Power coupling between gain-guided index-antiguided planar waveguides

Chih-Hsien Lai; Chang-I Hsieh

doi:10.1364/OE.24.028026

1. Introduction

In recent years, rapid development has been made in waveguide lasers [1‒5]. For high-power waveguide lasers, it is essential to have a large mode area (LMA) and operate with a single transverse mode. One of the techniques to meet both demands is the gain-guided index-antiguided (GGIAG) waveguide [6,7], in which the refractive index is lower in the core than in the cladding and a sufficient gain is introduced to the low-index core to help the fundamental transverse mode gain-guiding. This technique was initially implemented in fiber geometry, and single-mode lasing in GGIAG fibers was demonstrated with core diameters larger than 100 μm [8,9].

For planar waveguide geometry, the high aspect-ratio allows the heat resulting from the active region to be removed efficiently through the large faces, thereby reducing the thermal load. In addition, the planar geometry is well compatible with diode-array pumping and leads to a flexible degree of pumping arrangement, such as end pumping and edge pumping by laser diode bars or face pumping by laser diode stacks. These advantages make the planar waveguide a promising architecture for high-power solid-state lasers [10‒13]. Recently, the GGIAG approach has been extended to planar form and realization of the GGIAG planar waveguide laser was reported [14].

In practical applications, waveguide lasers may be placed in close proximity, for example, for the purpose of beam combining [15], photonic switching, or power splitting [16]. Hence, it is important to understand the power coupling behavior of the waveguides. Coupling behavior between two GGIAG fibers has been studied by solving the coupled-mode equations, and the coupling coefficient is found to be damped with the waveguide separation [17,18]. In this work, we investigate the power coupling between two GGIAG planar waveguides. A finite-difference beam propagation method (BPM) [19] and a finite-difference mode solver [20] are both used as the numerical tools for analysis. Numerical results show that strength of power coupling varies periodically with the gap between the two waveguide cores. This phenomenon is much different from that occurring in conventional index-guided (IG) waveguides where the coupling attenuates with the waveguide gap. It is also different from what observed in GGIAG fibers where the coupling is damped. Our analysis reveals that the gap between GGIAG planar waveguides plays a role like a Fabry-Perot etalon, which leads to the periodic coupling phenomenon. Moreover, it is found that the coupling length (L_c₁) at which one waveguide has the minimum power and the coupling length (L_c₂) at which another waveguide acquires the maximum power are different. Further investigation indicates that L_c₁ is independent of the gain, while L_c₂ increases with the gain. These novel findings discussed in this work will provide helpful information for the applications of GGIAG planar waveguide lasers and amplifiers when the waveguides are placed in proximity.

2. Structure for numerical modeling

Figure 1 plots the longitudinal cross-section of the structure containing two identical and parallel GGIAG planar waveguides, which is invariant along the y and z directions. The structure is similar to conventional optical directional couplers [21] except that the waveguides are GGIAG rather than IG. Transverse index profile of the two-waveguide structure is also shown in the figure. In this work, the following parameters are assumed: core index n_co = 1.5 + j(g/2k₀) where g is the small-signal material gain coefficient which is assumed uniform in the core and k₀ is the free-space wavenumber, cladding index n_cl = 1.6, core width w = 30 µm, and wavelength λ = 1.53 µm. Distance between the two waveguide cores, i.e., the waveguide gap, is denoted as d.

Fig. 1 Longitudinal cross-section and transverse index profile of the structure containing two identical and parallel GGIAG planar waveguides.

