Femtosecond laser direct writing (FsLDW) promotes the integration of photonic integrated circuits (PICs) from 2D into 3D by enabling arbitrary waveguide arrangement in three-dimensional (3D) space. Thus, FsLDW provides a straightforward solution to improve chip integration and a new physical degree of freedom to design and realize more complex on-chip photonic manipulation [1–7]. Moreover, FsLDW provides large amounts of technical support, especially in the emerging fields of quantum computing [8–12] and topological photonics [13–17] by adding refractive index and coupling modulation along the 3D waveguides. The 3D chip layouts were used to avoid waveguide crossings and realize oscillating potential so that the first on-chip photonic quantum Toffoli gate [12] and Floquet topological insulator [14] were exhibited.

In FsLDW-3D-PICs, the propagation constant (β) of waveguides relates to the effective refractive index (neff) and represents the phase change of light per unit propagation length, which is crucial for both compact integration and quantum simulating. On the one hand, the higher the neff, the smaller the size of the waveguide devices. The realization of a high refractive index was strongly related to the type of glasses. Currently, the record high refractive index change for FsLDW waveguides is ∼0.022 in fused silica, improving the integration of FsLDW PICs further [18]. On the other hand, the diagonal elements of the Hamiltonian matrix of a Hermitian system are generally filled with β of each waveguide. By constructing a Hamiltonian with a waveguide array, some quantum mechanics states and dynamic processes, challenging or impossible to observe directly, can be equivalently mapped onto optical systems due to the formal unity of Schrödinger’s equation and light wave evolution equation in the waveguide array [19]. During the past decades, plenty of quantum phenomena have been observed through optical and quantum simulation, including Bloch oscillations [20,21], Anderson localizations [22], Zener tunneling [23], and gravitational lensing effect [24], proving it a strong and versatile tool for visualizing some quantum phenomena.

As the demands for integration of PICs and the simulated system complexity have increased, efforts have been devoted to obtaining a larger β variation range of single-mode waveguides in FsLDW PICs. However, the methods to vary β are either by inducing periodic bending waveguides [20,21] or changing the writing parameters, such as the writing velocity and pulse energy [25], which is not compact and induces a lower Δneff (10−4–10−3) than that of silicon waveguides. These methods lead to a larger evolution length for quantum simulators [26] and a larger size of waveguide devices [27]. Cheng et al. increased the neff of FsLDW bending waveguides by inducing bend–loss–suppression walls (BLSWs) [28]. However, the additional structure will limit the device design in large-scale PICs. Thus, developing a simple and controllable method to enlarge the neff through β and finely regulate it is required to exploit the potentialities of 3D-FsLDW waveguide devices completely.

This paper presents a simple and efficient composite waveguide (CWG) method based on FsLDW technology to realize large propagation constant variation regardless of the type of glasses (Δβ−36cm−1 in Corning Eagle XG). The method enables the fine control of coupling and Hamiltonian reconstruction of waveguide arrays. A certain amount of propagation constant difference Δβ is induced by regulating the central distance between two circular sub-waveguides. The maximum of this varying amount is ∼2 times higher (Δneff=4.2×10−4) than that of varying the writing velocity (∼2.0×10−4). Additionally, based on the CWG method, a novel and compact straight-line directional coupler (SL-DC) is proposed and optimized. The pitch of the SL-DC is successfully minimized to 9 μm. Finally, with the CWG method, the Hamiltonian reconstruction of a waveguide array is theoretically and experimentally studied. The control of crosstalk between different layers is simultaneously shown visually. The CWG method we demonstrate is a useful tool to realize coupling control between waveguides in 3D scales, which will promote the development of 3D integrated devices and quantum simulation.

Jin et al. [29] proposed a double-track approach to deal with single waveguides (SWGs) restricted in terms of birefringence and mode-field shape. The approach has been widely applied to fine-regulate the birefringence of waveguides [30,31]. Nevertheless, the core idea of the proposed CWG method is to use two circular-cross section waveguides to construct a CWG with an elliptical mode profile to improve the control of the β of the waveguide. As illustrated in Fig. 1(a), the injected light oscillates at the coupling region of a symmetric directional coupler (DC), and the energy is fully coupled from the input waveguide to the adjacent waveguide at the coupling lengths only if two waveguides have the same β. While β1 greatly differs from β2, the energy can be completely limited in the input waveguide, and the evanescent coupling between two waveguides can be further controlled. Here, the central distance of two sub-waveguides arranged in the vertical (y) direction is fine-regulated to introduce a certain Δβ between the CWG in Fig. 1(c) and the SWG in Fig. 1(b) (the basic information of the single circular waveguide, the optical microscope images of cross-sections, and mode profiles of CWGs with different central distances are provided in Appendix A). By implementing CWGs in some lattice sites of the waveguide array, as illustrated in Fig. 1(d), the coupling between different layers is suppressed, even at a normal coupling distance due to the high Δβ, as if these two layers are distant. This approach leads to an equivalent transformation on the array arrangement and, hence, its Hamiltonian.

