## Grating superposition method: ultrafast electromagnetic numerical analysis for random structures

Optics Express, Vol. 16, Issue 11, pp. 8292-8299 (2008)

http://dx.doi.org/10.1364/OE.16.008292

Acrobat PDF (202 KB)

### 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

2. T. Dogaru, L. Collins, and L. Carin, “Optimal time-domain detection of a deterministic target buried under a randomly rough interface,” IEEE Trans. Antennas Propag. **49**, 313–326 (2001). [CrossRef]

4. K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal cooridinate system: Application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A **25**, 796–804 (2008). [CrossRef]

7. H. Ichikawa, “Subwavelength triangular random gratings,” J. Mod. Opt. **49**, 1893–1906 (2002). [CrossRef]

5. T. Kojima and T. Kawai, “Numerical analysis of detected signal characteristics from a blue laser optical disk model with random rough surfaces by FDTD method,” IEICE Trans. Electron. **E90-C**, 1599–1605 (2007). [CrossRef]

6. M.-Y. Ng and W.-C. Liu, “Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles,” Opt. Express **13**, 9422–9430 (2005). [CrossRef] [PubMed]

4. K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal cooridinate system: Application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A **25**, 796–804 (2008). [CrossRef]

*grating superposition method*(GSM) afterwards.

## 2. Scattering problem considered

*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.

*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

*W*=200

_{x}*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.

*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

7. H. Ichikawa, “Subwavelength triangular random gratings,” J. Mod. Opt. **49**, 1893–1906 (2002). [CrossRef]

9. H. Ichikawa and H. Kikuta, “Dynamic guided-mode resonant grating filter with quadratic electro-optic effect,” J. Opt. Soc. Am. A **22**, 1311–1318 (2005). [CrossRef]

*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,

*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,

*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.

*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.

## 4. Theory of the GSM

*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.

*N*=9 gratings with -4≤

*j*≤+4. The horizontal position

*d*and vertical height

_{j}*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.

*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

*θ*=sin

^{-1}(

*mλ*/

*n*

_{2}

*d*). For reflection,

_{j}*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.

*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

*n*

_{1}=1.5,

*n*=1.0,

_{2}*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).

*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.

## 6. Discussion

*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.

## 7. Conclusion

^{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.

## Appendix

*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*,

*b*affects the power density, but not the width of power spectrum.

## References and links

1. | C. F. Bohren, “Scattering by particles,” in |

2. | T. Dogaru, L. Collins, and L. Carin, “Optimal time-domain detection of a deterministic target buried under a randomly rough interface,” IEEE Trans. Antennas Propag. |

3. | F. D. Hastings, J. B. Schneider, and S. L. Shira, “A Monte-Carlo FDTD technique for rough surface scattering,” IEEE Trans. Antennas Propag. |

4. | K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal cooridinate system: Application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A |

5. | T. Kojima and T. Kawai, “Numerical analysis of detected signal characteristics from a blue laser optical disk model with random rough surfaces by FDTD method,” IEICE Trans. Electron. |

6. | M.-Y. Ng and W.-C. Liu, “Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles,” Opt. Express |

7. | H. Ichikawa, “Subwavelength triangular random gratings,” J. Mod. Opt. |

8. | J. Turunen, “Diffraction theory of microrelief gratings” in |

9. | H. Ichikawa and H. Kikuta, “Dynamic guided-mode resonant grating filter with quadratic electro-optic effect,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(290.5880) Scattering : Scattering, rough surfaces

(050.5745) Diffraction and gratings : Resonance domain

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 26, 2008

Revised Manuscript: May 2, 2008

Manuscript Accepted: May 20, 2008

Published: May 22, 2008

**Citation**

Hiroyuki Ichikawa, "Grating superposition method: ultrafast electromagnetic numerical analysis for random structures," Opt. Express **16**, 8292-8299 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-8292

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### References

- C. F. Bohren, "Scattering by particles," in Handbook of Optics I, M. Bass, ed., (McGraw-Hill, New York, 1995), pp. 6.1-6.21.
- T. Dogaru, L. Collins, and L. Carin, "Optimal time-domain detection of a deterministic target buried under a randomly rough interface," IEEE Trans. Antennas Propag. 49,313-326 (2001). [CrossRef]
- F. D. Hastings, J. B. Schneider, and S. L. Shira, "A Monte-Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag. 43, 1183-1191 (1995).
- K. Edee, B. Guizal, G. Granet, and A. Moreau, "Beam implementation in a nonorthogonal cooridinate system: Application to the scattering from random rough surfaces," J. Opt. Soc. Am. A 25, 796-804 (2008). [CrossRef]
- T. Kojima and T. Kawai, "Numerical analysis of detected signal characteristics from a blue laser optical disk model with random rough surfaces by FDTD method," IEICE Trans. Electron. E 90-C, 1599-1605 (2007). [CrossRef]
- M.-Y. Ng and W.-C. Liu, "Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles," Opt. Express 13, 9422-9430 (2005). [CrossRef] [PubMed]
- H. Ichikawa, "Subwavelength triangular random gratings," J. Mod. Opt. 49, 1893-1906 (2002). [CrossRef]
- J. Turunen, "Diffraction theory of microrelief gratings" in Micro-Optics, H. P. Herzig, ed., (Taylor & Francis, London, 1997).
- H. Ichikawa and H. Kikuta, "Dynamic guided-mode resonant grating filter with quadratic electro-optic effect," J. Opt. Soc. Am. A 22, 1311-1318 (2005). [CrossRef]

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