diagonal terms. Therefore, there is no transition between the adiabatic states during the evolution. To satisfy the adiabatic condition, smoothly changing the coupling strength and detuning is required. Specifically, completed light transfer by employing STIRAP demands longer device length. Due to the existence of scattering, the light energy in the waveguide dissipates along with its propagation. Therefore, the loss of light energy is very large by manipulating STIRAP and the STIRAP waveguide coupler could be of much longer length than the parallel configuration, which is harmful fabrication of an integrated optical device.

To overcome this detriment of STIRAP, a beneficial approach is to exploit STA based on STIRAP to achieve shorter waveguide length. In this paper, we consider three waveguides with the same material and same reflection index (see Fig. 1). Thus, there is no detuning between these three waveguides. Based on these configurations, the three-waveguide coupler can be similarly written as an effective two waveguides coupler,^27,28

Therefore, it is obvious to find that the adding Hamiltonian H_cd by using the counterdiabatic term is given by H_cd = −i∑_n|∂_zϕ_a(z)⟩⟨ϕ_a(z)|, where |ϕ_a(z)⟩ are the adiabatic states of the effective two-waveguide coupler. Thus,

where $Ωa=θ̇/2$ and $θ̇$ is the derivative of mixing angle θ, where $tan⁡θ=Ω1(z)Ω2(z)$⁠. Thus, the total Hamiltonian of the effective two-waveguide coupler H = H_eff + H_cd, and we set the phase φ as $tan⁡φ=2ΩaΩ1$⁠. We introduce the unitary matrix to transfer to a new basis, where

We can obtain the effective Hamiltonian of the two-waveguide coupler with STA based on STIRAP,

where $Ω1eff=(Ω1)2+4Ωa2$ and $Ω2eff=Ω2−2φ̇$⁠. Therefore, we can obtain the effective coupling strengths of the three-waveguide coupler.

At the beginning, we employ an example to illustrate the function of our device. For the typical STIRAP, two Gaussian pulses are applied to two coupling strengths as Ω₁ and Ω₂,^11,12 with

where 2σ = z_f/3 is the full width at half-maximum of the pulses and τ = z_f/8. In order to derive two coupling strengths with STA of STIRAP, we utilize Eq. (8) to obtain the coupling strengths with STA of STIRAP. In this example, we establish the parameters as Ω₀ = 10 mm⁻¹, and the device length is 4 mm.

Two Gaussian pulses for coupling strengths of STIRAP (Ω₁ and Ω₂) are given by the dashed lines, and the corresponding coupling strengths of STA based on STIRAP (Ω_1eff and Ω_2eff) are given by the solid lines in Fig. 2(a). Based on these coupling strengths, the evolution of light energy transfers from the input to output waveguide for both STIRAP (shown in dashed line) and STA based on STIRAP (shown in solid line) in Fig. 2(b). The results show that the transmission rate of the waveguide coupler with STA based on STIRAP is much higher than STIRAP. The fidelity of the waveguide employing STIRAP (with the coupling strengths Ω₁ and Ω₂) can only reach nearly 0.65. Contrastively, the fidelity of our device (with the coupling strengths Ω_1eff and Ω_2eff) can arrive nearly 0.95, with Ω₀ = 10 mm⁻¹ and the device length being 4 mm.

The key point of implementation of STIRAP and STA based on STIRAP in the waveguide coupler is to achieve the coupling strengths for the STIRAP and STA based on STIRAP in the waveguide coupler fabrication. Established on the coupled mode theory (CMT), the coupling strengths have the exponential relationship with distances between two waveguides (see Fig. 3). Therefore, we can set the distance function between input and middle waveguides (middle and output waveguides) to obtain appropriate Ω₁ (Ω₂) for STIRAP and Ω_1eff (Ω_2eff) for the STA of STIRAP. In order to obtain the specific coupling strengths for STIRAP (Ω₁ and Ω₂) and STA of STIRAP (Ω_1eff and Ω_2eff), we analogize the technique of ion implantation¹¹ and realize the specific coupling strengths by engineering the special-designed distance of two waveguides in order to vary the coupling strengths.

