Spatial-mode-coupling-based dispersion engineering for integrated optical waveguide

Yu Li; Jiachen Li; Yuandong Huo; Minghua Chen; Sigang Yang; Hongwei Chen

doi:10.1364/OE.26.002807

1. Introduction

Dispersion engineering on chip is crucial for many integrated nonlinear-based applications, such as monolithic optical frequency comb [1–3], supercontinum generation [4], entangled photons generation for quantum information processing [5] and so on. For these physical systems, dispersion results in the phase mismatch around the light waves in different wavelength, which ultimately determines the efficiency of the nonlinear process. The common approach to engineering dispersion in integrated optical waveguide is to change the cross-sectional size and shape of the waveguide by tailoring the height and width simultaneously [6–9]. It has been widely used but still has two major problems. On the one hand, it is difficult to deposit a film with an expected thickness for some kinds of materials in the fabrication. For example, it's hard to deposit a silicon nitride film with a thickness of over 300 nm by LPCVD process but broadband anomalous dispersion at telecommunication band only occurs in the region where the height of the silicon nitride waveguide is more than 700 nm [10]. On the other hand, the integrated photonic technology has been developed to mature over the past decade and the access to the state-of-art fabrication process is open by many foundries in the world [11–13]. But for the multi-project wafer (MPW) users, they are unable to change the thickness of the waveguide in most cases. For instance, the thickness of the TriPleX^TM double-strip waveguide in the standard process design kit (PDK) offered by LioniX is 840 nm, which is now the major optical waveguide platform offering the low loss Si₃N₄\SiO₂ waveguide technology [13]. But on this height, anomalous dispersion can never be realized at the wavelength of 1550 nm no matter what the width of the waveguide is. Hence, the dispersion engineering method by adjusting the width and height of the waveguide is invalid in some cases especially in the chip fabrication through the foundry approach, due to the design freedom of the height is limited in these situations. In addition to the above dispersion engineering method, some other approaches are also studied. For example, a novel vertical slot waveguide has been proposed and can achieve near zero group velocity dispersion (GVD) over hundreds of nanometers numerically [14]. But it is very sensitive to the thickness of the waveguide and the experimental demonstration has not been reported yet. Another method that utilizes the control of waveguide sidewall angle for dispersion engineering inspired by the dispersion control in fiber was studied and experimentally realized wide and flatted anomalous dispersion [15]. However, these approaches are not universal for integrated waveguide under the planar semiconductor technology and cannot deal with the above two problems.

In this paper, to tailor the dispersion of the waveguide with fixed height without inducing extra fabrication process, a dispersion control technique based on the spatial modes coupling is proposed. The mode coupling induced dispersion change has been observed and utilized in the optical frequency comb generation. Sven Ramelow et al. observed mode coupling between the TE00 and TM00 modes in thick silicon nitride waveguide with cross sections of 725 nm x 1100 nm and 725 nm x 900 nm. This mode coupling caused anomalous dispersion over about 50 nm [16]. Yang Liu et al. observed anomalous dispersion with a wavelength range of about 5 nm in the silicon nitride waveguide caused by the mode coupling between TE1 and TE2 modes [17]. However, not just the polarization mode coupling but the different-order TE modes coupling in the two studies both appear in a single waveguide, which highly depend on the geometrical shape of the waveguide and can only occur in specific waveguide. So the two kinds of mode coupling in these studies are difficult to extend to other waveguide platforms. In the proposed technique, we employ the spatial mode coupling by combining two waveguides to achieve dispersion control. It has no special need to the waveguide geometry and can change the waveguide from normal dispersion to anomalous dispersion by only changing the width of the waveguides. By introducing the method to the TriPleX^TM double-strip waveguide platform, we numerically realize anomalous dispersion over 70 nm and experimentally demonstrate anomalous dispersion with wavelength range of about 25 nm on a micro-ring resonator with high Q of 0.8 million. To our best knowledge, this is the first time to realize anomalous dispersion over 25 nm wavelength range on the low loss TriPleX^TM double-strip waveguide platform.

