Abstract

The Nomarksi differential interference contrast microscope is examined as a tool for determination of metallic mirror surface topography. This discussion includes the development of an optical model for the Nomarski system, an examination of the key results of the model’s application to sloped sample surfaces, and recommended procedures for implementation. The functional relationship is developed between image intensity and the component of surface slope along the Nomarski shear direction, the fixed parameters in the Nimarksi system, and the adjustable phase shifts related to Nomarski prism position. Equations are also developed to allow the determination of surface slope from relative image intensity when sample reflectively is uniform and slopes are small.

© 1979 Optical Society of America

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References

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  1. H. E. Bennett and J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [Crossref]
  2. B. P. Hildebrand, R. L. Gordon, and E. V. Allen, “Instrument for measuring the roughness of supersmooth surfaces,” Appl. Opt. 13, 177–180 (1974).
    [Crossref] [PubMed]
  3. D. Beaglehole and O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
    [Crossref]
  4. J. M. Bennett, “Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer,” Appl. Opt. 15, 2705–2721 (1977).
    [Crossref]
  5. G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9S–13S (1955).
  6. G. Nomarski and A. R. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall. 52, 121–134 (1955).
  7. J. Demarq and J. Rosch, in Advanced Optical Techniques, edited by A.C.S. Van Heel (North Holland, Amsterdam, The Netherlands, 1967), 393.
  8. U. Bertocci and T. S. Noggle, “Interference contrast employed to measure slopes on metallographie specimens,” Rev. Sci. Instum. 37, 1750–1751 (1966).
    [Crossref]
  9. Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1970), 4th ed., Chap. 14.
  10. Georg Hass, in American Institute of Physics Handbook, edited by Dwight E. Gray (McGraw Hill, New York, 1957), Chap. 6, Table 6k, pp. 6–104.

1977 (1)

1974 (1)

1970 (1)

D. Beaglehole and O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[Crossref]

1966 (1)

U. Bertocci and T. S. Noggle, “Interference contrast employed to measure slopes on metallographie specimens,” Rev. Sci. Instum. 37, 1750–1751 (1966).
[Crossref]

1961 (1)

1955 (2)

G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9S–13S (1955).

G. Nomarski and A. R. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall. 52, 121–134 (1955).

Allen, E. V.

Beaglehole, D.

D. Beaglehole and O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[Crossref]

Bennett, H. E.

Bennett, J. M.

Bertocci, U.

U. Bertocci and T. S. Noggle, “Interference contrast employed to measure slopes on metallographie specimens,” Rev. Sci. Instum. 37, 1750–1751 (1966).
[Crossref]

Born, Max

Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1970), 4th ed., Chap. 14.

Demarq, J.

J. Demarq and J. Rosch, in Advanced Optical Techniques, edited by A.C.S. Van Heel (North Holland, Amsterdam, The Netherlands, 1967), 393.

Gordon, R. L.

Hass, Georg

Georg Hass, in American Institute of Physics Handbook, edited by Dwight E. Gray (McGraw Hill, New York, 1957), Chap. 6, Table 6k, pp. 6–104.

Hildebrand, B. P.

Hunderi, O.

D. Beaglehole and O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[Crossref]

Noggle, T. S.

U. Bertocci and T. S. Noggle, “Interference contrast employed to measure slopes on metallographie specimens,” Rev. Sci. Instum. 37, 1750–1751 (1966).
[Crossref]

Nomarski, G.

G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9S–13S (1955).

G. Nomarski and A. R. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall. 52, 121–134 (1955).

Porteus, J. O.

Rosch, J.

J. Demarq and J. Rosch, in Advanced Optical Techniques, edited by A.C.S. Van Heel (North Holland, Amsterdam, The Netherlands, 1967), 393.

Weill, A. R.

G. Nomarski and A. R. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall. 52, 121–134 (1955).

Wolf, Emil

Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1970), 4th ed., Chap. 14.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Phys. Radium (1)

G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9S–13S (1955).

