1. Introduction

The nonlinear switching (NLS) properties of waveguide arrays and multicore fiber couplers have recently attracted considerable attention for mode-locked fiber laser applications [1–5]. In the linear regime, where optical power is low, neighboring waveguides exchange optical power periodically; the linear coupling is caused by the modal overlap of adjacent waveguides and is most efficient when adjacent modes have identical propagation constants [6]. In the nonlinear regime, where optical power is high, nonlinear effects alter the refractive index of each waveguide and consequently detune the effective propagation constants of the modes, reducing the power exchange efficiency between adjacent cores [7]. By only retaining the light in the launch waveguide at the output, it is possible to achieve power-dependent transmission and intensity discrimination required for mode-locked operation [1, 2].

Recently, it was shown that nonlinear multimodal interference in a graded-index multimode fiber has intensity discrimination properties as well and can be used for NLS [8]. However, it was observed that a much higher power is required for NLS based on the nonlinear multi-modal interference in multimode fibers compared to the NLS based on the nonlinear mode coupling in multicore fibers. Moreover, the NLS quality of the multimode fiber was shown to be inferior to those reported for multicore fibers.

The objective of this manuscript is to highlight the differences between the NLS behavior in multimode versus multicore waveguides. Several key questions are addressed: Why is higher power required to achieve NLS in multimode versus multicore waveguides? What is the main reason behind the inferior NLS quality in multimode waveguides? Is it because of the cross-phase modulation (XPM) and four-wave mixing (FWM) terms? Or because of the way the modes are excited in the multimodal junctions?

In order to present the arguments in a concrete model, NLS is explored for the Transverse Electric (TE) polarization in a pair of identical one-dimensional waveguides shown in Fig. 1(a). The design parameters are such that each waveguide supports only a single TE-polarization mode, with n_cl = 1.5, n_co = 1.506, and a = 3λ, where λ is the optical wavelength. We refer to the spatial transverse mode profile of a “single” waveguide as F_w(x) throughout this paper.

 

Fig. 1 (a) The refractive index profile of the double-waveguide nonlinear coupler is shown. (b) For weakly coupled waveguides (d = 10λ), the even and odd mode profiles F_e(x) and F_o(x) in Picture-𝒜 are sketched above the index profile and F₁(x) and F₂(x) in Picture-ℬ are sketched below the index profile. (c) is similar to (b), except the waveguides are strongly coupled for d = 2λ. Results are shown for the TE polarization, where the transverse electric field vector is pointing in the vertical direction in these figures.

Download Full Size | PDF

In the linear regime, light propagation through this double-waveguide is commonly treated using the standard coupled-mode theory, which is valid as long as the waveguides are weakly coupled [6]. In Fig. 1(b) where the separation between the waveguides is d = 10λ, the standard coupled mode theory can be reliably applied to the individual modes of the waveguides whose profiles are identified as F₁(x) and F₂(x). The overlap between these modal profiles determines their coupling. However, a rigorous approach, which is applicable to both weak and strong coupling, is based on directly solving for the even and odd supermodes of the double-waveguide and treating the light propagation as a modal interference problem. The even and odd eigenmodes of the full index profile are sketched as F_e(x) and F_o(x) in Fig. 1(b). When the waveguides are only weakly coupled, the two approaches can be identically mapped to one another by $F_{1,2}=(F_e\pm F_o)/\sqrt{2}$. We will take this relationship as the definition of F₁(x) and F₂(x). Therefore, for the case of the weakly coupled waveguides, we can write F₁ ≡ F₂ ≡ F_w, where ≡ implies similarity up to a shift in the x-coordinate.

The case of the strongly coupled waveguides with d = 2λ is shown Fig. 1(c), where F_e(x) and F_o(x) profiles are sketched as the exact eigenmodes of the system. The standard coupled mode theory is not applicable here, and F₁(x) and F₂(x) profiles obtained from $F_{1,2}(x)=(F_e\pm F_o)/\sqrt{2}$ are far from the eigenmodes of the individual waveguides. In this case, F₁ ≢F₂ ≢ F_w.

In this paper, the NLS behavior of the double-waveguide of Fig. 1(a) is studied as a function of the waveguide separation. This analysis provides an interpolation from the multicore to the multimode setup as the two waveguides are brought closer together and eventually merged. The NLS problem is initially treated using the generalized nonlinear Schrödinger equation (GNLSE) applied to the even and odd supermodes of the double-waveguide (F_e and F_o), which are the exact eigenmodes of the Helmholtz equation. This treatment is rigorous and is referred to as Picture-𝒜. The nonlinear propagation problem is then transformed to the language of the F₁(x) and F₂(x) profiles, referred to as Picture-ℬ. Although, F₁(x) and F₂(x) are not eigen-modes of the individual waveguides and take their meaning only from F_e(x) and F_o(x), the nonlinear propagation problem can be recast in their language with no loss of generality and both Pictures are equally valid. However, it will be seen that Picture-ℬ is more suitable for the purposes of this paper, as the role of the XPM/FWM versus self-phase modulation (SPM) terms and the initial modal excitations are more clear in this Picture. This holds especially true in the limit of weakly coupled waveguides, where the XPM/FWM terms are completely absent in the nonlinear propagation equation in Picture-ℬ.