2. Principle and design

The coupled-mode theory manifests that the relatively strong mode coupling between two modes would bring out mode splitting and avoided crossing of the propagation constant, which would change the dispersion of the two modes significantly at certain conditions [18]. As the fundamental principle, we utilize it to develop a novel dispersion engineered method. In the proposed scheme, two waveguides in different widths are combined together at a specific distance to form a new type of waveguide, which is called by us the dispersion engineered waveguide in this paper. The widths of the two waveguides are designed to realize the mode coupling between the TE0 mode of the narrower waveguide and the TE1 mode of the other one at the target wavelength. The two coupled modes would evolve into two hybrid modes. By altering the distance of the two waveguides, the coupling strength can be tuned so that the dispersion of the two modes can be engineered precisely. In this paper, we use the TriPleX^TM waveguide platform to demonstrate the proposed dispersion engineering technique in detail.

TriPleX^TM waveguide technology is one of the major integrated photonic platforms for Si₃N₄\SiO₂ waveguide and compatible with CMOS fabrication equipment [19]. The double-strip waveguide is one of the three typical TriPleX^TM waveguides, which is formed by three layers of Si₄N₃, SiO₂ and Si₄N₃ with the thickness of 170 nm, 500 nm, and 170 nm respectively. As shown in the inset of Fig. 1(a), the width of the standard double-strip waveguide is 1.2 μm and it only supports the TE0 mode and TM0 mode at the wavelength of 1550 nm. As the edge coupler of the TriPleX^TM waveguide platform is specially designed for TE0 mode with a polarization rejection ratio of 20 dB, the TE0 mode is the one that used in most of the applications. The numerical calculated effective index and the GVD parameter D of the TE0 mode are 1.535 at the wavelength of 1550 nm and around −700 ps/nm/km over the wavelength range of 1500 nm to 1600 nm, as shown in Fig. 1(a). By changing the width of the two waveguides individually, we find that there exits an intersection on the effective index at the wavelength of 1580 nm between the TE0 mode of the waveguide with width of 1.2 μm and the TE1 mode of the waveguide with width of 3.1 μm, as shown in Fig. 1(c). The calculated D of the TE1 mode is about −550 ps/nm/km across the wavelength range of 1550 nm to 1600 nm. The two spatial modes would interact with each other and split into two new hybrid modes (we call them hybrid mode 1 and hybrid mode 2 respectively in this paper) at the wavelength of 1580 nm after the two waveguides are combined together to form the dispersion engineered waveguide, which is shown in the inset of Fig. 1(c). The evolution of the corresponding effective index before and after the coupling are shown in Fig. 1(c) and the phenomena of avoided crossing is observed. In the coupled-mode theory, the two hybrid modes are the symmetry hybrid mode and the antisymmetric hybrid mode, composed of the two modes before coupling with the same phase and opposite phase respectively, of which the mode fields are shown in Fig. 1(c).

Fig. 1 (a) The characteristic of TE0 mode of the standard waveguide (in blue line) and the corresponding calculated dispersion (in orange line). Insets show the cross section of the waveguide and the corresponding mode profile at wavelength of 1550 nm respectively. The different colors standing for SiO₂ refer to different deposition technologies. Please see [19] to get the details about the fabrication process. (b) Measured transmission spectrum (blue line) and the FSR (orange points) of the TE0 mode of the standard micro-ring resonator. The calculated D of the TE0 mode (black line) is −750 ± 30 ps/nm/km across the entire measured wavelength range. Inset shows the loaded Q value of the TE0 mode is about 0.65 million at the wavelength of 1571 nm. (c) The red line and blue line are the effective index of waveguides with width of 1.2 μm and 3.1 μm individually. The black line and the black dotted line are the effective index of the two hybrid modes of the dispersion engineered waveguide respectively. Insets show the cross section of the dispersion-engineered waveguide and the mode profiles respectively. (d) The D of the two hybrid modes at different gaps.

