Transmission Characteristics Analysis of a Twin-Waveguide Cavity

Abstract

The transmission spectrum of a twin-waveguide cavity is systematically analyzed based on coupled mode theory, using the transfer matrix method (TMM). The results show that the traveling-wave transmission spectra of the twin-waveguide cavity is entirely determined by the coherent coupling effect involving the parameters of the effective refractive indices of the upper and lower waveguides, the coupling length Lc, and the ratio of the cavity length L to the coupling length (L/Lc). Filters with single, double, or triple-notch filtering could be obtained by choosing an appropriate L/Lc value. When the facet reflection is taken into consideration, the traveling-wave transmission spectrum is modified by the Fabry––Perot (FP) resonance, making it a standing-wave transmission spectrum. As a result, resonance splitting has been observed in the transmission spectrum of twin-waveguide resonators with high facet reflectivity. Further analysis shows that such an abnormal resonance phenomenon can be attributed to the destructive interference between the two FP resonance modes of the upper and lower waveguide through coherent coupling. In addition, narrow bandwidth amplification has also been observed through asymmetric facet reflections. Undoubtedly, all these unique spectral characteristics should be beneficial to the twin-waveguide cavity, achieving many more functions and being widely used in photonic integration circuits (PICs).

1. Introduction

PICs, which use photons as information carriers, have found applications in the information society, due to their advantages, such as parallel processing capabilities, multiple functions, high integration density, low power consumption, and small footprint, especially in optical computing [1], sensing [2], and communication [3,4]. Similar to electronic integrated circuits comprising a number of electric units, a PIC, which integrates several functional devices into one chip, typically consists of two types of components: active and passive units. Active units with narrower bandgap semiconductor materials usually provide gain for the entire device, while passive units with wider bandgap materials are responsible for signal transmission and processing. Up to now, a variety of technologies have been developed to implement PICs, including hybrid integration, heterogeneous integration, and monolithic integration. Monolithic integration is undoubtedly the most attractive candidate compared to the other two technologies because it allows the integration of passive and active components on a single substrate by tailoring the bandgap of III-V semiconductor materials to the functional needs [5], thus giving such PICs a unique advantage in terms of their stability, reliability, volume, weight, power consumption, and cost [6].

To realize monolithic integrated devices with high quality, several monolithic integration techniques have been developed [7,8,9,10,11]. Therein, the twin-waveguide technique is well-known, as it allows one to separately optimize the waveguide design and has a simple fabrication process (i.e., one-time epitaxial growth) [12]. Several optical functional devices and PICs have been reported based on this twin-waveguide technique, such as lasers [13], detectors [14], couplers [15,16,17,18,19], and packet optical switching [20], etc. Here, most devices are realized by utilizing the energy transfer effect [13,14,15,16,17,18,19,20]. However, in addition to the energy transfer function, the twin-waveguide cavity also has a wavelength filtering effect, which is a unique feature compared to the other monolithic integration techniques. Although such a wavelength filtering effect was first predicted as early as the 1970s [21,22], it was not experimentally demonstrated in semiconductor lasers [23,24,25] and filters [26,27,28,29,30,31] until the 1990s. Since then, few experimental demonstrations regarding this feature have been reported, except for the experimental observation of a wavelength filtering effect narrowing the spectral gain in a twin-waveguide-based semiconductor laser in 2021 [32]. As far as theoretical simulations go, the resonant conditions of a twin-waveguide cavity were derived in 1996 [33], and its excellent mode discrimination capability over conventional FP resonators was described. However, the refractive index dispersion relationship of the twin-waveguide cavity was not considered in [33], which was not suitable for the actual situation studied. In 1998, an extensive and rigorous analytical formulation was proposed [34] to analyze the coupling phenomena (i.e., the forward and backward coupling) in such a parallel waveguide (including the twin-waveguide cavity) structure. It was mentioned that only forward coupling needs to be considered [34] in such a twin-waveguide cavity for most practical purposes. To date, there have been no theoretical reports or analyses on the transmission spectrum of a twin-waveguide cavity, which undoubtedly hinders the widespread use of twin-waveguide cavities in PICs.

