The light scattered in a waveguide can be measured as both radiated substrate-cover (or substrate-air) and substrate modes, and also as radiation deflected in the forward direction. In all cases, a correct mathematical description of the resulting light distribution follows from solving the direct and inverse scattering problems. Here we consider a planar optical waveguide containing 2D irregularities.^{1–5}

We are dealing with a well-defined problem if the solution both exists and is single-valued. It must also depend on the input data in a continuous fashion. This latter requirement implies that small variations in the data lead to only small variations in the outcome. This, in turn, means that the result is stable.^{6} Problems that do not comply with these conditions are poorly defined. For instance, the so-called inverse problems in diffusion theory are classified as ill-posed in mathematical physics.^{1–3} We propose a novel approach to solving the scattering problems based on a combination of classical regularization^{6} and quasi-optimal filtering.^{1,2}

First, the scatter diagram is measured by a point photodetector in the far field. Subsequently, the digitized intensity values are analyzed to determine an approximate solution to the inverse diffusion problem in the presence of noise.^{1} One can now robustly reconstruct the second-order statistical characteristics of the irregularities,^{3} including the spectral density function (SDF) and the autocorrelation function (ACF). The important output parameters, particularly the rms deviation, σ, and the correlation radius, *r*, can now be determined with record high resolution.

The 2D problem of light scattering in the far field is formulated as

Here, *A* is a linear integral operator with kernel Φ, β' is the longitudinal component of the propagation constant of the radiating modes, and *F* is the SDF of the statistical irregularities. Furthermore, *P*_{W} = P + W, where *P* is the power scattered in the waveguide (the scatter pattern), and *W* is the intensity of the statistical additive real noise in the interval spanned by the observable radiating modes. Finally, γ is the effective refraction index, and the use of < … > implies averaging of an ergodic ensemble of statistically identical systems. The kernel Φ is the so-called waveguide optical factor, which is some type of optical transfer function. The linearity of the problem is reflected by the linear dependence of *P* on *F*.

For differential diffusion in the presence of noise, the ACF, ~R(u, γ), is restored as

The scatter pattern, *P*, is measured at fixed times, for instance, at CCD readout, the ergodic ensemble of which is then averaged. Therefore, we can exclude the time dependence in the equations. *P* is registered in the far-field (Fraunhofer) zone, or in the equivalent Fourier plane. The measurements are done over a certain bound interval defined by the real propagation constants, β, of the radiating modes (given by a continuous spectrum) that correspond to the observed radiative substrate-cover modes, the substrate modes of an asymmetric waveguide, and the observed cover modes of a symmetric waveguide.

In addition to estimating ∼*R(u*, γ), the classical regularization method^{6} is applied, as

where *u = z* − z', and *z* and *z*' are coordinates in the plane of the 2D irregularities. Here, β_{0} = kγ is the propagation constant of the wave mode, where *k* = 2π/ λ and λ is the wavelength of the laser. In addition, Φ* is the complex conjugate of Φ, μ is the regularization parameter, and the elementary stabilizers of order *p* are defined as *M*_{1} = β^{2p} and *M*_{2} = (β_{0} - β)^{2p}, where *p* ≥ 0 is the regularization order.

The accuracy of the extrapolation of the restored SDF to beyond the range of wave numbers of the observable radiating substrate-air and substrate modes is limited. In the presence of noise, the error remains large. In this case, a quasi-optimal regularization^{1,2} is applied to derive an approximately correct solution to the inverse diffusion problem,^{1–3}

Here, *E* is a smoothing function, which allows for constructing a linear filter that weighs the noisy diffusion pattern at the sampling points, and *mu* < *L* is a nonfixed interval, where *L* is the full length of the irregular region of interest.

Figure 1 demonstrates the application of the classical regularization method to obtain a solution to the inverse diffusion problem. The TE_{0} guided scattering mode is propagated in the irregular symmetric waveguide, with as refraction index of the waveguide layers n_{1} = n_{3} = 1.46 (quartz plates), and *n*_{2} = 1.59 for a wavelength of λ = 0.63μm, as for a He-Ne laser.^{1,2} The geometric parameters of the quartz-substrate surface roughness are σ = 5 and *r* = 30nm. The effective refraction index γ = 1.479, 1.525, 1.556, and 1.571 for curves 1, 2, 3, and 4, respectively. In Figure 1 the specific Gaussian ACF, *R*(*u*) (curve 5), and restored ACFs, *R(u*, γ)_{reg} (curves 1–4), are shown (in arbitrary units). Obviously, the specific Gaussian ACF is restored with a large error: in the *L*_{2} metric the error associated with the restored ACF is about 150%.

**Figure 1.** ACFs,

*R(u, γ)*_{reg}, restored using Equation (

3). The adopted parameters are μ = 0.5 and

*p* = 0.6, in the absence of noise.

In Figure 2 the restored Gaussian ACF, *R*(*u*, γ)_{sm}, illustrates the promise of employing quasi-optimal regularization using the sampling theorem. The error inherent in the restoration of the initial Gaussian ACF is less than 27%, and can be further reduced by either selection of another smoothing function, or a proper choice of the parameters determining the inverse scattering problem. Our method allows us to achieve a record high resolution for the correlation radii. The initial correlation radius is determined with an error of less than 7%, or 40 times smaller than for the solution shown in Figure 1.

**Figure 2.** ACFs,

*R(u, γ)*_{sm}, restored using Equation (

4). The adopted parameters are μ = 1,

*p* = 0.8, and

*m* = 12, for

*E* ∝ sin(mx)/mx. The average signal-to-noise ratio 〈

*S*/

*N*〉 = 7. The remaining parameters are as for Figure

1.

We have proven the existence of a solution to the inverse problem of waveguide light scattering based on the existence of regularizing operators for the integral equations of the first kind.^{1–6} We search for an approximately correct solution among the class of functions that are integrable with respect to the square of the modulus of *L*_{2}. Its uniqueness is taken as that of the statistically averaged solution of either an ensemble or a single relatively long realization. Stability is obtained by restricting the set of solutions to a compact set of functions that depend only on a finite number of parameters.^{1,2}

Alexandre Egorov

General Physics Institute

Computational Physics and Mathematical Modeling

Peoples’ Friendship University of Russia

Moscow, Russian Federation

Alexandre A. Egorov is a professor at the Peoples’ Friendship University of Russia. He received his PhD and DSc degrees in physics and mathematics in 1992 and 2006, respectively. Since 1993, he has also been a senior research fellow at the General Physics Institute of the Russian Academy of Sciences. He has published more than 130 scientific and technical papers. His main research interests include laser physics, integrated optics, statistical optics, heterodyne microscopy, physical ecology, and computer modeling.