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The numerical results of the relationship between the D and the gap are shown in Fig. 1(d). It shows that the D of the hybrid mode 2 can be altered from normal dispersion (−300 ps/nm/km) to anomalous dispersion (230 ps/nm/km) at the wavelength of 1575 nm while the gap is changed from 1 μm to 1.6 μm. As a valuable case, the wavelength range of the anomalous dispersion region is about from 1540 nm to 1610 nm, which is useful for many nonlinear applications such as optical frequency comb and super-continuum spectrum generation. Also, by slightly altered the width of the two waveguides, the intersection point can be tuned so that the anomalous dispersion region would be shifted to shorter wavelength or longer wavelength. It seems that the proposed dispersion engineered waveguide is similar to the slot waveguide [20], but they're quite a bit different with each other. For the slot waveguide, the two parts are placed so close that the strong coupling between the same spatial modes exists across a very wide wavelength range, which makes the modes changed fundamentally. While for the proposed waveguide, the strong coupling between different spatial modes only exists at certain wavelength so that the mode splitting just appears near the wavelength.

It is worth mentioning that the proposed method can also be applied to other spatial modes such as the coupling between the TE1 mode and the TE2 mode or the coupling between the TE0 mode and the TM0 mode. Under this case, some issues need to be considered carefully. First, the widths of the two waveguides should be redesigned to let the two selected modes couple with each other at target wavelength with relatively strong mode coupling. That’s why we use the TE0 mode and the TE1 mode rather than the TM0 mode, as the coupling between the TE0 mode and the TM0 mode is very low through the wavelength range of 1500 nm to 1600 nm. In addition, if the two selected modes don’t include the fundamental mode (TE0 mode in our presented case), an extra mode converter need to be designed to transfer optical power from the fundamental mode to one of the two selected modes.

3. Experiment

Compared to straight waveguide, it is easier to measure the dispersion of the proposed waveguide and verify the dispersion engineered method on micro-ring resonator by measuring the free spectral range (FSR). Therefore, we fabricated a micro-ring resonator with the proposed dispersion engineered waveguide as the ring waveguide on the TriPleX^TM waveguide platform, of which the schematic and the micrograph are shown in Fig. 2(a) and Fig. 2(b) respectively. For convenience, we name the resonator as dispersion engineered resonator in this paper. The standard waveguide with width of 1.2 μm is used as the bus waveguide and designed to be bend waveguide with curve radius of 650 μm. The purpose to use bend waveguide is to reduce excess loss and improve the Q value of the resonator [21]. In our previous study [22], we find that the existence of bend and the material difference between the fabrication and the numerical study may cause the deviation between the experiment and the theoretical design. In this experiment, the widths and internal gap of the dispersion engineered waveguide are designed to be 1.2, 3.2 and 1.8 μm respectively and the radius of the ring waveguide is designed to be 500 μm. The radius is defined as the distance between the center of the ring and the center of the waveguide with width of 3.2 μm, which is marked as the symbol R in Fig. 2(a). The distance between the bus waveguide and the ring waveguide is 2 μm. As a contrast, we fabricated a micro-ring resonator with the standard waveguide both as the ring waveguide and the bus waveguide, and its radius is also designed to be 500 μm.

Fig. 2 (a) The schematic of a ring resonator with the proposed dispersion engineered waveguide as the ring waveguide; (b) Micrograph of the fabric1ated dispersion engineered ring resonator with radius of 500 μm; (c) Measured transmission spectrum of the dispersion engineered ring resonator. Inset (left) shows that the hybrid mode 2 has a loaded Q of 0.86 million at the wavelength of 1571 nm; Inset (right) is the zoom view of the measured result plotted in the black dotted box, where the two hybrid modes have nearly the same extinction ratio; (d) The measured FSR of the two hybrid modes versus the wavelength are shown as points. The step-like form of the FSR is due to the resolution of the OVA is 160 MHz. The blue line and orange line are the calculated D of the hybrid mode 2 and hybrid mode 1 respectively.