Therefore, in this paper, we propose a theoretical model of the transmission spectrum of a twin-waveguide cavity based on coupled mode theory, using TMM. Based on this, we systematically study the influence of both the different structural parameters and facet reflectivities on the transmission spectra of twin-waveguide cavities. The results show that coherent coupling plays a dominant role in both traveling- and standing-wave transmission spectra, resulting in either a notch filtering effect or narrow bandwidth amplification in twin-waveguide cavities with zero facet reflectivities or one-side weak facet reflectivity. Moreover, the resonance splitting phenomenon has been observed in twin-waveguide cavities with high facet reflectivities, which can be attributed to the destructive interference between the upper and lower waveguide resonant modes through coherent coupling. We believe that such comprehensive investigations on the transmission spectrum of twin-waveguide cavities will benefit this specific optical unit that is being widely used in PICs.

2. Simulation

2.1. Transmission Model

A typical twin-waveguide cavity structure, as is schematically shown in Figure 1, is composed of two waveguides (i.e., upper and lower waveguides) and four facet reflectors. Assuming the facet reflection coefficients of the lower and the upper waveguides are r₁, r₂, r₃, and r₄, respectively, then the corresponding facet power reflectivities are R₁, R₂, R₃, and R₄. Here, a₂(0) is the amplitude of the incident wave at the left port and b₂(0) is that of the reflected wave. Meanwhile, a₂(L) is the amplitude of the output wave and b₂(L) is that of the incident wave at the right port of the twin-waveguide cavity, where L is the cavity length. The blue dashed line represents the energy transfer of the light power in this twin-waveguide cavity, during its propagation from the left side to the right side of the waveguide.

The whole transfer matrix of the twin-waveguide cavity, which is based on a directional coupler, is obtained using the TMM method. The directional coupler is well-known as a four-port network. By only considering the facet reflectors of the upper waveguide (i.e., r_3,4 ≠ 0 and r_1,2 = 0), this four-port network can be transformed into a two-port scattering network. According to coupled mode theory, the scattering matrix expression of this two-port scattering network is as follows [35]:

$$S=\left[\begin{array}{c}{S}_{11}\\ S_{21}\end{array}\left.\begin{array}{c}{S}_{12}\\ S_{22}\end{array}\right]\right.$$

(1)

where S₁₁, S₁₂, S₂₁, and S₂₂ are expressed as:

$$S_{11}=-{\displaystyle \frac{c^2{r}_4\exp \left(-2j\overline{\beta }L\right)}{1-r_3{r}_4\left(1-c^2\right)\exp \left(-2j\overline{\beta }L\right)}}$$

(2)

$$S_{12}=S_{21}={\displaystyle \frac{\sqrt{1-c^2}}{\exp \left(j\overline{\beta }L\right)}}-{\displaystyle \frac{c^2{r}_3{r}_4\sqrt{1-c^2}\exp \left(-3j\overline{\beta }L\right)}{1-r_3{r}_4\left(1-c^2\right)\exp \left(-2j\overline{\beta }L\right)}}$$

(3)

$$S_{22}=-{\displaystyle \frac{c^2{r}_3\exp \left(-2j\overline{\beta }L\right)}{1-r_3{r}_4\left(1-c^2\right)\exp \left(-2j\overline{\beta }L\right)}}$$

(4)

where $\overline{\beta }=({\beta }_1+{\beta }_2)/2$. β₁ and β₂ are the propagation constants of the isolated upper and lower waveguides, respectively. The effective refractive index is defined as ${n_{eff}\left(\lambda \right)}_{\mathrm{1,2}}={\beta }_{\mathrm{1,2}}/k$, where k is the wave number and is defined as $k=2\pi /\lambda$. Factor c is the power coupling coefficient and is expressed as $c={\displaystyle \frac{\sqrt{{\kappa }_{12}·{\kappa }_{21}}}{s}}\sin \left(sL\right)$, where $s=\sqrt{{\left({\displaystyle \frac{{\beta }_1-{\beta }_2}{2}}\right)}^2+{\kappa }_{12}{\kappa }_{21}}$. Here, the coupling coefficient κ₁₂ represents the coupling from the upper waveguide to the lower waveguide and κ₂₁ represents the coupling from the lower waveguide to the upper waveguide.

