Transverse localization of light and its dependence on the phase front curvature of the input beam in a disordered optical waveguide lattice

The concept of transverse localization of light [1] in disordered, one-dimensional (1D) and two-dimensional (2D) discrete optical systems has attracted a great deal of interest in view of its underlying interesting physics, and potential novel applications [2, 3]. It is known that, with deliberately introduced disorder, light confinement occurs only in a plane perpendicular to the direction of light transport, both in temporarily realized [4] and in permanently formed lattices [5]. With the ongoing intensive research on photonic bandgap structures and discrete photonic systems, study on light localization has emerged as an important field of contemporary research in the context of disordered optical structures [4–11]. In a recent study, we have shown that this phenomenon of light localization is independent of the input beam shape, taking the example of an evanescently coupled 1D disordered waveguide lattice [7]. Interactions between the initially chosen transverse phase of the propagating beam and multiple scattering that takes place with propagation along the disordered sample could significantly change the interference property, which is the key to light localization, and hence we expect that an input beam with a curved phase front could play an important role in the localization of light. However, no study in this direction has been reported to date.

In this paper, we numerically investigate the effect of the input beam having a finite phase front curvature on its localization in a disordered coupled waveguide lattice.

We consider an evanescently coupled waveguide lattice consisting of a large number (N) of unit cells, and in which all the waveguides spaced equally apart are buried inside a medium of constant refractive index n₀ [12, 13]. The overall structure is homogeneous in the longitudinal (z) direction along which the optical beam is assumed to propagate, as shown in figure 1(a). The perturbation in the refractive index Δn(x) (over the uniform background of n₀) due to disorder in this 1D waveguide lattice is assumed to be of the form

$$\begin{eqnarray} &&\Delta n \hspace{0.167em} (x)=\Delta {n}_{\mathrm{p}}\hspace{0.167em} (H(x)+C \delta (x)) \end{eqnarray} \tag{ 1 }$$

where C is a dimensionless constant, whose value governs the level/strength of disorder; the periodic function H(x) takes the value 1 inside the higher index regions and is zero otherwise; Δn (x) consists of a deterministic periodic part Δn_p of spatial period Λ and a spatially periodic random component δ (uniformly distributed over a specified range varying from 0 to 1). This particular choice of randomly perturbed refractive indices in the high index as well as low index layers enables us to model the diagonal and off-diagonal disorders in order to study the localization of light [5]. In our modeling, we have ignored any disorder in the spatial periodicity [7, 14, 15]. Wave propagation through the lattice is governed by the standard scalar Helmholtz equation, which under paraxial approximation could be written as

$$\begin{eqnarray} &&\mathrm{i}\frac{\partial A}{\partial z}+\frac{1}{2 k}\hspace{0.167em} \left (\frac{{\partial }^{2}A}{\partial {x}^{2}}\hspace{0.167em} \right )+\frac{k}{{n}_{0}}\Delta n \hspace{0.167em} (x)A=0 \end{eqnarray} \tag{ 2 }$$

where A(x,z) is the amplitude of an input CW optical beam having its electric field described as E(x,z,t) = Re[A(x, z)e^i(kz−ωt)]; k = n₀ω/c. To study the effect of having a curved phase front of an input beam on the phenomenon of transverse localization of light, the initial field amplitude of such an input Gaussian beam centered at x₀ is assumed to be of the form

$$\begin{eqnarray} &&A(x)={A}_{0}\hspace{0.167em} \exp [-(1-\mathrm{i}B)\hspace{0.167em} ((x-{x}_{0})/{\omega }_{0})^{2}] \end{eqnarray} \tag{ 3 }$$

where the parameter B represents the phase front curvature, and ω₀ is the characteristic spot size of the Gaussian beam. In the case of a CW beam having a plane phase front as the input, for which B = 0, equation (2) yields localized exponential solutions [1, 7].

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We solve equation (2) with the scalar beam propagation method which we have implemented in Matlab^® and consider input beams characterized with different values of the B-parameter. Details of the modeling methodology have been discussed in [7]. Our chosen waveguide array consists of 150 evanescently coupled waveguides, each of 7 μm width and separated from each other by 7 μm (i.e. the center-to-center spacing is 14 μm). Such a waveguide array is realizable through laser inscription in glass for direct observation of localized light [14]. We deliberately choose a relatively small refractive index contrast along with a relatively long unit-cell period compared to the wavelength (Λ ≫ λ) to ensure that the bandgap effects remain negligible.

