Abstract

A light beam generated by two laser-illuminated circular apertures in a plane screen and two properly adjusted phase-shifting glass plates exhibits a central irradiance minimum that makes it favorable for alignment applications. The far-field transverse irradiance distribution in such a beam is here determined theoretically and experimentally to facilitate appraisal of its suitability for this purpose. Criteria aiding optimization of the beam and factors that degrade the beam and produce alignment errors are discussed. Alignment accuracy depends upon correspondence of a reference beam to a mathematically ideal, perfectly symmetrical shape. It is shown that a beam closely approaching the ideal can be easily produced by this method. Alignment accuracy is then found to be limited chiefly by air turbulence and stray light, which in the experimental investigation limited precision to ±10 μm at 8.2 m.

© 1972 Optical Society of America

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References

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  1. A. C. S. van Heel, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), vol. 1, p. 289.
    [CrossRef]
  2. A.C.S. van Heel, in Advanced Optical Techniques, A.C.S. van Heel, Ed. (North-Holland Publishing Co., Amsterdam, 1967), p. 447.
  3. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954).
    [CrossRef]
  4. W. H. Steel, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, New York, 1960) p. 181.
  5. J. Dyson, in Optics in Metrology, P. Mollet, Ed. (Pergamon PressNew York, 1960), p. 169.
  6. A.C.S. van Heel, J. Opt. Soc. Am. 40, 809 (1950).
    [CrossRef]
  7. A. Kastler, Rev. Opt. 29, 304 (1950).
  8. A. Kastler, Compt. Rend. 230, 1052 (1950).
  9. H. Wolter, Ann. Phys. 7, 341 (1950).
    [CrossRef]
  10. H. Wolter, Z. Naturforsch. 5a, 139 (1950).
  11. H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), vol. 1, p. 155.
    [CrossRef]
  12. J. H. Dunn, D. D. Howard, Electronics 33, 50 (22April1960).
  13. R. M. Page, I.R.E. Convention Record 8, 132 (1955).
  14. H. D. Betz, Appl. Opt. 8, 1007 (1969).
    [CrossRef] [PubMed]
  15. J. F. Ramsay, Marconi Rev. 10, 17, 41 (1947).
  16. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).
  17. G. B. Airy, Trans. Cambridge Philos. Soc. 5, 283 (1834).

1969 (1)

1960 (1)

J. H. Dunn, D. D. Howard, Electronics 33, 50 (22April1960).

1955 (1)

R. M. Page, I.R.E. Convention Record 8, 132 (1955).

1954 (1)

1950 (5)

A.C.S. van Heel, J. Opt. Soc. Am. 40, 809 (1950).
[CrossRef]

A. Kastler, Rev. Opt. 29, 304 (1950).

A. Kastler, Compt. Rend. 230, 1052 (1950).

H. Wolter, Ann. Phys. 7, 341 (1950).
[CrossRef]

H. Wolter, Z. Naturforsch. 5a, 139 (1950).

1947 (1)

J. F. Ramsay, Marconi Rev. 10, 17, 41 (1947).

1834 (1)

G. B. Airy, Trans. Cambridge Philos. Soc. 5, 283 (1834).

Airy, G. B.

G. B. Airy, Trans. Cambridge Philos. Soc. 5, 283 (1834).

Betz, H. D.

Dunn, J. H.

J. H. Dunn, D. D. Howard, Electronics 33, 50 (22April1960).

Dyson, J.

J. Dyson, in Optics in Metrology, P. Mollet, Ed. (Pergamon PressNew York, 1960), p. 169.

Howard, D. D.

J. H. Dunn, D. D. Howard, Electronics 33, 50 (22April1960).

Kastler, A.

A. Kastler, Rev. Opt. 29, 304 (1950).

A. Kastler, Compt. Rend. 230, 1052 (1950).

McLeod, J. H.

Page, R. M.

R. M. Page, I.R.E. Convention Record 8, 132 (1955).

Ramsay, J. F.

J. F. Ramsay, Marconi Rev. 10, 17, 41 (1947).

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).

Steel, W. H.

W. H. Steel, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, New York, 1960) p. 181.

van Heel, A. C. S.

A. C. S. van Heel, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), vol. 1, p. 289.
[CrossRef]

van Heel, A.C.S.