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To investigate the power coupling behavior occurring between the two planar GGIAG waveguides, a Gaussian beam having the form of $\exp (- x^{2} / w_{0}^{2})$ with unity power P_in = 1 is launched into the first waveguide. The beam width is assumed to be 2w₀ = 22 µm. This value is determined according to a preliminary calculation, so that for a single GGIAG planar waveguide the modal power of the fundamental mode excited by the Gaussian beam is the maximum with this optimal beam width. Here, only the transverse electric (TE) polarization is considered. The transverse magnetic (TM) polarization has similar behavior as that of the TE polarization. BPM [19] is then utilized to simulate the power evolution along the two-waveguide structure, and the powers in the first and the second waveguides are denoted as P₁ and P₂, respectively. Since the GGIAG waveguides will be leaky if the material gain is not large enough, we take the cut-off approximation [22] and define P₁ and P₂ as the powers in the core regions of waveguide 1 and waveguide 2, respectively. That is to say, when calculating P₁ and P₂, integration of the squared modulus of the electric field (|E|²) is performed over the transverse direction (x-direction) only in core 1 and core 2, respectively. According to the coupled-mode theory [23], along the propagation distance, the power launched into the first waveguide will gradually transfer to the second waveguide. The distance at which total power transfer occurs is called the coupling length L_c. Coupling length is an important property of the two-waveguide structure, and is the major concern of this work. It can be taken as a measure of the coupling strength. Strong coupling leads to a short coupling length, and vice versa. For conventional IG waveguides where the core index is higher than the cladding index, owing to the power conservation, the distance at which P₁ reaches the minimum (zero) and the distance at which P₂ reaches the maximum are the same.

3. Numerical results

3.1 Propagating beam analysis

We first consider the case where the material gain coefficient is g = 0.75 cm^-1. Power evolution along the two GGIAG planar waveguides with gap d = 3.0 μm is simulated by BPM and is shown in Fig. 2. Owing to the material gain provided in the low-index waveguide cores, obviously, the total power (P₁ + P₂) grows along the propagation. Moreover, it is found that the distance at which P₁ reaches the minimum (denoted as L_c₁) and the distance at which P₂ reaches the maximum (denoted as L_c₂) are not the same. Therefore, we may assume that there are two coupling lengths, one is referred to as L_c₁ and another L_c₂. As stated previously, for conventional IG waveguides, the two coupling lengths are the same, i.e., L_c₁ = L_c₂. While for the GGIAG case studied here, the behavior is different and L_c_{1 $\neq$}L_c₂. From Fig. 2, L_c₁ = 3.16 cm and L_c₂ = 3.57 cm. Such a difference in coupling length arises owing to the modal gain of the GGIAG waveguides. The explanation will be provided later.

Fig. 2 Power evolution along the two GGIAG planar waveguides with g = 0.75 cm^-1 and d = 3.0 μm. In this case, the distance at which P₁ reaches minimum (denoted as L_c₁) and the distance at which P₂ reaches maximum (denoted as L_c₂) are different.

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We further examine the coupling behavior by varying the gap value between the two GGIAG planar waveguides. Figure 3 shows the simulated coupling length as a function of gap d. As can be seen, no matter what the gap value is, L_c₁ (blue circle) is smaller than L_c₂ (red triangle). Moreover, it is interesting to find that, both coupling lengths L_c₁ and L_c₂ vary periodically with d. This phenomenon is much different from what happens in conventional IG waveguides, where the coupling length increases with the gap between the waveguide cores [23]. This phenomenon is also different from what is observed in GGIAG fibers, in which the coupling is damped with the gap [17,18].

Fig. 3 Coupling length as a function of gap between waveguide cores with g = 0.75 cm^-1. L_c₁ and L_c₂ are obtained from BPM simulation. For the notation, “R” represents the resonant condition, and “AR” represents the antiresonant condition.

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The periodic behavior of coupling length shown in Fig. 3 can be explained by treating the gap between the two waveguide cores as a Fabry-Perot etalon. For a fixed wavelength, the thickness of the Fabry-Perot etalon plays an important role in determining whether the etalon is resonant or not. When the gap is resonant, i.e., the thickness of the gap meets the resonant condition, the gap becomes transparent because the transmission is 100%. As a result, the power launched to the first waveguide core can penetrate the gap and transfer to the second waveguide core more easily, leading to a shorter coupling length. On the other hand, when the gap is antiresonant, the transmission becomes very small and high reflection occurs. Therefore, it is difficult for the optical power to transfer from the first waveguide core, through the gap, to the second waveguide core, which makes the coupling length longer. Since the resonant and antiresonant conditions appear periodically with the thickness of the gap, the coupling length as a function of gap also exhibits a periodic feature.

