Abstract

The application of the Ronchi ruling beam characterization method to axially symmetric optical beams is analyzed. Specific results are derived for the Airy and focused annulus diffraction patterns. Plots of the ratio of minimum to maximum transmitted optical power vs the first null radius of the beam functions show that for the Airy pattern and other focused annuli with obscuration ratios smaller than ~0.30, the method should be as useful as with Gaussian beams.

© 1990 Optical Society of America

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References

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  1. L. D. Dickson, “Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 18, 70–75 (1979).
  2. E. C. Broockman, L. D. Dickson, R. S. Fortenberry, “Generalization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 22, 643–647 (1983).
    [CrossRef]
  3. M. A. Karim, “Measurement of Gaussian Beam Diameter Using Ronchi Rulings,” Electron. lett. 21, 427–429 (1985).
    [CrossRef]
  4. M. A. Karim et al., “Gaussian Laser-Beam-Diameter Measurement Using Sinusoidal and Triangular Rulings,” Opt. Lett. 12, 93–95 (1987).
    [CrossRef] [PubMed]
  5. D. K. Cohen, B. Little, F. S. Luecke, “Techniques for Measuring 1-μm Diam Gaussian Beams,” Appl. Opt. 23, 637–640 (1984).
    [CrossRef] [PubMed]
  6. R. Csomor, “Techniques for Measuring 1-μm Diam Gaussian Beams: Comment,” Appl. Opt. 24, 2295–2298 (1985).
    [CrossRef] [PubMed]
  7. J. Ebert, E. Kiesel, “Measurement of Laser-Induced Damage with an Unstable Resonator-Type Laser,” Appl. Opt. 23, 3759–3761 (1984).
    [CrossRef] [PubMed]
  8. R. M. O’Connell, R. A. Vogel, “Abel Inversion of Knife-Edge Data from Radially Symmetric Pulsed Laser Beams,” Appl. Opt. 26, 2528–2532 (1987).
    [CrossRef]
  9. W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
    [CrossRef]
  10. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, NJ, 1981), p. 106.
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 395–418.
  12. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, U.P.New York, 1944), p. 405.
  13. R. G. Stanton, Numerical Methods for Science and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1961), p. 116.
  14. Samuel M. Selby, editor, Standard Mathematical Tables (CRC Press, Cleveland, 1965), p. 350.
  15. W. T. Welford, “Use of Annular Apertures to Increase Focal Depth,” J. Opt. Soc. Am. 50, 749–753 (1960).
    [CrossRef]
  16. B. L. Mehta, “Total Illumination in an Aberration Free Annular Aperture,” Appl. Opt. 13, 736–737 (1974).
    [CrossRef] [PubMed]
  17. H. F. A. Tschunko, “Imaging Performance of Annular Apertures,” Appl. Opt. 13, 1820–1823 (1974).
    [CrossRef] [PubMed]

1987 (2)

1985 (2)

R. Csomor, “Techniques for Measuring 1-μm Diam Gaussian Beams: Comment,” Appl. Opt. 24, 2295–2298 (1985).
[CrossRef] [PubMed]

M. A. Karim, “Measurement of Gaussian Beam Diameter Using Ronchi Rulings,” Electron. lett. 21, 427–429 (1985).
[CrossRef]

1984 (2)

1983 (1)

E. C. Broockman, L. D. Dickson, R. S. Fortenberry, “Generalization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 22, 643–647 (1983).
[CrossRef]

1979 (1)

L. D. Dickson, “Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 18, 70–75 (1979).

1974 (2)

1969 (1)

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

1960 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 395–418.

Broockman, E. C.

E. C. Broockman, L. D. Dickson, R. S. Fortenberry, “Generalization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 22, 643–647 (1983).
[CrossRef]

Cohen, D. K.

Csomor, R.

Dickson, L. D.

E. C. Broockman, L. D. Dickson, R. S. Fortenberry, “Generalization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 22, 643–647 (1983).
[CrossRef]

L. D. Dickson, “Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 18, 70–75 (1979).

Ebert, J.

Fortenberry, R. S.

E. C. Broockman, L. D. Dickson, R. S. Fortenberry, “Generalization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 22, 643–647 (1983).
[CrossRef]

Karim, M. A.

M. A. Karim et al., “Gaussian Laser-Beam-Diameter Measurement Using Sinusoidal and Triangular Rulings,” Opt. Lett. 12, 93–95 (1987).
[CrossRef] [PubMed]

M. A. Karim, “Measurement of Gaussian Beam Diameter Using Ronchi Rulings,” Electron. lett. 21, 427–429 (1985).
[CrossRef]

Kiesel, E.

Krupke, W. F.

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

Little, B.

Luecke, F. S.

Mehta, B. L.

O’Connell, R. M.

Sooy, W. R.

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

Stanton, R. G.

R. G. Stanton, Numerical Methods for Science and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1961), p. 116.

Tschunko, H. F. A.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, NJ, 1981), p. 106.

Vogel, R. A.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, U.P.New York, 1944), p. 405.

Welford, W. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 395–418.

