Abstract

I show how to simulate the scattering generated by a scratch on the surface of high-quality optics and their elements. This is accomplished by first describing how the present cosmetic scratch standards tend to be used in the optics industry. Second, I derive from first principles, using the scalar model for electromagnetic radiation, the first-order scattering coefficients for the far-field radiation due to a particular scratch pattern. There are approximations made to get these coefficients. The results allow construction of a set of secondary scratch standards. These are a pattern of rectangular grooves that can be made precisely reproducible during the manufacturing phase. Appropriate selection from this set can provide the same range of scattering power and character as is present in the current scratch standards, which are not easily reproducible. Because the method for construction of these new secondary standards is nonrandom, to guarantee the reproducible construction between these standards it is necessary to restrict the observation range 5– 10° from the direct beam.

© 1983 Optical Society of America

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References

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  1. Military specification MIL-0-13830A, 1963. For the use of present standards.
  2. J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 7.
    [CrossRef]
  3. M. Young, “Objective measurement and characterization of scratch standards,” in Proc. Soc. Photo-Opt. Instrum. Eng. 362, 86 (1982);“Can you describe optical surface quality with one or two numbers” Proc. Soc. Photo-Opt. Instrum. Eng.406 (1983), in press.
  4. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chaps. 5–8.
  5. P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

1982 (1)

M. Young, “Objective measurement and characterization of scratch standards,” in Proc. Soc. Photo-Opt. Instrum. Eng. 362, 86 (1982);“Can you describe optical surface quality with one or two numbers” Proc. Soc. Photo-Opt. Instrum. Eng.406 (1983), in press.

Bennett, H. E.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 7.
[CrossRef]

Bennett, J. M.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 7.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

Elson, J. M.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 7.
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chaps. 5–8.

Moon, P.

P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chaps. 5–8.

Spencer, D. E.

P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

Young, M.

M. Young, “Objective measurement and characterization of scratch standards,” in Proc. Soc. Photo-Opt. Instrum. Eng. 362, 86 (1982);“Can you describe optical surface quality with one or two numbers” Proc. Soc. Photo-Opt. Instrum. Eng.406 (1983), in press.

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

M. Young, “Objective measurement and characterization of scratch standards,” in Proc. Soc. Photo-Opt. Instrum. Eng. 362, 86 (1982);“Can you describe optical surface quality with one or two numbers” Proc. Soc. Photo-Opt. Instrum. Eng.406 (1983), in press.

Other (5)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chaps. 5–8.

P. Moon, D. E. Spencer, Field Theory Handbook (Springer, Berlin, 1961).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 1.

Military specification MIL-0-13830A, 1963. For the use of present standards.

J. M. Elson, H. E. Bennett, J. M. Bennett, “Scattering from Optical Surfaces,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 7.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Nominal observation conditions for scattered radiation from a scratch.

Fig. 2
Fig. 2

Example of groove pattern for inducing scattering in an artifact.

Fig. 3
Fig. 3

Effects of changes in incoherence and wavelength.

Fig. 4
Fig. 4

Showing an alpha–beta change and a groove width change.

Fig. 5
Fig. 5

Lowest range of scattering strength, 4.6 × 10−6 to 2.3 × 10−5.

Fig. 6
Fig. 6

Midrange of scattering strength, 2.5 × 10−5 to 9.0 × 10−5.

Fig. 7
Fig. 7

Highest range of scattering strength, 5.8 × 10−5 to 1.24 × 10−3.

Fig. 8
Fig. 8

New range of scattering strength with reduced alpha and beta.

Fig. 9
Fig. 9

Showing the effects of a factor of 10 reduction in the incoherence parameter pf.

Tables (1)

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Table I Region Specification

Equations (51)

