Abstract

Simple null polarimetry is introduced to measure the refractive index n of transparent and semitransparent materials. For a linearly polarized incident light, the reflected polarization can be used to determine n with the Fresnel equations. This method is fast, convenient, and versatile enough to provide accurate results on small laboratory samples. Possible errors in experiments are analyzed, and ways to eliminate these errors are given. With our instrument, the uncertainty in the deduced n is ±0.0004. The accuracy is still within 0.1% for materials with high n and with the extinction coefficient of the order of 0.001. The method is also sufficiently accurate for characterizing the homogeneity of transparent materials.

© 1993 Optical Society of America

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References

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  1. S. F. Nee, H. E. Bennett, “Ellipsometric characterization of optical constants for transparent materials,” in Properties and Characteristics of Optical Glass, A. J. Marker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.970, 62–69 (1988).
    [Crossref]
  2. S. F. Nee, H. E. Bennett, “Optical characterization of transparent materials using ellipsometry,” in Laser-Induced Damage in Optical Materials: 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 10–23 (1989).
  3. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,”J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [Crossref]
  4. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23, 1980–1985 (1984).
    [Crossref] [PubMed]
  5. S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1441, 31–37 (1990).
    [Crossref]
  6. S.-M. F. Nee, “Error reductions for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
    [Crossref]
  7. S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1746, 119–127 (1992).
    [Crossref]
  8. J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
    [Crossref]
  9. D. E. Aspnes, B. Theeten, F. Hottier, “Investigation of effective medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
    [Crossref]
  10. D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
    [Crossref]
  11. S.-M. F. Nee, “Ellipsometric analysis for surface roughness and texture,” Appl. Opt. 27, 2819–2831 (1988).
    [Crossref] [PubMed]
  12. D. E. Aspnes, “The accurate determination of optical properties by ellipsometry,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 89–112.
  13. D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), pp. 799–846.
  14. J. C. Maxwell-Garnett, “Colours in metal glasses in metallic films and in solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
    [Crossref]
  15. D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
  16. H. R. Philipp, “Silicon dioxide (SiO2) Glass,” in Handbook Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 749–763.

1991 (1)

1988 (1)

1984 (1)

1982 (1)

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
[Crossref]

1979 (1)

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

1974 (1)

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[Crossref]

1965 (1)

1935 (1)

D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

1906 (1)

J. C. Maxwell-Garnett, “Colours in metal glasses in metallic films and in solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
[Crossref]

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

D. E. Aspnes, “The accurate determination of optical properties by ellipsometry,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 89–112.

D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), pp. 799–846.

Bennett, H. E.

S. F. Nee, H. E. Bennett, “Ellipsometric characterization of optical constants for transparent materials,” in Properties and Characteristics of Optical Glass, A. J. Marker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.970, 62–69 (1988).
[Crossref]

S. F. Nee, H. E. Bennett, “Optical characterization of transparent materials using ellipsometry,” in Laser-Induced Damage in Optical Materials: 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 10–23 (1989).

S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1441, 31–37 (1990).
[Crossref]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Chang, E. C.

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[Crossref]

Dodge, M. J.

Hottier, F.

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

Malitson, I. H.

Marton, J. P.

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[Crossref]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in metal glasses in metallic films and in solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[Crossref]

Nee, S. F.

S. F. Nee, H. E. Bennett, “Optical characterization of transparent materials using ellipsometry,” in Laser-Induced Damage in Optical Materials: 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 10–23 (1989).

S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1441, 31–37 (1990).
[Crossref]

S. F. Nee, H. E. Bennett, “Ellipsometric characterization of optical constants for transparent materials,” in Properties and Characteristics of Optical Glass, A. J. Marker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.970, 62–69 (1988).
[Crossref]

Nee, S.-M. F.

S.-M. F. Nee, “Error reductions for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
[Crossref]

S.-M. F. Nee, “Ellipsometric analysis for surface roughness and texture,” Appl. Opt. 27, 2819–2831 (1988).
[Crossref] [PubMed]

S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1746, 119–127 (1992).
[Crossref]

Philipp, H. R.