Phys. Rev. B (1)

D. Beaglehole and O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[Crossref]

Rev. Metall. (1)

G. Nomarski and A. R. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall. 52, 121–134 (1955).

Rev. Sci. Instum. (1)

U. Bertocci and T. S. Noggle, “Interference contrast employed to measure slopes on metallographie specimens,” Rev. Sci. Instum. 37, 1750–1751 (1966).
[Crossref]

Other (3)

Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1970), 4th ed., Chap. 14.

Georg Hass, in American Institute of Physics Handbook, edited by Dwight E. Gray (McGraw Hill, New York, 1957), Chap. 6, Table 6k, pp. 6–104.

J. Demarq and J. Rosch, in Advanced Optical Techniques, edited by A.C.S. Van Heel (North Holland, Amsterdam, The Netherlands, 1967), 393.

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Figures (9)

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

FIG. 3
FIG. 3

Ray schematic for Nomarski prism, objective lens, and reflecting specimen in Nomarski reflection microscopy.

FIG. 4
FIG. 4

Ray bundles from light source L in the focal plane of a converging lens to an image point L′ in the same plane.

FIG. 5
FIG. 5

Ray diagrams for reflection of ray LAB with surface tilt zero degrees (LABEGB′) and ψ degrees (LABCDB′), where surface rotation ψ is in the plane of the Nomarski shear.

FIG. 6
FIG. 6

Notation and coordinate system for a reflecting surface with negative fall line rotated by angle ϕ from the shear direction. (a) Angle identification and (b) identification of coordinate axis systems and significant unit vectors.

FIG. 7
FIG. 7

Ray paths near a stepped reflecting surface for a split by the Nomarski prism for the x′ oscillation (ACDE), the y′ oscillation (AB′GHE), and the ŷ oscillation if the surface had been level rather than stepped (AB′C′D′E).

FIG. 8
FIG. 8

(a) Calculated image intensity in differential interference contrast microscopy from reflecting specimens with tilt ψ = 3.781° as a function of rotation angle ϕ, with effects of phase change and absorption at the reflecting specimen included, (b) The intensities of (a) vertically shifted and scaled so that their maxima and minima coincide. Assumed parameters are in Table I.

FIG. 9
FIG. 9

(a) Calculated image intensity in differential interference contrast microscopy from reflecting specimens as a function of tilt angle ψ, with surface negative fall line at angle ϕ = 0 to the shear direction, (b) The intensities of (a) shifted and scaled so that their maxima and minima coincide. The high-conductivity mirror has complex index of refraction n ¯ = 10 4 + i 10 4. Other assumed parameters are in Table I.

Tables (1)

Tables Icon

TABLE I Parameters for image intensity calculation in Nomarski reflection microscopy of tilted planar metal facets (Fig. 8).

Equations (57)

