## Numerical demonstration of the validity of the Rayleigh hypothesis

Optics Express, Vol. 17, Issue 19, pp. 17102-17117 (2009)

http://dx.doi.org/10.1364/OE.17.017102

Acrobat PDF (205 KB)

### Abstract

The Rayleigh hypothesis and the related method of diffraction analysis are revisited. It is shown that the Rayleigh method can be applied to deep grating modeling without numerical problems and that it gives any desired accuracy whatever the groove depth. This proves the validity of the Rayleigh hypothesis and rehabilitates the Rayleigh method.

© 2009 Optical Society of America

## 1. Introduction

2. Lord Rayleigh and J. W. Strutt, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A **79**, 399–416 (1907).
[CrossRef]

4. B. A. Lippmann, “Note on the theory of gratings,” J. Opt. Soc. Am. **43**, 408 (1953).
[CrossRef]

5. J. L. Uretski, “The scattering of plane waves from periodic surfaces,” Ann. Phys. **33**, 400–427 (1965).
[CrossRef]

10. A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta **27**, 1671–1692 (1980).
[CrossRef]

## 2. Diffraction problem formulation

_{1}, a superstrate halfspace of permittivity ε

_{2}, and a corrugated zone in-between. The corrugation is expressed as a periodical surface undulation of period

*d*:

*ζ*(

*x*) is continuous and piecewise differentiable. It can be represented by a Fourier series:

*K*=2π/

*d*is the corrugation wavenumber.

*x*and

*z*wave vector components

*k*

^{x}

_{0}and

*k*

^{z}_{20}(the temporal term exp(-

*jωt*) is omitted for sake of brevity). A 1D grating under non-conical incidence is considered. The optogeometrical parameters and the fields do not depend on the

*y*coordinate. Therefore, the TE and TM problems can be considered independently. The electric field of TE waves as well as the magnetic field of TM waves is directed along the y axis. Therefore, the corresponding

*y*field component characterizes all fields completely. The resulting formulae are different for different polarizations but the analysis is similar. The transverse field of the incident wave is

**F**

_{inc}=

**ŷ**exp(

*jk*

^{x}_{0}

*x*-

*jk*

^{z}_{20}z), where

*F*can either be the electric

*E*, or the magnetic

*H*field. Then, in the regions above the grating,

*z*>max

*ζ*(

*x*), and below the grating,

*z*<min

*ζ*(

*x*), the scattered field is represented by the sum of diffracted plane and evanescent waves:

*a*

_{1m}and

*a*

_{2m}are the constant amplitudes of the waves diffracted into the lower and upper media, respectively,

*k*, projections

_{p}*k*are chosen according to the rule:

^{z}_{pm}*a*

_{1m}and

*a*

_{2m}.

## 3. Rayleigh method

*z*=

*ζ*(

*x*). This gives the first infinite set of equations on the unknown amplitudes

*a*

_{1m}and

*a*

_{2m}:

*jk*

^{x}_{0}

*x*). This allows for the development of both sides of Eq. (9) into series ∑∞

*q*=-∞

*F*exp(

_{yq}*jk*). To determine coefficients

^{x}_{q}x*F*, we first multiply both sides of Eq. (9) by exp(-

_{yq}*jk*), then integrate the product over one grating period. Finally, we get the first infinite set of linear equations for unknown amplitudes

^{x}_{q}x*a*

_{1m}and

*a*

_{2m}:

*I*

^{p±}

_{qm}represent the integrals:

*ζ*(

*x*)=σsin

*Kx*, for example, the integrals of Eq. (11) are represented by Bessel functions:

*χ*means permeability µ for the TE polarization and permittivity ε for the TM polarization,

*∂/∂n*means the derivative in the direction normal to the grating surface. Taking the derivatives yields the second equation:

*a*

_{1m}and

*a*

_{2m}:

*k*=0 does not lead to any difficulty since the corresponding coefficient writes:

^{z}_{pm}## 4. Far field calculation

*d*=1µm, the wavelength λ=0.6328 µm. The incidence is from the air side (

*n*

_{2}=1) under angle θ=arcsin(1/3). Two types of substrates are considered: a dielectric (

*n*

_{1}=2.5) and a lossless metal (

*n*

_{1}=0+i·5). For economy of place, we present the modeling results of the TE wave diffraction on a dielectric grating and of the TM wave diffraction on a metal grating only. The first case allows for possible comparison with any rigorous slicing techniques, the second case is the most critical since there are few possible reference techniques.

