## Consolidated series for efficient calculation of the reflection and transmission in rough multilayers |

Optics Express, Vol. 22, Issue 4, pp. 4499-4515 (2014)

http://dx.doi.org/10.1364/OE.22.004499

Acrobat PDF (901 KB)

### Abstract

Fresnel reflectance and transmittance coefficients of a thin film system consisting of an arbitrary number of layers are expressed explicitly in the form of a power series of Fresnel coefficients for individual boundaries and phase terms for the individual films. The series is based on the evaluation of all possible paths light can pass through the system. However, the series is written as consolidated, i. e. all paths corresponding to the same powers are represented using a single term in the series, with multiplicity which is a simple product of binomial coefficients. This result is used to express the normal reflectance of a thin film system with arbitrarily correlated randomly rough boundaries and it is shown that such approach can be computationally efficient in practice.

© 2014 Optical Society of America

## 1. Introduction

1. I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in *Progress in Optics*, E. Wolf, ed. (Elsevier, 2000), vol. 41, pp. 181–282. [CrossRef]

6. A. W. Crook, “The reflection and transmission of light by any system of parallel isotropic films,” J. Opt. Soc. Am. **38**, 954–963 (1948). [CrossRef] [PubMed]

7. D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B **20**, 3292–3302 (1979). [CrossRef]

9. L. Névot, B. Pardo, and J. Corno, “Characterization of X-UV multilayers by grazing incidence X-ray reflectometry,” Rev. Phys. Appl. **23**, 1675–1686 (1988). [CrossRef]

1. I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in *Progress in Optics*, E. Wolf, ed. (Elsevier, 2000), vol. 41, pp. 181–282. [CrossRef]

10. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. **53**, 1389–1394 (1963). [CrossRef]

17. J. M. Zavislan, “Angular scattering from optical interference coatings: scalar scattering predictions and measurements,” Appl. Opt. **30**, 2224–2244 (1991). [CrossRef] [PubMed]

18. C. Amra, J. H. Apfel, and E. Pelltier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. **31**, 3134–3151 (1992). [CrossRef] [PubMed]

26. T. Herffurth, S. Schröder, M. Trost, and A. D. A. Tünnermann, “Comprehensive nanostructure and defect analysis using a simple 3D light-scatter sensor,” Appl. Opt. **52**, 3279–3287 (2013). [CrossRef] [PubMed]

32. D. Franta and I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. **45**, 903–934 (1998). [CrossRef]

1. I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in *Progress in Optics*, E. Wolf, ed. (Elsevier, 2000), vol. 41, pp. 181–282. [CrossRef]

30. R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. **26**, 704–712 (1987). [CrossRef] [PubMed]

15. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. **18**, 104–115 (1979). [CrossRef]

*a/b*〉 ≈ 〈

*a*〉/〈

*b*〉 in the recursive formulae for reflectance of a multi-layer thin film [35

35. I. Ohlídal and F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. **46**, 2043–2062 (1999). [CrossRef]

36. M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, and H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. **52**, 583–602 (2005). [CrossRef]

11. I. Ohlídal, K. Navrátil, and F. Lukeš, “Reflection of light on a system of non-absorbing isotropic film–non-absorbing isotropic substrate with rough boundaries,” Opt. Commun. **3**, 40–44 (1971). [CrossRef]

12. I. Ohlídal, K. Navrátil, and F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. **61**, 1630–1639 (1971). [CrossRef]

37. I. Ohlídal and F. Lukeš, “Ellipsometric parameters of rough surfaces and of a system substrate-thin film with rough boundaries,” Opt. Acta **19**, 817–843 (1972). [CrossRef]

38. I. Ohlídal, F. Lukeš, and K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. **45**, 91–116 (1974). [CrossRef]

12. I. Ohlídal, K. Navrátil, and F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. **61**, 1630–1639 (1971). [CrossRef]

40. L. D. Killough, “The on-line encyclopedia of integer sequences,” http://oeis.org/A011782. Sequence A011782.

44. N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A024175. Sequence A024175.

45. N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A000108. Sequence A000108.

46. M. Kildemo, O. Hunderi, and B. Drévillon, “Approximation of reflection coefficients for rapid real-time calculation of inhomogeneous films,” J. Opt. Soc. Am. A **14**, 931–939 (1997). [CrossRef]

