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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 26 — Dec. 26, 2005
  • pp: 10854–10864
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Elimination of polarized light scattered by surface roughness or bulk heterogeneity

Claude Amra, Carole Deumie, and Olivier Gilbert  »View Author Affiliations


Optics Express, Vol. 13, Issue 26, pp. 10854-10864 (2005)
http://dx.doi.org/10.1364/OPEX.13.010854


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Abstract

An interferential technique is described to eliminate polarized scattering from optical substrates and coatings. Conditions of annulment are respectively given for surface roughness and for bulk heterogeneity, at each direction of space. At low-level scattering, the method offers a complete discrimination of surface and bulk effects, whatever the micro-structural parameters. Arbitrary scattering levels can be treated in a similar way, but require the knowledge of microstructure.

© 2005 Optical Society of America

1. Introduction

Elimination of light scattering [1

1 . J.M. Elson , “ Diffraction and diffuse scattering from dielectric multilayers ,” J. Opt. Soc. Am. A 69 , 682 – 694 ( 1976 )

,2

2 . J. M. Elson , J. P. Rahn , and J. M. Bennett , “ Light scattering from multilayer optics: comparison of theory and experiment ,” Appl. Opt. 19 , 669 – 679 ( 1980 ). [CrossRef] [PubMed]

] constitutes a challenge for most applications involving imaging systems, optical communication and characterization techniques. Numerous efforts have been performed in the thin film community to minimize scattering levels in optical coatings, thanks to polishing and cleaning techniques, as well as new deposition processes … However a relevant question lies in the possibility to reduce scattering whatever the materials, polishing processes or deposition technologies. As an example, and concerning optical multilayers whose surfaces are known to scatter most of the incident light, special designs have been emphasized to shift the stationary electric field [3

3 . J. H. Apfel , “ Optical coating design with reduced electric field intensity ,” Appl. Opt. 16 , ( 1977 ). [CrossRef] [PubMed]

,4

4 . C. Amra , “ Minimizing scattering in multilayers: technique for searching optimal realization conditions ,” Proceedings of Laser induced damage in optical materials , 756 , 265 – 271 , ( 1987 ).

] within the bulk of materials, with limited success. Another way to reduce scattering has consisted in the phase opposition of waves scattered by internal surfaces of the stack, with large success confirmed by experiment [5

5 . P. Roche , E. Pelletier , and G. Albrand , “ Antiscattering transparent monolayers : theory and experiment ,” J. Opt. Soc. Am. A 1 , 1032 , ( 1984 ). [CrossRef]

,6

6 . C. Amra , G. Albrand , and P. Roche , “ Theory and application of antiscattering single layers: antiscattering antireflection coatings ,” Appl. Opt. 25 , 2695 ( 1986 ). [CrossRef] [PubMed]

,7

7 . H. Giovannini and C. Amra , “ Scattering reduction effect with overcoated rough surfaces : theory and experiment ,” Appl. Opt. , 36 , pp. 5574 – 5579 ( 1997 ). [CrossRef] [PubMed]

]. However this technique can only be applied to single layers and narrowband filters, not to mirrors where the scattered waves are in phase. Moreover, it only works for correlated stacks [8

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

]. Lastly, it cannot work for bare substrates where only one surface is responsible for scattering.

In this work we present a procedure to eliminate angle-resolved polarized light scattering from optical substrates and coatings under monochromatic polarized illumination. The method is based on the angular behavior of the polarization of light scattering, a topic that was addressed by several authors [9–15

9 . J.M. Elson , J.P. Rahn , and J.M. Bennett , “ Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties ,” Appl. Opt. 22 , 3207 – 3219 ( 1983 ) [CrossRef] [PubMed]

] and that has led to the introduction of ellipsometric techniques [16–20

16 . C. Deumié , H. Giovannini , and C. Amra , “ Angle-resolved ellipsometry of light scattering: discrimination of surface and bulk effects in substrates and optical coatings ,” Appl. Opt. , 41 , n° 16, p 3362 – 3369 , ( 2002 ). [CrossRef] [PubMed]

] in the scattering pattern. These polarization techniques have shown that the sensitivity of scattering to specific sources or scatterers could be largely improved to provide a discrimination of surface and bulk effects [16–18