*h*=2σ=0.15µm. The results are summarized in Tables 1 and 2. One can conclude that in all cases the results converge fast with the number of orders

*M*. The energy balance corresponds well to the accuracy reached in the calculation of diffraction efficiencies. Increasing the number

*M*enables the grating calculation with any desired accuracy. Such behavior is not surprising since it is recognized that the RM leads to reliable results if the groove depth is below

*K*σ<0.448 [6]

*M*. The Rayleigh method still gives results within 1e-15 accuracy even when the grating grooves become deeper and exceed twice the well-known limit

*K*σ<0.448 [6].

*h*=2σ=0.6 µm,

*K*σ=1.885 (Table 8). Nevertheless, increasing the processor precision improves the convergence and finally leads to exact results whatever the groove depth and the prescribed accuracy.

*h*=2σ=2 µm,

*K*σ=6.283 (to be compared with the pretended validity limit of

*K*σ=0.448), with an accuracy better than 1e-15. They confirm the relevance of the RM for deep grating calculation and, most importantly, establish the validity of the RH well beyond its pretended limitations.

## 5. Numerical experiment on the near field calculation

29. P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. **71**, 1224–1229 (1981).
[CrossRef]

*a*

_{1m}and

*a*

_{2m}. Then, in the regions above the grating,

*z*≥max

*ζ*(

*x*), and below,

*z*≤min

*ζ*(

*x*), the scattered field is calculated by sum (3). Such approach reveals however to be numerically unstable when applied to calculate the fields in the grating region min

*ζ*(

*x*)<

*z*<max

*ζ*(

*x*). The reason of such instability lies in the fast exponential growth of high-order evanescent diffraction orders. One can conclude that although the diffracted near field can be represented by a superposition of outgoing plane and evanescent waves, such representation is not the best from a physical point of view. Alternative electromagnetic solutions, as for instance that using grating modes in the modal method [25

25. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B **26**2907–2916 (1982).
[CrossRef]

*q*in formula (19) is formally over all the harmonics taken into account. In practice, higher order harmonics are not well conditioned and taking them into account leads to numerical instabilities. Therefore, we made the summation in formula (19) over N central harmonics. The results of numerical modeling of the near field in a deep sinusoidal grating

*h*=2σ=1µm,

*K*σ=3.142 are shown in Tables 11 and 12. The transverse y-components of the electric field are calculated directly at the grating interface. The best results are obtained when the number of central harmonics N taken for summation in formula (19) is slightly less than half the total number of orders M.

30. E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: Application to highly conducting gratings,” J. Opt. Soc. Am. A **21**, 199–206 (2004).
[CrossRef]

## 6. Discussion and concluding remarks

*M*=45 orders is performed in 0.10 s, whereas the same calculation with (2×64 bits) precision takes 1.25 s; (5×64 bits) precision requires 2.87 s, and (25×64 bits) precision 33.15 s, respectively. The needed computer memory increases proportionally to the desired precision.

32. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1392 (1982).
[CrossRef]

33. A. G. Ramm, “Modified Rayleigh conjecture and applications,” J. Phys. A **35**, L357–L361 (2002).
[CrossRef]

35. N. Garcia, G. Armand, and J. Lapujoulade, “Diffraction intensities in helium scattering; Topographic curves,” Surf. Sci. **68**, 399–407 (1977).
[CrossRef]

## Acknowledgements

## References and links

1. | Lord Rayleigh and J. W. Strutt, “On the incidence of aerial and electromagnetic waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and the passage of electric waves through a circular aperture in a conducting screen,” Phil. Mag. |

2. | Lord Rayleigh and J. W. Strutt, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A |

3. | L. N. Deryugin, “Equations for coefficients of wave reflections from a periodically uneven surface,” Dokl. Akad. Nauk SSSR |

4. | B. A. Lippmann, “Note on the theory of gratings,” J. Opt. Soc. Am. |

5. | J. L. Uretski, “The scattering of plane waves from periodic surfaces,” Ann. Phys. |

6. | R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur, C. R. Acad. Sci. Paris |

7. | R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface, II,” Proc. Cambridge Philos. Soc. |

8. | J. Pavageau, “Sur la méthode des spectres d’ondes planes dans les problèmes de diffraction,” C. R. Acad. Sci. Paris |

9. | R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory and Tech. |

10. | A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta |

11. | K. Yasuura and H. Ikuno, “On the modified Rayleigh hypothesis and the mode-matching method,” in Summaries Int. Symp. Antennas and Propagation, Sendai, Japan, 173–174, (1971). |

12. | P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. |

13. | J. B. Davies, “A least-squares boundary residual method for the numerical solution of scattering problems,“ IEEE Trans. Microwave Theory and Tech. |