For the gap, whether its value d meets the resonant or antiresonant conditions can be determined according to the following equation [24]:

d = \frac{m λ}{4 \sqrt{n_{c l}^{2} - n_{c o, r}^{2}}},

where n_co_,_r is the real part of n_co, and m is a positive integer. When m is even, the gap meets the resonant condition; otherwise, when m is odd, the gap is under the antiresonant condition. Substituting m = 2, 4, and 6 into Eq. (1), the resultant d values are 1.374, 2.748, and 4.122 µm, respectively. These values which make the gap resonant are shown in Fig. 3 as black dashed lines. Clearly, at these d values, both the coupling lengths L_c₁ and L_c₂ are the shortest, confirming the aforementioned statement that power transfer is much easier to take place when the gap is under the resonant condition. On the other hand, taking m = 3, 5, and 7 in Eq. (1), the d values are 2.061, 3.435, and 4.809 µm, respectively. These values at which the gap is antiresonant are shown as black dotted lines in Fig. 3. It is obvious that the dotted lines and the positions where the longest coupling length appears agree very well. This indicates that power is difficult to transfer when the gap is under the antiresonant condition, and thus leads to a longer coupling length.

3.2 Modal analysis

The coupling characteristics between GGIAG planar waveguides can be further investigated with the modal analysis. To do this, the two-waveguide structure is treated as a whole guiding system, then modal distribution and propagation constant are calculated by using the mode solver [20]. Figure 4(a) plots the effective indices of the even and odd system modes of the two-waveguide structure as functions of waveguide gap. The effective index is defined as Re(β)/k₀, where β is the complex propagation constant of the system mode and k₀ is the free-space wavenumber. In Fig. 4(a), the values at which the gap becomes resonant, i.e., d = 1.374, 2.748, and 4.122 µm, are also shown as black dashed lines for reference. Although the antiresonant gap values are not explicitly pointed out in the figure, their locations (d = 2.061, 3.435, and 4.809 µm) can be easily found which lie midway between adjacent resonant ones.

Fig. 4 (a) Effective index as a function of waveguide gap for the even and odd system modes. For the notation, “R” represents the resonant condition. (b) Field distributions of the system modes corresponding to the points designated in (a). The dotted lines indicate the positions of core-cladding interfaces. The inset shows the field distribution zoomed in the gap region.

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From Fig. 4(a), it is clear that when the gap is antiresonant, there are two system modes existing in the structure. One is even and another is odd. When the Gaussian input with unity power is launched to the first waveguide, the two system modes would be equally excited with each one having a half power. For example, when d = 2.061 μm, excited modal powers of the even (designated as A in the figure) and the odd (designated as B) system modes, calculated according to the overlap integrals, are 0.491 and 0.480, respectively. Field distributions of the two system modes A and B are shown in Fig. 4(b). According to the coupled-mode theory [23], as the two system modes beat along the propagation, total power transfer occurs when the two modes are out of phase, and the coupling length L_c can be obtained by

L_{c} = \frac{π}{| Re (β_{e}) - Re (β_{o}) |} = \frac{π}{k_{0} | n_{e} - n_{o} |},

where β_e and β_o are the complex propagation constants and n_e and n_o are the effective indices of the even and the odd system modes, respectively. In Fig. 4(a), since the difference between n_e and n_o has the smallest magnitude when the waveguide gap is antiresonant, the longest coupling length occurs according to Eq. (2).

On the other hand, when the gap meets the resonant condition, three modes appear in the structure. Thus power transfer between the two GGIAG planar waveguides becomes a three-mode-beating behavior [25,26]. For example, when d = 2.748 µm, there are two even system modes (designated as C and E) and one odd system mode (designated as D) existing in the structure. Excited modal powers of the C, D, and E system modes are 0.342, 0.481, and 0.144, respectively. Field distributions of the three system modes can be found in Fig. 4(b). Because the difference between the effective indices of the even and the odd system modes are the largest when the gap is resonant, the coupling length becomes the shortest accordingly. In Fig. 4(a), it can be observed that the effective-index-difference between the even and odd system modes varies periodically with the waveguide gap. As a result, the coupling length has a periodic variation as well. Therefore, from above discussions, the phenomenon shown in Fig. 3 that the coupling lengths L_c₁ and L_c₂ vary periodically with the waveguide gap is indeed attributed to the Fabry-Perot-like property of the gap.

