Abstract

The application of reflective Nomarski differential interference contrast microscopy for the determination of quantitative sample topography data is presented. The discussion includes a review of key theoretical results presented previously plus the experimental implementation of the concepts using a commercial Nomarski microscope. The experimental work included the modification and characterization of a commercial microscope to allow its use for obtaining quantitative sample topography data. System usage for the measurement of slopes on flat planar samples is also discussed. The discussion has been designed to provide the theoretical basis, a physical insight, and a cookbook procedure for implementation to allow these results to be of value to both those interested in the microscope theory and its practical usage in the metallography laboratory.

© 1980 Optical Society of America

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References

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  1. D. L. Lessor, J. S. Hartman, R. L. Gordon, J. Opt. Soc. Am. 69, 357 (1979).
    [Crossref]
  2. U. Bertocci, T. S. Noggle, Rev. Sci. Instrum. 37, 1750 (1966).
    [Crossref]
  3. W. Lang, “Nomarski Differential Interference—Contrast Microscopy. IV. Applications,” Zeiss Information 77 (1971), pp. 22–27.
  4. Reference to a company or product name does not imply approval or recommendation from the Pacific Northwest Laboratory or the U.S. Department of Energy to the exclusion of others that may be suitable.
  5. Equation (8) in Ref. 1 is erroneous. The correct equation is given by Eq. (3), this work. The authors regret the error.
  6. B. P. Hildebrand, R. L. Gordon, E. V. Allen, Appl. Opt. 13, 177 (1974).
    [Crossref] [PubMed]
  7. D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). Program NLIN by T. Baumeister, J. A. Sheldon, R. M. Stanley, revised by D. W. Marquardt, R. M. Stanley, share distribution 3094-01 (Aug.1966).
    [Crossref]

1979 (1)

1974 (1)

1966 (1)

U. Bertocci, T. S. Noggle, Rev. Sci. Instrum. 37, 1750 (1966).
[Crossref]

1963 (1)

D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). Program NLIN by T. Baumeister, J. A. Sheldon, R. M. Stanley, revised by D. W. Marquardt, R. M. Stanley, share distribution 3094-01 (Aug.1966).
[Crossref]

Allen, E. V.

Bertocci, U.

U. Bertocci, T. S. Noggle, Rev. Sci. Instrum. 37, 1750 (1966).
[Crossref]

Gordon, R. L.

Hartman, J. S.

Hildebrand, B. P.

Lang, W.

W. Lang, “Nomarski Differential Interference—Contrast Microscopy. IV. Applications,” Zeiss Information 77 (1971), pp. 22–27.

Lessor, D. L.

Marquardt, D. W.

D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). Program NLIN by T. Baumeister, J. A. Sheldon, R. M. Stanley, revised by D. W. Marquardt, R. M. Stanley, share distribution 3094-01 (Aug.1966).
[Crossref]

Noggle, T. S.

U. Bertocci, T. S. Noggle, Rev. Sci. Instrum. 37, 1750 (1966).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Soc. Ind. Appl. Math. (1)

D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). Program NLIN by T. Baumeister, J. A. Sheldon, R. M. Stanley, revised by D. W. Marquardt, R. M. Stanley, share distribution 3094-01 (Aug.1966).
[Crossref]

Rev. Sci. Instrum. (1)

U. Bertocci, T. S. Noggle, Rev. Sci. Instrum. 37, 1750 (1966).
[Crossref]

Other (3)

W. Lang, “Nomarski Differential Interference—Contrast Microscopy. IV. Applications,” Zeiss Information 77 (1971), pp. 22–27.

Reference to a company or product name does not imply approval or recommendation from the Pacific Northwest Laboratory or the U.S. Department of Energy to the exclusion of others that may be suitable.

Equation (8) in Ref. 1 is erroneous. The correct equation is given by Eq. (3), this work. The authors regret the error.

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Figures (13)

Fig. 1
Fig. 1

Nomarski microscope system with translating wedge for phase adjustment. Angular orientation of components is measured from the Nomarski shear plane.

Fig. 2
Fig. 2

Nomarski microscope schematic showing (a) the plane of apparent beam splitting and (b) the beam offset and shear direction.

Fig. 3
Fig. 3

Comparison of Nomarski prism motion for the standard Zeiss and modified configurations: (a) Zeiss configuration causes simultaneous prism translation and rotation. (b) Micrometer drive screw in the modified mechanism results in only translation of prism. (c) Orientation of prism optical axes remain fixed as prism is translated.

Fig. 4
Fig. 4

Exploded view of modified Nomarski objective assembly including micrometer drive screw for precise prism translation.