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[ x 2 + z 2 + k 2 ] E ( x , z ) = 0
[ x 2 + z 2 + k 2 n 2 ] E ( x , z ) = 0
E a ( x , z ) = + ds exp ( i s x ) [ C a U ( a , z ) + D a V ( a , z ) ] ,
U ( a , z ) exp ( i k a z )
V ( a , z ) exp ( i k a z )
[ E 1 E 2 ] | z = h ( x ) = 0 .
{ z x [ h ( x ) ] x } [ E 1 E 2 ] | z = h ( x ) = 0 .
[ E 2 E 3 ] | z = z 1 = 0 .
z [ E 2 E 3 ] | z = z 1 = 0 .
C 2 = U 3 ( g 1 U 2 C 3 + g 2 U 4 D 3 ) ,
D 2 = U 1 ( g 2 U 2 C 3 + g 1 U 4 D 3 ) .
D ̂ 1 C ̂ 2 D ̂ 2 = ds [ M 2 ( s 1 , s ) C 2 + M 3 ( s 1 , s ) D 2 M 1 ( s 1 , s ) D 1 ] ,
k ̂ 1 D ̂ 1 + k ̂ 2 ( D ̂ 2 C ̂ 2 ) = k ds { G 2 [ C 2 M 2 ( s 1 , s ) D 2 M 3 ( s 1 , s ) ] + G 1 [ D 1 M 1 ( s 1 , s ) ] } .
g 1 ( 1 + k 1 / k 2 ) / 2 , g 2 ( 1 k 1 / k 2 ) / 2 , G 1 ( k 2 s 1 s ) / [ k 1 ( s ) k ] , G 2 ( k 2 n 2 s 1 s ) / [ k 2 ( s ) k ] , U 1 U ( 2 , z 1 ) , U 2 U ( 3 , z 1 ) , U 3 V ( 2 , z 1 ) , and U 4 V ( 3 , z 1 ) .
R D U 4 g 1 g 2 ( U 3 U 1 ) / [ U 2 ( g 2 2 U 1 g 1 2 U 3 ) ] ,
T D U 1 U 3 U 4 ( g 2 2 g 1 2 ) / ( g 2 2 U 1 g 1 2 U 3 ) .
M r ( s 1 , s ) d x 2 π exp [ i ( s s 1 ) x ] [ F r 1 ] ,
F 1 V [ 1 , h ( x ) ] , F 2 U [ 2 , h ( x ) ] , and F 3 V [ 2 , h ( x ) ] .
k 1 = k ̂ 1 = k , k 2 = k ̂ 2 = k n , g 1 = ( 1 + n ) / 2 n ,
g 2 = ( n 1 ) / 2 n , G 1 = 1 , G 2 = n , F 1 = exp [ ikh ( x ) ] ,
F 2 = exp [ iknh ( x ) ] , F 3 = 1 / F 2 , U 1 = exp [ ikn z 1 ] ,
U 3 = 1 / U 1 , U 2 = exp ( i k z 1 ) , and U 4 = 1 / U 2 .
Z ̂ 1 ds [ M 2 g 2 + M 1 g 1 M 1 ] T ,
Z ̂ 2 d s [ G 2 ( g 2 M 2 g 1 M 3 ] + G 1 M 1 ] T ,
Y ̂ 1 = ( U ̂ 3 g ̂ 1 + U ̂ 1 g ̂ 2 ) U ̂ 2 ,
Y ̂ 2 = ( g ̂ 2 U ̂ 1 U ̂ 3 g ̂ 1 ) k ̂ 2 U ̂ 2 / k .
S ̂ 1 Y ̂ 1 S ̂ 3 = Z ̂ 1 ,
k ̂ 1 S ̂ 1 / k + Y ̂ 2 S ̂ 3 = Z ̂ 2 ,
[ Y ̂ 2 k ̂ 1 Y ̂ 1 / k ] S ̂ 1 = Y ̂ 2 Z ̂ 1 + Y ̂ 1 Z ̂ 2 .
S ̂ 1 = n dsT ( s ) [ g 2 2 U 1 ( M 1 M 2 ) / g 1 + ( M 3 M 1 ) g 1 ] ,
T ( s ) T 0 d s 0 exp [ ( s s 0 ) 2 / p 2 ] B ( s 0 ) ,
T 0 D 0 U 1 U 3 U 4 ( g 2 2 g 1 2 ) / ( g 2 2 U 1 g 1 2 U 3 ) .
D = D 0 d s 0 exp [ ( s s 0 ) 2 / p 2 ] B ( s 0 ) ,
B ( s 0 ) = 0 ,
B ( s 0 ) B * ( s ¯ 0 ) = δ ( s 0 s ¯ 0 ) f ( s 0 ) .
f ( s 0 ) = exp [ s 0 2 / p f 2 ] ,
W ( s 1 , s ) = n g 1 w π [ M 3 ( s 1 , s ) M 1 ( s 1 , s ) ] .
S 1 ( s 1 ) = T 0 d s 0 W ( s 1 , s 0 ) B ( s 0 ) ,
T = T 0 d s 0 B ( s 0 ) exp [ s 0 2 / p 2 ] .
F ( s 1 ) = d s 0 W W * T 0 T 0 * exp [ s 0 2 / p f 2 ] .
G ( 0 ) = T 0 T 0 * w x π ,
1 / w x 2 = 1 / p f 2 + 2 / p 2 .
R ( s 1 ) F ( s 1 ) / G ( 0 ) = d s 0 exp [ s 0 2 / p f 2 ] W W * / ( w x π ) .
x b = λ 0 b ( α + β b ) ,
W = W a W b W c ,
W a = c 2 n p π 2 π g 1 [ exp ( i k 0 z 2 n ) exp ( i k 0 z 2 ) ] ,
W b = b exp [ i ( s 0 s 1 ) x b ] ,
W c = { sin [ ( s 0 s 1 ) c / 2 ( s 0 s 1 ) c / 2 ] } .
S N ( s 1 ) = W c 2 Y * b , b 1 cos [ s 1 ( x b x b 1 ) ] × exp { ( x b x b 1 ) p f / 2 ] 2 } .
S f W a W a * Y ( p f / w x ) .
R ( s 1 ) = S f * S N ( s 1 ) .

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