16 . C. Deumié , H. Giovannini , and C. Amra , “ Angle-resolved ellipsometry of light scattering: discrimination of surface and bulk effects in substrates and optical coatings ,” Appl. Opt. , 41 , n° 16, p 3362 – 3369 , ( 2002 ). [CrossRef] [PubMed]

], localized defects and others … Moreover, they were used to extract roughness and cross-correlation parameters in single layers, as shown by numerous results given by Germer [19–20

19 . T.A. Germer , “ Characterizing interfacial roughness by light scattering ellipsometry ,” in “ Characterization and Metrology for ULSI Technology: 2000 International Conference ”, Proc. AIP 550 , 186 – 190 , ( AIP, New York , 2001 ) [CrossRef]

]. In a general way, all these methods are valid for low scattering levels that can be calculated with perturbation theories that predict a polarization ratio not dependent on micro-structural parameters such as roughness and heterogeneity.

The last section of the paper addresses the case of high scattering levels that originate from arbitrary surface roughness and bulk heterogeneity. Because an equivalent polarization can still be defined and measured [17

17 . O. Gilbert , C. Deumié , and C. Amra , “ Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks ,” Opt. Express , 13 , 2403 – 2418 ( 2005 ). [CrossRef] [PubMed]

] in the scattering pattern of samples with diffuse reflectance, the procedure can be directly again generalized. However in this case the annulment conditions depend on the structural parameters (topography and heterogeneity …), which rises additional difficulties that we emphasize.

2. Principles

2.1 Incident illumination

As usual in the ellipsometric techniques, the method requires a polarized illumination of the sample, so that interferences may occur between waves scattered by each incident polarization. We consider (Fig. 1) an incident plane wave given as:

E+=ES++EP+=(AS++AP+)exp(jk+.ρ)
(1)

where ρ = (x,y,z) is the spatial coordinates, and k + the incident wave vector:

k+=k0(sin(i),0,cos(i))
(2)

with λ the wavelength, n0 and n1 the refractive index of superstrate and substrate, i the incidence angle and k = 2πn0/λ.

Fig. 1. Incident polarized plane wave on a sample with normal z

In relation (1) AS+ and AP+ respectively designate the proper polarization states of the electric field, whose complex amplitudes are given as:

AS+=AS+exp(jηS)
(3-a)
AP+=AP+exp(jηP)
(3-b)

with ηS and ηP the polarization phase terms. Relations (1–3) are valid for elliptical (ηS ≠ ηP) or linear (ηS = ηP) polarizations, depending on the relative values of the phase terms.

2.2 Polarized scattered waves

Whatever the origin of scattering (surface or bulk), the polarized fields scattered in the far field (Fig. 2) by each polarization of the incident beam can be written at direction (θ,ϕ) of space as:

ESS±=ASS±exp(jk±.ρ)
(4-a)
ESP±=ASP±exp(jk±.ρ)
(4-b)
EPS±=APS±exp(jk±.ρ)
(4-c)
EPP±=APP±exp(jk±.ρ)
(4-d)

where the superscript (±) is for reflection (-) or transmission (+), and k ± the scattered wave vector:

k±=k(sinθcosϕ,sinθsinϕ,±cosθ)
(4-e)

with k = 2πn0/λ (reflection hemisphere) or k = 2πn1/λ (transmission hemisphere). The subscripts UV designate the V polarization of the wave scattered by the U polarization of the incident beam.

Fig. 2: Scattered wave vectors k± at directions (θ0,ϕ) by reflection, and (θ1,ϕ) by transmission. The two wave vectors k- and k+ have the same tangential component which is the spatial pulsation σ = 2π ν, with ν the spatial frequency.

