1. Introduction

Grating-Assisted Couplers (GACs) are studied and used in optoelectronics as basic elements for a variety of optical applications such as input-output couplers [1], contra-directional couplers [2], optical wavelength filters [3] and wavelength multiplexing devices [4]. The guiding structure of the GAC consists of two parallel but unequal (asynchronous) waveguides in a close proximity to each other. The power transfer between these asynchronous waveguides is not efficient, and a gratings structure with a period related to the difference of the propagation constants of the optical modes facilitates the complete power transfer. While conventional GACs are reversible and symmetric, here we explore a unidirectional GAC, by employing a complex single sideband gratings structure. The unidirectionality of a power transfer is a desirable feature for many optical components, e.g., for adding light components to an optical channel without simultaneously dropping similar components (lossless combiner, wavelength independent “Add” etc.). Yet another benefit of the irreversible coupler – is length insensitivity, contrary to conventional couplers which exhibit periodical power exchange. Nonreciprocal Bragg gratings were analyzed using resonance mode expansion in Ref. [5].

In the coupler structure – few compound optical modes are supported (“compound mode” – a mode of the complete structure consisting of the two unequal waveguides within the embedding medium). The transfer of excitation from one compound mode to the other (“mode conversion”) can be achieved by introducing a periodical perturbation of the refractive index of the form: p(x,y,z)=Δ(x,y)f(z), where Δ(x,y) is a constant lateral profile and f(z) is a longitudinal perturbation function. It can be shown [6] that, to the first approximation, the inter-mode power conversion is proportional to the Fourier transform of the longitudinal perturbation function f(z):

where P_m,n is the power converted from compound mode n to m and β_m , β_n are the modal propagation constants. Two compound modes are coupled if the perturbation is purely sinusoidal, with the period matched to the difference of their propagation constants. The power conversion is symmetric in this case, since the Fourier transform of any real function (transparent optics) is either symmetric or antisymmetric: F{sin(Δβz)}=(δ(Ω-Δβ)-δ(Ω+Δβz))/2 i, where Ω stands for a spatial frequency. Such perturbation will convert an initially excited compound mode m to n and vice-versa, after the same propagation distance. A perturbation with a single spatial frequency is the complex function: f(z)=exp{±iΔβz}, that provides a unidirectional and time irreversible mode conversion [7]. The ‘+’ perturbation will convert power from compound mode m to n, while the ‘-’ one will match only the conversion from n to m. The realization of exp{±iΔβz}=cos(Δβz)±isin(Δβz), is by a simultaneous synchronized modulation of the real and imaginary (absorption/gain) parts of refractive index.

2. Analysis

We explore the mode evolution due to this single sideband perturbation in a GAC which can be implemented by standard technologies: e.g., core height h=5µm, cladding index n_clad =1.445 and core index is n_core =1.455, all the calculations performed at the wavelength of λ=1.55µm. The directional coupler can be described by the notation w1×d×w2 where w1,w2 are the widths of the waveguides and d is their separation. The analysis is for the first (zero order) and second compound modes of the GAC. The slowly varying mode amplitudes c₁ , c₂ for a complex single sideband perturbation function f(z)=exp{iΔβz} can be derived from the coupled equations [6]:

where K_mn with m,n=1,2 are the coupling coefficients given by:

where ω is the frequency, ε ₀ the free space permeability, E_mt , E_nt the transverse modal fields and each mode is normalized to power 2P. The amplitudes c₁ , c₂ are related to forwards propagating modes, due to small relative changes of the refractive indices and long periods, which do not excite backward fields.

The long term contribution of the fast oscillating terms of frequencies Δβ, 2Δβ can be neglected, since their average over a period is zero to the first approximation. Averaging Eqs. (2) and (3) over one period results in:

The unidirectionality of the coupler is evident since Eqs. (5) and (6) are not symmetric. For the initial condition of the input power in compound modes 1 and 2 (which for the asynchronous coupler translates to light incident on waveguides 1 and 2) c1(0)=C01, c₂ (0)=C₀₂ the formal solution is:

The amplitude c₁ is constant for any choice of the initial conditions. However, if c₁ (0) is non-zero the amplitude of the second compound mode grows linearly with the propagation distance. We define this case as “conversion”. On the other hand when c₁ (0)=0 the power in both compound modes remains constant and there is no power transfer (“no-conversion”). For both c₁ (0),c₂ (0) non-zero, the total solution for c₂ (z) is the interference of the initial field of the second mode C₀₂ with the field which flows to the second mode. The total power in the system grows with the propagation but the conservation law is not violated since we are dealing with an “active” system and energy should be supplied to maintain optical gain. The reasons for the peculiar overall power growth are discussed in [7].