A.C.S. van Heel, J. Opt. Soc. Am. 40, 809 (1950).
[CrossRef]

A.C.S. van Heel, in Advanced Optical Techniques, A.C.S. van Heel, Ed. (North-Holland Publishing Co., Amsterdam, 1967), p. 447.

Wolter, H.

H. Wolter, Ann. Phys. 7, 341 (1950).
[CrossRef]

H. Wolter, Z. Naturforsch. 5a, 139 (1950).

H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), vol. 1, p. 155.
[CrossRef]

Ann. Phys. (1)

H. Wolter, Ann. Phys. 7, 341 (1950).
[CrossRef]

Appl. Opt. (1)

Compt. Rend. (1)

A. Kastler, Compt. Rend. 230, 1052 (1950).

Electronics (1)

J. H. Dunn, D. D. Howard, Electronics 33, 50 (22April1960).

I.R.E. Convention Record (1)

R. M. Page, I.R.E. Convention Record 8, 132 (1955).

J. Opt. Soc. Am. (2)

Marconi Rev. (1)

J. F. Ramsay, Marconi Rev. 10, 17, 41 (1947).

Rev. Opt. (1)

A. Kastler, Rev. Opt. 29, 304 (1950).

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, Trans. Cambridge Philos. Soc. 5, 283 (1834).

Z. Naturforsch. (1)

H. Wolter, Z. Naturforsch. 5a, 139 (1950).

Other (6)

H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), vol. 1, p. 155.
[CrossRef]

W. H. Steel, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, New York, 1960) p. 181.

J. Dyson, in Optics in Metrology, P. Mollet, Ed. (Pergamon PressNew York, 1960), p. 169.

A. C. S. van Heel, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), vol. 1, p. 289.
[CrossRef]

A.C.S. van Heel, in Advanced Optical Techniques, A.C.S. van Heel, Ed. (North-Holland Publishing Co., Amsterdam, 1967), p. 447.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949).

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Figures (4)

Fig. 1
Fig. 1

Schematic of two-aperture arrangement for producing a light beam with odd-symmetry: D = aperture diameter; 2a = aperture spacing.

Fig. 2
Fig. 2

Variation of the rate of change of the transverse field distribution at an axial point with normalized distance along the z axis, for three values of hole spacing.

Fig. 3
Fig. 3

Transverse irradiance distribution—diffraction pattern—produced by two circular apertures radiating out of phase: incident light wavelength = 632.8 nm; aperture diameter = 1.128 mm; aperture spacing = 2.752 mm; plane of observation at 8.196 m from aperture plane, corresponding to zN = 2.037.

Fig. 4
Fig. 4

Transverse irradiance distribution produced by two circular apertures radiating out of phase: —, calculated from amplitude distribution obtained from Eq. (4); …., measured; incident light wavelength = 632.8 nm; aperture diameter = 1.128 mm; aperture spacing = 2.752 mm; plane of observation 8.196 m from aperture plane, corresponding to zN = 2.037.

Equations (16)

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A 1 = 2 [ J 1 ( q 1 ) / q ] 1 e - j ϕ 1 ;
q 1 = ( π D / λ ) [ ( y - a ) / z ] ,
A 2 = 2 [ J 1 ( q 2 ) / q 2 ] e - j ϕ 2 ;
q 2 = ( π D / λ ) [ ( y + a ) / z ] ,
A = 2 { [ J 1 ( q 1 ) / q 1 ] e - j ϕ 1 + [ J 1 ( q 2 ) / q 2 ] e - j ( ϕ 2 + θ ) } .
A = 2 { [ J 1 ( q 1 ) / q 1 ] e - j ϕ 1 - [ J 1 ( q 2 ) / q 2 ] e - j ϕ 2 } .
ϕ 1 ( 2 π / λ ) [ ( a - y ) 2 / 2 z ]
ϕ 2 ( 2 π / λ ) [ ( a + y ) 2 / 2 z ]
A - j 2 k y [ J 1 ( q 2 ) / q 2 ] .
d A / d y y = 0 - j ( 8 / D ) J 1 ( π D a / λ z ) .
z min z f = 2 D 2 / λ .
z m = π D a / 3.83 λ .
a / D = 2.44.
z N = z / z f = z / ( 2 D 2 / λ ) ,
d A / d y y = 0 - j ( 8 / D ) J 1 [ ( a / D ) ( π / 2 z N ) ] .
maximum irradiance in measured distribution minimum irradiance at center of measured distribution

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