Abstract

A laser optical metrology system is described that remotely measures at high rate the presence and thickness of a thin-film lubricant on metallic cylinders with diameters on the order of 0.5 mm. Applications include remote measurement of hypodermic needle dielectric coating thickness in a clean room environment. High accuracy computer simulation of the electric field scattered from a coated cylinder by an incident laser beam is demonstrated using the condition numbers of the matrices defined by the boundary value matching equations derived from the eigenfunction expansion of the exact solution to Maxwell’s equations. Dielectric coatings from 1 μmto 50 μm are seen to be readily observed and accurately measured using a remotely placed CMOS array. Distinctive signatures are shown for film thicknesses in the range from 0 to 10 μm, and an appropriate location for CMOS detector placement is determined from the scattering patterns.

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References

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  1. C. A. Balanis, Advanced Engineering Electromagnetics (J. Wiley & Sons, 1989).
  2. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).
  3. J. J. Bowmann, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North Holland, 1969).
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  5. Handbook of Ellipsometry, H.G. Tompkins and E.A. Irene, Eds. (William Andrew, 2005).
  6. G. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, 2003).
  7. V. Borovikov and B. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, 1994).
  8. G. Strang, Linear Algebra and its Applications (Thompson Brooks/Cole, 2006).

Other

C. A. Balanis, Advanced Engineering Electromagnetics (J. Wiley & Sons, 1989).

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).

J. J. Bowmann, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North Holland, 1969).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

Handbook of Ellipsometry, H.G. Tompkins and E.A. Irene, Eds. (William Andrew, 2005).

G. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, 2003).

V. Borovikov and B. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, 1994).

G. Strang, Linear Algebra and its Applications (Thompson Brooks/Cole, 2006).

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Figures (12)

Fig. 1
Fig. 1

Optical metrology system schematic: C = detector array, D = computer

Fig. 2
Fig. 2

Logarithm of condition number c m of Λ m

Fig. 3
Fig. 3

logarithm of Fourier coefficient magnitude | A m H m (2) ( k 0 ρ) |

Fig. 4
Fig. 4

Normalized scattered field intensity | E scatt | 2 , dielectric thickness = 0

Fig. 6
Fig. 6

Normalized scattered field intensity | E scatt | 2 , dielectric thickness = 10 microns

Fig. 5
Fig. 5

Normalized scattered field intensity | E scatt | 2 , dielectric thickness = 4 microns

Fig. 7
Fig. 7

Backscattered energy intensity | E scatt | 2 , dielectric thickness = 1 micron

Fig. 8
Fig. 8

Scattered energy density, dielectric thickness = 0 microns

Fig. 9
Fig. 9

Scattered energy density, dielectric thickness = 4 microns

Fig. 10
Fig. 10

Scattered energy density, dielectric thickness = 0

Fig. 12
Fig. 12

Scattered energy density, dielectric thickness = 4 microns

Fig. 11
Fig. 11

Scattered energy density, dielectric thickness = 4 microns

Equations (17)

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E inc = E 0 e i k 0 x z ^ = E 0 e i k 0 ρcosϕ z ^ ,
E inc = E 0 ( i m J m ( k 0 ρ) e imϕ ) z ^ .
E scatt = E 0 ( A m H m (2) ( k 0 ρ) e imϕ ) z ^ .
E diel = E 0 ( ( B m J m ( k 1 ρ)+ C m Y m ( k 1 ρ) ) e imϕ ) z ^ ,
k 1 = n 1 n 0 k 0 .
C m = J m ( k 1 a) Y m ( k 1 a) B m .
H= 1 iωμ ×E
H ϕ = 1 iωμ E z ρ .
Λ m [ A m B m ]=[ i m J m ( k 0 b) i m J m ( k 0 b) ]
Λ m =[ H m (2) ( k 0 b) [ J m ( k 1 b) J m ( k 1 a) Y m ( k 1 a) Y m ( k 1 b) ] H m (2 ) ( k 0 b) k 1 k 0 [ J m ( k 1 b) J m ( k 1 a) Y m ( k 1 a) Y m ( k 1 b) ] ]
A m = i m det Λ m [ Λ m 22 J m ( k 0 b) Λ m 12 J m ( k 0 b) ].
E est scatt = E 0 ( M M A m H m (2) ( k 0 ρ) e imϕ ) z ^ ,
Λ m x m = y m ,
x m =[ A m B m ], y m =[ i m J m ( k 0 b ) i m J m ( k 0 b ) ].
Λ m ( x m +δ x m )= y m +δ y m .
| δ x m | | x m | c m | δ y m | | y m | .
E est scatt ( ϕ l )= E 0 ( M M A m H m (2) ( k 0 ρ) e iml(2π/2M) ) z ^ .

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