## 2. Construction of the series

### 2.1. Parametrisation

*j*= 0, 1,...,

*L*, with 0 denoting the ambient,

*L*denoting the last medium which is presumably the substrate. Index

*j*will be referred to as medium depth for brevity. The reflection and transmission coefficients and phase terms will be then denoted

*r̂*,

_{j}*r̂′*

_{j}*t̂*,

_{j}*t̂′*and exp(i

_{j}*X̂*) as displayed in Fig. 2, where The symbols

_{j}*N̂*,

_{j}*h*and

_{j}*ϑ̂*denote the complex refractive index, thickness, and complex angle of incidence corresponding to medium

_{j}*j*; and

*λ*is the wavelength. No specific relations among the coefficients will be assumed because it would not bring any significant simplification while it would prevent application of the results in other situations, namely when the layer boundaries are not simple ideal boundaries.

*m*of interactions at the boundaries. It is more useful to introduce path length

*p*using the formula The path length is non-negative and

*p*= 0 corresponds to a single reflection at the 0–1 boundary. The total power of all Fresnel coefficients in the term corresponding to this path is 2

*p*+ 1 and the total power of all phase terms is 2

*p*. Beside path length we will also define path depth

*d*which is equal to the depth of the deepest medium the path passes through (so

*d*≤

*L*). Any two paths that differ in either

*p*or

*d*are obviously different.

*d*is then formed by non-negative integers

*m*

_{0},

*m*

_{1}, . . . ,

*m*called medium counts and

_{d}*v*

_{0},

*v*

_{1}, . . .

*v*called visit counts.

_{d}*v*expresses how many times the light:

_{j}- visits the subsystem from medium
*j*downward, - passes through the (
*j*− 1)–*j*boundary from the side of medium*j*− 1, - passes through the (
*j*− 1)–*j*boundary from the side of medium*j*.

*j*− 1 and segments contained in media from

*j*downward. Visit count

*v*is the number of the latter segments. For all paths and all layer systems it holds

_{j}*v*

_{0}= 1.

### 2.2. Parameter ranges

*j*≤

*d*it holds while for

*j*>

*d*it holds

*v*=

_{j}*m*= 0. Once the light enters medium

_{j}*j*it must pass through it at least once which leads to the inequality

*v*≤

_{j}*m*. Finally, since visits of medium

_{j}*j*+ 1 must be separated by at least one pass through medium

*j*it must hold

*v*

_{j}_{+1}≤

*m*. The preceding two relations can be written summarily for

_{j}*v*or

_{j}*m*in a more compact form It is evident that conditions (4)–(5) are necessary, i. e. all paths must satisfy them. It may not be immediately evident that they are also sufficient, i. e. that each set of positive

_{j}*m*

_{1},

*m*

_{2}, . . . ,

*m*and accompanying set of

_{d}*v*

_{1},

*v*

_{2}, . . . ,

*v*bound by these conditions corresponds to at least one possible path.

_{d}- enters medium
*j*from medium*j*− 1 the same number of times as it leaves medium*j*to medium*j*− 1, - enters medium
*j*+ 1 from medium*j*the same number of times as it leaves medium*j*+ 1 to medium*j*, - leaves the (
*j*− 1)–*j*boundary downwards the same number of times as it is incident on the*j*–(*j*+ 1) boundary from the side of medium*j*, and - leaves the
*j*–(*j*+ 1) boundary upwards the same number of times as it is incident on the (*j*− 1)–*j*boundary from the side of medium*j*then it is always possible to ‘match the arrows’ in medium*j*close to the (*j*− 1)–*j*boundary with those close to the*j*–(*j*+ 1) boundary. Hence a path without any loose ends can be always formed provided that*m*−_{j}*v*and_{j}*m*−_{j}*v*_{j}_{+1}are non-negative – which corresponds exactly to conditions (5) (if the matching is done in order left to right no cycles are formed).

