Generalized Analytical Solutions for Nonlinear Positive-Negative Index Couplers

Abstract

We find and analyze a generalized analytical solution for nonlinear wave propagation in waveguide couplers with opposite signs of the linear refractive index, nonzero phase mismatch between the channels, and arbitrary nonlinear coefficients.

1. Introduction

By leveraging the capabilities of photonics (speed) and of electronics (compactness) it may be possible to realize high-performance integrated optoelectronic systems for high bandwidth signal processing or sensing applications. Such integration requires the availability of ultracompact, ultrafast, low-loss components that can be efficiently coupled to the rest of a network [1–3]. While some of the basic functionalities are presently available in the form of individual components, their integration is still challenging for a number of reasons, including their size, speed, and power consumption. Therefore, the development of optoplasmonic devices that could be integrated on a single chip and could bridge existing gaps in optoelectronic integration is of paramount importance.

One of the major requirements for the realization of efficiently performing optoelectronics circuits is the ability to buffer optical signals so that the data traffic jams are prevented. An optical buffer is a device that slows down (or even stops) light to store it for a certain period of time. Although several approaches to the realization of such structures have been demonstrated, a majority of slow light schemes based on various waveguide geometries are not easily scalable to a chip-size footprint [1].

It is well known that optical bistability, a phenomenon in which two different values of output power are possible for the same input power, finds numerous applications in optical memory and storage devices. Therefore, realized in compact configuration, it can provide a viable solution for all-optical on-chip storage applications.

Metamaterials (MMs) were shown to enable subwavelength waveguides and cavities—a property that fundamentally differentiates them from conventional-materials-based light wave components [4, 5]. Therefore, in this work, we investigate the most general solution for wave interactions in positive-negative index MM-based nonlinear optical couplers (shown in Figure 1). It should be mentioned that nonlinear optical couplers made of conventional positive index materials (PIMs) are not bistable (unless some additional components providing optical feedback are introduced) [6]. However, in MM-based couplers, bistability results from the effective feedback mechanism enabled by the opposing directionality of phase and energy velocities in negative index materials (NIMs). Moreover, such a coupler supports gap solitons—a feature commonly associated with periodic structures [7, 8]. These unusual properties of MM directional couplers form a basis for the development of all-optical processing applications, including wavelength converters, flip-flops, and mirrorless lasers.

Figure 1 

Positive-negative index nonlinear optical coupler.

Our previous studies focused on particular cases of phase-matched symmetric couplers with identical nonlinear properties and on asymmetric couplers with only one nonlinear channel. In this work, we found a generalized analytical solution in the presence of phase mismatch and for arbitrary values of nonlinear coefficients of both channels. The availability of such a solution enables novel optimized designs of such couplers.

2. Theoretical Model

As it has been mentioned above, in this work, the case of continuous wave propagation in a directional PIM-NIM coupler with third-order Kerr nonlinearity [9] is considered. From Maxwell’s equations in a slowly varying envelope approximation the following system of differential equations, which can describe continuous wave propagation in such optical system, can be derived [9, 10]:𝑖𝜎₁𝜕𝑎₁𝜕𝑧+𝜅₁₂𝑎₂exp(−𝑖𝛿𝑧)+𝛾₁𝑎₁²𝑎₁=0,𝑖𝜎₂𝜕𝑎₂𝜕𝑧+𝜅₂₁𝑎₁exp(𝑖𝛿𝑧)+𝛾₂𝑎₂²𝑎₂=0, (1) here 𝑎₁ and 𝑎₂ are the complex amplitudes of the waves in the PIM and NIM channels, respectively, 𝜅₁₂ and 𝜅₂₁ are coupling coefficients, 𝛾₁ and 𝛾₂ are normalized nonlinearity coefficients, 𝛿=𝛽₁−𝛽₂ is the mismatch between the propagation constants in the channels, and 𝜎_𝑖 is the sign of the refractive index. In the case of the PIM-NIM coupler 𝜎₁>0, 𝜎₂<0. It is worth noting that the magnitude of the coupling coefficients depends on the geometrical parameters, such as channel width and separation distance between the channels, as well as on the channels dielectric properties. In a particular situation, it may be difficult to fabricate two identical waveguides with different signs of the refraction indexes, but, without losing any generalization of the analytical solution, it can be assumed that coupling coefficients of both channels are equal.