According to Eq. (2), coupling length can be calculated from the effective indices n_e and n_o. For most situations, the two-waveguide structure contains only one even and one odd system modes, thus Eq. (2) can be directly applied. If there are three system modes, i.e., in the case that the gap is around resonant, the pair of even and odd system modes having a smaller difference in effective index is chosen for calculation. This is because the modes in this pair have larger excited modal powers than the third mode has. For example, in Fig. 4(a), when d = 2.748 µm, the difference of effective index between modes C and D is smaller than the difference between modes D and E, and the powers of modes C (0.342) and D (0.481) are larger than that of mode E (0.144). Calculated result according to Eq. (2) is plotted in Fig. 3 as the magenta solid line. Compared with L_c₁ and L_c₂ which are obtained from BPM simulation, it is interesting to find that the coupling length predicted by Eq. (2) agrees with L_c₁ very well, and is much different from L_c₂.

3.3 Influence of gain on coupling length

For the operation of GGIAG waveguides, it is important to provide a gain in the low-index core. In previous subsections, the material gain coefficient is assumed to be g = 0.75 cm^-1. Here, we examine the influence of gain by varying the gain coefficient.Figure 5 shows L_c₁ and L_c₂, obtained from BPM simulation, as functions of waveguide gap for several material gain coefficients. Results obtained according to Eq. (2) are also shown in the figure. As can be seen, for all the gain coefficients, both L_c₁ and L_c₂ vary periodically with the gap, in that the shortest coupling length occurs when the waveguide gap is resonant and the longest coupling length appears when the gap is antiresonant. The periods are all the same for these gain coefficients. In other words, material gain does not affect the condition, as shown in Eq. (1), under which the gap is resonant or antiresonant. Moreover, clearly, no matter what the gain coefficient is, L_c₁, the propagation distance at which P₁ becomes the minimum, has the same variation with d. In addition, L_c₁ agrees well with the result obtained from Eq. (2) regardless of gain coefficient. That is to say, behavior of L_c₁ is independent of material gain. However, the situation is different for L_c₂. From Fig. 5, it is found that L_c₂, the propagation distance at which P₂ reaches the maximum, increases with material gain coefficient.

Fig. 5 Coupling length as a function of gap for several material gain coefficients. (a) g = 0, (b) g = 0.25 cm^-1, (c) g = 0.5 cm^-1, (d) g = 0.75 cm^-1, (e) g = 1.0 cm^-1, and (f) g = 1.25 cm^-1.

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To have a more direct observation about the influence of gain on coupling length, it is illustrated with two examples in Fig. 6 where BPM results of L_c₁ and L_c₂ are shown as functions of material gain coefficient. Figure 6(a) is for the case where the gap is antiresonant (d = 2.1 μm), and Fig. 6(b) the case the gap is resonant (d = 2.75 μm). In Fig. 6(a), it is observed again that L_c₁ is independent of material gain. While for L_c₂, on the contrary, it is strongly affected by the material gain, in that it increases with gain coefficient g. Taking L_c₁ as a benchmark, it is found that when g < 0.5 cm^-1, L_c₂ < L_c₁; when g > 0.5 cm^-1, L_c₂ > L_c₁; and when g = 0.5 cm^-1, L_c₂≈L_c₁. The reason why L_c₂ is smaller or larger than L_c₁ is due to the modal loss or modal gain encountered by the system modes, and will be explained in the next subsection. In Fig. 6(b), which is the case the gap is resonant, L_c₂ is found to be very close to L_c₁. Even so, the trend appeared in Fig. 6(a) where the gap is antiresonant can still be observed here, i.e., when g < 0.5 cm^-1, L_c₂ is (slightly) smaller than L_c₁; and when g > 0.5 cm^-1, L_c₂ is (slightly) larger than L_c₁.

Fig. 6 Coupling length as a function of material gain coefficient. (a) d = 2.1 μm (the gap is antiresonant), and (b) d = 2.75 μm (the gap is resonant).