14. | V. I. Tatarskii, “Relation between the Rayleigh equation in diffraction theory and the equation based on Green’s formula, ” J. Opt. Soc. Am. |

15. | J. Wauer and T. Rother, “Considerations to Rayleigh’s hypothesis,” Opt. Commun. |

16. | L. Kazandjian, “Rayleigh methods applied to electromagnetic scattering from gratings in general homogeneous media,” Phys. Rev. E |

17. | V. A. Sychugov and A. V. Tishchenko, “Light emission from a corrugated dielectric waveguide,” Sov. J. Quantum Electron. |

18. | V. A. Sychugov and A. V. Tishchenko, “Propagation and conversion of light in corrugated waveguide structures,” Sov. J. Quantum Electron. |

19. | G. A. Golubenko, A. A. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. |

20. | J. Turunen, “Diffraction theory of dielectric surface relief gratings,” in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997). |

21. | A. M. Prokhorov, V. A. Sychugov, A. V. Tishchenko, and A. A. Khakimov, “Kinetics of the rippling of a germanium surface bombarded by an intense laser beam,” Sov. Tech. Phys. Lett. |

22. | V. A. Sychugov, A. V. Tishchenko, N. M. Lyndin, and O. Parriaux, “Waveguide coupling gratings for high-sensitivity biochemical sensors,” Sens. Actuators B |

23. | I. F. Salakhutdinov, V. A. Sychugov, A. V. Tishchenko, B. A. Usievich, and F. A. Pudonin, “Anomalous light reflection at the surface of a corrugated thin metal film,” IEEE J. Quantum Electron. |

24. | I. A. Avrutsky, V. A. Sychugov, and A. V. Tishchenko, “The study of excitation, radiation, and reflection processes in corrugated waveguides,” in Waveguide Corrugated Structures in Integrated and Fiber Optics, IOFAN Proc.34, 3–98 (Nauka, Moscow, 1991, in Russian). |

25. | P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B |

26. | J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Paris |

27. | M. Greenberg, |

28. | G. H. Golub and C. F. Van Loan, |

29. | P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. |

30. | E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: Application to highly conducting gratings,” J. Opt. Soc. Am. A |

31. | D. E. Knuth, |

32. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. |

33. | A. G. Ramm, “Modified Rayleigh conjecture and applications,” J. Phys. A |

34. | J. A. Fawcett, “Modeling acousto-elastic waveguide/object scattering with the Rayleigh hypothesis,” J. Acoust. Soc. Am. |

35. | N. Garcia, G. Armand, and J. Lapujoulade, “Diffraction intensities in helium scattering; Topographic curves,” Surf. Sci. |

36. | V. Pan, “Complexity of computations with matrices and polynomials,” SIAM Rev. |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(290.0290) Scattering : Scattering

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: April 13, 2009

Revised Manuscript: June 8, 2009

Manuscript Accepted: June 11, 2009

Published: September 11, 2009

**Citation**

Alexandre V. Tishchenko, "Numerical demonstration of the validity of the Rayleigh hypothesis," Opt. Express **17**, 17102-17117 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-17102