Let us represent complex amplitudes 𝑎₁ and 𝑎₂ in terms of real amplitudes 𝐴_𝑖 and phases 𝜙_𝑖:𝑎₁=𝐴₁exp𝑖𝜙₁,𝑎₂=𝐴₂exp𝑖𝜙₂. (2)

Substituting (2) into (1) and separating the real and imaginary parts lead to the following system of equations: 𝜕𝐴₁𝜕𝑧=𝜅₁₂𝐴₂sin(𝜃),𝜕𝐴₂𝜕𝑧=𝜅₂₁𝐴₁sin(𝜃),𝜕𝜃=𝜅𝜕𝑧¹²𝐴₂𝐴₁+𝜅₂₁𝐴₁𝐴₂cos(𝜃)+𝛾₁𝐴²₁+𝛾₂𝐴²₂+𝛿, (3) here 𝜃=𝜙₁−𝜙₂+𝛿𝑧.

If the light is initially launched into channel 1, then the boundary conditions take the following form:𝐴₁(0)=𝐴₀,𝐴₂(𝐿)=0, (4) where 𝐴₀ is the amplitude of the input signal and 𝐿 is the length of the coupler. Assuming that coupling coefficients 𝜅₁₂=𝜅₂₁=𝜅, we can rewrite (3) as follows:𝜕𝐴₁𝜕𝑧=𝜅𝐴₂sin(𝜃),𝜕𝐴₂𝜕𝑧=𝜅𝐴₁sin(𝜃),𝜕𝜃𝐴𝜕𝑧=𝜅²𝐴₁+𝐴₁𝐴₂cos(𝜃)+𝛾₁𝐴²₁+𝛾₂𝐴²₂+𝛿. (5)

Next, we found that (5) have two integrals of motion:𝑃₁−𝑃₂=𝐶,Γ=𝐴²₂𝛾₂−𝛾₁𝐴²₂+2𝛾₁𝐴²₁+2𝛿+4𝜅𝐴₁𝐴₂cos(𝜃), (6) where 𝑃₁=𝐴²₁ and 𝑃₂=𝐴²₂. From the boundary conditions (4) one can conclude that 𝐶=𝐴²₁(𝐿) and Γ=0. From these integrals of motion, we can derive an expression for cos(𝜃):(𝜃)𝐴cos=−²²𝛾₁+𝛾₂𝐴²₂+2𝛿+𝛾₁𝐶4𝜅𝐴₁𝐴₂. (7) Substituting (7) into the second equation of (5), we derive an equation for power evolution in the second channel:𝜕𝑃₂𝜕𝑧²=4𝜅²𝑃₁𝑃₂−𝑃²₂𝛾𝑃₂+𝛿, (8)

where 𝛾=(𝛾₁+𝛾₂)/2 and 𝛿=𝛿+𝛾₁𝐶. Using the first integral of motion, we can rewrite this equation in terms of 𝑃₂: 𝜕𝑃₂𝜕𝑧²=−𝛾𝑃⁴₂−2𝛾𝛿𝑃³₂+4𝜅²−𝛿𝑃²₂+4𝜅²𝐶𝑃₂. (9) The solutions for 𝑃₁ and 𝑃₂ are found in terms of the Weierstrass elliptic function ℘(𝑧;𝑔₂,𝑔₃):𝑃₁(𝑧)𝜅=𝐶+²𝐶℘𝐿−𝑧;𝑔₂,𝑔₃−(1/12)4𝜅²−𝛿²,𝑃₂(𝑧)=𝜅²𝐶℘𝐿−𝑧;𝑔₂,𝑔₃−(1/12)4𝜅²−𝛿². (10) The invariants 𝑔₂ and 𝑔₃ are defined as follows:𝑔₂=2𝛾𝛿𝜅²𝐶+4𝜅²−𝛿²,𝑔12₃=−𝛾𝛿𝜅²𝐶4𝜅²−𝛿6−4𝜅²−𝛿6³+𝛾𝜅⁴𝐶². (11) The parameter 𝐶 can be found by solving the following transcendental equation:𝐴²₀𝜅=𝐶+²𝐶℘𝐿;𝑔₂,𝑔₃−(1/12)4𝜅²−𝛿². (12)

