Abstract

The intensity distribution of the beam from a laser operated in the TEM00 mode is Gaussian. In some applications it is desirable to have a uniform intensity over a certain region in space. For example, when a Gaussian beam is incident on a smooth surface containing small isolated defects the amount of light scattered by a defect will depend on the position of the defect relative to the center of the beam. In the past, several techniques have been devised to convert a Gaussian intensity profile into a uniform intensity over a specified region in space. In the present work a method of normalization is described that makes direct use of the Gaussian intensity distribution of the TEM00 mode. By this method, the amount of light scattered by a defect can be normalized to the value that would be observed if the defect were located at the center of the beam for a defect small in size compared with the 1/e2 diameter of the Gaussian profile. Experimental data were obtained that verify the theory.

© 1980 Optical Society of America

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References

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  1. C. S. Ih, Appl. Opt. 11, 694 (1972).
    [CrossRef]
  2. P. E. Klingsporn, Appl. Opt. 15, 2355 (1976).
    [CrossRef] [PubMed]
  3. S. S. Charschan, Ed., Lasers in Industry (Van Nostrand Reinhold, New York, 1972), pp. 481–483.
  4. J. L. Kreuzer, “Coherent Light Optical Systems Yielding an Output Beam of Desired Intensity Distribution at a Desired Equiphase Surface,” U.S. Patent3,476,463 (4November1969).

1976 (1)

1972 (1)

Ih, C. S.

Klingsporn, P. E.

Kreuzer, J. L.

J. L. Kreuzer, “Coherent Light Optical Systems Yielding an Output Beam of Desired Intensity Distribution at a Desired Equiphase Surface,” U.S. Patent3,476,463 (4November1969).

Appl. Opt. (2)

Other (2)

S. S. Charschan, Ed., Lasers in Industry (Van Nostrand Reinhold, New York, 1972), pp. 481–483.

J. L. Kreuzer, “Coherent Light Optical Systems Yielding an Output Beam of Desired Intensity Distribution at a Desired Equiphase Surface,” U.S. Patent3,476,463 (4November1969).

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Figures (5)

Fig. 1
Fig. 1

Arrangement in which a collimated laser beam of 1/e2 diameter 2a is incident normally on a reflective surface. The retroreflected light from the surface is diverted by a beam splitter and brought to focus at point P by a lens. A small defect on the surface will scatter light outside the focused spot at P, in the x′-z′ plane, but the amount of scatter for a given defect is dependent on the position of the defect relative to center O of the Gaussian intensity distribution.

Fig. 2
Fig. 2

Mapping of illuminated spots on the surface of Fig. 1 showing the geometry for the case of minimum overlap (at the 1/e2 diameter of 2a) so that no part of the surface is missed. Defect at point Pd would be detected by two spots S1 and S2.

Fig. 3
Fig. 3

Defect at point Pd is illuminated by four adjacent spots S1, S2, S3, and S4. Centers O1, O2, O3, and O4 of the illuminated spots form a square of side L. Defect is at distances r1, r2, and r3 from the centers of spots S1, S2, and S3, respectively.

Fig. 4
Fig. 4

Mapping of illuminated spots on surface of Fig. 1 for the case in which centers of adjacent spots do not lie on the vertices of a square. Distance between adjacent spots on a given scan is L, and adjacent scans are also separated by distance L, but centers on adjacent scans are offset by the distance h.

Fig. 5
Fig. 5

Mapping of centers of illuminated spots for the case in which there is an offset h. Signals obtained from the spots with centers at points O1, O2, and O3 are used for calculating the normalization from Eq. (17a). The signal from an illuminated spot with center at point ON,M can be calculated from Eqs. (19)(21), where u is the distance from the defect at point Pd to point ON,M. If N = M = 0, the value of u is the distance to point O4.

Tables (3)

Tables Icon

Table I Comparison of Observed and Calculated Values of W4 for Six Different Defects a

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Table II Comparison of Observed and Calculated Values of WN,M for Three Different Defects a

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Table III Comparison of Normalized Values W0 Determined with Two Different Distances L Between Adjacent Illuminated Spots for Each of Five Different Defects

Equations (23)

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I ( r ) = I 0 exp ( - 2 r 2 / a 2 ) ,
W 1 = W 0 exp ( - 2 r 1 2 / a 2 ) W 2 = W 0 exp ( - 2 r 2 2 / a 2 ) } ,
W 1 = W 0 exp ( - 2 r 1 2 / a 2 ) ,
W 2 = W 0 exp ( - 2 r 2 2 / a 2 ) ,
W 3 = W 0 exp ( - 2 r 3 2 / a 2 ) ,
r 2 2 - r 1 2 = 1 2 a 2 ln ( W 1 / W 2 ) ,
r 3 2 - r 1 2 = 1 2 a 2 ln ( W 1 / W 3 ) .
r 1 cos β + r 2 cos ψ = L ,
r 3 cos α + r 1 sin β = L ,
r 1 sin β = r 2 sin ψ ,
r 1 cos β = r 3 sin α .
2 L r 3 cos α - A = L 2 + r 3 2 - r 2 2 ,
L 2 - A = 2 L r 3 sin α ,
r 2 2 - A = r 3 2 - B ,
A = 1 2 a 2 ln ( W 1 / W 2 ) ,             B = 1 2 a 2 ln ( W 1 / W 3 ) .
r 3 2 = ( L 2 - A ) 2 + ( L 2 + B ) 2 4 L 2 .
W 0 = W 3 exp [ ( L 2 - A ) 2 + ( L 2 + B ) 2 2 a 2 L 2 ] .
r 1 cos β = r 3 sin α - h ,
r 3 2 = [ ( L + h ) 2 - A - h 2 ] 2 + { B + L 2 + h 2 - h L [ ( L + h ) 2 - A - h 2 ] } 2 4 L 2 ,
W 0 = W 3 exp [ [ ( L + h ) 2 - A - h 2 ] 2 + { B + L 2 + h 2 - h L [ ( L + h ) 2 - A - h 2 ] } 2 2 a 2 L 2 ] ;
W N , M = W 0 exp [ - 2 ( u x 2 + u y 2 ) / a 2 ] ,
u x = M L + L - N h - r 3 sin α ,             u y = N L + r 3 cos α ,
α = arcsin [ ( L + h ) 2 - A - h 2 2 L r 3 ] .

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