Then, the transmission relationship of the whole twin-waveguide cavity with four facet reflectors can be described by the TMM; the problem could be solved by multiplying the T-matrices, which can be expressed as:

$$\left[\begin{array}{c}{a}_2\left(0\right)\\ b_2\left(0\right)\end{array}\right]=T\left[\begin{array}{c}{a}_2\left(L\right)\\ b_2\left(L\right)\end{array}\right]$$

(5)

where $T=T_1·T_2·T_3$. T₁ and T₃ are the transfer matrices for two dielectric interfaces, respectively, and the transfer matrix T₂ of the two-port network can be obtained from Equation (1). Therefore, the complete transfer matrix T of the twin-waveguide cavity can be obtained. When the light is only incident from the left port (b₂(L) = 0), the power transmission of this twin-waveguide cavity is given by:

$$T_s={\left|{\displaystyle \frac{a_2\left(L\right)}{a_2\left(0\right)}}\right|}^2$$

(6)

It is worth mentioning that compared with the method proposed in [33], the above-described model can not only be used to simulate twin-waveguide cavities with identical lengths, but can also be further extended to analyze twin-waveguide cavities with nonidentical upper and lower cavity lengths, the inconsistent case, allowing for the probing of some new properties (e.g., vernier effect) of this twin-waveguide cavity and expanding its potential applications in optoelectronic devices. In addition, by taking the boundary conditions (i.e., facet reflections) of a twin-waveguide cavity into account, the effect of the interaction between the forward and backward fields on the transmission spectrum can be investigated.

2.2. Parameters

To investigate the transmission spectrum of this twin-waveguide cavity structure correctly and objectively, the refractive index dispersion relationship should be considered. It can be obtained by scanning the effective refractive indices at different wavelengths on a specific twin-waveguide cavity material structure. As shown in

Table 1. InP-based twin-waveguide cavity structure.

Material	Refractive Index	Thickness (µm)
InGaAsP	3.3548	0.2921
InP	3.169	2
InGaAsP	3.2887	0.45

, an InP-based epitaxial structure with a central wavelength of 1.55 µm is composed of two InGaAsP materials with different refractive indices. The upper waveguide is the InGaAsP with a higher refractive index of 3.3548 and a thickness of 0.2921 µm. The lower waveguide is the InGaAsP with a lower refractive index of 3.2887 and a thickness of 0.45 µm. The space layer between the upper and lower waveguides is the InP with a refractive index of 3.169 and a thickness of 2 µm.

To clearly describe the physical picture of the transmission spectrum of the twin-waveguide cavity, all nonlinear dispersion and absorption effects, as well as the material dispersion, are ignored, and only the first-order waveguide dispersion is considered. As a result, the effective refractive index of each waveguide can be approximated as a first-order linear function of the wavelength over a narrow wavelength range, which is expressed as:

$${n_{eff}\left(\lambda \right)}_{\mathrm{1,2}}=n_{eff}^0+\left(\lambda -{\lambda }_0\right){\displaystyle \frac{{d{n}_{eff}}_{\mathrm{1,2}}}{d\lambda }}=n_{g\mathrm{1,2}}+\lambda {\displaystyle \frac{{d{n}_{eff}}_{\mathrm{1,2}}}{d\lambda }}$$

(7)

where $n_{eff}^0$ is the effective refractive index at the center of the wavelength λ₀, and n_g1 and n_g2 are the group refractive indices of the upper and lower waveguides, respectively. Based on this, the dispersion relationships between the upper and lower waveguides obtained from the twin-waveguide structure, shown in Table 1, are shown in Figure 2. It can be seen that these two dispersion relationships intersect at 1.55 µm, with an effective index of 3.211, corresponding to the λ₀ and $n_{eff}^0$, respectively. A fit, as shown in Figure 2, using Equation (7), we further find that n_g1 and n_g2 are 3.339 and 3.298, respectively. For the convenience of the analysis in the following simulation, n_g2 of the lower waveguide is fixed at 3.298. The central wavelength and its refractive index are kept unchanged, which means that the effective refractive indices of the upper and lower waveguides will always intersect at λ₀, as shown in Figure 2.

