Abstract

The angular distributions of light scattered by gold-coated and aluminum-coated gratings with amplitudes of ∼90 nm and periods of 6.67 µm were measured and calculated for light incident from a He–Ne laser at an angle of 6°. Experimental results are compared with predictions of Beckmann’s scalar theory and Rayleigh’s vector theory. The measured scattering pattern has a background of scattered light due mainly to residual surface roughness. Also the power in the higher-order peaks is larger by several orders of magnitude than the computed one, which can be attributed mainly to the low-order contributions of the harmonics in the profile.

© 2000 Optical Society of America

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References

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  1. E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).
  2. E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).
  3. E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
    [CrossRef] [PubMed]
  4. B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
    [CrossRef]
  5. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.
  6. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, Bristol, 1991), Chap. 4, pp. 80–84.
  7. Ref. 6, p. 42.
  8. J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990), p. 219.
  9. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 16–17.
  10. Ref. 5, p. 81 or Ref. 6, p. 89.
  11. Ref. 8, p. 60.
  12. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 76–81.
  13. G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
    [CrossRef]
  14. Certain commercial equipment is identified to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by NIST or does it imply that the equipment is necessarily the best available for the purpose.
  15. C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
    [CrossRef]
  16. T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
    [CrossRef]
  17. J. E. Harvey, C. L. Vernold, A. Krywonos, P. L. Thompson, “Diffracted radiance: a fundamental quantity of nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
    [CrossRef]
  18. E. D. Palik, Handbook of Optical Constants (Academic, San Diego, Calif., 1985), pp. 286–295, 369–406.
  19. Ref. 8, pp. 112–115.
  20. A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt. Soc. Am. 73, 1028–1041 (1983).
    [CrossRef]
  21. D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).
    [CrossRef]

1999

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

J. E. Harvey, C. L. Vernold, A. Krywonos, P. L. Thompson, “Diffracted radiance: a fundamental quantity of nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[CrossRef]

1995

1983

1973

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
[CrossRef]

Asmail, C. C.

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

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.

Content, D. A.

D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).
[CrossRef]

Cromer, C. L.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Germer, T. A.

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

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
[CrossRef]

Harvey, J. E.

Hsia, J. J.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Jiracek, G. R.

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
[CrossRef]

Krywonos, A.

Lettieri, T. R.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

Marx, E.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
[CrossRef]

Maystre, D.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 76–81.

McIntosh, M.

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, Bristol, 1991), Chap. 4, pp. 80–84.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants (Academic, San Diego, Calif., 1985), pp. 286–295, 369–406.

Park, B. C.

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
[CrossRef]

Petit, R.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 16–17.

Proctor, J. E.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990), p. 219.

Thompson, P. L.

Vernold, C. L.

Vorburger, T. V.

E. Marx, T. R. Lettieri, T. V. Vorburger, “Light scattering by sinusoidal surfaces: illumination windows and harmonics in standards,” Appl. Opt. 34, 1269–1277 (1995).
[CrossRef] [PubMed]

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
[CrossRef]

Wirgin, A.

Appl. Opt.

IEEE Trans. Antennas Propag.

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. 21, 393–396 (1973).
[CrossRef]

J. Opt. Soc. Am.

Rev. Sci. Instrum.

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

Other

D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).
[CrossRef]

E. D. Palik, Handbook of Optical Constants (Academic, San Diego, Calif., 1985), pp. 286–295, 369–406.

Ref. 8, pp. 112–115.

Certain commercial equipment is identified to specify adequately the experimental procedure. In no case does such identification imply recommendation or endorsement by NIST or does it imply that the equipment is necessarily the best available for the purpose.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

E. Marx, T. R. Lettieri, T. V. Vorburger, M. McIntosh, “Sinusoidal surfaces as standards for BRDF instruments,” in Optical Scattering: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 15–21 (1991).

E. Marx, T. V. Vorburger, “Windowing effects on light scattered by sinusoidal surfaces,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 2–14 (1993).

B. C. Park, T. V. Vorburger, T. A. Germer, E. Marx, “Scattering from sinusoidal gratings,” in Scattering and Surface Roughness, Z.-H. Gu, A. Maradudin, eds., Proc. SPIE3141, 65–77 (1997).
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chaps. 4 and 5.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institute of Physics, Bristol, 1991), Chap. 4, pp. 80–84.

Ref. 6, p. 42.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990), p. 219.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 16–17.

Ref. 5, p. 81 or Ref. 6, p. 89.

Ref. 8, p. 60.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 76–81.

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Figures (7)

Fig. 1
Fig. 1

Sample configuration and sign convention for angles of the wave vectors of incident and scattered waves.

Fig. 2
Fig. 2

Partial profile of the aluminum-coated grating measured with a 1-µm tip and PSD of the full profile: (a) modified profile; (b) residual roughness computed from the Fourier transform of the profile, truncated below a spatial frequency of 0.33 µm-1; (c), (d) corresponding PSD’s.

Fig. 3
Fig. 3

Measured BRDF for the scattering of s- and p-polarized light by the aluminum sample. The peak power relative to the power in the specular peak is in Table 1.

Fig. 4
Fig. 4

Measured BRDF for the scattering of s- and p-polarized light by the gold sample. The peak power relative to the power in the specular peak is in Table 1.

Fig. 5
Fig. 5

Instrument signature.

Fig. 6
Fig. 6

Measured BRDF (s polarization) and BRDF computed from measured profiles by use of the Kirchhoff approximation for the aluminum sample. The peak power relative to the power in the specular peak is in Table 1.