Combining the FsLDW with a high-precision 3D linear stage (see Section 5 for fabrication details), the central distance between two sub-waveguides can be fine-regulated. A series of DCs, consisting of CWGs with varying central distance and SWGs, were fabricated with an interaction distance of 10 μm. As shown in Fig. 2(a), all transmissions of DCs with a certain CWG central distance are sine-squared. As the central distance increases, the maximum transmission of the oscillation curves decreases from ∼1 to ∼0.05 in Fig. 2(b), showing a significant negative correlation between the maximum transmissions and the central distance. The transmission of light in the coupling region of DCs can be described by the coupled mode theory (CMT) using Eqs. (1) and (2) as follows: Trans=κ2κ2+Δ2sin2(κ2+Δ2·l+φ),(1)Δ=Δβ2,(2)where κ is the coupling coefficient. In addition, l and φ denote the interaction length and the initial phase induced by arc-bending waveguides near the coupling region, respectively. Hence, Δβ can be obtained from the transmission curves in Fig. 2(a) using Eqs. (1) and (2). Figure 2(c) shows that Δβ increases nearly linearly with the central distance. Based on maintaining the CWGs single-mode, the maximum central distance reaches 3.6 μm under a writing velocity of 40 mm/s and induces a Δβ of ∼36cm−1. This value is ∼2 times higher than those obtained by solely changing the writing velocity of SWGs under the same pulse energy (the pulse energy of a single circular waveguide is 390 nJ; see Section 5). This result indicates the feasibility of the method to expand the variation range of Δβ while being assisted with the change of velocity and single-pulse energy. Figure 2(d) exhibits the crosstalk between the 3.6 μm central-distance CWGs and SWGs of different waveguide distances (dc). As the dc increases from 9.5 to 12.5 μm, the crosstalk decreases from −12.02dB to a negligible −20.12dB, demonstrating a good anti-coupling effect (the mode profile images under different interaction lengths when CWGs are placed near SWGs are provided in Appendix A). This result implies that the 3.6 μm central-distance CWGs (all CWGs described in the following text correspond to the 3.6 μm central distance) can be used in 3D PICs to prevent optical signal crosstalk.

Note that excessive loss to the original circuits by CWGs must be avoided to exploit the CWGs in PICs to control the evolution of the light wave. Here, the propagation loss of CWGs and mode-mismatch loss between CWGs and SWGs were studied. At first, the propagation loss of the CWGs is accurately measured to be 0.264 dB/cm using the method introduced in Ref. [32]. Then, by inserting a different number of CWG segments into a single SWG, as illustrated in the insets of Fig. 3(a), the total propagation loss (TPL) through the entire waveguide is given as follows: TPL=N·[(αCWG−αSWG)·l+2·αML]+αSWG·L,(3)where αCWG, αSWG, and αML are the propagation loss of CWGs, SWGs, and the mode-mismatch loss between CWGs and SWGs, respectively. Additionally, N, l, and L are the number of segments of CWGs, the length of a single CWG segment, and the total length of all waveguides. As l is designed to be 3 mm, αSWG and αML can be obtained from the intercept and slope of the curve in Fig. 3(a), which is 0.273 dB/cm and 0.193 dB/facet, respectively. Note that the propagation losses of SWGs and CWGs are similar; however, the mode-mismatch loss is high, which is the main factor reducing the practical use of CWGs.

Furthermore, a designed taper was added between the CWG and the SWG to reduce the mode-mismatch loss between CWGs and SWGs. Before the taper, the SWG (or CWG) consisted of two single circular waveguides. Then, two circular waveguides were apart from each other (or merged together) in the y direction to form the taper. As Fig. 3(b) shows, the mode-mismatch loss decreases while the length of the taper waveguide increases. With a taper length of 1.5 mm, the mode-mismatch loss is negligible, approximately 0.04 dB/facet. This result indicates that no excessive loss is introduced with the taper waveguides, and the CWGs with taper waveguides can be directly implanted in the PICs to manipulate the evolution of the light wave effectively.