The value of Δn_p was chosen to be 0.001 over and above that of the background material of refractive index n₀ = 1.46. To appreciate transverse localization of light in a disordered optical medium, we present in figure 1 results for the propagation of an input CW beam with a plane phase front at an operating wavelength of 980 nm. figure 1(a) depicts the array of waveguide lattices chosen for simulation study while figures 1(b) and (c) respectively correspond to C = 0 (indicating the absence of disorder) and C = 0.2. The input beam is assumed to cover a few lattice sites around the central unit cell at the input plane; the chosen beam width ω₀ (FWHM) was 12 μm ( > width (7 μm) of an individual lattice site). From figure 1(c), a clear signature of a transition to a localized state after an initial ballistic mode of propagation can be seen as the beam propagates along the length through the disordered waveguide lattice.

In order to investigate quantitatively the effect of having a finite phase front curvature of the input beam on its localization in a disordered medium, we have studied the beam dynamics for different lengths of the lattice and for different values of the B-parameter (incorporated in equation (3)). A measure of the localization is assumed to be quantifiable through decrease in the average effective width (ω_eff; as defined in [7])

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle P\equiv \left [\int I(x,L)^{2}\mathrm{d}x\right ]\biggl /\left [\int I(x,L)\hspace{0.167em} \mathrm{d}x\right ]\hspace{0.167em} ^{2}\\ \displaystyle {\omega }_{\mathrm{eff }}=\langle P\rangle ^{-\hspace{0.167em} 1} \end{array} \end{eqnarray} \tag{ 4 }$$

of the propagating beam after including the statistical nature of the localization phenomenon in a finite system; here 〈⋯〉 represents a statistical average over several realizations of the same level of disorder. After a certain propagation distance, depending upon the strength of disorder and lattice aspect ratio, ω_eff is almost unchanged with characteristic statistical fluctuations. It is worth pointing out that physically as we increase the value of the C-parameter, the total number of localized eigenstates supported by a particular realization of the disordered lattice increases for a given length [5], which eventually favors a smaller ω_eff value. Naturally, a larger value of ω_eff in the localized regime as well as an increase in the localization length [4, 7] of an eigenstate in a disordered medium would imply degradation in the quality of localization. The results of such an effect due to a finite curvature of the input beam's phase front are illustrated in figure 2. This plot of ω_eff (ensemble averaged over 200 different realizations of a particular level of disorder and normalized with respect to ω₀) with L clearly reveals degradation in the quality of localization with propagation for a finite value of the parameter B in the chosen disordered waveguide lattice.

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For a waveguide lattice with a given strength of disorder, the localized eigenchannels supported by the lattice are given. A light beam having a finite input phase front curvature would preferably excite one of the extended (non-localized) natural eigenstates of a finite disordered lattice along with the localized states in a few cases while performing the ensemble averaging over different realizations of the disorder. Hence, figure 2 essentially depicts the interplay between the strength of disorder and the phase front curvature of an input beam in the disordered lattice of finite length. We have plotted, as a sample result, the beam dynamics for three different B-parameters when the values of C are set at 0.2, 0.4 and 0.6. It may be noted that a similar trend was seen in our numerical simulations for negative values of the B-parameter.

To allow appreciation of the above-mentioned degradation in the evolution of localization, in figure 3 we have plotted the ensemble averaged (over 100 realizations) output intensity profiles from a 15 mm long disordered waveguide lattice of the above kind for B = 0 as well as a finite B-parameter (both positive and negative) when the level of disorder is set at C = 0.2. The plots in figure 3 correspond to an intermediate regime of propagation, in which both diffraction and localization are simultaneously present, before it attains a localized state. If we compare the plots corresponding to finite B relative to the plot for B = 0, this particular trend also indicates that the finite B has a detrimental effect on the simultaneous suppression of the ballistic side lobes to achieve characteristic localized tails. Hence, it can be concluded that an input beam with a plane phase front would favor a faster transition to localization in comparison to a beam with a finite phase front curvature.