Fig. 7
Fig. 7

Measured BRDF (s polarization) and BRDF computed from measured profiles by use of the Kirchhoff approximation for the gold sample. The peak power relative to the power in the specular peak is in Table 1.

Tables (3)

Tables Icon

Table 1 Measured and Computed Diffraction Peak Intensities Relative to the Specular Peak Intensity

Tables Icon

Table 2 Computed and Measured Power in the Peaks Relative to the Incident Power

Tables Icon

Table 3 Background Scattering Ratio

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

hx=a sinKx+α,
ρθi, θs=Fθi, θs/L-½L½Ldx×expixvxθi, θs+ihxvzθi, θs,
Fθi, θs=1+cos θi cos θs-sin θi sin θs/cos θicos θi+cos θs.
expiΔ sin ξ=n=- JnΔexpinξ,
ρθi, θs=Fθi, θs/L-½L½Ldx expixvxθi, θs×n=- JnΔθi, θsexpinKx+α.
ρθi, θs=Fθi, θsn=-expinαJnΔθi, θs×sinc½Lvxθi, θs+nK.
sin θn-sin θi=nλ/D,  n=-N1, -N1+1, , N2.
|ρθi, θs|2=Fθi, θs2n=-JnΔθi, θs2×sinc2½Lvxθi, θs+nK,
Pθi, θs  n=-N1N2Fθi, θn2Jn2Δθi, θn×rectθs-θn/θA,
hx=m=1M am sinKmx+αm,  Km=mK.
ρθs=Fθs/L-½L½Ldx expixvxθs+im=1M am sinKmx+αmvzθs=Fθs/L-½L½Ldx expixvxθs×m=1M expiam sinKmx+αmvzθs,
ρθs=Fθs/L-½L½Ldx expixvxθs×m=1Mnm=- JnmΔmθsexpinmKmx+αm,
ρθs=Fθsn1=-n2=-  nM=- expiανJn1Δ1θs×Jn2Δ2θsJnMΔMθssinc½Lvxθs+NνK,
αν=αn1, , nM=m=1M nmαm,  Nν=Nn1, , nM=m=1M mnm.
ρθs=Fθsn=- sinc½Lvxθs+nKρnθs,
ρnθs=n1=-n2=-  nM=- δnNν expiανJn1Δ1θs×Jn2Δ2θsJnMΔMθs.
|ρθs|2=Fθs2n=- sinc2½Lvxθs+nK|ρnθs|2.
Pθs  n=-N1N2Fθn2 rectθs-θn/θA|ρnθn|2.
ρθs=Fθs-dxWxexpixvxθs+ihxvzθs.
Wx=exp-αx2/L2-x2|x|L0elsewhere,
Pn=βn/β|Bn|2,  βn=k2-αn21/2,  αn=k sin θi+Kn,  β=β0=k cos θi,
n=- amnBn=hm,  m=0, ±1, ±2, .
amn=im-nDJm-nβnaif βn20-1m-nDIm-n|βn|aif βn2<0,
hm=--imDJmβa,
amn=im-nπaαn-Jm-n-1βna-Jm-n+1βna+DβnJm-nβnaif βn20-1m-niπaαnIm-n-1|βn|a-Im-n+1|βn|a+DβnIm-n|βn|aif βn2<0,
hm=-1m+1imπaα0Jm-1βa+Jm+1βa+DβJmβa,
hx=a sinKx+α+hrx,
ρθs=Fθs-dxWxexpixvxθs+ia sinKx+αvzθs+ihrxvzθs.
Wx=2π1/2w-1 exp-x2/2w2,
|ρθs|2=Fθs2n=-n=- expin-nαJnΔθs×JnΔθs1/2πw2-dx -dx×exp-x2+x2/2w2+ix-xvxθs+Knx-nxexpihrx-hrxvzθs,
expivzhrx-hrx=exp-gexpgCR,
|ρθs|2Fθs2n=- Jn2Δθs1-g×exp-w2vxθs+nK2+gT/2w×exp-¼T2vxθs+nK2.
Sfx=-dR exp-2πifxRσ2CR=-dR exp-2πifxRσ2 exp-R2/T2=σ2π T exp-π2fx2T2.
|ρθs|2n=- Jn2Δθs1-gexp-w2vxθs+nK2+vzθs2/2π wSfnx,
fnx=vxθs+nK/2π=sin θn-sin θs/λ.
hx, y=a sinKx+α+hrx, y.
ρθs, φs=F3θs, φs--dxdyWx, y×expixvxθs, φs+yvyθs, φs+a sinKx+αvzθs+hrx, yvzθs,
F3θs, φs=1+cos θi cos θs-sin θi sin θs cos φs/cos θicos θi+cos θs.
Wx, y=2πw2-1 exp-x2+y2/2w2.
|ρθs, φs|2F3θs, φs2n=- Jn2Δθs1-g×exp-w2vxθs, φs+nK2-w2vyθs, φs2+gTxTy/4w2exp-¼Tx2vxθs, φs+Kn2-¼Ty2vyθs, φs2,
|ρθs, φs|2F3θs, φs2n=- JnΔθs21-g×exp-w2vxθs, φs+nK2-w2vyθs, φs2+vzθs2/4πw2Sfnx, fy,
Rback=0f1dfyS2fx2+fy21/20dfyS2fx2+fy21/2.
hx½A0+m=1MAm cosmKx+Bm sinmKx=½A0+m=1M am sinmKx+αm,
An=KπNp0LdxhxcosnKx,  n=0, 1, 2, ,
Bn=KπNp0LdxhxsinnKx,  n=1, 2, ,
BRDF=radianceirradiancedPs/dΩsPi cos θsPs/ΩsPi cos θs,

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