Abstract

We present a general method for extracting optical constants n and k plus film thickness t of weakly absorbing materials that exhibit film thicknesses in the micrometer range. This method utilizes the simultaneous fit of multiple transmission measurements of different film thicknesses and employs the constraint that the frequencies of the corresponding measured interference patterns in the nonabsorbing wavelength region have to be matched by means of n, its derivative n′ at a certain value E 0, and t. Applying this method in practice, we calculate, in two different, independent ways, the optical constants of muscovite mica at 330–800 nm for thicknesses that range from ∼5 to ∼160 μm.

© 2004 Optical Society of America

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References

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  1. Y. Le Calvez, G. Stephan, S. Robin, “Monochromateur à réseau concave pour l’U.V. lointain. Determination des constantes optiques complexes de cristaux anisotropes,” Opt. Acta 15, 583–594 (1968).
    [CrossRef]
  2. A. T. Davidson, A. F. Vickers, “The optical properties of mica in the vacuum ultraviolet,” J. Phys. C 5, 879–887 (1972).
    [CrossRef]
  3. A. J. Atkins, D. L. Misell, “Electron energy loss spectra for members of the mica group and related sheet silicates,” J. Phys. C 5, 3153–3160 (1972).
    [CrossRef]
  4. U. Buechner, “The dielectric function of mica and quartz determined by electron energy losses,” J. Phys. C 8, 2781–2787 (1975).
    [CrossRef]
  5. E. B. Singleton, C. T. Shirkey, “Optical constants in the IR from thin film interference and reflectance: the reststrahlen region of muscovite mica,” Appl. Opt. 22, 185–189 (1983).
    [CrossRef] [PubMed]
  6. A. I. Bailey, S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39–46 (1965).
    [CrossRef]
  7. B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485–487 (1998).
    [CrossRef]
  8. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).
  9. This approximation is used only to illustrate the physical difficulties. The actual calculations are carried out by use of the precise equations.
  10. T. Fritz, J. Hahn, H. Boettcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
  12. Note that the interference frequency has to be evaluated by use of the plot transmission versus energy and not versus wavelength.
  13. R. Nitsche, T. Fritz, “Correct interpretation of spectral interference measurements of weakly absorbing thin films,” Opt. Lett. 29, 938–940 (2004).
    [CrossRef] [PubMed]
  14. Equation (10) already implies the condition k ≪ n, so the first term of the product in Eq. (10) depends on n only.
  15. Note that Eq. (16) is, strictly speaking, valid only when R122 is small, so the assumption of one slope A for the complete thickness regime holds.

2004 (1)

1998 (1)

B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485–487 (1998).
[CrossRef]

1989 (1)

T. Fritz, J. Hahn, H. Boettcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

1983 (1)

1975 (1)

U. Buechner, “The dielectric function of mica and quartz determined by electron energy losses,” J. Phys. C 8, 2781–2787 (1975).
[CrossRef]

1972 (2)

A. T. Davidson, A. F. Vickers, “The optical properties of mica in the vacuum ultraviolet,” J. Phys. C 5, 879–887 (1972).
[CrossRef]

A. J. Atkins, D. L. Misell, “Electron energy loss spectra for members of the mica group and related sheet silicates,” J. Phys. C 5, 3153–3160 (1972).
[CrossRef]

1968 (1)

Y. Le Calvez, G. Stephan, S. Robin, “Monochromateur à réseau concave pour l’U.V. lointain. Determination des constantes optiques complexes de cristaux anisotropes,” Opt. Acta 15, 583–594 (1968).
[CrossRef]

1965 (1)

A. I. Bailey, S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39–46 (1965).
[CrossRef]

Atkins, A. J.

A. J. Atkins, D. L. Misell, “Electron energy loss spectra for members of the mica group and related sheet silicates,” J. Phys. C 5, 3153–3160 (1972).
[CrossRef]

Bailey, A. I.

A. I. Bailey, S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39–46 (1965).
[CrossRef]

Boettcher, H.