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### References

- Lord Rayleigh (J. W. Strutt), "On the incidence of aerial and electromagnetic waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and the passage of electric waves through a circular aperture in a conducting screen," Phil. Mag. 44, 28-52 (1897).
- Lord Rayleigh (J. W. Strutt), "On the dynamical theory of gratings," Proc. R. Soc. London Ser. A 79, 399-416 (1907). [CrossRef]
- L. N. Deryugin, "Equations for coefficients of wave reflections from a periodically uneven surface," Dokl. Akad. Nauk SSSR 87,913-916 (1952).
- B. A. Lippmann, "Note on the theory of gratings," J. Opt. Soc. Am. 43,408 (1953). [CrossRef]
- J. L. Uretski, "The scattering of plane waves from periodic surfaces," Ann. Phys. 33, 400-427 (1965). [CrossRef]
- R. Petit, M. Cadilhac, "Sur la diffraction d’une onde plane par un réseau infinement conducteur, C. R. Acad. Sci. Paris 262B, 468-471 (1966).
- R. F. Millar, "On the Rayleigh assumption in scattering by a periodic surface, II," Proc. Cambridge Philos. Soc. 69,217-225 (1971). [CrossRef]
- J. Pavageau, "Sur la méthode des spectres d’ondes planes dans les problèmes de diffraction," C. R. Acad. Sci. Paris 266B, 135-138 (1968).
- R. H. T. Bates, "Analytic constraints on electromagnetic field computations," IEEE Trans. Microwave Theory and Tech. MTT-23, 605-623 (1975). [CrossRef]
- A. Wirgin, "On Rayleigh’s theory of sinusoidal diffraction gratings," Opt. Acta 27, 1671-1692 (1980). [CrossRef]
- K. Yasuura and H. Ikuno, "On the modified Rayleigh hypothesis and the mode-matching method," in Summaries Int. Symp. Antennas and Propagation, Sendai, Japan, 173-174, (1971).
- P. C. Waterman, "Scattering by periodic surfaces," J. Acoust. Soc. Am. 57, 791 (1975). [CrossRef]
- J. B. Davies, "A least-squares boundary residual method for the numerical solution of scattering problems," IEEE Trans. Microwave Theory and Tech. MTT-21, 99-103 (1973). [CrossRef]
- V. I. Tatarskii, "Relation between the Rayleigh equation in diffraction theory and the equation based on Green’s formula, " J. Opt. Soc. Am. 12, 1254-1260 (1995). [CrossRef]
- J. Wauer, T. Rother, "Considerations to Rayleigh’s hypothesis," Opt. Commun. 282, 339 (2009). [CrossRef]
- L. Kazandjian, "Rayleigh methods applied to electromagnetic scattering from gratings in general homogeneous media," Phys. Rev. E 54, 6802-6815 (1996). [CrossRef]
- V. A. Sychugov and A. V. Tishchenko, "Light emission from a corrugated dielectric waveguide," Sov. J. Quantum Electron. 10, 186-189 (1980). [CrossRef]
- V. A. Sychugov and A. V. Tishchenko, "Propagation and conversion of light in corrugated waveguide structures," Sov. J. Quantum Electron. 12, 923-926 (1982). [CrossRef]
- G. A. Golubenko, A. A. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985). [CrossRef]
- J. Turunen, "Diffraction theory of dielectric surface relief gratings," in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997).
- A. M. Prokhorov, V. A. Sychugov, A. V. Tishchenko, and A. A. Khakimov, "Kinetics of the rippling of a germanium surface bombarded by an intense laser beam," Sov. Tech. Phys. Lett. 8, 605-606 (1982).
- V. A. Sychugov, A. V. Tishchenko, N. M. Lyndin, and O. Parriaux, "Waveguide coupling gratings for high-sensitivity biochemical sensors," Sens. Actuators B 38-39 360-364 (1997). [CrossRef]
- I. F. Salakhutdinov, V. A. Sychugov, A. V. Tishchenko, B. A. Usievich, and F. A. Pudonin, "Anomalous light reflection at the surface of a corrugated thin metal film," IEEE J. Quantum Electron. 34, 1054-1060 (1998). [CrossRef]
- I. A. Avrutsky, V. A. Sychugov, and A. V. Tishchenko, "The study of excitation, radiation, and reflection processes in corrugated waveguides," in Waveguide Corrugated Structures in Integrated and Fiber Optics, IOFAN Proc. 34, 3-98 (Nauka, Moscow, 1991, in Russian).
- P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 262907-2916 (1982). [CrossRef]
- J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. Paris 11, 235-241 (1980).
- M. Greenberg, Advanced Engineering Mathematics, 2nd ed., (Prentice Hall, 1998).
- G. H. Golub, C. F. Van Loan, Matrix computations, 3rd ed., (Johns Hopkins, Baltimore, 1996).
- P. M. van den Berg, "Reflection by a grating: Rayleigh methods," J. Opt. Soc. Am. 71, 1224-1229 (1981). [CrossRef]
- E. Popov, B. Chernov, M. Nevière, and N. Bonod, "Differential theory: Application to highly conducting gratings," J. Opt. Soc. Am. A 21, 199-206 (2004). [CrossRef]
- D. E. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 3rd ed. (Reading, MA: Addison-Wesley, 1998).
- M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982). [CrossRef]
- A. G. Ramm, "Modified Rayleigh conjecture and applications," J. Phys. A 35, L357-L361 (2002). [CrossRef]
- J. A. Fawcett, "Modeling acousto-elastic waveguide/object scattering with the Rayleigh hypothesis," J. Acoust. Soc. Am. 106, 164-168 (1999). [CrossRef]
- N. Garcia, G. Armand, and J. Lapujoulade, "Diffraction intensities in helium scattering; Topographic curves," Surf. Sci. 68, 399-407 (1977). [CrossRef]
- V. Pan, "Complexity of computations with matrices and polynomials," SIAM Rev. 34, 225-262 (1992). [CrossRef]

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