*P*denotes the maximum path length which can be either specified explicitly or determined implicitly by a termination condition, e. g. by terminating the series once the absolute value of the entire length-

*p*term falls below a certain small value. The vectors

**m**and

**v**were introduced as shorthand for

*m*

_{1},

*m*

_{2},...,

*m*

_{d}_{−1}and

*v*

_{2},

*v*

_{3},...,

*v*. The

_{d}**m**-summation is then while the

**v**summation is The symbol

*M*denotes the sum of medium counts for media from 1 to

_{j}*j*and

*m*=

_{d}*p*−

*M*

_{d}_{−1}which follows from Eq. (3). No summation is thus carried out over

*m*because it is uniquely determined by the path length and other

_{d}*m*. Similarly, no summation is carried out over

_{j}*v*

_{1}because

*v*

_{1}≤ min(

*m*

_{0},

*m*

_{1}) = 1, hence

*v*

_{1}= 1.

*m*appeared in Eq. (7). If this requirement is removed the upper limits of

_{j}*m*summations can be chosen differently, e. g. each can be chosen independently based on physical considerations. Therefore, a criterion based on the behaviour of individual reflection and refraction coefficients can be used, taking into account more reflections in specific selected layers.

_{j}*p*= 0,

*d*= 0). It could be formally included in the scheme but only at the cost of impairing its clarity. Since it would still represent a special case, only made more obscure, the result would not translate well into a computer implementation. Hence it was excluded and must be added separately.

### 2.3. Coefficient and phase term powers

*j*contributes the factor to the sum term. For reference, we will write out explicit forms of the summation terms for depths from 0 to 3 that occur in the series for systems consisting of up to three layers.

*v*

_{1}= 1 was used to obtain the latter expression from the former which corresponds formally to Eq. (10). The following formulae will already be written with this simplification.

### 2.4. Multiplicities

*j*contributes to the term multiplicity with an independent factor that will be denoted

*F*(

*v*,

_{j}*m*,

_{j}*m*

_{j}_{−1}), i. e. and the entire multiplicity

*C*for a depth-

_{d}*d*summation term can be obtained simply by multiplying these factors for all

*d*layers: For reference, the multiplicities will be again listed for depths from 0 to 3:

### 2.5. Second consolidation and summation algorithm

*Q̂*(

**m**) is a homogeneous polynomial of degree 2

*p*+1 in Fresnel coefficients. The ranges of

*m*, which are omitted here for brevity, are the same as in Eq. (6) where they were written out explicitly. The polynomial

_{j}*Q̂*(

**m**) is constructed using the tail part of Eq. (6), expression (10) with phase terms excluded, and formula (17):

*v*. Conversely, terms

_{j}*j*conceptually but depend on

*v*

_{j}_{+1}, not on

*v*. The product from

_{j}*j*= 2 to

*d*should be understood as empty (equal to 1) for

*d*= 1.

*p*instead of the

*p*! growth which corresponds to the fully expanded sum. If the further mathematical operations can be only carried out with complete terms, the expansion is straightforward: the summations over

^{d}/d*v*are performed sequentially, in a nested manner indicated in Eq. (6), instead of factoring the sums. In paragraph 2.6 where the number of terms is discussed, the latter, more pessimistic, variant will be assumed. It should be noted that both variants coincide for a two-layer system because only one

_{j}*v*summation (over

_{j}*v*

_{2}) is done.

### 2.6. Number of terms

^{p−1},

*F*

_{2p+1}and (3

*p*− 1)/2+1, respectively, where

*F*denotes the

_{n}*n*-th Fibonacci number [40

40. L. D. Killough, “The on-line encyclopedia of integer sequences,” http://oeis.org/A011782. Sequence A011782.

42. C. Mallows, N. J. A. Sloane, S. Plouffe, and R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051. Sequence A007051.

*p*. For paths of depth 2 and

*p*≥ 2 it immediately follows from parameter ranges given by Eqs. (4) and (5) that the number of terms is Here ⌊...⌋ denotes rounding down to the nearest integer (the ‘floor’ function). For paths of depth 3 the exact formula is convoluted, however, the number of terms grows as

*p*

^{4}/72. Therefore, the number of terms in the series including all paths with length up to

*p*, inclusively, grows as

*p*

^{3}/12 for a two-layer system and as

*p*

^{5}/360 for a three-layer system. Generally, for an

*L*-layer system it grows as

*p*

^{2}

^{L}^{−1}. This means that while the combinatorial explosion may be much slower asymptotically in the consolidated series than in the expanded series, it is still fast if the system consists of many layers.

*p*) if

*L*is large and if any contribution of the deeper layers is to be included because the

*j*-th medium contributes only to paths of length

*j*or more.