Since this is the most general solution for PIM-NIM couplers, first, we test it against previously considered particular cases [7, 8, 10]: (i) a linear phase-matched coupler with 𝛾₁=𝛾₂=𝛿=0, (ii) a nonlinear phase-matched coupler with 𝛾₁=𝛾₂=𝛾,𝛿=0, and (iii) phase-matched asymmetric couplers. In the first case, the discriminant of the Weierstrass function Δ=𝑔³₂−27𝑔²₃=0. Therefore, in this case we can rewrite (10) in terms of hyperbolic functions. So we get the following solution for the linear case of a PIM-NIM coupler [9]:𝑃₁(𝑧)=𝐶1+sinh(𝜅(𝐿−𝑧))²,𝑃₁(𝑧)=𝐶sinh(𝜅(𝐿−𝑧))². (13)

In the second case, by expanding the Weierstrass function ℘(𝑧;𝑔₂,𝑔₃) in terms of the Jacobi elliptic function one can derive the same expressions for 𝑃₁(𝑧) and 𝑃₂(𝑧) as in [7].

Next, we compare two asymmetric cases: (a) a phase-matched coupler in which the PIM channel is nonlinear and the NIM channel is linear, and (b) a phase-matched coupler in which the PIM channel is linear and the NIM channel is nonlinear. Figure 2 shows a numerical solution of (12), which determines the dependence of the output signal power 𝑃₁(𝐿) on the input field intensity 𝑃₁(0), where 𝜅=8, 𝛿=0, and 𝐿=1. The solid, black curve represents the case when 𝛾₁=5, 𝛾₂=0, that is, case (a); the blue, dashed curve shows the opposite case when 𝛾₁=0, 𝛾₂=5, that is, case (b).

Figure 2 

Output power 𝑃1(𝐿) as a function of input power 𝑃1(0) when 𝜅=8 and 𝛿=0. Solid black curve: 𝛾1=5,𝛾2=0; dashed blue curve:𝛾1=0,𝛾2=5.

These results show in both limiting cases that the phase-matched PIM-NIM coupler is bi- (or multi-) stable. Also, it is noteworthy that for given combinations of the linear coupling coefficient and the length of the coupler, the bistability threshold is higher only in the second (NIM) channel, which is nonlinear while light initially is pumped into the first (PIM) channel. This is due to the fact, that in this case, nonlinearity takes effect only if enough power is coupled to the second (nonlinear) channel.

3. Effect of Phase Mismatch

So far, we have considered the behavior of somewhat idealized PIM-NIM couplers—phase-matched couplers with 𝛿=0. However, from a practical realization viewpoint, it would be important to analyze the influence of the phase mismatch during wave propagation in such systems. In this section, we investigate the dependence of 𝑃₁(𝑧) and 𝑃₂(𝑧) on 𝛿.

Figure 3 represents the output power 𝑃₁(𝐿) as a function of input power 𝑃₁(0) when 𝜅=8, 𝛾₁=5 and 𝛾=0 (self-focusing nonlinearity only in the first channel) with different values of phase mismatch 𝛿: solid black curve: 𝛿=0; blue curve; 𝛿=10; dashed red curve: 𝛿=−10. It is noteworthy that the dependence of the output field intensity 𝑃₁(𝐿) on input field power 𝑃₁(0) depends on the sign of phase mismatch 𝛿. This can be understood by considering the dispersion relation for this system.