H. R. Philipp, “Silicon dioxide (SiO2) Glass,” in Handbook Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 749–763.

Theeten, B.

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

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, “Berechung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Appl. Opt. (2)

J. Appl. Phys. (1)

J. P. Marton, E. C. Chang, “Surface roughness interpretation of ellipsometer measurements using the Maxwell-Garnett theory,” J. Appl. Phys. 45, 5008–5014 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Philos. Trans. R. Soc. London (1)

J. C. Maxwell-Garnett, “Colours in metal glasses in metallic films and in solutions,” Philos. Trans. R. Soc. London 205, 237 (1906).
[Crossref]

Phys. Rev. B (1)

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

Thin Solid Films (1)

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89, 249–262 (1982).
[Crossref]

Other (7)

S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1746, 119–127 (1992).
[Crossref]

S. F. Nee, H. E. Bennett, “A simple high precision extinction method for measuring refractive index of transparent materials,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1441, 31–37 (1990).
[Crossref]

S. F. Nee, H. E. Bennett, “Ellipsometric characterization of optical constants for transparent materials,” in Properties and Characteristics of Optical Glass, A. J. Marker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.970, 62–69 (1988).
[Crossref]

S. F. Nee, H. E. Bennett, “Optical characterization of transparent materials using ellipsometry,” in Laser-Induced Damage in Optical Materials: 1989, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newman, M. J. Soileau, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1438, 10–23 (1989).

D. E. Aspnes, “The accurate determination of optical properties by ellipsometry,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 89–112.

D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), pp. 799–846.

H. R. Philipp, “Silicon dioxide (SiO2) Glass,” in Handbook Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 749–763.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the null polarimeter. The incident light is polarized at 45°, and the reflected polarization is examined by an analyzer.

Fig. 2
Fig. 2

Polarization vectors for the incident field E0, the reflected field Er, and the analyzer polarization A for θ < θB and θ > θB.

Fig. 3
Fig. 3

Two zone measurements for θ < θB. The polarizer is set at 45° and at 135° in zones 1 and 2, respectively. The two-zone average can eliminate the errors that are due to misalignment of P and A.

Fig. 4
Fig. 4

Separation between reflected beams from front surface and back surface for a sample thickness of 1 cm. Separation is smaller for larger n.

Fig. 5
Fig. 5

Deflection angle for the backsurface-reflected beam from the front-surface-reflected beam. The deflection is larger for larger n. Wedge angle α = 1°.

Fig. 6
Fig. 6

Variation of dn/dΨ versus θ for different n. dn/dΨ changes sign as θ crosses θB. The slope is flat near θB. Pair measurements near both sides of θB can eliminate unknown systematic errors.

Fig. 7
Fig. 7

Value of Δ′/k versus for k ≪ 1. Its value diverges near the Brewster angle.

Fig. 8
Fig. 8

Value of λk*/ñd versus n where k* is the equivalent extinction coefficient that is due to a rough surface.

Fig. 9
Fig. 9

NP-measured Ψ versus θ for fused silica samples at different λ.

Fig. 10
Fig. 10

Deviation of n at different θ from the average n for fused silica samples. The errors are small for a laser source at λ = 0.633 μm and a diffuse source at λ = 3.797 μm.

Tables (4)

Tables Icon

Table 1 NP Results for SiO2 at Various Wavelengths with Different Sources

Tables Icon

Table 2 Uniformity Test for Fused Silica Samplesa

Tables Icon

Table 3 Results Produced by NP and NE for a Silicon Wafer at λ = 1 μma

Tables Icon

Table 4 Least-Square-Fit Results for NP and NE Data with Use of Two- and Three-Phase Models for a Silicon Wafer at λ = 1 μm

Equations (31)

