Abstract

The phase difference between s and p polarization of the light reflected from a material is used for measuring the material’s complex refractive index. First, two phase differences that correspond to two different incidence angles are measured by heterodyne interferometry. Then these two phase differences are substituted into Fresnel’s equations, and a set of simultaneous equations is obtained. Finally, the equations are solved by use of a personal computer by a numerical analysis technique, and the complex refractive index of the material can be estimated.

© 1999 Optical Society of America

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References

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  1. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 69–87.
  2. I. Simon, “Spectroscopy in infrared by reflection and its use for highly absorbing substances,” J. Opt. Soc. Am. 41, 336–345 (1951).
    [CrossRef]
  3. D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. Landon Sect. B 65, 425–428 (1952).
    [CrossRef]
  4. S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based the Brewster angle,” Proc. Phys. Soc. 5, 949–957 (1961).
    [CrossRef]
  5. W. R. Hunter, “Error in using the reflectance vs angle of incidence method for measuring optical constants,” J. Opt. Soc. Am. 55, 1197–1204 (1965).
    [CrossRef]
  6. W. R. Hunter, “Optical constants of metals in the extreme ultraviolet. I. A modified critical-angle technique for measuring the index of refraction of metals in the extreme ultraviolet,” J. Appl. Phys. 34, 15–19 (1964).
  7. R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983).
    [CrossRef]
  8. J. Lekner, “Determination of complex refractive index and thickness of a homogeneous layer by combined reflection and transmission ellipsometry,” J. Opt. Soc. Am. A 11, 2156–2158 (1994).
    [CrossRef]
  9. G. D. Feke, D. P. Snow, R. D. Grober, P. J. De Groot, L. Deck, “Interferometric back focal plane microellipsometry,” Appl. Opt. 37, 1796–1802 (1998).
    [CrossRef]
  10. M.-H. Chiu, J.-Y. Lee, D.-C. Su, “Refractive-index measurement based on the effects of total internal reflection and the uses of heterodyne interferometry,” Appl. Opt. 36, 2936–2939 (1997).
    [CrossRef] [PubMed]
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.
  12. D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
    [CrossRef]
  13. M.-H. Chiu, C.-D. Chen, D.-C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1929 (1996).
    [CrossRef]
  14. Ref. 1, pp. 285 and 323.
  15. N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
    [CrossRef]
  16. J. M. De Freitas, M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
    [CrossRef]
  17. W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
    [CrossRef]
  18. A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
    [CrossRef]
  19. P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
    [CrossRef]

1998 (1)

1997 (1)

1996 (3)

M.-H. Chiu, C.-D. Chen, D.-C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1929 (1996).
[CrossRef]

P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
[CrossRef]

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

1994 (1)

1993 (2)

N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

J. M. De Freitas, M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

1992 (1)

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

1990 (1)

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

1983 (1)

1965 (1)

1964 (1)

W. R. Hunter, “Optical constants of metals in the extreme ultraviolet. I. A modified critical-angle technique for measuring the index of refraction of metals in the extreme ultraviolet,” J. Appl. Phys. 34, 15–19 (1964).

1961 (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based the Brewster angle,” Proc. Phys. Soc. 5, 949–957 (1961).
[CrossRef]

1952 (1)

D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. Landon Sect. B 65, 425–428 (1952).
[CrossRef]

1951 (1)

Apostol, D.

P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
[CrossRef]

Avery, D. G.

D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. Landon Sect. B 65, 425–428 (1952).
[CrossRef]

Azzam, R. M. A.

Bobroff, N.

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Chen, C. D.

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Chen, C.-D.

Chiu, M. H.

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Chiu, M.-H.

Damian, V.

P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
[CrossRef]

De Freitas, J. M.

J. M. De Freitas, M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

De Groot, P. J.

Deck, L.

Feke, G. D.

Grober, R. D.

Hetrick, P. S.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Hou, W.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based the Brewster angle,” Proc. Phys. Soc. 5, 949–957 (1961).
[CrossRef]

Hunter, W. R.

W. R. Hunter, “Error in using the reflectance vs angle of incidence method for measuring optical constants,” J. Opt. Soc. Am. 55, 1197–1204 (1965).
[CrossRef]

W. R. Hunter, “Optical constants of metals in the extreme ultraviolet. I. A modified critical-angle technique for measuring the index of refraction of metals in the extreme ultraviolet,” J. Appl. Phys. 34, 15–19 (1964).

Kramar, J. A.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Lee, J.-Y.

Lekner, J.

Logofatu, P. C.

P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
[CrossRef]

Oldham, N. M.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 69–87.

Player, M. A.

J. M. De Freitas, M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

Rosenbluth, A. E.

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Simon, I.

Snow, D. P.

Su, D. C.

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Su, D.-C.

Teague, E. C.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Tumbar, R.