According to coupled mode theory, when the upper and lower waveguides are phase matched at λ₀, a full energy transfer occurs between these two waveguides under a certain cavity length L, and this cavity length is referred to as the coupling length Lc, which is mainly controlled by the space layer in Table 1. The energy transfer process is illustrated in Figure 1 by the blue dashed line, according to which, one can see the definition of the coupling length Lc. The expression of Lc is defined as:

$$L_c\equiv {\displaystyle \frac{\pi }{2\sqrt{{\kappa }_{12}·{\kappa }_{21}}}}={\displaystyle \frac{\pi }{2\kappa }}$$

(8)

Table 2. Simulation parameters.

Twin-Waveguide Cavity	Meaning	Symbol	Value	Units
	Central wavelength	λ0	1.55	µm
	Effective refractive index at λ0	$n_{eff}^0$	3.211	-
	Coupling length	Lc	1000	µm
	Cavity length	L	1000	µm
Upper Waveguide	Group refractive index	ng1	3.339	-
	Facet reflection coefficient	r3, r4	0	-
	Facet power reflectivity	R3, R4	0	-
Lower Waveguide	Group refractive index	ng2	3.298	-
	Facet reflection coefficient	r1, r2	0	-
	Facet power reflectivity	R1, R2	0	-

lists the parameters used in our simulation. To avoid excessively long lists in Table 2, only the basic parameters are listed. Based on the above model and the parameters listed in Table 2, the transmission spectrum of this twin-waveguide cavity can be simulated.

3. Results and Discussion

3.1. Without Facet Reflection

When the twin-waveguide cavity has zero facet reflectivity (i.e., R₁ = R₂ = R₃ = R₄ = R = 0), its traveling-wave spectrum profile is entirely dependent on the power coupling c², which means that the transmission spectrum is just determined by three parameters, namely the Δn, the coupling length Lc, and the ratio of the cavity L to the Lc (L/Lc). Here, Δn is the difference between the effective refractive indices of the upper and lower waveguides at 1.6 μm, which is used to describe the difference in the dispersion relationship between the upper and lower waveguides in a twin-waveguide cavity. All these parameters depend on the actual waveguide profile and, in practice, it is straightforward to optimize these parameters by modifying such a twin-waveguide profile. Therefore, it is necessary to investigate the influence of all these parameters on the transmission spectrum of the twin-waveguide cavity.

Figure 3a shows the transmission spectra under different Δn, assuming that Lc = 1000 μm and L/Lc = 1. As is shown, all these spectra have the same transmission valley of zero at λ₀ for any Δn, which means that complete energy transfer happens here. However, when the wavelength is far away from λ₀, their coupling coefficient becomes less, and then the energy transfer becomes smaller. As a result, the transmission spectra shown in Figure 3a exhibit a series of grating-like notch filter characteristics as demonstrated in [22,23]. Meanwhile, Figure 3a also shows that a larger Δn will sharpen the filtering effect, even though the profile of the transmission spectra seems to be less dependent on Δn. In addition, to accurately describe the influence of the structure parameters of the twin-waveguide cavity (i.e., Δn, Lc, and L/Lc) on the transmission spectra, three characteristic parameters of the 3 dB bandwidth, coarse free spectral range (C-FSR), and side-lobe suppression ratio (SLSR) are defined and shown in the inset of Figure 3a.

The relationship of the 3 dB bandwidth on various Δn and Lc is shown in Figure 3b. One can see that an increase in both the Δn and Lc will lead to a large reduction in the 3 dB bandwidth. For example, the 3 dB bandwidth is only 7.21 nm when the Lc = 1000 μm and Δn = 8.5 × 10⁻³, compared to the bandwidth of 158.6 nm when Lc = 300 μm and Δn = 1.322 × 10⁻³. Based on this feature of the twin-waveguide cavity structure, it is possible to realize a grating-free single longitudinal mode laser by simultaneously increasing the Δn and extending the Lc. However, simultaneously increasing the Δn and extending the Lc may degrade their discrimination capability because of a reduction in the C-FSR, as shown in Figure 3c. Therefore, the Δn and Lc must be carefully designed to achieve a high-performance semiconductor laser using such a twin-waveguide cavity.