Let us now be interested in optical samples with low-level scattering, so that first-order electromagnetic theory [2

2 . J. M. Elson , J. P. Rahn , and J. M. Bennett , “ Light scattering from multilayer optics: comparison of theory and experiment ,” Appl. Opt. 19 , 669 – 679 ( 1980 ). [CrossRef] [PubMed]

, 21–24

21 . C. Amra , “ First order vector theory of bulk scattering in optical multilayers ,” J. Opt. Soc. Am. A 10 , 365 – 374 ( 1993 ). [CrossRef]

] can be used. The complex amplitudes of the scattered waves can furthermore be detailed as:

ASS±(θ,ϕ)=CSS±(θ,ϕ)f(θ,ϕ)AS+
(5-a)
ASP±(θ,ϕ)=CSP±(θ,ϕ)f(θ,ϕ)AS+
(5-b)
APP±(θ,ϕ)=CPP±(θ,ϕ)f(θ,ϕ)AP+
(5-c)
APS±(θ,ϕ)=CPS±(θ,ϕ)f(θ,ϕ)AP+
(5-d)

2.3 The interferential zero condition

In order to project and align all polarizations of the scattered wave, we introduce an analyzer on the scattered beam at direction (θ,ϕ). The resulting wave is the interferential sum:

A=AScosψ+APsinψ
(6)

with:

AS=ASS+APS
(7-a)

and:

AP=APP+ASP
(7-b)

with ψ the position angle of the analyzer with respect to the tangential S (or TE) direction. Therefore a condition of zero scattering (A = 0) can be written at direction (θ,ϕ) of space as:

cosψ[CSS(θ,ϕ)AS++CPS(θ,ϕ)AP+]+sinψ[CPP(θ,ϕ)AP++CSP(θ,ϕ)AS+]=0
(8)

3. First-order solutions

Under the assumption of equal energy in each incident polarization (|AS+| = |AP+|), relation (8) can be rewritten as:

tgψ(θ,ϕ)=[CSS(θ,ϕ)exp(jΔη)+CPS(θ,ϕ)]/[CPP(θ,ϕ)+CSP(θ,ϕ)exp(jΔη)]
(9)

with Δη the polarization parameter of the incident beam:

Δη=ηSηP
(10)

Solutions of (9) require 2 conditions on modulus and argument at each scattering direction, whereas the only free parameter is the analyzer angle ψ. Though ψ(θ,ϕ) can be adjusted versus (θ,ϕ), an additional and controllable phase term Δη*(θ,ϕ) must be introduced to satisfy the argument condition. This objective may be reached thanks to a tunable phase retardation device. The fast axis of this device should be parallel to the S direction of the scattered wave, which is parallel to the average sample surface and perpendicular to the spatial pulsation σ (see fig. 2). With such a device introduced on the scattered beam, relation (9) is turned into:

tgψ=exp[η*][CSSexp(jΔη)+CPS]/[CPP+CSPexp(jΔη)]
(11)

with Δη* (θ,ϕ) the retardation phase term that describes an additional delay between the two polarization states of the scattered wave. Therefore at each direction of space, a condition for zero scattering can be reached by simultaneous matching of the analyzer angle ψ(θ,ϕ) and the phase term Δη*(θ,ϕ). The analyzer angle (ψ>0) is given by the modulus condition:

tgψ(θ,ϕ)=[CSS(θ,ϕ)exp(jΔη)+CPS(θ,ϕ)]/[CPP(θ,ϕ)+CSP(θ,ϕ)exp(jΔη)]
(12)

and the phase term is given as:

Δη*(θ,ϕ)=πArg{[CSS(θ,ϕ)exp(jΔη)+CPS(θ,ϕ)]/[CPP(θ,ϕ)+CSP(θ,ϕ)exp(jΔη)]}
(13)

Notice from Eqs. (12) and (13) that ψ(θ,ϕ) and Δη*(θ,ϕ) can be calculated independently at each scattering direction, which is practical from the point of view of experiment. It should not be the case if the retardation plate element were introduced on the incident beam rather than on the scattered beam. Indeed one can check that in this situation the phase term Δη of relation (9) would be replaced by a new Δδ term:

tgψ=[CSSexp(jΔδ)+CPS]/[CPP+CSPexp(jΔδ)]
(14-a)

with:

Δδ(θ,ϕ)=Δη+Δη*(θ,ϕ)
(14-b)

and Δη* the retardation phase term that describes an additional delay between the two polarization states of the incident wave. We obtain:

exp(jΔδ)=(tgψCPP+CPS)/(CSS+tgψCSP)
(15)

with the argument and modulus conditions:

Δδ=π+Arg[(tgψCPP+CPS)/(CSS+tgψCSP)]
(16)
1=(tgψCPP+CPS)/(CSS+tgψCSP)
(17)

We notice from relation (17) that the modulus condition cannot be satisfied in the general case, for which reason the retardation device should not be introduced on the incident beam.