T. Fritz, J. Hahn, H. Boettcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Buechner, U.

U. Buechner, “The dielectric function of mica and quartz determined by electron energy losses,” J. Phys. C 8, 2781–2787 (1975).
[CrossRef]

Davidson, A. T.

A. T. Davidson, A. F. Vickers, “The optical properties of mica in the vacuum ultraviolet,” J. Phys. C 5, 879–887 (1972).
[CrossRef]

Fritz, T.

R. Nitsche, T. Fritz, “Correct interpretation of spectral interference measurements of weakly absorbing thin films,” Opt. Lett. 29, 938–940 (2004).
[CrossRef] [PubMed]

T. Fritz, J. Hahn, H. Boettcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Gauthier-Manuel, B.

B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485–487 (1998).
[CrossRef]

Hahn, J.

T. Fritz, J. Hahn, H. Boettcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

Kay, S. M.

A. I. Bailey, S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39–46 (1965).
[CrossRef]

Le Calvez, Y.

Y. Le Calvez, G. Stephan, S. Robin, “Monochromateur à réseau concave pour l’U.V. lointain. Determination des constantes optiques complexes de cristaux anisotropes,” Opt. Acta 15, 583–594 (1968).
[CrossRef]

Misell, D. L.

A. J. Atkins, D. L. Misell, “Electron energy loss spectra for members of the mica group and related sheet silicates,” J. Phys. C 5, 3153–3160 (1972).
[CrossRef]

Nitsche, R.

Robin, S.

Y. Le Calvez, G. Stephan, S. Robin, “Monochromateur à réseau concave pour l’U.V. lointain. Determination des constantes optiques complexes de cristaux anisotropes,” Opt. Acta 15, 583–594 (1968).
[CrossRef]

Shirkey, C. T.

Singleton, E. B.

Stephan, G.

Y. Le Calvez, G. Stephan, S. Robin, “Monochromateur à réseau concave pour l’U.V. lointain. Determination des constantes optiques complexes de cristaux anisotropes,” Opt. Acta 15, 583–594 (1968).
[CrossRef]

Vickers, A. F.

A. T. Davidson, A. F. Vickers, “The optical properties of mica in the vacuum ultraviolet,” J. Phys. C 5, 879–887 (1972).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Appl. Opt. (1)

Br. J. Appl. Phys. (1)

A. I. Bailey, S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39–46 (1965).
[CrossRef]

J. Phys. C (3)

A. T. Davidson, A. F. Vickers, “The optical properties of mica in the vacuum ultraviolet,” J. Phys. C 5, 879–887 (1972).
[CrossRef]

A. J. Atkins, D. L. Misell, “Electron energy loss spectra for members of the mica group and related sheet silicates,” J. Phys. C 5, 3153–3160 (1972).
[CrossRef]

U. Buechner, “The dielectric function of mica and quartz determined by electron energy losses,” J. Phys. C 8, 2781–2787 (1975).
[CrossRef]

Meas. Sci. Technol. (1)

B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485–487 (1998).
[CrossRef]

Opt. Acta (1)

Y. Le Calvez, G. Stephan, S. Robin, “Monochromateur à réseau concave pour l’U.V. lointain. Determination des constantes optiques complexes de cristaux anisotropes,” Opt. Acta 15, 583–594 (1968).
[CrossRef]

Opt. Lett. (1)

Thin Solid Films (1)

T. Fritz, J. Hahn, H. Boettcher, “Determination of the optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Note that the interference frequency has to be evaluated by use of the plot transmission versus energy and not versus wavelength.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

This approximation is used only to illustrate the physical difficulties. The actual calculations are carried out by use of the precise equations.

Equation (10) already implies the condition k ≪ n, so the first term of the product in Eq. (10) depends on n only.

Note that Eq. (16) is, strictly speaking, valid only when R122 is small, so the assumption of one slope A for the complete thickness regime holds.