Figure 4a,b shows the transmission spectra with the ratios of L/Lc being odd and even, respectively. Here, Lc is assumed to be 1000 μm and n_g1 = 3.339. Clearly, Figure 4 shows that the profile of such transmission spectra is dependent on the ratio of L/Lc. One can see that the transmission spectra exhibit a single-notch filter profile when L is odd times the Lc and a double-notch filter profile when L is even times the Lc, which means that a single or double-notch filter can be achieved just by changing the cavity length of such a twin-waveguide cavity. Moreover, as the ratio of L/Lc increases, the notch filtering effects have a narrowed 3 dB bandwidth and a reduced C-FSR and SLSR. Such phenomena are attributed to the periodic coherent coupling of the two optical modes in the lower and upper waveguides. Therefore, to achieve a single or double-notch filter with a high level of performance, the cavity length of the twin-waveguide cavity must be delicately designed according to the requirements.

We have analyzed the transmission spectra for the waveguide cavity length L being an integer of the coupling length Lc. However, it is important to extend our analysis to the scenarios in which the ratio between the waveguide cavity length L and the coupling length Lc is an arbitrary number, since in the real world, it is quite hard to make the waveguide cavity length L an integer of the coupling length Lc, due to the cleaving error, as well as the uncertainty in the Lc caused by other fabrication tolerances.

Figure 5a–c shows the transmission spectra with the non-integer L/Lc under different Lc, which displays the evolution of the transmission spectral profiles from a single-notch filter to a double-notch filter when the L/Lc varies from 0.5 to 2. As we can see, the profile of all these transmission spectra seems to be more dependent on the L/Lc, but less on the Lc. When L/Lc is changing from 0.5 to 1.625, the spectra always exhibits a single-notch filter profile and the central notch valley becomes increasingly narrower and deeper. When L/Lc = 1.625, the central notch profile is identical to that of the two-side notch profile, which results in such a twin-waveguide cavity becoming a triple-notch filter. With a further increase in the L/Lc, the central notch becomes even shallower, while the neighboring side notches tend to be deeper until the central notch fully disappears at L/Lc = 2, where a double-notch filtering effect is observed eventually. Therefore, such evolution of the transmission spectral profile indicates that the twin-waveguide cavity can be used as a single, double, or even triple-notch filter by adjusting the L/Lc.

To further evaluate the influence of the L/Lc on the transmission spectrum, three characteristic parameters (i.e., 3 dB bandwidth, C-FSR, and SLSR) versus different L/Lc are shown in Figure 5d,e. As is seen, both the 3 dB bandwidth and C-FSR decrease monotonously with the L/Lc and Lc. This decreasing trend becomes slower with an increase in both the L/Lc and Lc. The same extreme point in the valley has been observed in the transmission spectra at an L/Lc of 0.8 and 1.2, and 0.5 and 1.5, respectively, leading to a nearly symmetrical distribution of the Lc-independent SLSR in Figure 5e. However, the 3 dB bandwidth and C-FSR are much smaller with the L/Lc being 1.2 and 1.5 compared with 0.8 and 0.5, as seen in Figure 5d,e, respectively. Therefore, when the notch filtering effect is utilized in a twin-waveguide cavity, the rough value of the L/Lc is initially determined according to the requirements of the notch filter profile, such as the single-notch filter, double-notch filter, or triple-notch filter. Moreover, a larger L/Lc should be selected to achieve a stronger notch-filtering effect. And then, the Lc should be determined based on the requirements of the 3 dB bandwidth and C-FSR.

3.2. With Facet Reflection

A twin-waveguide cavity with non-zero facet reflectivity, which makes it a resonator, should be a more general case in PIC applications, thus it is necessary to investigate the transmission spectrum in this situation.

Figure 6a shows the transmission spectra of twin-waveguide resonators, with the same cavity length L but different facet reflectivities R, in one-period wavelength domain from 1.542 μm to 1.558 μm. Here, Lc is assumed to be 1000 μm, L/Lc = 1, and n_g1 = 3.5643. In addition, the transmission spectrum with R = 0 is also added as a reference. As we can see, similar to a general FP resonator, the ripple of all the transmission spectra is enhanced with the increased R, and all the transmission spectra have an approximate resonance period (i.e., free spectral range, FSR), as the FSR is determined by the cavity length L and the group refractive indices (i.e., n_g1 and n_g2) of the upper and lower waveguides, which can also be derived from [33] with the expression:

$$FSR={\displaystyle \frac{{\lambda }^2}{\left(n_{g1}+n_{g2}\right)L}}$$

(9)

However, there are also some characteristics in the transmission spectra of twin-waveguide resonators that are different from those of general FP resonators. First, the transmission spectrum profiles are symmetrical and follow the counterpart reference one as shown in Figure 6a, which indicates that coherent coupling plays a dominant role in these transmission spectra. Second, a resonance reversal has been observed at the waist of each transmission spectrum of such twin-waveguide resonators. Furthermore, a resonance-splitting phenomenon is also observed at the waist of the transmission spectra when R is high. Moreover, the wavelength range of such resonance splitting broadens with an increase in R. Undoubtedly, such specific features of twin-waveguide resonators do not conform to general cognition.