2. The formulation and results

In what follows, only the continuous wave (CW) limit is considered, in order to reduce the complexity of the analysis and to ensure that the physics is not buried under phenomena that are not essential in conveying the intended message. However, the temporal effects can be easily included and the main observation will not be affected. The observations are general and equally apply, at least qualitatively, to Transverse Magnetic (TM) waves, and optical fibers.

In the coupled waveguide problem of Fig. 1(a), the lowest order modes are a pair of z-propagating bound states with the even and odd spatial profiles F_e(x) = F_e(−x) and F_o(x) = −F_o(−x) being invariant along the y-axis. The symmetries of these two profiles are dictated by the parity symmetry of the refractive index profile n(x) = n(−x) and its invariance along the y-axis. The GNLSE describing the CW evolution of light in the two-mode double-waveguide in Picture-𝒜 can be written as [9, 10]

In Fig. 2(a), δβ_e is plotted as a function of the normalized separation d/λ between the waveguides. At a large separation, the even and odd modes are nearly degenerate in the value of the propagation constant. However, δβ_e rapidly increases as the waveguides are brought closer together and the degeneracy between the even and odd modes is strongly broken.

 

Fig. 2 (a) The splitting between the propagation constants of the modes increases significantly as the waveguides are brought closer together. (b) The nonlinear coupling coefficients for the even and odd modes are shown as a function of the normalized separation, where their degeneracy is removed when the waveguides are strongly interacting.

Download Full Size | PDF

In Fig. 2(b), the non-zero nonlinear coupling coefficients of the even and odd modes in Picture-𝒜 are plotted as a function of the normalized waveguide separation. The nonlinear coupling coefficients f_eeee, f_eeoo and f_oooo are equal when the two waveguides are far apart and hence, weakly coupled. However, as the two waveguides get closer, the difference between the nonlinear couplings increases. Other elements of the f_μνκρ tensor not shown in Fig. 2(b) are related to the plotted elements by symmetry relationships of the tensor; e.g., f_eoeo = f_eeoo.

The large values of the XPM/FWM terms in Picture-𝒜 complicate the analysis of the NLS. In Picture-ℬ, the nonlinear propagation is more intuitive and the XPM/FWM terms are absent in the limit of weakly coupled waveguides. In Picture-ℬ, the field amplitudes are defined as ${\tilde{A}}_{1,2}:=(A_e\pm A_o)/\sqrt{2}$. The GNLSE Eq. (1) is transformed to

The non-zero elements of f_jklm are plotted in Figs. 3(a) and 3(b). The SPM nonlinear coupling terms f₁₁₁₁ and f₂₂₂₂ are larger than the XPM/FWM terms by more than an order of magnitude, even when the two waveguides are merged. This behavior is dictated by the relationships presented in Eq. (4) and the near degeneracy of f_μνκρ in Fig. 2(b), especially at weak coupling. Thus, in Picture-ℬ, the nonlinear switching is dominated by SPM, which is easier to analyze.

 

Fig. 3 The nonlinear coupling coefficients in Picture-ℬ are shown as a function of d/λ. The SPM coefficients in (a) are much stronger than the XPM/FWM coefficients in (b).

Download Full Size | PDF

The NLS behavior of the waveguide coupler is shown in Fig. 4. A multicore NLS configuration is sketched in Fig. 4(a), where the beam is injected into the two-core switch using an input coupler waveguide and is collected at an output coupler. The length of the nonlinear switch is equal to the half-beat-length L_h, so that at low powers, all the injected light is transferred to the second waveguide and no power is collected at the output. However, when the input power is increased, the nonlinearity detunes the coupling between the two waveguides and all the power is collected at the output port. This behavior is shown in Fig. 4(b), where the transmission τ is plotted as a function of the effective nonlinear coefficient defined as γ = n₂ω₀P̃(cδβ_e), where P̃ is the linear optical power density injected from the input coupler.

 

Fig. 4 (a) A sketch of the NLS device. (b) The relative power transmission is plotted as a function of γ. The solid (black) curve corresponds to weakly coupled waveguides with d/λ = 10. The dashed (red) and dotted (blue) NLS curves correspond to the strongly interacting merged waveguide with d/λ = 0, where in case-1 the injected and collected beam profiles are F₁(x), while in case-2, the injected and collected beam profiles are F_w(x).

