## Abstract

A very efficient numerical tool to electromagnetically analyze random structures is proposed. The principle is treating random structure as superposition of diffraction gratings. Then, influence of each component grating can be computed with any electromagnetic grating theory which is well established for its accuracy and computation speed. This article explains how to treat obtained data in detail. Applied to single-tiered scatteres, the proposed method gives comparable results with a standard way based on FDTD method in far shorter time.

© 2008 Optical Society of America

## 1. Introduction

Electromagnetic numerical analysis of optical properties of random structures is often a difficult task. If an object is solely composed of spherical particles, so-called Mie theory may be applied [1]. However, analyzing electromagnetic wave scattered by general random structures is not easy to treat. A typical problem of this kind may be found in microwave engineering; detecting an object buried in the earth [2]. Here, random scattering is rather background phenomenon and more important subject is the scattering by the object itself. Scattering by rough surfaces in the resonance domain has also been an important topic [3, 4]. In recent years, thanks to the development of fabrication technology, there appear some applications in which optics has to deal with electromagnetically scattered wave by random structures, which are a bit different from conventional rough surfaces. Those include randomization of local period of triangular diffraction gratings for antireflection in attempt to ease fabrication tolerances [7], structural irregularity of optical disks [5] and scattered silver nano-particles for intensification of surface plasmon resonance in super-RENS disk system [6]. Note that these particular examples contain random structures smaller than the wavelength of illuminating optical waves.

I would like to emphasize an important issue here. Analyzing a problem with random structures is much more than merely analyzing a certain fixed or deterministic random structures. The analyzing process needs to treat variety of random structures which may possibly be included in the problem. As a result, for electromagnetic analysis of general random structures the most frequently employed approach is using finite-difference time-domain (FDTD) method: (1) preparing many fixed random structures, (2) analyzing wave scattering by each of them with the FDTD method, and (3) statistically process the obtained results, e.g. considering average values as representative properties. Although the FDTD method is one of few options capable of treating general random structures, it is notorious for its slowness in computation. Even a single run of the FDTD method for a random structure takes long time and if we prepare the more structures pursuing more reliable results, the eventual computation time will no doubt be enormous even for one-dimensional structures. Alternatively, use of electromagnetic approach in the resonance domain such as the C method is proposed [4].

In this article, I propose a completely new concept of analyzing random structures. It is simple and extremely fast. The basis of the concept is that a random structure is regarded as superposition of periodic structures. Accepting this, we can employ electromagnetic grating theories in the frequency domain, e.g. Fourier modal method (FMM) [8], which are well established for their accuracy, speed and versatility. With this concept, well comparable results to those of conventional approach with the FDTD method were obtained with far shorter computation time. The proposed method is termed *grating superposition method* (GSM) afterwards.

## 2. Scattering problem considered

In order to investigate fundamental usefulness of the proposal, a simplest class of one-dimensional random structure is considered here.

Binary-phase surface relief between two dielectric materials is normally illuminated by a plane wave. A mound and a trench of the same width consists of a unit structure whose combined width *d* is named a local period. The value *d* is assumed randomly distributed in the *x* direction according to the normal distribution with an average *d*
_{a} and a standard deviation *σ*. Figure 1 depicts just five of such local periods.

In this article, *n*
_{1}=1.5, *n*
_{2}=1.0, *h*=*λ* and *d*
_{a}=2.5*λ* are used, where *λ* is the wavelength of light in vacuum. As a regular diffraction grating with the period of 2.5*λ* generates up to second diffraction orders in transmission, the employed structure here would be convenient one for comparing results on scattering properties by different analysis methods. Only TE polarization is treated here, because the concept of the GSM is independent of polarization state and equally applicable to TM polarization.

## 3. Preliminaries with the FDTD method

At first, I would like to summarize results obtained with the FDTD method for the scattering problem defined above. The sizes of analyzed area are *W _{x}*=200

*λ*and

*W*=5

_{z}*λ*with the corresponding cell sizes of Δ

*x*=

*λ*/25 and Δ

*z*=

*λ*/25.6 in the

*x*and

*z*directions, respectively (Fig. 2). Total field/scattered field incident condition is employed such that a plane wave propagating in the +

*z*direction is generated in the plane indicated by a dotted green line in Fig. 2. The standard perfectly matched layer (PML) is placed at the boundaries perpendicular to the

*z*direction, while at boundaries perpendicular to the

*x*direction specially prepared technique is employed so that there appear no unwanted reflection from the boundaries. This technique will be soon reported elsewhere.