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3.4 Clarifying the different behaviors of L_c1 and L_c2

To clarify the phenomenon observed from BPM simulation that L_c₂ behaves differently from L_c₁, propagation of the system modes is considered in this subsection. For simplicity, it is assumed that the gap is under the antiresonant condition. Therefore, there are one even system mode and one odd system mode propagating in the two-waveguide structure, and the two system modes are equally excited.

Assume the transverse fields of the even and the odd system modes are E_e(x) and E_o(x), respectively. Complex propagation constants of the two system modes are β_e = (β + Δβ/2) + jγ_e /2 and β_o = (β − Δβ/2) + jγ_o /2, where γ_e and γ_o are modal gain coefficients of the even and the odd system modes, β = k₀(n_e + n_o)/2 and Δβ = k₀(n_e − n_o). When the gap is antiresonant, the modal gain coefficients are almost equal for the two system modes, thus we can let γ_e = γ_o = γ. Note that if γ is negative, instead of modal gain, the system modes will suffer a modal loss. The total field in the two-GGIAG-waveguide structure E_t, assuming that the propagation is along the z direction, can be expressed as

E_{t} (x, z) = a_{e} E_{e} (x) e^{- j [(β + Δ β / 2) + j γ / 2] z} + a_{o} E_{o} (x) e^{- j [(β - Δ β / 2) + j γ / 2] z},

where a_e and a_o are excited modal amplitudes of the even and the odd system modes. Under the antiresonant condition, excited modal powers of the two system modes are nearly the same, thus we can let a_e = a_o = a. Refer to the inset of Fig. 7(a), where distributions of E_e(x) and E_o(x) at the input end (z = 0) of the two-GGIAG-waveguide structure are shown. At the input end, field magnitudes of the two system modes in the center of the first waveguide core (x = x₁) have the same value, i.e., E_e(x₁) = E_o(x₁) = E. Therefore, the field in the center of the first waveguide core E₁ can be expressed as

E_{1} (z) = E_{t} (x_{1}, z) = a E {e^{- j [(β + Δ β / 2) + j γ / 2] z} + e^{- j [(β - Δ β / 2) + j γ / 2] z}} .

On the other hand, at the input end, field magnitudes of the two system modes in the center of the second waveguide core (x = x₂) are E_e(x₂) = E and E_o(x₂) = −E, respectively. Thus the field in the center of the second waveguide core E₂ is given by

Fig. 7 Results calculated with d = 2.1 μm where the gap is antiresonant. (a) Modal gain coefficient of the system modes as a function of material gain coefficient. Inset: Transverse field distributions E_e(x) and E_o(x) of the even and the odd system modes at the input end (z = 0). (b) Coupling length as a function of material gain coefficient.

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E_{2} (z) = E_{t} (x_{2}, z) = a E {e^{- j [(β + Δ β / 2) + j γ / 2] z} - e^{- j [(β - Δ β / 2) + j γ / 2] z}} .

The power in the first waveguide core P₁ would be proportional to |E₁|². Then we can have

P_{1} (z) = C_{1} | E_{1} (z) |^{2} = 4 C_{1} | a |^{2} | E |^{2} e^{γ z} \cos^{2} (\frac{Δ β z}{2}),

where C₁ is a positive constant. Similarly, the power in the second waveguide core P₂ would be proportional to |E₂|², and has the form

P_{2} (z) = C_{2} | E_{2} (z) |^{2} = 4 C_{2} | a |^{2} | E |^{2} e^{γ z} \sin^{2} (\frac{Δ β z}{2}),

where C₂ is another positive constant.

To determine the distance L_c₁ at which P₁ becomes the minimum (zero), one simply solves P₁ = 0. According to Eq. (5)a, the solution is the root of cos²(Δβz/2), and is given by

L_{c 1} = \frac{π}{Δ β} .

Equation (6) is totally the same as Eq. (2). This explains why coupling length obtained from Eq. (2) agrees well with L_c₁ obtained from BPM simulation, as shown in Figs. 3 and 5. Moreover, from Eq. (6), it is clear that L_c₁ is independent of material gain coefficient g. Thus varying the value of g will not affect L_c₁, as evidenced in Fig. 6.