To understand the behavior of these transmission spectra, the resonance depth, which is defined as the resonance difference between the resonant valley and its left resonant peak, of the symmetrical transmission spectra is first extracted from Figure 6a over a half-period wavelength domain (1.542 μm to 1.550 μm) and is shown in Figure 6b. One can see that each transmission spectrum of the twin-waveguide resonator consists of two sets of resonance spectra, whose resonance depth varies with the energy coupling, i.e., the c². In detail, with the increases in the c², one resonance depth increases and reaches the maximum at c² = 1, while another resonance depth decreases from the maximum at c² = 0. When the two resonance spectra intersect, the transmission spectrum with R = 0.1 exhibits a nearly zero resonance depth, but the transmission spectra with R = 0.3 and 0.5 show a non-zero resonance depth and opposite resonance spectra variation trends. Moreover, the resonance intersecting range of the two sets of resonance spectra broadens from 3 nm to 5 nm when R is increased from 0.3 to 0.5, as displayed in Figure 6b. With a further look at Figure 6a,b, it can be found that the resonance splitting of these transmission spectra mainly occurs between the wavelengths where the two sets of resonance spectra overlap. For example, the two resonance spectra with R = 0.5 show that the wavelength range where resonance splitting occurs is between 1.5434 μm and 1.5481 μm. When such resonance splitting strength is defined as the resonance depth at the wavelength where the two sets of resonance spectra intersect, this resonance splitting strength increases with R as well.

To further reveal the underlying mechanism of such complex transmission spectra, the transmission spectra with R = 0 and 0.5, in Figure 6a, are selected and plotted as spectra A and D in Figure 6c. The transmission spectra of the two cases in which only the facet reflectivity of the lower or upper waveguide is considered are also supplemented in Figure 6c as transmission spectra B and C. Similarly, the resonance depth variations of transmission spectra B, C, and D in Figure 6c are extracted and shown in Figure 6d. As is seen from Figure 6c, spectra B and C exhibit a standing-wave property with their profiles following spectrum A, which implies a dominant role of the coherent coupling in these transmission spectra. In addition, unlike spectrum D, both spectra B and C have only one set of resonance spectra and no resonance splitting behavior in their transmission spectra. Moreover, the resonance intensity and depth of spectrum B and spectrum C exhibit different variation trends. In detail, the resonance intensity and depth are degraded in spectrum B but enhanced in spectrum C with the wavelength red-shifting trend, as seen in Figure 6c,d. Furthermore, the resonance peaks and valleys of spectra B and C are opposite. In other words, the resonance peak of spectrum B coincides with the resonance valley of spectrum C, as can be seen in Figure 6c. That is to say, a reverse resonance phase happens between the transmission spectra B and C, which means that the transmission spectrum B (C) originates from the separate lower (upper) waveguide FP cavity resonance. As the light output is located in the lower waveguide in our simulation model, thus the transmission spectrum C, which only considers the facet reflectivity of the upper waveguide, has experienced coherent coupling twice. It is the twice coherent coupling process of transmission spectrum C that results in its reverse resonance phase with spectrum B. More information can be obtained from the evolution of spectra B and C and their resonance depths, as shown in Figure 6c,d. The resonance depths of spectra B and C are similar to the resonance spectra D₁ and D₂ of spectrum D, respectively, which indicates that the interference between spectra B and C constitutes spectrum D. Moreover, one can see that the wavelength range of the resonance splitting that occurs in spectrum D is the region where the non-zero resonant depths of spectra B and C, respectively, tend to be minimized. These features, as shown in Figure 6c,d, reveal that the whole resonant spectra of such twin-waveguide resonators are the result of the interference between the two separate resonance modes from upper and lower waveguide resonators through their coherent coupling, and the resonance-splitting phenomenon is just the result of destructive interference in the above case.