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χ = ( 2 π / λ ) ( l x l y ) ( 2 π / λ ) ( n E n o ) [ ( s 1 + s 2 ) ( s 1 + s 2 ) ] ,
tan θ s 2 ( n E n o ) tan θ w ,
E = g 1 ( E o p ̂ p ̂ + E o q ̂ q ̂ ) ,
E o p ̂ + E o q ̂ = E o .
E N = g 2 ( E x ̂ e i χ x ̂ + E ŷ ŷ ) .
E A = g 3 ( E N â â + γ E N b ̂ b ̂ ) ,
E A E A * = | g 1 g 2 g 3 E o | 2 { ( x ̂ p ̂ + x ̂ q ̂ ) 2 ( x ̂ â ) 2 + ( ŷ p ̂ + ŷ q ̂ ) 2 ( ŷ â ) 2 + 2 ( x ̂ p ̂ + x ̂ q ̂ ) × ( ŷ p ̂ + ŷ q ̂ ) x ̂ â ŷ â cos χ } .
I = I o [ C + D ( 1 cos χ ) ]
= I MAX [ Q + 1 / 2 ( 1 cos χ ) ] ,
C = cos 2 ( A P ) + sin 2 ( A P ) + 2 sin 2 ( A P ) ,
D = sin 2 A [ ½ ( 1 2 ) sin 2 p + cos 2 P ] .
I MAX = I o ( C + 2 D ) ,
Q = C / ( C + 2 D ) .
χ = α + β .
ξ c = ξ B s tan ( B 1 + 2 ψ ) ,
ξ B = ( s f ) tan B 1 ,
ξ D = ξ C + ( f / s ) ( ξ B ξ C ) , = ξ C + ( f / s ) ( ξ A ξ C ) .
ξ D = f tan ( B 1 + 2 ψ ) .
s 1 = ( x o + f tan B 1 cos ϕ ) tan θ w ,
s 1 = W s 1 ,
s 2 = ( x o f tan ( B 1 + 2 ψ ) cos ϕ ) tan θ w ,
s 2 = W s 2 .
χ = 8 π λ ( n E n o ) { W / 2 + [ x o ( f / 2 ) tan 2 ψ cos ϕ ] tan θ w } .
β = β o + x o d β d x o ,
d β d x o = 8 π λ ( n E n o ) tan θ w ,
α = 1 2 f tan 2 ψ cos ϕ d β d x o .
β = β o + x d β d x ,
l ABCDE = l A B C D E .
l JHE = l C D E ,
l A B GHE l ABCDE = ( C G ) + ( G J ) = Δ ζ / cos B 1 + ( Δ ζ / cos B 1 ) cos 2 B 1 .
α = ( 2 π / λ ) ( l A B GHE l ABCDE ) ( 4 π / λ ) Δ ζ ,
α ( 4 π / λ ) [ ζ ( r + g ) ζ ( r ) ] ,
g = 2 ( n E n o ) f tan Θ w x ̂ ,
α = ( 4 π / λ ) g ζ ,
x ̂ = sin ψ cos ψ ( cos ϕ x ̂ + sin ϕ ŷ ) , ŷ = cos ϕ ŷ sin ϕ x ̂ , = cos ψ + sin ψ ( cos ϕ x ̂ + sin ϕ ŷ ) = n ̂ .
E x ( r ) = ( R / A ) E x ( i ) E y ( r ) = ( R / A ) E y ( i ) E z ( r ) = ( R / A ) E z ( i ) .
R / A = ( n ¯ cos θ i cos θ t ) / ( n ¯ cos θ i cos θ t ) , R / A = ( cos θ i n ¯ cos θ t ) / ( cos θ i + n ¯ cos θ t ) ,
θ i ψ
sin θ t = sin θ i / n ¯ , cos θ t = ( 1 sin 2 θ t ) 1 / 2 .
E ( i ) = ( E p x ̂ ) x ̂ e i β / 2 + ( E p y ) y ,
E N = ( E ( r ) x ̂ ) x ̂ e i ( α + β / 2 ) + ( E ( r ) ŷ ) ŷ .
E A E A * = | â E N | 2 .
I = I MAX { Q + ( 1 / 2 ) ( 1 Q ) [ 1 cos ( R T + β ) ] } ,
T = ( 1 / 2 ) tan 2 ψ cos ϕ .
T = ( 1 / R ) { β + arc cos [ 1 2 ( I / I MAX Q ) / ( 1 Q ) ] } .
π / 2 < R T < π / 2 .
( 1 / 2 ) tan 2 ψ = ( T 1 2 + T 2 2 ) 1 / 2 ,
cos ϕ 1 = T 1 / ( T 1 2 + T 2 2 ) 1 / 2
sin ϕ 1 = T 2 / ( T 1 2 + T 2 2 ) 1 / 2 .
0 < | R T | < π .
Q = I MIN / I MAX .
I = I MAX { Q + ( 1 / 2 ) ( 1 Q ) [ 1 cos ( β R + x d β / d x ) ] } .
I = I MAX { Q + 1 2 ( 1 Q ) [ 1 cos ( f d β d x o 1 2 × tan 2 ψ cos ϕ + β o + x d β d x ) ] }
d x o / d x = 1
n ¯ = 0.7684 + 2.4578 i
n ¯ = 0.175 + 3.294 i
n ¯ = 0.81667 + 5.981 i