On the other hand, to determine the distance L_c₂ at which P₂ reaches the maximum, we need to solve dP₂/dz = 0. The derivative of P₂ can be derived from Eq. (5)b as

\begin{matrix} \frac{d P_{2}}{d z} = 4 C_{2} | a |^{2} | E |^{2} \frac{d}{d z} [e^{γ z} \sin^{2} (\frac{Δ β z}{2})] \\ = 4 C_{2} | a |^{2} | E |^{2} \sqrt{Δ β^{2} + γ^{2}} e^{γ z} \sin (\frac{Δ β z}{2}) \cos (\frac{Δ β z}{2} - ϕ), \end{matrix}

where

\sin ϕ = \frac{γ}{\sqrt{Δ β^{2} + γ^{2}}} .

According to Eq. (7), L_c₂ can be determined from the root of the last term cos(Δβz/2 − ϕ), and is given by

L_{c 2} = \frac{π}{Δ β} + \frac{2 ϕ}{Δ β} = L_{c 1} + \frac{2 ϕ}{Δ β} .

From Eq. (9), it is clear that L_c₂ differs from L_c₁ with the term 2ϕ/Δβ. Moreover, L_c₂ will be influenced by modal gain coefficient γ via ϕ. If γ is negative, in this case the system modes suffer a modal loss rather than a modal gain, ϕ is negative, and then L_c₂ is smaller than L_c₁. On the contrary, if γ is positive, ϕ is positive, thus L_c₂ is larger than L_c₁. When γ is zero, ϕ is zero too, then L_c₂ and L_c₁ are equal.

Figure 7(a) plots modal gain coefficient γ as a function of material gain coefficient g. The value of γ are obtained by taking the average of γ_e and γ_o, and the results in Fig. 7(a) are calculated with d = 2.1 μm at which the gap is antiresonant. A black dashed line representing γ = 0 is also shown for reference. If γ falls below this line, the system modes suffer a modal loss; otherwise, the system modes acquire a modal gain if γ is above. Note that when g = 0, the GGIAG waveguides become purely index-antiguided (IAG) because the core index is lower than the cladding index and there is no material gain present in the core. In Fig. 7(a), the modal gain is −0.518 cm^-1 when g = 0. That is to say, the system modes under the IAG condition have a modal loss α_IAG = 0.518 cm^-1. To make the system modes gain-guiding, a sufficient material gain should be provided in the low-index core to compensate the IAG loss α_IAG. In fact, the relation between modal gain and material gain can be expressed as

γ = g - α_{I A G} .

Therefore, for the system modes to have a modal gain, i.e., γ > 0, the material gain coefficient should be larger than 0.518 cm^-1, as illustrated in Fig. 7(a).

After having the modal gain coefficient γ, we can further calculate L_c₂ according to Eq. (9). Calculated result of L_c₂ using Eq. (9) (blue solid line) and simulated result of L_c₂ using BPM (red triangles) are compared in Fig. 7(b). Obviously, results obtained by the two different methods agree well with each other. Thus effectiveness of Eq. (9) is validated. We also calculate L_c₁ according to Eq. (6), and the result is shown in Fig. 7(b) as a black dashed line. Since Eq. (6) and Eq. (2) are totally the same, and agreement between Eq. (2) and BPM for L_c₁ has been demonstrated in Figs. 3 and 5, it is needless to show the BPM-simulated result of L_c₁ in the figure for comparison. From Fig. 7(b), again, it is found that L_c₂ increases with material gain coefficient g. Moreover, when g < α_IAG (0.518 cm^-1), under which the system modes suffer a modal loss (γ < 0), L_c₂ < L_c₁. Oppositely, when g > α_IAG, in this case the system modes acquire a modal gain (γ > 0), L_c₂ > L_c₁. Particularly, when g≈α_IAG, for example, g = 0.5 cm^-1, in this case γ ≈0 and L_c₂≈L_c₁.

3.5 Power evolution along the propagation

Finally, we examine the power evolution along the propagation distance for different material gain coefficients. Here we assume g = 0.25, 0.5, and 0.75 cm^-1, which correspond to the three cases of g < α_IAG, g≈α_IAG, and g > α_IAG, respectively.