4. Numerical calculation

First of all we notice that the procedure is also valid for optical coatings, provided that all surfaces or bulks are identical within the multilayers. Under this assumption all cross-correlation coefficients are unity [8

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

,11

11 . C. Amra , “ Light scattering from multilayer optics. Part A: investigation tools ”, J. Opt. Soc. Am. A 11 , 197 – 210 ( 1994 ) C. Amra , “ Light scattering from multilayer optics. Part B: application to experiment ,” J. Opt. Soc. Am. A 11 , 211 – 226 ( 1994 ) [CrossRef]

,23

23 . C. Amra , C. Grèzes-Besset , and L. Bruel , “ Comparison of surface and bulk scattering in optical coatings ,” Appl. Opt. 32 , 5492 – 5503 ( 1993 ). [CrossRef] [PubMed]

] and relation (5) is not modified for substrates or multilayers, with equivalent optical factors CUV [8

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

]. In this section we limit ourselves to angle-resolved scattering by reflection in the incidence plane (ϕ=0), which is easier to investigate because of the absence of cross-polarized light. Moreover this configuration may easily fit experiment since most scatterometers work in the incidence plane [25

25 . C. Amra , D. Torricini , and P. Roche , “ Multiwavelength (0.45 – 10.6 μ m) angle-resolved scatterometer or how to extend the optical window ,” Appl. Opt. , 32 , 5462 – 5474 ( 1993 ). [CrossRef] [PubMed]

]. With a retardation plate on the scattered beam at direction θ, relation (11) is therefore reduced to:

tgψ=exp[j(Δη*+Δη)]CSS/CPP
(18)

so that ψ and Δη8 are given by:

tgψ(θ)=CSS(θ)/CPP(θ)
(19-a)
Δη*(θ)=πΔηarg[CSS(θ)/CPP(θ)]
(19-b)

In what follows we assume the incident polarization to be linear (Δη=0). Notice that in many situations the optical factors CUU are slightly different, in particular at low scattering angles and for single designs under normal illumination. For this reason optimal values of ψ and Δη* will often be close to π/4 and π, respectively. However these optimal values should be different for surface and bulk scattering, in order to eliminate one (surface roughness) or the other (bulk heterogeneity) scattering component. Therefore specific designs or illumination angles must be used to allow discrimination of effects.

Numerous results can be found in the literature [1–9

1 . J.M. Elson , “ Diffraction and diffuse scattering from dielectric multilayers ,” J. Opt. Soc. Am. A 69 , 682 – 694 ( 1976 )

, 23

23 . C. Amra , C. Grèzes-Besset , and L. Bruel , “ Comparison of surface and bulk scattering in optical coatings ,” Appl. Opt. 32 , 5492 – 5503 ( 1993 ). [CrossRef] [PubMed]

] concerning the angular behaviour of the modulus | CSS(θ)/CPP(θ) |. At the inverse, few studies [15–20

15 . C. Deumié , H. Giovannini , and C. Amra , “ Ellipsometry of light scattering from multilayer coating ,” Appl. Opt. 35 , No 28, 5600 – 5608 ( 1996 ). [CrossRef] [PubMed]

] concern the Δη*(θ) variations that must be controlled to eliminate scattering. These variations are calculated in Fig. 3–5 for substrates and correlated designs of single layers, mirrors and filters. They are given for surface and bulk scattering, respectively. The wavelength under study is 633nm. Figure 3 is calculated for a fused silica substrate illuminated at 56° incidence. At angles lower than 56°, the phase term is identical for surface and bulk origins, so that both components of scattering should be eliminated at the same time. At larger angles, the phase terms are different and differ from a π value, which allows to eliminate one (surface) or the other (bulk) scattering component. Notice that these differences in the angular behaviors of the bulk and surface phase factors are similar to what is observed for specular reflection after the Brewster angle. In other words, there is a sign change for the field scattered from surface roughness, but this is not the case for the field scattered from bulk heterogeneity.