Figure 3 

Output power 𝑃1(𝐿) as a function of input power 𝑃1(0) when 𝜅=8, 𝛾1=𝛾2=5 (self-focusing nonlinearity). Solid black curve: 𝛿=0; dashed blue curve: 𝛿=10; dot-dashed red curve 𝛿=−10.

Assuming that the solution of (1) has the following form:𝑎_1,2=𝑢_1,2𝛿exp𝑖𝑞𝑧exp±𝑖2𝑧, (14) we can find that 𝛿 and 𝑞 have the following forms:𝜅𝛿=−²¹+𝜅₁₂1−𝐶/𝑃₀±1−𝐶/𝑃₀−𝑃₀𝛾₁+𝛾₂𝐶1−𝑃₀,𝜅𝑞=−²¹−𝜅₁₂1−𝐶/𝑃₀±21−𝐶/𝑃₀−𝑃₀𝛾₂𝐶1−𝑃₀−𝛾₁, (15) where because of the boundary conditions 𝐶∈[0,𝑃₀].

In a linear PIM-NIM coupler, these dispersion relations show a bandgap [10]. Let us fix two values of detuning to be 𝛿_+,−=±𝛿₀ in the linear case such that the center of the bandgap corresponds to 𝛿=0. Next, from (15) we find that in the case of positive nonlinear coefficients (self-focusing Kerr nonlinearity) the “𝛿−𝑞” dispersion relations shift towards negative values of 𝛿 leading to asymmetry of the center of the bandgap with respect to 𝛿₊ and 𝛿₋ and, as a result, different dependences of the output field intensity on input field power on either side of the bandgap [7]. In contrast, in the case of negative nonlinear coefficients (self-defocusing Kerr nonlinearity) the “𝛿−𝑞” dispersion relations shift towards positive values of 𝛿 and the dependences of the output field intensity on input field power reverse. Indeed, if now we plot the output power 𝑃₁(𝐿) as a function of input power 𝑃₁(0) when 𝜅=8 and 𝛾₁=𝛾₂=−5 (self-defocusing nonlinearity only in the first channel) with different values of phase mismatch 𝛿: solid black curve: 𝛿=0; blue curve: 𝛿=10; dashed red curve: 𝛿=−10 (see Figure 4), we find that the curves corresponding to positive and negative detuning interchanged as compared to those in Figure 3.

Figure 4 

Output power 𝑃1(𝐿) as a function of input power 𝑃1(0) when 𝜅=8,  𝛾1=𝛾2=−5 (self-defocusing nonlinearity). Solid black curve: 𝛿=0; dashed blue curve: 𝛿=10; dot-dashed red curve 𝛿=−10.

4. Discussion

Although the analysis of the PIM-NIM coupler in this work was based on coupled-mode equations assuming effective medium parameters for the dielectric permittivity and magnetic permeability of MMs, in practice such an NIM channel (which is the most challenging part of the proposed device) can be realized in at least two configurations: (i) using double-negative resonant MMs and (ii) using strongly anisotropic MM waveguides. Such waveguides were shown to support negative-index propagating modes [11, 12]. For these modes, the wave propagation is in a direction opposite to the phase velocity. As a result, the waveguide behaves as a 2-dimensional counterpart of 3-dimensional negative index material. Such a waveguide can be designed using alternating metal and dielectric subwavelength layers with positive and negative permittivities, respectively. As a nonlinear optical material, we envision incorporate chalcogenide glasses or nonlinear polymers that possess relatively high nonlinear refractive indices. Theoretical design and experimental realization of such devices will be discussed elsewhere.

In conclusion, we found a generalized analytical solution for nonlinear wave interactions in PIM-NIM couplers in the presence of phase mismatch and for arbitrary values of nonlinear coefficients in both channels. These results offer a practical tool for designing novel MM-based couplers based on either double-negative or strongly anisotropic MMs that are likely to enable ultracompact optical storage and memory components for photonics on chip applications.

Acknowledgment

This work was supported by the US Army Research Office Award no. W911NF0910075.