Figure 8(a) shows the results for d = 2.1 μm, where the gap is under the antiresonant condition. From the evolutions of P₁ (the power in the first waveguide core) and P₂ (the power in the second waveguide core), the aforementioned phenomenon can be clearly observed, i.e., when g < α_IAG, L_c₂ < L_c₁; when g≈α_IAG, L_c₂ ≈L_c₁; when g > α_IAG, L_c₂ > L_c₁. Note that the power in the gap region, denoted as P_g, is shown in the figure as well. Obviously, P_g is zero when the gap is antiresonant. We also plot the total power P_t, where P_t is defined as P_t = P₁ + P₂ + P_g. As can be seen, P_t attenuates, slightly attenuates, and grows with propagation distance for g = 0.25, 0.5, and 0.75 cm^-1, respectively. This can be explained from the modal gain coefficient γ, which is shown in Fig. 7(a) for the case of d = 2.1 μm. The modal gain coefficients are −0.268, −0.018, and 0.232 cm^‒1 for g = 0.25, 0.5, and 0.75 cm^-1, respectively. The first two values are negative, therefore modal power loss will take place when g = 0.25 and 0.5 cm^-1. On the contrary, owing to the positive modal gain coefficient, power will grow with distance when g = 0.75 cm^-1. Although not displayed in the figure, our simulation shows that the curves of P_t in Fig. 8(a) agree well with the curves of exp(−γz) for the three g values.

Fig. 8 Power evolution along the propagation distance for g = 0.25, 0.5 and 0.75 cm^-1. (a) d = 2.1 μm (the gap is antiresonant), and (b) d = 2.75 μm (the gap is resonant). P₁, P₂, and P_g are the powers in the first waveguide core, the second waveguide core, and the gap between the waveguide cores. The total power P_t = P₁ + P₂ + P_g.

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Power evolutions for d = 2.75 μm are shown in Fig. 8(b). In this case, the gap is resonant. It is obvious that, under the resonant condition, L_c₁ and L_c₂ are very close. However, the minimum of P₁ is different from what is observed in Fig. 8(a) for the case of d = 2.1 μm where the gap is antiresonant. In the antiresonant case, the minimum value of P₁ is zero. While in the resonant case as shown in Fig. 8(b), because there are three system modes beating in the structure, P₁ will never reach zero and is slightly above. In addition, it is found that there is a small amount of power existing in the gap region. This can be understood from the modal field distributions. As shown in Fig. 4(b), when the gap is resonant, two of the system modes (modes C and E) have fields oscillating strongly in the gap region, which cause P_g not zero along the propagation. For the total power P_t, it still attenuates with propagation distance when g = 0.25 cm^-1 and grows when g = 0.75 cm^-1. While when g = 0.5 cm^-1, P_t becomes to grow slightly with distance when the gap is under the resonant condition.

4. Conclusion

We have investigated the characteristics of power coupling between two identical and parallel GGIAG planar waveguides. It is found that the coupling is periodic with the gap between the two waveguide cores, because the gap behaves like a Fabry-Perot etalon. Strongest coupling occurs when the gap is resonant, which leads to the shortest coupling length. On the contrary, when the gap is under the antiresonant condition, the coupling length becomes the longest. It is also found that the coupling length L_c₁ at which the first waveguide power is minimum and the coupling length L_c₂ at which the second waveguide power is maximum are different, owing to the material gain provided in the waveguide cores. Further investigation reveals that L_c₁ is independent of material gain, while L_c₂ increases with the gain. If the material gain is sufficient to compensate the IAG loss, the system modes will acquire a modal gain, which makes L_c₂ > L_c₁. Otherwise, the system modes suffer a modal loss and L_c₂ < L_c₁. This work provides useful information for future applications of GGIAG planar waveguide lasers and amplifiers when they are placed in close proximity.

Funding

Ministry of Science and Technology of the Republic of China (MOST) (102-2221-E-224-069-MY3).

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Power coupling between gain-guided index-antiguided planar waveguides

Abstract

1. Introduction

2. Structure for numerical modeling