*p*in systems consisting of two to five layers (paths of any depth are counted, not just those that of maximum possible depth). It can be seen that twofold reduction occurs if paths of length at least 5 are considered so the presented scheme does not lead to a significant simplification if only a handful of terms is taken into account. For paths longer than 5, however, the gain from consolidation is considerable and it increases rapidly with increasing

*p*.

### 2.7. Transmission

*d*=

*L*so summation over

*d*is avoided in formulae such as (6) and

*d*is replaced with

*L*. The number of terms in the consolidated series grows somewhat less rapidly, though still as

*p*

^{2}

^{L}^{−1}.

*m*and

_{j}*v*changes slightly: they still denote the numbers of downward passes and refractions, however, the numbers of upwards passes and refractions are now

_{j}*m*− 1 and

_{j}*v*− 1, respectively. Figure 3 remains largely unchanged, only with upward arrow counts replaced with

_{j}*v*− 1 and

_{j}*v*

_{j}_{+1}− 1. Definition (3) of path length

*p*can be retained, however, the total number of interactions is 2

*p*+ 1 −

*L*now and the total power of phase terms is similarly 2

*p*−

*L*. Condition (4) holds for all 0 ≤

*j*≤

*L*+ 1 and condition (5) remains unchanged.

*r̂*

_{1}disappears.

*v*− 1 visits can be freely chosen among the remaining

_{j}*m*

_{j}_{−1}− 1 incidences and the factor

*F*(

*v*,

_{j}*m*,

_{j}*m*

_{j}_{−1}), expressed by Eq. (16) for reflection, becomes With these modification it is possible to utilise the same summation approach as for reflection.

## 3. Application to scalar diffraction theory for normal incidence

35. I. Ohlídal and F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. **46**, 2043–2062 (1999). [CrossRef]

47. I. Ohlídal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. **71**, 323–326 (1989). [CrossRef]

48. I. Ohlídal, “Approximate formulas for the reflectances, transmittances, and scattering losses of nonabsorbing multilayers systems with randomly rough boundaries,” J. Opt. Soc. Am. A **10**, 158–170 (1993). [CrossRef]

*R*is expressed where

**u**=

**u**(

*x,y*) is the vector with

*L*+ 1 components representing the heights of irregularities of boundaries 1, 2,...,

*L*+1 at the Cartesian coordinates (

*x,y*) in the sample plane;

*u*

_{1}is the first component of

**u**. The symbol

*r̂*(

**u**) denotes the Fresnel reflection coefficient of the corresponding system of smooth films with thicknesses

*h*replaced with

_{j}*h*+

_{j}*u*−

_{j}*u*

_{j}_{+1}(putting

*u*

_{0}≡

*u*

_{L}_{+1}≡ 0 to simplify the notation). The thickness

*h*must be now understood as the mean thickness of the

_{j}*j*-th film. The factor

*v*is equal to where

*N*

_{0}is refractive index of the ambient. The averaging over

**u**, denoted with angle brackets 〈...〉, is carried out within the illuminated area. Under the assumption that the roughness corresponds to a stationary random process (in the wide sense), the sample-plane averaging can be replaced by statistical averaging with the multi-dimensional probability density function of

**u**, which will be denoted

*ρ*(

**u**): For Gaussian roughness, the probability density of

**u**is where subscripts

^{T}and

^{−1}denote transposition and matrix inversion, respectively. The matrix

**S**is a symmetrical positive semi-definite matrix with components where is the rms value of height irregularities of the

*j*-th boundary and the cross-correlation coefficients between

*i*-th and

*j*-th boundary are equal to [35

35. I. Ohlídal and F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. **46**, 2043–2062 (1999). [CrossRef]

**f**is an arbitrary vector) and the following series is obtained: For the normal incidence cos

*ϑ̂*= 1 and thus formula (1) reduces to The factor

_{j}*Ĥ*(

*m*) is given by where Note this expression makes use of the relations

*m*

_{0}≡ 1 and

*m*= 0 for

_{j}*j*>

*d*to include the edge terms

*D̂*

_{1}and

*D̂*

_{d}_{+1}in a single expression. For a single layer-system the sum (32) coincides with the well-known sum [12

12. I. Ohlídal, K. Navrátil, and F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. **61**, 1630–1639 (1971). [CrossRef]