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In the measurement, we utilized the optical vector analyzer (OVA) with a resolution of 1.25 pm to measure the transmission spectrum of the two fabricated devices. The relationship between D and FSR of the micro-ring resonator is as follow:

D (λ_{1}) = \frac{1}{C} \cdot \frac{d n_{g}}{d λ} = \frac{1}{C} \cdot \frac{n_{g} (λ_{1}) - n_{g} (λ_{2})}{λ_{1} - λ_{2}} = \frac{1}{2 π R \cdot (λ_{1} - λ_{2})} \cdot \frac{F S R (λ_{2}) - F S R (λ_{1})}{F S R (λ_{1}) \cdot F S R (λ_{2})}

where c, n_g, λ and R are the speed of light in vacuum, group index, wavelength in vacuum and the radius of the micro-ring resonator respectively. Figure 1(b) shows the transmission spectrum of the micro-ring based on the standard waveguide, and the D of the TE0 mode is measured to be about −750 ± 30 ps/nm/km across the wavelength range of 1520 nm to 1605 nm, which agrees well with the numerical study. As shown in Fig. 2(c), the dispersion engineered resonator supports two distinct spatial modes and the two modes have almost the same rejection ratio near the wavelength of 1578 nm. This is the region where the avoided crossing appears. We used linear function to fit the measured FSR and calculated the D of the two spatial modes based on (1), which is shown in Fig. 2(d). The hybrid mode 2 shows anomalous dispersion with D of 1300 ± 200 ps/nm/km across a wavelength band of about 25 nm while the hybrid mode 1 shows relative large normal dispersion of −2800 ± 300 ps/nm/km at the same wavelength band, which is about four times larger than the standard waveguide. At longer wavelength of more than 1580 nm, the two modes both appear normal dispersion. The wavelength region of the anomalous dispersion and the variation trend of the FSR agree well with the theoretical study. But the hybrid mode 2 seems to be near the zero dispersion at the wavelength of 1600 nm. This will be studied further using the instrument with higher resolution. It's worth mentioning that the step-like data of FSR comes from the fact that the resolution of the OVA is 160 MHz.

On this basis, we further changed the width of the wider sub-waveguide of the dispersion engineered waveguide from 3.2 μm to 3.3 μm without other parameters changed to verify the tunability of the anomalous dispersion region. Figure 3(a) shows that the extinction ratio of the hybrid mode 2 (marked as blue triangle) is obviously larger than the hybrid mode 1. According to the trend of the hybrid mode 1, it is extrapolated that the intersection point of the two modes should be at the wavelength of more than 1605 nm. As a result, the measured anomalous dispersion is redshifted to the wavelength range of from 1582 nm to 1605 nm or to even longer wavelength with the D of 800 ± 100 ps/nm/km as shown in Fig. 3(a). The results demonstrated that by increasing the width of the wider sub-waveguide of the dispersion engineered waveguide, the anomalous dispersion of the hybrid mode 2 can be tuned to longer wavelength. Based on the same principle, the anomalous dispersion can also be tuned to shorter wavelength just by reducing the width of the wider sub-waveguide.

Fig. 3 Measured transmission spectrum (blue line) of the two fabricated micro-ring resonators. The hybrid mode 2 is marked as blue triangle on the transmission spectrum. The FSR and the D of the hybrid mode 2 are shown with blue points and orange line respectively. (a) The resonator has the same parameters as the resonator described in Fig. 2(c) except for the wider sub-waveguide of the dispersion engineered waveguide is changed from 3.2 μm to 3.3 μm; (b) The micro-ring resonator has a radius of 700 μm.

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To validate the conjecture that the bend has effect on the dispersion even at such large radius of 500 μm, we fabricated another micro-ring resonator with the radius of 700 μm. Other parameters are the same as previous study that the widths and internal gap of the dispersion engineered waveguide are 1.2, 3.1 and 1.6 μm respectively. Figure 3(b) suggests that the hybrid mode 2 (marked as blue triangle) is stimulated significantly at such radius of 700 μm with anomalous dispersion region of over 25 nm at the central wavelength of 1580 nm. Comparing to the previous fabricated resonator, we can conclude that the bend effect is important to the dispersion engineering and should be considered carefully in the design.

