Abstract

The phase shift on reflection at a double-layer system is treated by using a modified Malé nomogram. For given materials, layer thicknesses are calculated so as to provide a perfect optical-impedance match with zero reflectance. With the system in this state, the phase shift is an extremely sensitive function of small changes in the optical parameters. An experimental application is proposed which uses the effect for the detection and comparison of changes in the optical parameters of a double-layer system.

© 1962 Optical Society of America

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References

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  1. P. Rouard, Ann. phys. 7, 291 (1937). Further references are given in this paper.
  2. D. Malé, J. phys. radium 11, 332 (1950).
    [Crossref]
  3. A. Vasicek, J. phys. radium 11, 342 (1950).
    [Crossref]
  4. R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948).
    [Crossref]
  5. P. J. Leurgens, J. Opt. Soc. Am. 41, 714 (1951).
    [Crossref]
  6. O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications, Ltd., London, 1955).
  7. J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 514 (1960).
  8. J. M. Bennett, Radiative Transfer from Solid Materials (The Macmillan Company, New York, 1962).
  9. J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 49, 466 (1959).
    [Crossref]
  10. J. M. Bennett (private communication).

1960 (1)

J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 514 (1960).

1959 (1)

1951 (1)

1950 (2)

D. Malé, J. phys. radium 11, 332 (1950).
[Crossref]

A. Vasicek, J. phys. radium 11, 342 (1950).
[Crossref]

1948 (1)

1937 (1)

P. Rouard, Ann. phys. 7, 291 (1937). Further references are given in this paper.

Bennett, J. M.

J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 514 (1960).

J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 49, 466 (1959).
[Crossref]

J. M. Bennett, Radiative Transfer from Solid Materials (The Macmillan Company, New York, 1962).

J. M. Bennett (private communication).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications, Ltd., London, 1955).

Koehler, W. F.

J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 50, 514 (1960).

J. M. Bennett and W. F. Koehler, J. Opt. Soc. Am. 49, 466 (1959).
[Crossref]

Leurgens, P. J.

Malé, D.

D. Malé, J. phys. radium 11, 332 (1950).
[Crossref]

Muchmore, R. B.

Rouard, P.

P. Rouard, Ann. phys. 7, 291 (1937). Further references are given in this paper.

Vasicek, A.

A. Vasicek, J. phys. radium 11, 342 (1950).
[Crossref]

Ann. phys. (1)

P. Rouard, Ann. phys. 7, 291 (1937). Further references are given in this paper.

J. Opt. Soc. Am. (4)

J. phys. radium (2)

D. Malé, J. phys. radium 11, 332 (1950).
[Crossref]

A. Vasicek, J. phys. radium 11, 342 (1950).
[Crossref]

Other (3)

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications, Ltd., London, 1955).

J. M. Bennett (private communication).

J. M. Bennett, Radiative Transfer from Solid Materials (The Macmillan Company, New York, 1962).

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Figures (7)

Fig. 1
Fig. 1

Geometry used in the discussion of the double-layer system.

Fig. 2
Fig. 2

Vector diagram in the complex plane used in the evaluation of the phase shift on reflection at a double-layer system.

Fig. 3
Fig. 3

Vector diagram in the complex plane used in the determination of the layer thicknesses in the double-layer system. The intersection of the logarithmic spiral and the image circle represents the perfect optical-impedance match.

Fig. 4
Fig. 4

The change Δβ in the phase shift β at a double-layer system with variations in the thickness of the middle layer. Shown are the curves for four initial thicknesses d2 near the thickness for perfect matching. Δβ = 0 corresponds to the phase shift in the reflectance minimum which occurs at a different wavelength λ for each different initial thickness.

Fig. 5
Fig. 5

Calibration curve for Example 1. The gradient of the phase shift β with variations in the thickness of the middle layer is calculated as a function of the minimum reflectance as determined by initial values of d2. The interpretation of an observed change in the phase shift in terms of a particular parameter is read from the ordinate, once the minimum reflectance r0r0min* for the mismatched system is determined experimentally.

Fig. 6
Fig. 6

The phase shift β on reflection at a double layer as a function of the index of refraction n of the middle layer around the wavelength λ = 5893 Å, at which the reflectance goes through a minimum for the true value of the index of refraction.

Fig. 7
Fig. 7

The phase shift β on reflection at a double layer as a function of the optical constants n and k of the substrate. The curve “n constant” plots the phase shift β when k only is changed (lower scale), “k constant” represents the phase shift β when n only is changed (upper scale). These curves were plotted for λ = 5797 Å, the wavelength at which the reflectance minimum occurs for the given thicknesses d1 and d2.

Equations (15)

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r = r 2 + r 3 exp ( 2 i δ 2 ) 1 + r 2 r 3 exp ( 2 i δ 2 ) ,
r 0 = r 1 + r exp ( 2 i δ 1 ) 1 + r 1 r exp ( 2 i δ 1 ) ,
r 0 = 1 r 1 r [ r 1 exp ( 2 i δ 1 ) ] r 1 r [ exp ( 2 i δ 1 ) / r 1 ] = 1 r 1 ( r N ) ( r D ) = ρ 0 exp ( i β ) ,
exp ( 2 i δ 1 ) = exp [ ( 2 π / λ ) i n 1 d 1 ] exp [ ( 2 π / λ ) k 1 d 1 ]
r = a z + b / c z + d ,
z 0 = r 2 r 2 * | r 3 | 2 1 | r 2 | 2 | r 3 | 2 ,
ρ = | r 3 | | ( 1 r 2 2 ) | 1 | r 2 | 2 | r 3 | 2 .
exp ( 2 i δ 1 ) = r / r 1 ,
exp ( 2 i δ 2 ) = r 3 ( 1 r r 2 ) / ( r r 2 ) .
n 0 = 1.000 ; Top layer : n 1 = 2.085 ; Middle layer : n 2 = 1.465 ; Substrate : n 3 = 1.160 ; k 3 = 1.230.
d 1 = 330 Å ; d 2 = 1358 Å .
n 0 = 1.000 ; Top layer : n 1 = 2.017 ; Middle layer : n 2 = 2.407 ; Substrate : n 3 = 1.790 ; k 3 = 1.860.
d 1 = 1461 Å ; d 2 = 98 256 Å .
n 0 = 1.000 ; Top layer : n 1 = 0.261 ; k 1 = 2.730 ; Middle layer : n 2 = 17.5 ; Substrate : n 3 = 5.565 ; k 3 = 1.700.
d 1 = 123 Å ; d 2 = 4400 Å .