 

Fig. 1. Influence of the self-coupling coefficients in the case of conversion. In each simulation the inter-modal coupling was K12=K21=K=1.15·10-3µm-1. and the self-coupling coefficients were 1. K11=K22=0; 2. K11=K, K22=0; 3. K11=0, K22=K; and 4. K11=K22=K. Calculations were performed by numerical integration of Eqs.(2) and (3).

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The averaging of Eqs. (2) and (3) deemphasized some oscillating terms proportional to K₁₁ , K₂₂ which can be defined as self-coupling coefficients of the compound modes. To elucidate their impact, we performed a numerical solution of Eqs. (2) and (3) shown for the case of “conversion” in Fig. 1. The values for a period and inter-modal coupling coefficients were chosen Λ=500µm, K₁₂ =K₂₁ =K=1.15·10^-3µm^-1, respectively. The self-coupling coefficients varied in each simulation according to: 1. K₁₁ =K₂₂ =0; 2. K₁₁ =K, K₂₂ =0; 3. K₁₁ =0, K₂₂ =K; and 4. K₁₁ =K₂₂ =K. It follows that K₁₁ can increase slightly the conversion efficiency for positive value of K₁₁ (or decrease for negative) due to the dependence of the derivative of c₂ on this coupling coefficient mediated by c₁ . By a similar reasoning, in the case of “no-conversion” K₁₁ has no influence. The oscillations of c₁ (not shown) are mainly attributed to the term proportional to K₁₂ exp{i2Δβz}, so they have halved periodicity. The value of K₂₂ is in charge of the undesirable power oscillations of the second compound mode. K₂₂ is multiplied by c₂ which in both cases (conversion and no-conversion) is much larger than c₁ , thus it is the critical parameter which enhances undesirable oscillations in the system.

It is worth noting that if modes are either symmetric or antisymmetric and the lateral profile of the perturbation is antisymmetric then it follows from Eq. (4) that the self-coupling coefficients will be zero. This can be a key design rule for eliminating the undesirable oscillations from the system.

In order to obtain the unidirectional inter-waveguide power transfer we should use highly asynchronous waveguides for the coupler to have the compound modes confined each to a single waveguide. In that case the compound modes provide good approximation for the modes of the single waveguides. Figures 2(a) and (b) shows the fundamental modes of the single waveguides and Figs. 2(c) and (d) shows the compound modes of the coupler 6µm×6µm×3µm which is far from synchronism. The effective indices of the compound modes are n_eff1=1.449849 and n_eff2=1.447640 corresponding to a difference period of Λ≈705µm. The overlap integrals between the single and compound modes are 99.8% and 99.4% for the first and second modes respectively. In order to achieve a substantial coupling coefficient and consequently short device length, the gratings should be located where both compound modes have significant power. This optimization for a given structure can be performed using Eq. (4). Our analysis showed the best position of the perturbation is within the core of the waveguides, but slightly decentered to compensate for the asymmetry. It is possible to eliminate K₂₂ which is responsible for the undesirable oscillations if the perturbation is located only in the core of the larger waveguide (w1=6µm). With the perturbation strength of Δ=0.001 the inter-modal coupling coefficient is K₁₂ =2.7510^-3µm^-1. K₂₂ is negligible since the second mode has only a small fraction of power in this waveguide.

Numerical simulations to validate the closed form results were performed with the scalar finite-difference algorithm – scalar beam propagation method (SBPM) [8]. Since the index contrast is small the results are almost identical to the full-vector calculations. The results are shown in Fig. 3 together with the results obtained by closed form solutions and direct numerical integration of Eqs. (2),(3). The small discrepancy between the results is due to: 1. In the closed form analytical solution we neglected the terms proportional to K₁₁ which enhances the conversion efficiency; 2. The period for simulations by the SBPM can be slightly out of resonance. (The dependence of the system on detuning from resonant condition is out of scope of current paper).