Fig. 5
Fig. 5

Dependence of image intensity on relative phase difference between beam polarization components. Scales illustrate total phase effect (upper scale) and the effect of prism retardation for the special case of the sample perpendicular to the microscope axis, α = 0 (lower scale).

Fig. 6
Fig. 6

Dependence of theoretical Nomarski image intensity on sample surface slope ψ for selected system operating conditions: (a) effect of different prism retardation values β for a 40× objective assembly; (b) effect of objective lens focal length differences for 16, 40, and 80× assemblies operated at β = π/2.

Fig. 7
Fig. 7

Dependence of image intensity on the Nomarski prism position for a flat sample orthogonal to the microscope axis (α = 0). Experimental data (open squares) and the results of a nonlinear regression analysis (solid curve) based on Eq. (4) are shown for the 40× objective assembly.

Fig. 8
Fig. 8

Calculated dependence of relative prism phase shift β on the Nomarski prism position based on a linear least squares analysis of recorded image intensity data using Eq. (10) for a 40× objective assembly.

Fig. 9
Fig. 9

Plot of independent phase shift contributions of Eq. (16) for three distinct cases.

Fig. 10
Fig. 10

Nomarski image intensity as a function of rotation angle on the microscope stage for fixed values of prism retardation β and sample slope ψ. Points are experimental data, and solid curves are the optical model predictions. (a) Sample slope ψ = 3.645°, and prism retardation values are β = 0.498π (closed circles) and β = 1.49π (open circles). (b) Sample slope ψ = 2.712°, and prism retardation is β = 0.08π.

Fig. 11
Fig. 11

Nomarski image intensity as a function of stage rotation angle for a sample with surface slope ψ = 3.781°. Experimental data (open circles) are shown with theoretical predictions for two selected values of prism parameter β0.

Fig. 12
Fig. 12

Rotational intensity data (points) for three values of surface slope recorded for β = 0.488π. Solid curves are corresponding predictions of a Nomarski system optical model.

Fig. 13
Fig. 13

Relationship between the smallest useful value of rotational signal modulation and the corresponding minimum value of surface slope ψ that can be calculated reliably. Curves are shown for both 40 and 16× assemblies operated at β = π/2.

Tables (5)

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Table I Alignment Sequence for Nomarski Microscope Components

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Table II Comparison of Results from Linear and Nonlinear Regression

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Table III Results of Translation Experiments 16× Objective

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Table IV Results of Translation Experiments 40× Objective

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Table V Tilt Angles Determined by Nonlinear Regression on Intensity Data

Equations (28)

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β = β 0 + x ( d β / d x ) ,
d β / d x = ± ( 8 π / λ ) ( n E n O ) tan θ w ,
χ = α + β .
I = I max [ Q + 1 2 ( 1 Q ) ( 1 cos χ ) ] .
I min = I max Q .
I = I max ( 1 Q ) ,
I = I min + I ( 1 cos χ ) / 2 .
I = I min + I [ 1 cos ( β 0 + x d β d x ) ] / 2 .
Q = I min / I max ,
β ( x ) = cos 1 { 1 2 [ I ( x ) I min ] / I } .
β = ( 2 n 1 2 ) π ,
α = ( f / 2 ) ( d β / d x ) tan 2 ψ cos ϕ ,
χ = α + β = 2 π δ λ + β ,
I ( β , δ ) = I [ 1 cos ( β + 2 π δ λ ) ] ,
I ( β , δ ) I ( β , 0 ) = [ 1 cos ( β + 2 π δ / λ ) 1 cos β ] .
I ( β , δ ) δ = 2 π I λ sin ( β + 2 π δ / λ ) .
β ( x 0 ) f 2 ( d β d x ) tan 2 ψ cos ϕ .
Case I : | β I | < | f 2 tan 2 ψ d β d x | .
Case II : | β II | = | f 2 tan 2 ψ d β d x | .
Case III : | β III | > | f 2 tan 2 ψ d β d x | .
I ( 0 , β ) = I min + 1 2 I [ 1 cos ( s f d β d x + β ) ] ,
I ( π , β ) = I min + 1 2 I [ 1 cos ( s f d β d x + β ) ] ,
s = 1 2 tan 2 ψ .
I + = I ( 0 , β ) + I ( π , β ) 2 I min 2 [ 1 cos ( s f d β d x ) cos β ] ,
I = I ( π , β ) I ( 0 , β ) 2 sin ( s f d β d x ) sin β .
I = ( I + + I ) / 2 .
I max I min I max γ ,
ψ min = 1 2 tan 1 { 2 sin 1 ( γ 2 γ ) f d β d x } .

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