As we can see, the coupling strength of STA based on STIRAP is relatively tedious, which requires a complicated curve of the input/output waveguide. To overcome this engineering problem, we can employ ion implantation technology to engineer the arbitrary complicated curves of input/output waveguides.¹² In order to obtain the coupling strength function against gap between waveguides with ion implantation technique and the mode analysis of the gradient-index waveguide is complicated, we employ the software simulator to calculate the coupling strength between adjacent waveguides. We simulate two parallel waveguides having gradient-index, and the maximum core channel width is 3 µm. The substrate of the waveguides is the Er:Yb-doped glass substrate, and the refractive index in our design is 1.5218. The height of the substrate is 6 µm, and the lengths and width of the substrate depend on the design.¹² Considering that the lateral and in-depth diffusion of ions has good linearity within the range of 1.522–1.538, we take the refractive index step (d_n) as a constant d_n = 0.002 when building the simulated model. We change the distance between the two waveguides to obtain the coupling strength function against gap between two adjacent waveguides [see Fig. 3(a)]. We take some different wavelengths and perform similar calculations from 680 nm to 1080 nm at intervals of 100 nm. From our results [Fig. 3(a)], it is easy to obtain that the coupling strength between two adjacent waveguides decays exponentially with the increase in distance, which is consistent with CMT. Subsequently, according to Fig. 3(a), we can transfer the coupling strength of our device [see Fig. 2(a)] to the geometrical structure, as shown in Fig. 3(b). Figure 3(b) demonstrates the distance functions of input/middle and middle/output waveguides. Based on this geometrical structure, we can easily fabricate our device with the ion implementation technique.

To compare STIRAP and STA based on STIRAP in optics coupler design, we take the coupling strength Ω₁(z) and Ω₂(z) of STIRAP as two Gaussian pulses, shown in Eq. (9) and corresponding to the coupling strengths of STA based on STIRAP. First, we show that our device (by applying STA of STIRAP) is shorter than the device of STIRAP. Furthermore, we demonstrate that our device is robust against geometry parameters.

Subsequently, we vary the maximum coupling strength Ω₀ and device length to obtain the final transmitted energy from the input waveguide to output waveguide (shown in Fig. 4). From the results of Fig. 4, we can easily obtain that our device can achieve complete transfer of light power with much shorter device length with STA based on STIRAP, compared with STIRAP. The waveguide coupler with STA based on STIRAP has shorter device length (roughly shorter 3–5 times) than the STIRAP waveguide coupler. This feature can be performed with both smaller and larger maximum coupling strengths Ω₀. When the device length is longer, the performance of STA based on the STIRAP waveguide coupler is increasingly similar to the STIRAP technique. In addition, compared with STIRAP, the most powerful advantage of using STA based on STIRAP is that it not only shortens the length of the device but also maintains robustness. In order to verify the robustness of our device, we fluctuate the two coupling strengths by changing τ and varying full width at half maximum (FWHM) by tuning σ, shown in Fig. 5(a). Furthermore, we demonstrate that our device is also robust against the maximum coupling strengths of Ω₁ and Ω₂ [see Eq. (9)], which makes the fabrication of our device much more convenient, as shown in Fig. 5(b). In Fig. 5(b), we plot the transmission efficiency with varying different maximum coupling strengths of Ω₁ and Ω₁ from 0.2 mm⁻¹ to 20 mm⁻¹. Thereby, we can claim that our device is robust with against the maximum coupling strengths of Ω₁ and Ω₁ due to the same feature in STIRAP.^21,30

In our paper, we achieve complete light transfer with much shorter device and keeping the robustness against perturbations of the geometrical structure, conducting STA based on STIRAP. The shortcomings of this method are as follows: (i) STA based on STIRAP requires more coupling strengths compared with STIRAP, and it requires smaller distance between the waveguides. (ii) Due to non-trivially coupling strengths from STA based on STIRAP, it requires a complicated geometrical structure. To overcome two issues, thus, we choose the ion implantation technique to precisely control the distances between input/middle and middle/output waveguides.

In this paper, we present a novel design for the complete and robust three-waveguide coupler by employing shortcut to adiabaticity (STA) based on stimulated Raman adiabatic passage (STIRAP). We control the coupling strength between two adjacent waveguides by changing the distance between them based on the above theoretical derivation and simulation calculation. By employing STA based on STIRAP, light can transfer completely with the shorter device length, and the energy loss reduces during light transmission. Furthermore, we verify that our device is robust against perturbations on device geometry parameters. The smaller device size makes this design a promising application in integrated optics.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This work was supported by funding of the National Science and Technology Major Project (Grant No. 2017ZX02101007-003), the National Natural Science Foundation of China (Grant Nos. 61565004, 6166500, and 61965005), the Natural Science Foundation of Guangxi Province (Grant Nos. 2017GXNSFBA198116 and 2018GXNSFAA281163), the Science and Technology Program of Guangxi Province (Grant No. 2018AD19058), and the GUET postgraduate outstanding dissertation project (Grant No. 18YJPYSS24). W.H. acknowledges for funding from the Guangxi oversea 100 talent project and W.Z. acknowledges funding from the Guangxi distinguished expert project.