Download Full Size | PDF

In Fig. 4(b), the solid (black) line shows the NLS behavior of the double-waveguide coupler when the normalized separation is d/λ = 10. In this case, the injected and collected field profiles are assumed to be those of a single waveguide identical to the waveguides in the nonlinear coupler. The relative transmitted power can be obtained from [8]

When the waveguides are brought closer together, F₁(x) can no longer be approximated as F_w(x), as can also be seen in Fig. 1(c). In Fig. 4(b), the dashed (red) line shows the NLS behavior of the double-waveguide coupler, when the normalized separation is d/λ = 0, which is the limit of a standard single-core multimode coupler. In this case (case-1), the BCs are assumed to be ${\tilde{A}}_1(0)/\sqrt{\tilde{P}}=1$ and ${\tilde{A}}_2(0)/\sqrt{\tilde{P}}=0$. However, given the difference between F₁(x) and F_w(x) in this strong coupling regime, it is clear that these BCs cannot be achieved using an input coupler waveguide identical to those used in the nonlinear switch. In fact, the injected and collected beam profiles must be modified by additional optics (such as a spatial light modulator) or must come from special waveguides in order to mode-match to F₁(x). Even if F₁(x) is used for the injection, it might be more desirable to collect the beam directly from the nonlinear coupler in the form of F_w(x). In this case, it is possible to slightly change the device length away from L_h to improve the coupling efficiency at high power at the expense of a perfect attenuation at low power (linear case). This is a reasonable trade-off because efficient transmission at high power is desired for mode-locked lasers while some level of compromise in nonlinear modulation depth is acceptable.

The difference between the dashed (red) line of d/λ = 0 compared with the solid (black) line of d/λ = 10 is caused by the XPM/FWM terms; if the XPM/FWM terms in Fig. 3(b) are artificially set to zero, the dashed (red) line in Fig. 4(b) falls almost exactly on the solid (black) line. The XPM/FWM terms nearly double the switching threshold in units of γ. However, given that δβ_e is nearly 555 times larger for d/λ = 0 than d/λ = 10 [see Fig. 2(a)], the switching threshold in the merged waveguides will become larger by the same factor when expressed in terms of power density instead of γ, because a factor of δβ_e is embedded in the definition of γ. The merged waveguides will have to be shorter by a factor of δβ_e, because in Picture-ℬ, δβ_e plays the role of the coupling between Ã₁ and Ã₂ as can be seen in Eq. (3), and because L_c must equal the half-beat-length L_h.

Injecting the double-waveguide coupler with a complex beam profile of F₁(x) to obtain the ${\tilde{A}}_1(0)/\sqrt{\tilde{P}}=1$ and ${\tilde{A}}_2(0)/\sqrt{\tilde{P}}=0$ BCs may not be possible. In many situations, it is more practical to inject the beam from a waveguide that is identical to one of the waveguides in the multicore coupler. For the merged waveguides (d/λ = 0), this means that F_w(x) is used as the injected and collected beam, instead of F₁(x). We refer to this as case-2 and the NLS curve is shown as dotted (blue) in Fig. 4(b). Because the injected and collected beams are not properly mode-matched to the input and output couplers, respectively, both modes are excited (with different powers) and we need to use ${\tilde{A}}_1(0)/\sqrt{\tilde{P}}\approx 0.95$ and ${\tilde{A}}_2(0)/\sqrt{\tilde{P}}\approx 0.29$. Moreover, approximately 1.7% of the injected power is coupled to radiation modes. The modulation depth in the NLS curve in Fig. 4(b) for case-2 in dotted (blue) is lower because of the injection beam profile, resulting in a lower quality NLS.

We note that a linear switch can fully transfer the power from one waveguide to another if L_c = nL_h, where n is an odd integer. Therefore, all such linear switches are equivalent. However, in the nonlinear case the quality of the NLS is reduced for n > 1 as shown in Fig. 5 and undesired oscillations are observed before γ is sufficiently large for full power switching.

 

Fig. 5 The relative power transmission is plotted as a function of γ for (a) L_c = L_h versus L_c = 3L_h and (b) for L_c = 15L_h. The NLS quality for L_c = 3L_h and L_c = 15L_h are lower.

Download Full Size | PDF

3. Conclusions

It is argued that the switching power threshold in nonlinear multimode junctions is larger than in multicore junctions, mainly because the value of δβ_e is substantially larger in the former case. The XPM/FWM terms also play a role in setting the power threshold but their importance is orders of magnitude lower than that of δβ_e. When expressed in proper dimensionless parameters, the NLS curves are almost identical in a multimode and a multicore junction if the XPM/FWM terms are artificially switched off.

The presence of the XPM/FWM terms do not seem to play any essential role in the quality of the nonlinear switching curve of the multimode or multicore systems, especially in the value of the modulation depth. Rather, injection of the right combination of the modes at the input and the collection of the right combination of the modes at the output are the main factors behind the quality of the NLS curves. Creating the proper injection profile is often trivial in a weakly-coupled multicore waveguide; however, obtaining the right combination of the modes in a multimode junction is difficult, if not impossible, to achieve. Therefore, multimode junctions show lower quality NLS curves. This finding agrees with the observations reported in Ref. [8] and provides an intuitive explanation for those results.

Finite element method has been used to calculate the spatial profiles and propagation constants of the modes. Embedded nonlinear differential equation solvers in Mathematica have been used for the NLS simulations.