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To provide a deeper appreciation of this effect on the degree of localization, we estimate the so called localization length (l_C) characteristic of a localized state in a particular disordered lattice [4]. To obtain l_C, we averaged 100 output intensity profiles for a given value of C and then performed a three-point moving average to smooth further the resulting profile, as mentioned in [7]. As a particular localized state carries the signature of a corresponding disordered optical system, in general the state is not symmetric on either side of the peak. The corresponding variation of l_C (fitted on either side of the peak of the output intensity profile) for various B-parameters is shown in figure 4. These plots also confirm the advantage of a plane phase front over a curved one from the localization point of view. It is obvious that the larger the absolute value of the initial phase front curvature, the stronger the degradation in the quality of localization. These results should form useful guidelines when undertaking experimental studies on the localization of light in a disordered dielectric.

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3.1. The effect of the input beam width

Until now, we have chosen an input beam with a width ω₀, which covers nearly four sites (high and low index) of the lattice, to excite the eigenmodes supported by the same disordered waveguide lattice. Naturally, the input excites only those eigenmodes that are localized near the spatial location of the excitation. In this subsection, we investigate the influence of the width of an input excitation while investigating the effect of a finite phase front curvature on the phenomenon of transverse localization of light. For this study we choose two different input beam widths that cover a single site and nearly 18 sites of the lattice. As before, the values of the B-parameter were chosen to be 0, ±0.6, and the level of disorder was set at 20%. This choice of relatively small C (less than the threshold for localization) was made so as to visualize the evolution of the effective beam width with propagation for different values of B in the intermediate regime of transition from ballistic to a localized state. The influence of a finite phase front curvature of the input beam is prominent in this regime. In figure 5, we have plotted the variation in ω_eff (normalized with respect to ω₀) as a function of the propagation length for the two different input beam widths. figure 5(a) corresponds to the case when the input beam width ω₀ (3 μm) covers only one waveguide of the lattice; it could be seen that after the propagation of only a few mm of the sample length, the ω_eff corresponding to the B-parameter (±0.6) approaches nearly the same final value, whereas for B = 0 the beam evolves to a ω_eff which is much smaller relative to the previous value. Next, we choose a much broader input beam of ω₀ (60 μm) that covers nearly 18 lattice sites and the corresponding propagation dynamics is shown in figure 5(b). Interestingly, to explain the behavior depicted in figure 5, we may need to analyze the results in terms of the transverse wavenumber (k_⊥) which is inversely proportional to the width (ω₀) of the input beam [4]. This k_⊥ is large when the input beam is narrow and simultaneously if we introduce the phase front curvature as defined in equation (3), the phase of the input beam strongly oscillates along the transverse profile of a localized mode. As a result, the input beam does not excite the localized modes efficiently. Apparently, the effect of this inefficient coupling as compared to the case for a plane phase front is likely to increase the ω_eff of the beam. In contrast, according to equation (3), the phase is more likely to be homogeneous for an input beam with relatively large ω₀ and closer to the case for a plane phase front. Hence, we observe a different propagation dynamics of the beam as compared to figure 5(a). We have also verified that for an intermediate stage of the above two cases, when we consider an input beam with ω₀ (22 μm) which covers nearly seven lattice sites, there is a trend similar to what are shown in figures 5(a) and (b). We may mention that the overall trend of ω_eff variations for the three different cases (as shown in figure 5) remains almost unchanged with increasing number of averagings and the nature of variations of individual curves establishes the fact that the transverse wavenumber plays a key role when studying the effect of phase front curvature, whereas the tall error bars present in the plot carry the signature of the beam dynamics in the regime of transition from a ballistic to a localized state. Thus figure 5 presents the influence of the input beam width when studying the effect of a finite phase front curvature and reveals that a sufficiently broad input beam can lessen the adverse effect of input phase front curvature.

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We have studied the significance of a finite initial phase front curvature of an optical beam in the context of transverse localization of light in a 1D coupled disordered waveguide lattice. The results of our extensive numerical simulations reveal that on an average, the presence of an initial phase front curvature tends to degrade the effect of transverse localization. These results should be of interest in designing experiments related to the study of the transverse localization of light in a disordered medium. Also, an appropriate choice of the initial phase front can enable us to control the flow of light inside the disordered discrete photonics structures. Hence, we envisage that the initial phase front of a light beam will introduce a new degree of freedom for manipulating the confinement of light in disordered optical systems.

This work relates to the Department of the Navy Grant N62909-10-1-7141 issued by the Office of Naval Research Global. The United States Government has a royalty-free license throughout the world in all copyrightable material contained herein.