Figure 4 is given for a single half-wave thin film SiO2 layer of refractive index 1.49, with similar conclusions. Fig. 5 is calculated for a narrow-band filter of design HLHLH(6L)HLHLH, where H and L designate high (TiO2) and low (SiO2) index thin film materials with quarter-wave optical thicknesses. The refractive index of TiO2 is 2.15 at wavelength 633nm. For this coating the separation of surface and bulk effects is effective in practically the whole angular range.

In most situations illumination and observation conditions will be found to enhance discrimination of phase terms characteristic of surface and bulk effects. The reason is connected with the fact that surface scattering originates from electric and magnetic currents located at surfaces, while bulk scattering originates from electric currents within volumes [23

23 . C. Amra , C. Grèzes-Besset , and L. Bruel , “ Comparison of surface and bulk scattering in optical coatings ,” Appl. Opt. 32 , 5492 – 5503 ( 1993 ). [CrossRef] [PubMed]

].

Fig. 3. Angular variations of the phase term Δθ*(θ) (see text) calculated for bulk and surface scattering at wavelength 633nm. The illumination incidence is 56°. The sample is a fused silica substrate (n1 = 1.50).
Fig. 4. Angular variations of the phase term Δη*(θ) (see text) calculated for bulk and surface scattering at wavelength 633nm. The illumination incidence is 56°. The sample is a single thin film half-wave SiO2 layer.
Fig. 5. Angular variations of the phase term Δη*(θ) (see text) calculated for bulk and surface scattering at wavelength 633nm. The illumination incidence is 56°. The sample is a narrowband filter of design HLHLH (6L) HLHLH (see text).

5. Case of arbitrary roughness and heterogeneity

cos(ψ)[ASS+APS]+sin(ψ)[APP+ASP]=0
(20)

A first difference lies in the presence of cross-polarization terms (AUV) that do not vanish in the incidence plane. But the key difference results from the fact that the scattered fields AUV are no longer proportional to the Fourier Transform of defects (surface profile or random permittivity), so that relation (5) cannot be used any more. Indeed all fields are now connected with microstructure (roughness or heterogeneity) via complex relationships such as integral equations [26

26 . P. A. Martin and P. Ola , “ Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle ,” Proc. Roy. Soc. Edinburgh , 123 A, pp. 185 – 208 , ( 1993 ). [CrossRef]

,27

27 . M. Saillard and A. Sentenac , “ Rigorous solution for electromagnetic scattering from rough surfaces ,” Waves in Random Media , 11 , 103 – 137 ( 2001 ). [CrossRef]

] or others. The result is that the zero condition will also be dependent on microstructure at each scattering direction, which is less practical. With a phase retardation device on the scattered beam, we obtain:

tg(ψ)=exp(jΔη*)[ASS+APS]/[APP+ASP]
(21)
tg(ψ)=exp(jΔη*)[νSSexp(jΔη)+νPS]/[νPP+νSPexp(jΔη)]
(22)

where ν UV are scattering coefficients that are micro-structural dependent:

ASS(θ,ϕ)=νSS(θ,ϕ)AS+
(23-a)
ASP(θ,ϕ)=νSP(θ,ϕ)AS+
(23-b)
APP(θ,ϕ)=νPP(θ,ϕ)AP+
(23-c)
APS(θ,ϕ)=νPS(θ,ϕ)AP+
(23-d)

The analyzer angle and additional phase term are again given by the modulus and argument conditions at each scattering direction (θ,ϕ) as:

tg(ψ)=[νSSexp(jΔη)+νPS]/[νPP+νSPexp(jΔη)]
(24)
Δη*=πArg{[νSSexp(jΔη)+νPS]/[νPP+νSPexp(jΔη)]}
(25)

Relations (24–25) offer ψ and Δη* values for elimination of surface or bulk scattering (or both) at each direction. However these values can be calculated only if the specific sample microstructure is known, so that the scattering coefficients ν UV may be predicted with a rigorous theory [26

26 . P. A. Martin and P. Ola , “ Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle ,” Proc. Roy. Soc. Edinburgh , 123 A, pp. 185 – 208 , ( 1993 ). [CrossRef]

,27

27 . M. Saillard and A. Sentenac , “ Rigorous solution for electromagnetic scattering from rough surfaces ,” Waves in Random Media , 11 , 103 – 137 ( 2001 ). [CrossRef]

] involving structural data.