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E 0 = E 0 ( p ^ + s ^ ) / 2 ,
E r = E 0 ( p ^ r p + s ^ r s ) / 2 .
ρ r p / r s = tan Ψ exp ( i Δ ) .
E r = - E r ( p ^ sin Ψ cos Δ o + s ^ cos Ψ ) / 2 .
A = p ^ cos Ψ - s ^ sin Ψ cos Δ o .
I = I 0 [ 1 - cos 2 P cos 2 Ψ + cos 2 A ( cos 2 P - cos 2 Ψ ) + sin 2 A sin 2 P sin 2 Ψ cos Δ ] .
tan 2 A = sin 2 P sin 2 Ψ cos Δ / ( cos 2 P - cos 2 Ψ ) .
Ψ = ( A 1 + A 2 ) / 2.
Ψ = { - A 1 / cos Δ o for P = 45° A 2 / cos Δ o for P = 135° .
Ψ ¯ = ( A 2 - A 1 ) / 2 cos Δ o .
δ A = δ P sin 2 Ψ cos Δ o / ( 1 - cos 2 P cos 2 Ψ ) .
ζ = ( t sin 2 θ ) / ( n 2 - sin 2 θ ) 1 / 2 ,
θ = sin - 1 [ sin θ cos 2 α + sin 2 α ( n 2 - sin 2 θ ) 1 / 2 ]             for θ < θ c ,
θ c = sin - 1 [ cos 2 α - sin 2 α ( n 2 - 1 ) 1 / 2 ] .
δ θ 2 α sec θ ( n 2 - sin 2 θ ) 1 / 2 .
n 2 = sin 2 θ [ 1 + tan 2 θ ( 1 - tan Ψ cos Δ o 1 + tan Ψ cos Δ o ) 2 ] .
δ n = - 2 δ Ψ cos Δ o ( n 2 - sin 2 θ ) / n cos 2 Ψ .
δ n - 2 δ Ψ cos Δ o n 3 / ( n 2 + 1 )             for θ θ B .
( n + i k ) 2 = sin 2 θ { 1 + tan 2 θ [ 1 - tan Ψ exp ( i Δ ) 1 + tan Ψ exp ( i Δ ) ] 2 } .
Δ = - n k / n ˜ 2 tan 2 Ψ cos Δ o = 2 n k sin 2 θ cos θ / n ˜ ( n 2 cos 2 θ - sin 2 θ ) ,
n ˜ = ( n 2 - sin 2 θ ) 1 / 2 .
δ Ψ Δ 2 sin 4 Ψ / 8.
δ Ψ n 2 k 2 cos 2 2 Ψ / 4 n ˜ 4 tan 2 Ψ .
n δ n - k δ k = D [ 2 δ Ψ cos 2 Ψ cos Δ - δ Δ sin 2 Ψ × ( sin Δ + 2 n k cos 2 Ψ cos Δ / n ˜ 2 ) ] ,
k δ n + n δ k = D [ 2 δ Ψ ( sin Δ + 2 n k cos 2 Ψ cos Δ / n ˜ 2 ) + δ Δ sin 2 Ψ cos 2 Ψ cos Δ ) ] ,
D = - n ˜ 2 / ( cos 2 2 Ψ cos 2 Δ + sin 2 Δ ) .
δ n = 2 n D δ Ψ cos 2 Ψ cos Δ + D δ Δ sin 2 Ψ × [ - n sin Δ + k ( 1 - 2 n 2 / n ˜ 2 ) cos 2 Ψ cos Δ ] ,
δ k = n D δ Δ sin 2 Ψ cos 2 Ψ cos Δ + 2 D δ Ψ × [ n sin Δ - k ( 1 - 2 n 2 / n ˜ 2 ) cos 2 Ψ cos Δ ] ,
D = D / ( n 2 + k 2 ) .
δ n = n k 2 ( 1 + sin 2 θ / n 2 ) / n ˜ 2 - n k 2 cos Δ o × ( 1 + cos 2 2 Ψ / 2 ) / n ˜ 2 sin 2 Ψ .
δ Δ = 4 π d cos θ n 2 ( n 2 - n s 2 ) ( n s 2 - 1 ) λ n s 2 ( n 2 - 1 ) ( n 2 cot 2 θ - 1 ) .

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