The unidirectional power conversion is length insensitive in the sense that if the first compound mode is launched into the system (light incident onto w1=6µm waveguide) the power of the second mode is growing constantly (contrary to the case of real gratings where the power is exchanged periodically from one compound mode to the other). The longer the device is, the more power is converted to the second waveguide (until saturation). However if light is incident onto the w2=3µm waveguide, almost no power (less than -26dB) is transferred to the w1=6µm guide.

 

Fig. 2. (a) Fundamental mode of the single waveguide w=6µm, (b) fundamental mode of the single waveguide w=3µm. (c) and (d) first and second compound modes of the coupler structure 6µm×6µm×3µm. The overlap integrals between the powers of the corresponding single and compound modes are 99.8% and 98.4%.

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Fig. 3. (Movies 834 KB and 157 KB) Modal power for 6µm×6µm×3µm coupler with the complex perturbation f(z)=exp{iΔβz} located in w=6µm waveguide, perturbation strength Δ=0.001. (a) Initial conditions c₁ (0)=1, c₂ (0)=0 and (b) Initial conditions c₁ (0)=0, c₂ (0)=1. Maximum relative power of the first mode does not exceed -26dB.

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3. Separation of real and imaginary perturbations

The modulation of the real and the imaginary parts of the refractive index was performed simultaneously in the same spatial location. It is technologically challenging, so separation of the modulation – e.g., modulation of refractive index in one waveguide, whereas modulation of the loss/gain in the other – can make this device more feasible. The underlying equations will than take a form:

where ${K^r}_{\mathit{\text{mn}}}$ , ${K^i}_{\mathit{\text{mn}}}$ are coupling coefficients of the real and imaginary perturbations. For the asynchronous coupler, no single transverse perturbation profile can result in equal real and imaginary coupling constants. This problem is mitigated by tuning the depth of the real and imaginary perturbations to obtain ${K^r}_{\mathit{12}}$ =${K^r}_{\mathit{21}}$ =${K^i}_{\mathit{12}}$ =${K^i}_{\mathit{21}}$ . Yet, significant oscillations appear since ${K^r}_{\mathit{11}}$ ≠${K^i}_{\mathit{11}}$ , ${K^r}_{\mathit{22}}$ ≠${K^i}_{\mathit{22}}$ , and moreover the self-coupling coefficients are larger than the inter-modal coupling. Fig. 4 depicts these characteristics for the 6µm×6µm×3µm coupler with the real and imaginary perturbations in different waveguides. Even though the unidirectionality of the coupler is still evident, we loose the device length insensitivity – it should now be tuned to the center of the beat period of the compound modes.

The time irreversibility of the complex coupling is exemplified by tracing the light which is back launched from the output ports of the coupler. The perturbation that the back reflected light is experiencing is the inverse transformation f(z′)=exp{-iΔβz′}, z′=-z, thus power is back converted from the second to the first mode. However – it is obvious that the original initial conditions (e.g., power only in waveguide 1) – will not be regenerated here in contrary to all conventional time reversal optical elements.

 

Fig. 4. Modal power for the 6µm×6µm×3µm coupler with the complex gratings f(z)=exp{iΔβz}, gratings depth Δ=0.001. Real and imaginary perturbations are located in different waveguides. (a) Initial conditions c₁ (0)=1, c₂ (0)=0. The power of the second mode exhibits significant oscillations. (b) Initial conditions c₁ (0)=0, c₂ (0)=1. Maximum relative power of the first mode does not exceed -30[dB]. Calculations were performed by SBPM.

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6. Summary

An asynchronous unidirectional GAC as a realization of the irreversible coupling by a single sideband perturbation was discussed. A number of possible implementations were shown. Breaking the reversibility of the optical system was achieved here by introducing a synchronized modulation of the refractive index and absorption/gain. The unidirectionality and “length insensitivity” of the complex coupler are desirable features that can be exploited in integrated optics components. The suggested scheme is designed to be potentially realizable: - the separation of passive and active waveguides (e.g. using selective growth or regrowth) is well known; The codirectional coupling requires long period gratings such that real and imaginary gratings synchronization requires only low resolution lithography; Suitable possible materials are semiconductor and doped glass based waveguide amplifiers – in our simulation we used modal gain of 70cm^-1 (~twice of a reasonable value) – but it was done only to reduce simulation time – reducing further the gain will result only in increasing the device length by the same factor; The accurate matching between the modulation depth of the real and imaginary gratings is achievable since (at least) the gain-loss gratings is actively controlled by injection and thus can be tuned to the proper value.