4. Analysis and discussion

The main purpose to propose the dispersion engineered technique is for the nonlinear application. Not only the dispersion is critical in the nonlinear optics, but the loss and the nonlinearity parameter of the waveguide are also very important. In this section, we further study the effect of the proposed method on the loss and the nonlinearity parameter. For the propagation loss of the dispersion engineered waveguide, the key question is that weather the proposed method aggravate the loss compared to the standard waveguide. We study the issue through experiment and theoretical study. By fitting the measured transmission spectrum of the hybrid mode 2 with Lorentz function, we can get the propagation loss and the Q value of the ring resonator [23]. For the three fabricated dispersion engineered ring resonators in this paper, the measured propagation loss and Q value of the hybrid mode 2 are all about 0.3 dB/cm and 0.8 million respectively across the entire anomalous dispersion region, as shown in the inset of the Fig. 2(c). This low loss and high Q value is at an equivalent level with many works of on-chip optical frequency comb [3,7,9,24]. As a comparison, the loss and Q value of the ring resonator with standard waveguide are measured to be 0.36 dB/cm and 0.65 million respectively at the same wavelength range as shown in Fig. 1(b). We can see that the dispersion engineered waveguide has the similar low loss with the standard TriPleX^TM waveguide and even a bit lower. For the TriPleX^TM waveguide, the bend loss is negligible for the TE0 mode of the waveguide with the width of 1.2 μm and for the TE1 mode with the width of 3.1 μm when the bend radius is more than 125 μm. Hence, the interface scattering induced by the sidewall roughness of the waveguide is the main contribution to the propagation loss. In general, the scattering loss is proportional to the ratio of the optical power at the sidewalls to the total power of the mode. So that the propagation loss of TE1 mode of the wider waveguide is a bit lower than the TE0 of the narrower waveguide due to the power ratio at the sidewalls of the former is lower than the latter. Considering the propagation loss of the hybrid mode should be the weighted mean of the TE1 mode and the TE0 mode, it's naturally inferenced that the two hybrid modes have lower loss than the TE0 mode of the standard waveguide.

In order to compare the propagation loss between the dispersion engineered waveguide and the standard TriPleX^TM waveguide quantitatively, we utilize a three-dimensional method to calculate the interface scattering induced loss [25]. In this model, the roughness sidewalls are regarded as the radiation units. The detailed principle is discussed in the reference. In the calculation, the two widths and the internal gap of the dispersion engineered waveguide are 1.2 μm, 3.1 μm and 1.6 μm respectively. The propagation loss of the TE0 mode of the standard waveguide and the hybrid mode 2 of the dispersion engineered waveguide are calculated based on the model. The ratio of the two calculated losses (standard waveguide is the denominator) over the wavelength range of 1500 nm to 1620 nm is plotted in orange line in Fig. 4(a). To better understand the loss relationship between the two modes, the corresponding simulated D of the hybrid mode 2 is also shown in Fig. 4(a). The result shows the hybrid mode 2 has lower loss than the TE0 mode of the standard waveguide across the entire wavelength range. And the ratio monotonically decreases versus the wavelength as the mode field of the hybrid mode 2 tends to be like TE1 mode. In addition, the result also shows the propagation loss of the hybrid mode 2 is about 0.9~0.7 to the standard waveguide across the anomalous dispersion region. All the results shown in the Fig. 4(a) agree well with the theoretical speculation and the experiment. We can conclude that the two hybrid modes in the proposed dispersion engineered method have a bit lower propagation loss than the standard waveguide. In other words, if the proposed dispersion engineered technique is applied on the low loss waveguide platform such as the TriPleX^TM waveguide platform, the dispersion engineered waveguide can keep the characteristic of low loss.

Fig. 4 Numerical study on the propagation loss and the nonlinear parameter of the hybrid mode 2. The blue line is the D of the hybrid mode 2. The orange line is the ratio of the hybrid mode 2 of the engineered waveguide to the TE0 mode of the standard waveguide regarding to (a) the propagation loss and (b) the nonlinear parameter $γ$ respectively. Insets are the mode evolution of the hybrid mode 2 versus the wavelength.

