Abstract

This paper is a theoretical presentation of a two wavelength holographic method of measuring the roughness of a surface. This method extends the range of surface roughness which can be measured to arbitrarily rough surfaces. This paper also discusses the factors that limit the accuracy of the proposed method.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. B. Ribbens, Appl. Opt. 11, 807 (1972).
    [CrossRef] [PubMed]
  2. B. P. Hildebrand, K. A. Haines, J. Opt. Soc. Am. 57, 155 (1967).
    [CrossRef]
  3. J. Zalenka, J. Varner, Appl. Opt. 8, 1431(1969).
    [CrossRef]
  4. J. R. Varner, “Multiple-Frequence Holographic Contouring,” Ph.D. Thesis, Electrical Engineering, The University of Michigan (1971).
  5. G. L. Lazik, “The Application of Coherent Fourier Optics to Physical Surface Analysis,” Ph.D. Thesis, Electrical Engineering, The University of Michigan (1970).

1972

1969

1967

Haines, K. A.

Hildebrand, B. P.

Lazik, G. L.

G. L. Lazik, “The Application of Coherent Fourier Optics to Physical Surface Analysis,” Ph.D. Thesis, Electrical Engineering, The University of Michigan (1970).

Ribbens, W. B.

Varner, J.

Varner, J. R.

J. R. Varner, “Multiple-Frequence Holographic Contouring,” Ph.D. Thesis, Electrical Engineering, The University of Michigan (1971).

Zalenka, J.

Appl. Opt.

J. Opt. Soc. Am.

Other

J. R. Varner, “Multiple-Frequence Holographic Contouring,” Ph.D. Thesis, Electrical Engineering, The University of Michigan (1971).

G. L. Lazik, “The Application of Coherent Fourier Optics to Physical Surface Analysis,” Ph.D. Thesis, Electrical Engineering, The University of Michigan (1970).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Definition of roughness function

Fig. 2
Fig. 2

Two wavelength holographic interferometry configuration.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

Z ( x y ) = Z ( x y ) + z ( x , y ) ,
Z = 1 X Y 0 Y 0 X Z ( x y ) d x d y ,
1 Y X 0 Y 0 X z ( x y ) d x d y = 0 ,
r = [ 1 X Y 0 Y 0 X z 2 ( x y ) d x d y ] 1 / 2 .
U 0 ( λ n ) = ρ s U a exp [ j ϕ o n ( x 0 y 0 ) ] ,
U i n ( x i y i ) = U i n exp [ j ϕ i n ( x i y i ) ] ,
U r = k ( U i 1 + U i 2 ) ,
I r = k 2 U i 1 + U i 2 2
= k 2 U i 1 2 exp [ ( j ϕ i 1 ) + ρ exp ( j ϕ i 2 ) 2 .
I r 0 = I r / ( k 2 U i 1 2 ) = exp ( j ϕ ) + ρ 2 ,
ϕ = ϕ i 1 - ϕ i 2
= 2 π Z [ ( 1 / λ 1 ) - ( 1 / λ 2 ) ]
= 2 π Z / λ eff .
[ 2 π Z ( x y ) ] / λ eff ( 2 N π ) in the aperture X Y .
ϕ = [ 2 π ( Z + z ) ] / λ eff = 2 N π + ( 2 π z / λ eff ) .
exp ( j ϕ ) = exp j [ 2 N π + ( 2 π z / λ eff ) ]
= exp j ( 2 π z / λ eff ) 1 + j ( 2 π z / λ eff ) .
I r 0 ( N π ) ( 1 + ρ ) 2 + ( 2 π z / λ eff ) 2 .
2 π Z / λ eff = ( 2 N + 1 ) π .
I r 0 [ ( 2 N + 1 ) π ] - ( 1 - ρ ) 2 + ( 2 π z / λ eff ) 2 .
i = k d 0 Y 0 X I r 0 d x i d y i ,
i l = k d X Y [ ( 1 + ρ ) 2 + 1 X Y ( 2 π λ eff ) 2 0 Y 0 X z 2 d x i d y i ] .
i l = k d X Y [ ( 1 + ρ ) 2 + ( 2 π r / λ eff ) 2 ] .
i d = k d X Y [ ( 1 - ρ ) 2 + ( 2 π r / λ eff ) 2 ]
R = i l / i d = ( 1 + ρ ) 2 + ( 2 π r / λ eff ) 2 ( 1 - ρ ) 2 + ( 2 π r / λ eff ) 2 .
r = λ eff / [ π ( R - 1 ) 1 / 2 ] .

Metrics