At last we notice in this section that the phase term is expected to show rapid and large angular variations in the scattering pattern. In a recent work [17

17 . O. Gilbert , C. Deumié , and C. Amra , “ Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks ,” Opt. Express , 13 , 2403 – 2418 ( 2005 ). [CrossRef] [PubMed]

] we showed that the polarization state could still be perfectly predicted and measured in the far field of the speckle pattern, so that all phase terms can still be measured with high-angle resolution, and then used for the annulment procedure.

6. Conclusion

We have first shown how surface or bulk scattering can be eliminated in low-loss optical coatings and substrates. The method is based on polarized interferences and requires both a controllable analyzer and a retardation phase device. Under these conditions, angle-resolved scattering can be eliminated at each particular scattering direction. The annulment condition does not depend on the specific sample microstructure, but only on the scattering origins. This result allows a complete discrimination of surface and bulk effects. It is therefore possible to probe bulks after elimination of surface scattering, or to probe surfaces after elimination of bulk scattering.

The procedure can be directly extended to samples with arbitrary surface roughness or bulk heterogeneity. The annulment conditions are given in a similar way, but the major difference lies in the fact that these conditions depend on the specific sample microstructure, which should be preliminary measured. In all cases the annulment of scattering can be reached by scanning all analyzer and retardation plate positions.

References and links

1 .

J.M. Elson , “ Diffraction and diffuse scattering from dielectric multilayers ,” J. Opt. Soc. Am. A 69 , 682 – 694 ( 1976 )

2 .

J. M. Elson , J. P. Rahn , and J. M. Bennett , “ Light scattering from multilayer optics: comparison of theory and experiment ,” Appl. Opt. 19 , 669 – 679 ( 1980 ). [CrossRef] [PubMed]

3 .

J. H. Apfel , “ Optical coating design with reduced electric field intensity ,” Appl. Opt. 16 , ( 1977 ). [CrossRef] [PubMed]

4 .

C. Amra , “ Minimizing scattering in multilayers: technique for searching optimal realization conditions ,” Proceedings of Laser induced damage in optical materials , 756 , 265 – 271 , ( 1987 ).

5 .

P. Roche , E. Pelletier , and G. Albrand , “ Antiscattering transparent monolayers : theory and experiment ,” J. Opt. Soc. Am. A 1 , 1032 , ( 1984 ). [CrossRef]

6 .

C. Amra , G. Albrand , and P. Roche , “ Theory and application of antiscattering single layers: antiscattering antireflection coatings ,” Appl. Opt. 25 , 2695 ( 1986 ). [CrossRef] [PubMed]

7 .

H. Giovannini and C. Amra , “ Scattering reduction effect with overcoated rough surfaces : theory and experiment ,” Appl. Opt. , 36 , pp. 5574 – 5579 ( 1997 ). [CrossRef] [PubMed]

8 .

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

9 .

J.M. Elson , J.P. Rahn , and J.M. Bennett , “ Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation-length, and roughness cross-correlation properties ,” Appl. Opt. 22 , 3207 – 3219 ( 1983 ) [CrossRef] [PubMed]

10 .

S. Maure , G. Albrand , and C. Amra , “ Low-level scattering and localized defects ,” Appl. Opt. 35 , n°28, pp. 5573 – 5582 , October 1, 1996 [CrossRef] [PubMed]

11 .

C. Amra , “ Light scattering from multilayer optics. Part A: investigation tools ”, J. Opt. Soc. Am. A 11 , 197 – 210 ( 1994 ) C. Amra , “ Light scattering from multilayer optics. Part B: application to experiment ,” J. Opt. Soc. Am. A 11 , 211 – 226 ( 1994 ) [CrossRef]

12 .

T.A. Germer , C. Asmail , and B. W. Scheer , “ Polarization of out of plane scattering from microrough silicon ,” Opt. Lett. 22 , 1284 ( 1997 ) [CrossRef]

13 .

T.A. Germer , T. Rinder , and H. Rothe , “ Polarized light scattering measurements of polished and etched steel surfaces ”, in “ Scattering and Surface Roughness III ,” Proc. SPIE 4100 , 148 – 155 ( 2000 ) [CrossRef]

14 .