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The nonlinearity parameter $γ$ quantifies the $χ$ ⁽³⁾ process induced extra phase shift of the light wave in the propagation. The third-order nonlinearity is the primary nonlinear process for the material with inversion symmetry, such as silicon, silicon nitride, fused silica and so on. The expression to calculate the nonlinearity parameter of the mode is as follow:

γ = \frac{2 π}{λ} \cdot \frac{\int^{​} n_{2} \cdot {[(E \times H^{*}) \cdot k]}^{2} \cdot d A}{{[\int^{​} (E \times H^{*}) \cdot k \cdot d A]}^{2}}

where n₂ is the Kerr coefficient of the corresponding material, k is the unit vector in the propagation direction, E and H are the electric field and the magnetic field of the optical mode respectively. The integration is applied to the entire range of the cross section. This equation indicates that the nonlinear parameter

γ

is inversely proportional to the size of the mode field. The nonlinear parameter of the TE0 mode of the standard waveguide and the hybrid mode 2 of the dispersion engineered waveguide are calculated and the ratio of the two results (standard waveguide is the denominator) in different wavelength is plotted with orange line in Fig. 4(b). The evolution of the mode filed of the hybrid mode 2 versus the wavelength and the corresponding D are also shown in Fig. 4(b) for better understanding. In the calculation, the parameters of the dispersion engineered waveguide is the same as the calculation about the loss. From the view of the mode evolution, the mode field of the hybrid mode 2 is composed of the TE0 mode and TE1 mode. As the wavelength to be longer, the ratio of the TE1 mode increases while the TE0 mode decreases. So that the nonlinear parameter of hybrid mode 2 is from nearly the same as the TE0 mode down to the TE1 mode, that's why the ratio is from 0.9 to 0.4 over the wavelength range. At the anomalous dispersion region, the nonlinear parameter of hybrid mode 2 is about 0.7~0.4 to the TE0 mode of standard TriPleX^TM waveguide. It seems that this is the imperfection of the proposed dispersion engineered technique. But if we consider the determinant of the parametric gain, we can get another insight. The parametric gain coefficient in the four-wave mixing is determined by both the nonlinear parameter and the dispersion as follow [26]:

g = \sqrt{γ P_{p u m p} Δ β - {(Δ β / 2)}^{2}}

where P_pump is the power of the pump light,

Δ β = 2 β_{p u m p} - β_{s i g n a l} - β_{i d l e r}

is the propagation constant difference in the four-wave mixing process. From the equation, we can get that only in the anomalous dispersion region (

Δ β

>0), the nonlinearity induced phase shift can be compensated by the mismatch between the pump, signal and idler light. In the normal dispersion region, the four-wave mixing is observed but in a narrow band with very low efficiency [27]. So it's nearly impossible for the standard TriPleX^TM waveguide with the D of −750 ps/nm/km to achieve observable parametric gain. Therefore, it's worth sacrificing part of the pump efficiency to improve the parametric gain significantly.

5. Conclusion

In conclusion, we have proposed and experimentally demonstrated a dispersion engineering method by combining two waveguides with different width based on spatial mode coupling between the TE0 mode and TE1 mode. This approach provides the independently controlling of the position and the bandwidth of the anomalous dispersion by only tailoring the width and the gap of the two waveguides. It has been employed on a low loss waveguide platform to realize anomalous dispersion in the micro-ring resonator with high Q value. Because it just needs the design freedom in the horizontal direction of the waveguide without introducing extra fabrication process, it is especially suited for dispersion engineering in the waveguide platform through foundry approach based on the standard process design kit, or in the occasion where it is difficult to deposit a chick waveguide for anomalous dispersion. Besides, the measured very large normal dispersion in the proposed engineering waveguide may be useful for other on-chip optical applications, such as on-chip time-frequency mapping.

Funding

This work was partially supported by NSFC under Contract 61335002 and 61771285.

Acknowledgment

The authors thank LioniX BV for offering the TriPleX^TM waveguide technology.

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Spatial-mode-coupling-based dispersion engineering for integrated optical waveguide

Abstract

1. Introduction

2. Principle and design

3. Experiment

4. Analysis and discussion

5. Conclusion

Funding

Acknowledgment

References and links

Cited By

Figures (4)

Equations (3)

Optics Express