The conventional DCs comprise three parts: in/output ports, S bends, and coupling region. Only when two waveguides are close enough can the evanescent fields of two waveguides overlap, leading to the energy exchange between them. At the top of Fig. 4(a), S bends are generally designed to avoid the evanescent wave coupling effect outside the coupling region, which needs to separate the input and output ports to at least 24 μm in FsLDW PICs, resulting in chip space waste. Although two parallel straight waveguides of small interaction distance can also demonstrate coupling effects, the exchange of energy exists as long as the two waveguides are not separated. Because the CWGs do not couple with SWGs for high Δβ, the S bends can be equivalently replaced by CWGs, as illustrated at the bottom of Fig. 4(a), which minimizes the pitch between two ports of DCs in the x direction to that size of the coupling region. Specifically, the CWGs and SWGs are connected by taper waveguides in Fig. 4(b) to lower the mode-mismatch losses. We set Ltaper to 1.5 mm and the central distance, dw, to 3.6 μm. As the CWG approaches the SWG, the sectional optical microscope images in Fig. 4(b) suggest that the CWG adiabatically turns into the SWG. Subsequently, the transmission of SL-DCs with different coupling distances was studied with the vertically polarized (V polarization) laser injected through the SWG port [more detailed characterization of the transmission of SL-DCs under V and horizontal (H) polarization is given in Appendix B]. As the coupling distance varies from 9 to 12 μm, the coupling coefficient (κ) ranges from 7.59 to 1.93cm−1. The sine-squared transmission in Fig. 4(c) indicates that the SL-DCs can perform the coupling function similar to the traditional DCs but with only one-third of the space. Compared with other FsLDW DCs [32–35], the SL-DCs are the shortest input/output pitch. Moreover, the results are significant in at least two major respects. First, waveguide devices comprising traditional DCs, such as MZIs, CNOT gates, and even 3D tritters, can be realized in a form containing straight waveguides, offering a new geometric design to waveguide devices. Second, by exploiting this design, the integration of PICs based on FsLDW technology can be improved.

The values of κ and β of each waveguide are determined after the fabrication of the waveguide array. Thus, its Hamiltonian is determined under linear light. In particular, the only way to transform the Hamiltonian of an established array is to inject nonlinear light. This study demonstrated the Hamiltonian reconstruction of a waveguide array under linear light by applying the CWG method.

First, two 3D straight waveguide arrays were proposed, of which the schematics are shown in Figs. 5(a) and 5(b). One comprised SWGs, as illustrated in Fig. 5(c), while the SWGs at a deeper layer were precisely substituted by CWGs in the other array, as shown in Fig. 5(d). Two arrays, named SWG array and composite array, were fabricated; the cross-section microscope images are shown in Figs. 5(e) and 5(f). Because of the depth difference d1 between these two layers, in the SWG array, the κ of the shallower layer κ1 was experimentally determined as ∼0.30cm−1 higher than that of the deeper layer κ3. In contrast, the diagonal coupling coefficient κ2 was ∼0.40cm−1 lower than κ1.

The normalized intensity of each waveguide in the array with different inputs can be obtained using the CMT (details in Appendix C). The optical axis of all waveguides is vertical (in the y direction), which means that the light evolution for the H and V polarizations in the two arrays was basically the same except for the oscillation periods. Here, the experimental results of V polarization were considered for demonstration purposes. In particular, the laser was coupled into waveguides W4 and W1 of each array. Figures 6(a) and 6(b) show that as the laser is injected through W4 and W1, respectively, the experimental data points at different propagation distances agree well with the theory results, represented by lines. Additionally, the experimental cross-sectional mode profile images at the propagation distances of 3, 5, and 7 mm are given in Fig. 6(c). Note that the laser energy can be close to fully coupled from the input (W4 or W1) to other waveguides in the SWG array due to the negligible Δβ between SWGs. The light evolution in this SWG array was controlled and determined solely through a particular Hamiltonian. In the composite array, the corresponding Hamiltonian Hc can be given as follows: Hc=(β1κ1κ2κ20κ1β10κ2κ2κ20β2κc0κ2κ2κcβ2κc0κ20κcβ2),(4)where κ1, κ2, and κc are 3.3, 3.1, and 2.2cm−1, respectively. Although the non-zero κ2, the coupling between CWGs and SWGs is negligible due to the high Δβ. Hence, κ2 can be replaced by 0, and Hc is equivalent to He=(β1κ1000κ1β100000β2κc000κcβ2κc000κcβ2).(5)