T.A. Germer and C. Asmail , “ Polarization of light scattered by microrough surfaces and subsurface defects ,” J. Opt. Soc. Am. A 16 , 1326 ( 1999 ) [CrossRef]

15 .

C. Deumié , H. Giovannini , and C. Amra , “ Ellipsometry of light scattering from multilayer coating ,” Appl. Opt. 35 , No 28, 5600 – 5608 ( 1996 ). [CrossRef] [PubMed]

16 .

C. Deumié , H. Giovannini , and C. Amra , “ Angle-resolved ellipsometry of light scattering: discrimination of surface and bulk effects in substrates and optical coatings ,” Appl. Opt. , 41 , n° 16, p 3362 – 3369 , ( 2002 ). [CrossRef] [PubMed]

17 .

O. Gilbert , C. Deumié , and C. Amra , “ Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks ,” Opt. Express , 13 , 2403 – 2418 ( 2005 ). [CrossRef] [PubMed]

18 .

T.A. Germer and C. C. Asmail , “ Goniometric optical scatter instrument for out-of-plane ellipsometry measurements ,” Rev. Sci. Instrum 70 (9), 3688 – 3695 ( 1999 ). [CrossRef]

19 .

T.A. Germer , “ Characterizing interfacial roughness by light scattering ellipsometry ,” in “ Characterization and Metrology for ULSI Technology: 2000 International Conference ”, Proc. AIP 550 , 186 – 190 , ( AIP, New York , 2001 ) [CrossRef]

20 .

T.A. Germer , “ Polarized light scattering by microroughness and small defects in dielectric layers ,” J. Opt. Soc. Am. A 18 , 1279 ( 2001 ) [CrossRef]

21 .

C. Amra , “ First order vector theory of bulk scattering in optical multilayers ,” J. Opt. Soc. Am. A 10 , 365 – 374 ( 1993 ). [CrossRef]

22 .

S. Kassam , A. Duparré , K. helm , P. Bussemer , and J. Neubert , “ Light scattering from the volume of optical thin films: theory and experiment ,” Appl. Opt. 31 , 1304 – 1313 ( 1992 ). [CrossRef] [PubMed]

23 .

C. Amra , C. Grèzes-Besset , and L. Bruel , “ Comparison of surface and bulk scattering in optical coatings ,” Appl. Opt. 32 , 5492 – 5503 ( 1993 ). [CrossRef] [PubMed]

24 .

11. P. Bussemer , K. Hehl , and S. Kassam , “ Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack ,” Waves in Random Media , 1 , 207 – 221 ( 1991 ). [CrossRef]

25 .

C. Amra , D. Torricini , and P. Roche , “ Multiwavelength (0.45 – 10.6 μ m) angle-resolved scatterometer or how to extend the optical window ,” Appl. Opt. , 32 , 5462 – 5474 ( 1993 ). [CrossRef] [PubMed]

26 .

P. A. Martin and P. Ola , “ Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle ,” Proc. Roy. Soc. Edinburgh , 123 A, pp. 185 – 208 , ( 1993 ). [CrossRef]

27 .

M. Saillard and A. Sentenac , “ Rigorous solution for electromagnetic scattering from rough surfaces ,” Waves in Random Media , 11 , 103 – 137 ( 2001 ). [CrossRef]

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.5820) Instrumentation, measurement, and metrology : Scattering measurements
(240.0240) Optics at surfaces : Optics at surfaces
(240.5770) Optics at surfaces : Roughness
(290.5880) Scattering : Scattering, rough surfaces
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Research Papers

Citation
Claude Amra, Carole Deumie, and Olivier Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13, 10854-10864 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-26-10854


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References

  1. J.M. Elson, "Diffraction and diffuse scattering from dielectric multilayers," J. Opt. Soc. Am. A 69, 682-694 (1976).
  2. J. M. Elson, J. P. Rahn, and J. M. Bennett, "Light scattering from multilayer optics: comparison of theory and experiment," Appl. Opt. 19, 669-679 (1980). [CrossRef] [PubMed]
  3. J. H. Apfel, "Optical coating design with reduced electric field intensity," Appl. Opt. 16, (1977). [CrossRef] [PubMed]
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