Although computation is carried out in the entire analyzed area, electromagnetic field values only in the central portion with the width of *W*
_{0}=40*λ*, i.e. in the dotted red line for transmission and in the dotted blue line for reflection, are used in actual data analysis. This is a further measure to avoid any possible unwanted boundary effects. The method of data analysis is similar to the ones in Refs. [7, 9] and yields a power spectrum for transmitted and reflected waves. Here is its brief summary. Let an electromagnetic field component at position (*i*,*k*) at time *n* be *F ^{n}*(

*i*,

*k*), where (

*i*,

*k*) and

*n*denote integer-numbered space and time indices defined by

*x*=

*i*Δ

*x*,

*z*=

*k*Δ

*z*and

*t*=

*n*Δ

*t*. After discrete-Fourier-transforming

*F*(

^{n}*i*,

*k*) in the time domain,

its first order *C*
_{1}(*i*,*k*) gives a complex amplitude at the position (*i*,*k*). Then, discrete-Fourier-transforming it in the space domain, i.e. in the *x* direction,

gives an angular spectrum in the plane *z*=*k*Δ*z*, where an integer *l* denotes discrete spatial frequency. The value *A _{l}*(

*k*) represents scattering properties we are seeking. Obviously, the plane where

*A*(

_{l}*k*) is evaluated corresponds to the dotted red line for transmission and the dotted blue line for reflection in Fig. 2.

In actual analysis, many different random structures defined in Fig. 1 must be considered. Figure 3 (a) shows averaged power spectra of transmitted waves through random structures with local periods within the range of *d*=2.5*λ*±*λ*, which corresponds to *σ*=0.25*λ*. Here, each color denotes the numbers of random structures considered in the computation. In each random structure, the obtained scattered power is normalized to the incident power. It is interesting to note that computation with even ten random structures provides an almost similar result with a hundred random structures. Also shown in Fig. 3 (b) is standard deviation of the power spectra. Bearing in mind that the vertical scales for the both Figs. 3 (a) and (b) are the same, fluctuation between the random structures is quite large. Nevertheless the average power spectra seem to provide reliable tendency of the scattering properties, because the effect of the number of random structure is negligible.

However, we should remember a nature of random problems: random distribution is so fluctuating that average values themselves do not indicate the direct properties of one certain random structure, but do indicate probable properties. For example, a high average power means that scattering is propably intense at the particular direction.

## 4. Theory of the GSM

The essence of the GSM is that considering a random structure as superposition of *N* diffraction gratings with equally spaced grating periods. The question is how to superpose. The dotted curve in the Fig. 4 shows the normal distribution of the local periods of a random structure, which has the average value of *d*
_{a}. I consider the range of *d*
_{a} ±4*σ*, which covers 99.994% of probability.

The vertical red bars indicate the component gratings. Here shows an example of a random structure represented by *N*=9 gratings with -4≤*j*≤+4. The horizontal position *d _{j}* and vertical height

*γ*denote the grating period and power of incident wave for each component grating, respectively. The value

_{j}*γ*behaves as a weighting factor for contribution of the

_{j}*j*th grating. Selection of the number of gratings

*N*within the range of 8

*σ*, i.e. grating period space Δ

*d*=8

*σ*/(

*N*-1), is quite important for analyzing results. This issue is discussed in detail in the later sections.

Each component grating provides a power spectrum such as in Fig. 5, where *η _{m}*(

*d*) denotes power of

_{j}*m*th diffraction order of the

*j*th component grating. As diffraction angle depends on the grating period, superposition is straightforward for non-zeroth diffraction orders. On the other hand, contribution of each component grating is simply summed for the zeroth order. Then, entire power spectrum for transmission is given by

where *θ*=sin^{-1}(*mλ*/*n*
_{2}
*d _{j}*). For reflection,

*n*

_{2}is simply replaced by

*n*

_{1}. Processing the computed values in this way, a sum of the total scattered power in transmission and reflection is kept identical to the incident power, when the two media are lossless.

Then, a core part of the GSM is computation of *η _{m}*(

*d*), which is achieved much faster by well established electromagnetic grating theories in the frequency domain including the FMM than the FDTD method.

_{j}## 5. Results

As an example, a random structure with *n*
_{1}=1.5, *n _{2}*=1.0,

*h*=

*λ*,

*d*

_{0}=2.5

*λ*and

*σ*=0.25

*λ*is investigated and the results are compared with the FDTD method in Fig. 6. Here, GSM1 and GSM2 denote computation with Δ

*d*=0.17

*λ*,

*N*=13.0 and Δ

*d*=0.076

*λ*,

*N*=27.4, respectively. The truncation order for the FMM is 40. Please note that

*N*is not necessarily an integer. FDTD denotes averaged values for 10 different deterministic random structures analyzed by the FDTD method. The data are identical to the black curve in Fig. 3 (a).

At first glance, results of the GSM and the FDTD method are similar except for the absolute value of the first order peak. Notice that in the GSM there appear two data points at some scattering angles, e.g. around *θ*=60 deg, where two diffraction orders of component gratings overlap, but this causes negligible effect in this particular example. The value Δ*d* for GSM1 is chosen such that sampling space in the angular domain corresponds to the value of the FDTD method around the first order peak. It is seen that not only horizontal spacing but also vertical positions of red circles coincide with those of the FDTD method around the first order diffraction peak. The same happens to GSM2 around the second order peak. The difference in scattering angles between the GSM and the FDTD method is due to the difference in sampling methods: for the GSM sampling is linear to period of component gratings, while for the FDTD method it is linear to spatial frequency. In addition, we should note that zeroth order power is almost the same for the three computations, GSM1, GSM2 and FDTD. Reflected power in this structure is a bit too low to show clear comparison, but overall tendencies of the GSM and the FDTD method look similar.