While linear light was injected into W1 and W4 of the composite array, the experimental normalized intensity of different waveguides was measured. The results are shown in Figs. 6(d) and 6(e). Simultaneously, the theoretical evolution of the light wave under the equivalent Hamiltonian He is depicted as colored lines in Figs. 6(d) and 6(e). The experimental cross-section mode-profile images at 1, 5, and 9 mm propagation distances under Hc fit well with the theoretical results of He. This result indicates that the array of Hc can govern the evolution of light wave as same as that of He without a large d1 to minimize κ2 (the equivalent SWG array demonstrating the same evolution of light is provided in Appendix C). Under the linear light propagation, the Hamiltonian of the composite array can be written as Hc or He, implying that the reconstruction of a waveguide array was realized. Furthermore, a comparison of the results in Figs. 6(c)–6(f) shows that the light propagating in one layer will not interfere with another layer with the CWG method, which is practical to control the crosstalk among the waveguides at different layers in the large-scale 3D PICs and increase integration.

In summary, we proposed a flexible CWG method based on FsLDW technology to induce a controllable amount of Δβ between the normal SWGs and the CWGs by regulating the central distance of two circular sub-waveguides. The maximum Δβ achieved by the CWG method is approximately twice that achieved solely by regulating the scanning velocity under the same processing power. Thus, the proposed method provides a new degree of freedom to enlarge Δβ regardless of the types of glasses. The measured propagation loss of CWGs and SWGs was as low as ∼0.27dB/cm, while the mode-mismatch loss between each other was minimized to 0.02 dB/facet with the taper waveguides. By exploiting CWGs in the SWGs, the most compact FsLDW DC, the SL-DC, was demonstrated and investigated at interaction distances of 9 to 12 μm, exhibiting the same functions as traditional DCs with S bends while having a more integrated arrangement in the chip. The developed straight-line coupling devices provide insights that can help develop integrated waveguide devices such as SL-MZIs and SL-CONT gates. Moreover, Hamiltonian reconstruction under linear light was achieved by replacing some of the SWGs in an array with CWGs without the geometric changes in the arrangement of arrays.

This study demonstrated the full ability and relevancy of the CWG method in coupling control and Hamiltonian reconstruction. In particular, the CWG method could regulate the coupling ability between waveguides and promote the increased integration of 3D PICs. Moreover, the larger Δβ benefits the construction of a more complicated Hamiltonian, thereby simulating a more complicated physical system and observing the quantum dynamics in the optical system. In the future, by combining the 3D processing ability and the CWG method, the FsLDW PICs can open new ways to photonic simulations and integrated optics.

Sample Fabrication. At first, the femtosecond (fs) laser pulses used in this study were generated by a commercial integrated fs-laser system (CARBIDE, Light Conversion Ltd.). The central wavelength, duration, and repetition rate of the pulses were 1030 nm, 290 fs, and 1 MHz, respectively. The fs laser pulses were focused by an objective lens (40×, NA=0.75, f=0.51mm; UPlanFL N, Olympus). The circular sub-waveguides were fabricated inside the borosilicate glass samples (EAGLE XG, Corning) with a pulse energy of 390 nJ and a writing velocity of 40 mm/s. The geometric center of all waveguide structures [including the structures in Figs. 1(b), 1(c), Figs. 2(b)–2(d), Fig. 3, Figs. 4(b), 4(c), Figs. 5(a)–5(c) and Figs. 6(a)–6(c)] were set at a depth of 190 μm beneath the surface of the substrates. The writing tracks were controlled accurately using an air-bearing X–Y–Z linear stage (ABL1000, Aerotech Inc., USA). Subsequently, the cross-sections of the fabricated photonic chips were finely polished for characterization using an automatic precision polishing machine (UNIPOL-802, Shenyang Kejing Auto-instrument Co., Ltd.).

Experimental Measurement. The laser was launched with a wavelength of 808 nm (MDL-III-808L, Changchun New Industries Optoelectronics Tech, Co., Ltd.) and butt-coupled into the waveguides through an objective lens (10×, NA=0.25, f=17.7mm). The polarization orientation of the light was controlled by combining a polarization beam splitter and a rotatable half-wave plate. A similar metallurgical objective was used to collect the light emitted from the waveguide cross-section. Subsequently, the power of the emitted laser and mode profiles were measured using a power meter (PM100D, Thorlabs Inc.) and a laser beam analyzer (SP928, Spiricon Inc.), respectively.