**S**[38

38. I. Ohlídal, F. Lukeš, and K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. **45**, 91–116 (1974). [CrossRef]

49. I. Ohlídal, K. Navrátil, and M. Ohlídal, “Scattering of light from multilayer systems with rough boundaries,” in *Progress in Optics*, E. Wolf, ed. (Elsevier, 1995), Vol. 34, pp. 249–331. [CrossRef]

## 4. Numerical example

**46**, 2043–2062 (1999). [CrossRef]

_{3}N

_{4}(150 nm), SiO

_{2}(130 nm) and Si

_{3}N

_{4}(100 nm). The substrate is formed by crystalline silicon. Optical constants of all materials were taken from standard tables [51

51. H. R. Philipp, “Silicon dioxide (SiO_{2}) (glass),” in *Handbook of Optical Constants of Solids*, E. Palik, ed. (Academic, 1985), Vol. I, pp. 749–763. [CrossRef]

53. C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. **83**, 3323–3336 (1998). [CrossRef]

*p*for which the entire

*p*-length term falls below a prescribed value.

*x*

^{2})-like factors in the terms decrease more rapidly with increasing roughness. For integration, however, increased roughness means larger ranges of

**u**and this implies more abscissa points are required to achieve the same precision of the numeric quadrature. This effect is illustrated in Fig. 7, where the computation efficiencies of both methods are compared in dependence of the roughness magnitude. The computations were performed for rough thin film systems that were identical as in the preceding example, except for the overall roughness magnitude. The overall roughness magnitude was varied by scaling all

*σ*values by the same factor. The value of the lowermost roughness

*σ*

_{L}_{+1}, which is plotted on the abscissa, was varied in the interval [1.25, 15]nm. The mean computation times plotted in the figure correspond to times needed for obtaining the entire reflectance spectra with precision at least 10

^{−6}. Again, the precision was determined as the difference to the values calculated using the series to the full double precision. From Fig. 7 it can be seen that although the efficiency of the consolidated series is decreasing with decreasing roughness, the series is more efficient for all roughness values above approximately 2 nm for this thin film system. Of course, this threshold value depends on the structure and optical constants of the thin film system and would differ for other systems. For zero roughness, i. e. a smooth system, the series is not very efficient. However, for such system, one would use the matrix formalism or recursive formulae in practice.

## 5. Conclusion

## Acknowledgments

## References and links

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37. | I. Ohlídal and F. Lukeš, “Ellipsometric parameters of rough surfaces and of a system substrate-thin film with rough boundaries,” Opt. Acta |

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40. | L. D. Killough, “The on-line encyclopedia of integer sequences,” http://oeis.org/A011782. Sequence A011782. |

41. | N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A001519. Sequence A001519. |

42. | C. Mallows, N. J. A. Sloane, S. Plouffe, and R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051. Sequence A007051. |

43. | H. Bottomley, “The on-line encyclopedia of integer sequences,” http://oeis.org/A080937. Sequence A080937. |

44. | N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A024175. Sequence A024175. |

45. | N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A000108. Sequence A000108. |

46. | M. Kildemo, O. Hunderi, and B. Drévillon, “Approximation of reflection coefficients for rapid real-time calculation of inhomogeneous films,” J. Opt. Soc. Am. A |

47. | I. Ohlídal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. |

48. | I. Ohlídal, “Approximate formulas for the reflectances, transmittances, and scattering losses of nonabsorbing multilayers systems with randomly rough boundaries,” J. Opt. Soc. Am. A |

49. | I. Ohlídal, K. Navrátil, and M. Ohlídal, “Scattering of light from multilayer systems with rough boundaries,” in |

50. | A. H. Stroud and D. Secrest, |

51. | H. R. Philipp, “Silicon dioxide (SiO |

52. | H. R. Philipp, “Silicon nitride (Si |

53. | C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(240.0310) Optics at surfaces : Thin films

(240.5770) Optics at surfaces : Roughness

(310.6805) Thin films : Theory and design

**ToC Category:**

Thin Films

**History**

Original Manuscript: September 24, 2013

Revised Manuscript: January 10, 2014

Manuscript Accepted: January 13, 2014

Published: February 20, 2014

**Citation**

David Nečas and Ivan Ohlídal, "Consolidated series for efficient calculation of the reflection and transmission in rough multilayers," Opt. Express **22**, 4499-4515 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4499

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