One may suppose that the GSM and the FDTD method give almost the same results, if sampling space for the GSM is nonlinear such that discrete scattering angles for the GSM is exactly the same as the FDTD method. However, this is neither possible in principle nor justified, because such attempt requires ignoring some diffraction orders of component gratings.

Summarizing this section, as far as sampling space is comparable, the GSM provides scattering properties well comparable to the FDTD method. Therefore, the GSM can substitute for the FDTD method in analyzing scattering problems by random structures.

## 6. Discussion

Let us consider effects of sampling space, which is in fact a fundamental issue in interpreting obtained results. Figure 6 shows two results by the GSM: Δ*d*=0.17 and 0.076. Then, a question must arise: “which is a true one ?” As Δ*d* becomes smaller, resolution of the power spectrum becomes finer and at the same time the power level becomes lower in proportion to the value of Δ*d* for scattering angles *θ*≠0 in orders to satisfy the energy conservation law, while the power for *θ*=0 is almost the same. This somewhat awkward phenomenon is due to the nature of discrete data processing both in the GSM and the FDTD method under a condition that the total scattered power is kept identical to the incident one. As a result of sampling, a data point at scattering angle *θ* represents integrated power of neighboring area around *θ*. Then, a ratio of power at *θ*=0 and the total power for *θ*≠0 is constant whatever the sampling space is and the absolute power level at *θ*≠0 is determined by the sampling space.

This interpretation may be well compared with a hypothetical experiment shown in Fig. 7, which may simulate what we will observe in actual experiments. A beam of finite width, e.g. a Gaussian beam, normally illuminates a one-dimensional random structure we have considered in this article and transmitted angular power spectrum is measured by a detector with a slit narrower than the beam width. If finite beam size covers sufficient number of local periods, a scattered power spectrum would be regarded as the same one obtained with plane wave illumination.

While keeping the slit width constant, how does the beam width affect power spectrum ? Here, we try to answer the question under the condition that angular divergence owing to the finite beam width (marked by blue curves) is much narrower than angular divergence owing to randomness of local periods (marked by red curves). Remember that the directly observed value at a certain angle is power density and an integral of the power density over a certain angular extent is power.

What will be observed is qualitatively depicted in Fig. 8. Black broken curves denote angular divergence of scattered light, which is due to the normal distribution as shown in Fig. 4. In considering Gaussian beam profile, a power spectrum is obtained as a convolution of the angular divergence profile with the Gaussian profile (see Appendix). When the beam size of an illuminating beam becomes doubled (from red to blue curves in Fig. 8), observed power density is doubled for the first order. On the other hand, power density is the same for the zeroth order. However, the ratio of power to the zeroth and first orders is constant. This means that the ratio of power density of the zeroth and first orders depends upon the beam size and no way to determine.

From this hypothetical experiment, we know that (1) a beam size affects power density ratio of zeroth and higher orders, and (2) a beam size does not affect the profile shape of power spectrum of non-zeroth orders.

## 7. Conclusion

A totally novel approach to analyze random structures with the FMM is proposed and is shown that the obtained results are comparable with frequently used technique based on the FDTD method with far shorter time. For an example in this article, the GSM can be an order of 10^{4} faster, though the value heavily depends on parameter setting. One of advantages of the GSM is that any numerical grating theories in the frequency domain can be used depending on the analyzed structures and materials.

As discussed in detail in the previous section, comparing obtained results between zeroth and non-zeroth order scattering requires special care on influence of sampling spaces.

Some readers may question fundamental validity of considering a random structure as superposition of periodic structures. I believe that this treatment is justified, because it is not amplitude, but power or intensity which a conventional and reasonable approach with the FDTD method statistically treats. In fact, the GSM treats incoherent superposition of light waves.

So far, I have confirmed the applicability of the GSM only for rectangular surface relief structures, i.e. single-tiered scatterers, because the principle of the present GSM does not cope with multiple scattering. Thus, I shall work on modifying the GSM to extend its applicability to wide variety of electromagnetic random scattering problems involving, e.g. thick, complicated or multi-layered structures.

## Appendix

Suppose, just as an example, that angular divergence of the first order is expressed as Gaussian function

and the beam profile also has a Gaussian profile

where *a*≫*b*. Normalization of *p* and *q* for simplicity does not affect the physical meaning here. Then, power spectrum is a convolution of *p* with *q*,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\simeq b\sqrt{\pi}\mathrm{exp}\left(\frac{-{x}^{2}}{{a}^{2}}\right).$$

This means that the beam width *b* affects the power density, but not the width of power spectrum.

## References and links

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