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Figures (5)

Fig. 1
Fig. 1

Surface plot of the objective function of the transmission of a hypothetical example with n = 1.6 and with k = 3 × 10-5 at 550 nm and k = 3 × 10-4 at 350 nm for two film thicknesses, 25 and 160 μm. (a), (b) t = 25 μm, relative objective function for λ = 550 and 350 nm, respectively. (c), (d) t = 160 μm, relative objective function for λ = 550 and 350 nm, respectively; (e) unscaled sum of (a) and (c); (f) unscaled sum of (b) and (d); (g) scaled sum (see text) of (a) and (c); (h) scaled sum (see text) of (b) and (d).

Fig. 2
Fig. 2

Comparison of measured and fitted transmission curves of mica for several film thicknesses. Solid curves, smoothed experimental transmission data for mica sheets of various thicknesses (Table 2; curves with higher transmission values correspond to lower sheet thicknesses). Dashed curves, Lorentz model fits to the experimental data. Inset, zoom of the nonabsorbing region (for clarity, sample 2 has been omitted).

Fig. 3
Fig. 3

Optical constants of mica. (a) Index of refraction n of mica from two different approaches: solid curve, Lorentz model fit; dashed-dotted curve, point-by-point fit; dashed curves, upper and lower error limits of the point-by-point fit (single standard error). (b) Absorption coefficient k based on the Lorentz model. Inset, difference between k of the point-by-point calculation and k of the Lorentz model.

Fig. 4
Fig. 4

Comparison of measured and fitted transmission spectrum of sample 1, exhibiting the interference patterns. Solid curve, experimentally observed coherent transmission spectrum of sample 1; dashed curve, coherent transmission spectrum of sample 1 calculated by use of the optical constants from the Lorentz fit and the corresponding improved sheet thickness of Table 2.

Fig. 5
Fig. 5

Transmission spectra of two mica sheets that exhibiting similar interference frequencies (inset) and absolute sheet thicknesses. Note the rather large spectral differences.

Tables (2)

Tables Icon

Table 1 Final Parameters of the Lorentz Model Fit Describing the Average Optical Constants of Micaa

Tables Icon

Table 2 Start Sheet Thicknesses t start and Improved Thicknesses t fit of the Lorentz Model and the Point-By-Point Fit

Equations (18)

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T=16k2+n2k2+n+122×exp-Δk1+r124 exp-2Δk-2r122 exp-ΔkcosΔn-2δ,
Δn=4πntλ, Δk=4πktλ, δ=-arctan2kn2+k2-1, r12=1-nˆ1+nˆ,
T16n2n+14exp-Δk1-2r122 exp-ΔkcosΔn-2δ.
fMλi=j=1NTcalcλi, tj, n, k-Texpλi, tjTexpλi, tj2.
fMλi=i=λstartλendj=1N1wλi, tj×Tcalcλi, tj, n, k-Texpλi, tjTexpλi, tj2,
wλi, tj=¼fMλi, nmax, kmax+fMλi, nmax, kmin+fMλi, nmin, kmax+fMλi, nmin, kmin.
ω0=4πnt/C,
ω0=2πCt2Cn+tE0n+n021/2+tE0n+n0,
ω0=4πE0n+n0tC+2πnn0.
Tincoh=16n2n+14expΔkexp2Δk-R122=TnλiexpΔkexp2Δk-R122,
ODλi, tj=-Δkλi, tjloge-logTnλiexp2Δkλi, tj-R122.
ODλi, tj-logTn+logeΔk-logeR122exp-2Δk.
ODλi, tj-logTn+logeΔk,
ODλi, tj-logTn-logeR122+logeΔk2R122+1,
-logTn11-R122,
ODλi, tj=-logTn11-R122+logeΔkλi, tj.
kλi=Aλiλi4π loge, nλi=1-10-2Bλi1/2+110-Bλi.
n=n0+l=1MAlE2-El2E2-El22+EΓl2, k=l=1MAlΓlEE2-El22+EΓl2,

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