P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
[CrossRef]

Wilkening, G.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Appl. Opt. (2)

J. Appl. Phys. (1)

W. R. Hunter, “Optical constants of metals in the extreme ultraviolet. I. A modified critical-angle technique for measuring the index of refraction of metals in the extreme ultraviolet,” J. Appl. Phys. 34, 15–19 (1964).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (2)

P. C. Logofatu, D. Apostol, V. Damian, R. Tumbar, “Optimum angles for determining the optical constants from reflectivity measurements,” Meas. Sci. Technol. 7, 52–57 (1996).
[CrossRef]

J. M. De Freitas, M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

Precis. Eng. (4)

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

N. M. Oldham, J. A. Kramar, P. S. Hetrick, E. C. Teague, “Electronic limitations in phase meter for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Proc. Phys. Soc. (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based the Brewster angle,” Proc. Phys. Soc. 5, 949–957 (1961).
[CrossRef]

Proc. Phys. Soc. Landon Sect. B (1)

D. G. Avery, “An improved method for measurements of optical constants by reflection,” Proc. Phys. Soc. Landon Sect. B 65, 425–428 (1952).
[CrossRef]

Other (3)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 69–87.

Ref. 1, pp. 285 and 323.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

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Figures (7)

Fig. 1
Fig. 1

Reflection at the surface of an absorbing material.

Fig. 2
Fig. 2

Schematic diagram of measurement of the phase differences owing to reflection at (a) an absorbing material and (b) a beam splitter. Other abbreviations are defined in text.

Fig. 3
Fig. 3

Δϕ r versus ϕ for several values of θ r .

Fig. 4
Fig. 4

Δϕ m versus ϕ′ for a polarizer with the extinction ratio shown.

Fig. 5
Fig. 5

Measurement resolution versus θ1 for Δn and Δk of (a) nickel and (b) copper at θ2 = 80°.

Fig. 6
Fig. 6

Constant n and k as functions of ϕ at θ1 = 60° and θ2 = 80°, where the values of n and k are 0.2 to 4 in steps of 0.2.

Fig. 7
Fig. 7

Constant n and k as functions of Δn and Δk at θ1 = 60° and θ2 = 80°, where the values of n and k are 0.5 to 4 in steps of 0.5.

Tables (1)

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Table 1 Experimental Conditions and Measurement Resultsa

Equations (23)

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rs=cos θ-u+ivcos θ+u+iv=|rs|expiδs,
rp=N2 cos θ-u+ivN2 cos θ+u+iv=|rp|expiδp,
u2=½n2-k2-sin2 θ+n2-k2-sin2 θ2+4n2k21/2,
v2=½-n2-k2-sin2 θ+n2-k2-sin2 θ2+4n2k21/2,
δs=tan-12v cos θu2+v2-cos2 θ,
δp=tan-12v cos θn2-k2-2u2u2+v2-n2+k22 cos2 θ,
ϕ=δs-δp=tan-1ad-bcac+bd,
a=2v cos θ,  b=u2+v2-cos2 θ,  c=2v cos θn2-k2-2u2,  d=u2+v2-n2+k22 cos2 θ.
ϕ1=ϕ1n, k, θ1,
ϕ2=ϕ2n, k, θ2
Ir=½1+cos2πft-ϕBS,
It=½|rs|22+|rp|22+|rsrp|cos2πf-ϕ,
ϕ=ϕ-ϕBS
Δnϕ2k|Δϕ1|+ϕ1k|Δϕ2|ϕ1nϕ2k-ϕ2nϕ1k,
Δkϕ1n|Δϕ2|+ϕ2n|Δϕ1|ϕ1nϕ2k-ϕ2nϕ1k,
Δϕp=ffc×360°,
Δϕr=tan ϕsec 2θr-11+sec 2θr tan2 ϕ
ExEy=A expi2πft/2+β exp-i2πft/2B exp-i2πft/2+α expi2πft/2=|A|expi2πft/2+iϕA+|β|exp-i2πft/2+iϕβ |B|exp-i2πft/2+iϕB+|α|expi2πft/2+iϕα,
Ir=|Ar|2+|Br|2+|αr|2+|βr|2+2|Arαr|cosϕAr-ϕαr+2|Brβr|cosϕBr-ϕβr+2|ArBr|cos2πft+ϕAr-ϕBr+2|Arβr|cos2πft+ϕAr-ϕβr+2|αrBr|cos2πft+ϕαr-ϕBr+2|αrβr|cos2πft+ϕαr-ϕβr.
Ir=|Ar|+|αr|2+|Br|+|βr|2+2|ArBr|+|Arβr|+|αrBr|+|αrβr|cos2πft+ϕBS.
It=|At|2+|Bt|2+|αt|2+|βt|2+2|Atαt|+|Btβt|cos ϕ+2|AtBt|+|αtβt|cos ϕ+|Atβt|+|Btαt|2+|AtBt|-|αtβt|2 sin2 ϕ1/2 cos2πft-ϕ,
ϕ=tan-1|AtBt|-|αtβt|sin ϕ|Atβt|+|Btαt|+|AtBt|+|αtβt|cos ϕ,
Δϕm=ϕ-ϕ.

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