An investigation of the transmission spectrum of twin-waveguide resonators with asymmetric facet reflectivity is carried out and the results are shown in Figure 7, where the right-side facet reflectivity (R₂ = R₄) varies, while the left-side facet reflectivity is fixed at 0.1% (R₁ = R₃ = 0.1%). The low left-side facet reflectivity corresponds to the case of antireflection coating devices. As we can see, in this situation, the resonant ripples are always quite low, regardless of the right-side facet reflectivity. The resonance depth is only enhanced by about 0.4 dB, even though the right-side facet reflectivity is increased from 1% to 70%, which is much lower than that of general FP resonators in the same situation. This result means that a single-side facet antireflection coating can effectively suppress the resonance of twin-waveguide resonators. Meanwhile, with the increase in the right-side facet reflectivity, the SLSR decreases from 5.9 dB to about 0.6 dB, but the 3 dB bandwidth is only increased to 0.38 nm. Therefore, it is possible to achieve a narrow bandwidth integrated traveling amplifier with such a twin-waveguide resonator through suitable facet coating.

The transmission spectra of general twin-waveguide resonators with different L/Lc but the same R are shown in Figure 8. The corresponding transmission spectrum with R = 0 is also added. It can be seen that the profiles of the transmission spectra under different L/Lc demonstrate the dominant role of coherent coupling in twin-waveguide resonators, once again. Meanwhile, the resonance-splitting phenomenon still exists for L/Lc = 0.8 and 1.2, as shown in Figure 8a. The resonance depth variations in these transmission spectra under different L/Lc over half-period wavelength domains are shown in Figure 8b. We can see that each transmission spectrum has two sets of resonance depths. For the sake of description and discussion, the resonance depths increasing and decreasing with the wavelength red-shift trend are defined as resonance depths A and B, respectively. Such a phenomenon is similar to the foregoing section. For resonance depth A, the resonance depth with L/Lc = 0.8 is always higher than that with L/Lc = 1.2 until they are the same at the center of the wavelength, while the resonance depth B shows an opposite variation trend. Moreover, it can be seen from Figure 8a,b that the resonance range and resonance-splitting range of the spectrum with L/Lc = 0.8 is wider than that with L/Lc = 1.2, which is because of a wider 3 dB bandwidth with L/Lc = 0.8. Further observations of Figure 8b show that the resonance depth at which two sets of resonance spectra intersect is almost the same for each L/Lc, but the wavelength intersection is continuously red-shifted with the increase in the L/Lc. Therefore, L/Lc plays an important role in the resonance-splitting position and range that happens in the transmission spectrum of the twin-waveguide resonator.

4. Conclusions

To summarize, the transmission spectra of a twin-waveguide cavity with zero and non-zero facet reflectivity were theoretically and systematically analyzed. Therein, the traveling-wave transmission spectrum exhibits a notch-filter characteristic, which is determined by three structure parameters (i.e., Δn, Lc, and L/Lc). Based on this specific feature, a twin-waveguide cavity can be used as a single-notch filter, double-notch filter, or triple-notch filter, depending on the ratio of L/Lc. Moreover, the design principle of a twin-waveguide cavity has been briefly summarized. When the facet reflectivity is taken into account, the transmission spectrum exhibits an FP-like resonance behavior. Meanwhile, the transmission spectra of the twin-waveguide resonator are attributed to the interference of the two resonance modes of upper and lower waveguide resonators through the coherent coupling effect and are exhibited in the following three aspects: (1) each transmission spectral profile follows the corresponding traveling-wave transmission spectrum; (2) the resonance reversal phenomenon was observed in each transmission spectrum of the twin-waveguide resonator; and (3) the resonance-splitting phenomenon was observed when the facet reflectivity was higher. Further investigation indicates that this resonance splitting can be attributed to the destructive interference between the upper and lower waveguide resonant modes through coherent coupling. Moreover, the influence factors of such resonance splitting (resonance splitting depth, range, and position) are also investigated. By applying an antireflective coating on the one-side facet, a narrow bandwidth traveling-wave amplifier can be realized using such a twin-waveguide cavity. Overall, these fascinating features of twin-waveguide cavities demonstrated in this paper should be beneficial in designing various integrated photonic devices for PICs.