Appl. Opt. (6)

Electron. lett. (1)

M. A. Karim, “Measurement of Gaussian Beam Diameter Using Ronchi Rulings,” Electron. lett. 21, 427–429 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

L. D. Dickson, “Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 18, 70–75 (1979).

E. C. Broockman, L. D. Dickson, R. S. Fortenberry, “Generalization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter,” Opt. Eng. 22, 643–647 (1983).
[CrossRef]

Opt. Lett. (1)

Other (5)

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, NJ, 1981), p. 106.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 395–418.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, U.P.New York, 1944), p. 405.

R. G. Stanton, Numerical Methods for Science and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1961), p. 116.

Samuel M. Selby, editor, Standard Mathematical Tables (CRC Press, Cleveland, 1965), p. 350.

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Figures (3)

Fig. 1
Fig. 1

Geometry used to analyze the characterization of an arbitrary axially symmetric beam with an ideal Ronchi ruling: (a) for the calculation of Pmin; (b) for the calculation of Pmax.

Fig. 2
Fig. 2

Geometry of an annular aperture (inset) and normalized cross sections of two axially symmetric beam functions: (a) the Airy pattern [Eq. (6)] or focused annulus [Eq. (4)] with = 0.0; (b) the focused annulus [Eq. (4)] with = 0.5.

Fig. 3
Fig. 3

Plots of K vs r1/L obtained using Rc = 200 in Eq. (8) and M = 5 in Eq. (12) for various focused annuli: (a) = 0.0 (the Airy pattern); (b) = 0.25; (c) = 0.50.

Equations (19)

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P = A I ( r ) d A .
P = x R r I ( r ) d r 0 cos - 1 ( x / r ) d θ = x R r I ( r ) cos - 1 ( x / r ) d r .
ϕ ( x ) = x R r I ( r ) cos - 1 ( x / r ) d r .
P min = 4 [ ϕ ( L / 2 ) - ϕ ( 3 L / 2 ) + ϕ ( 5 L / 2 ) - ϕ ( 7 L / 2 ) + ] = 4 m = 1 { ϕ [ ( 4 m - 3 ) L / 2 ] - ϕ [ ( 4 m - 1 ) L / 2 ] } , P max = 4 [ ϕ ( 0 ) - ϕ ( L / 2 ) + ϕ ( 3 L / 2 ) - ϕ ( 5 L / 2 ) + ϕ ( 7 L / 2 ) + ] = 4 ( ϕ ( 0 ) - ϕ ( L / 2 ) + m = 1 { ϕ [ ( 4 m - 1 ) L / 2 ] - ϕ [ ( 4 m + 1 ) L / 2 ] } ) .
K = m = 1 { ϕ [ ( 4 m - 3 ) L / 2 ] - ϕ [ ( 4 m - 1 ) L / 2 ] } ϕ ( 0 ) - ϕ ( L / 2 ) + m = 1 { ϕ [ ( 4 m - 1 ) L / 2 ] - ϕ [ ( 4 m + 1 ) L / 2 ] } ,
I ( r ) = I 0 / ( 1 - 2 ) 2 [ 2 J 1 ( r c ) r c - 2 2 J 1 ( r c ) r c ] 2 ,
c = 2 π α / f λ .
I ( r ) = I 0 [ 2 J 1 ( r c ) / r c ] 2 .
ϕ ( x c ) = 4 I 0 / ( 1 - 2 ) 2 c 2 y = x c = R c [ J 1 ( y ) - J 1 ( y ) ] 2 × ( 1 / y ) cos - 1 ( x c / y ) d y .
Θ ( z ) = ϕ ( z = x c ) 4 I 0 / ( 1 - 2 ) 2 c 2 = z R c [ J 1 ( y ) - J 1 ( y ) ] 2 ( 1 / y ) cos - 1 ( z / y ) d y .
K = m = 1 M { Θ [ ( 4 m - 3 ) L c / 2 ] - Θ [ ( 4 m - 1 ) L c / 2 ] } π ( 1 - 2 ) / 4 - Θ ( L c / 2 ) + m = 1 M { Θ [ ( 4 m - 1 ) L c / 2 ] - Θ [ ( 4 m + 1 ) L c / 2 ] } ,
Θ ( 0 ) = π 2 0 [ J 1 ( y ) - J 1 ( y ) ] 2 ( 1 / y ) d y = π 2 [ 0 J 1 2 ( y ) y d y + 2 0 J 1 2 ( y ) y d y - 2 0 J 1 ( y ) J 1 ( y ) y d y ] = π 2 ( 1 2 + 2 2 - 2 ) = π ( 1 - 2 ) 4 ,
0 J n ( a t ) J n ( b t ) t d t = ( b / a ) n 2 n
c = g ( ) / r 1 .
J 1 ( r c ) - J 1 ( r c ) = 0 ,
K = m = 1 M { Θ [ ( 4 m - 3 ) g ( ) L / 2 r 1 ] - Θ [ ( 4 m - 1 ) g ( ) L / 2 r 1 ] } π ( 1 - 2 ) / 4 - Θ [ g ( ) L / 2 r 1 ] + m = 1 M { Θ [ ( 4 m - 1 ) g ( ) L / 2 r 1 ] - Θ [ ( 4 m + 1 ) g ( ) L / 2 r 1 ] } ,
r 1 / L ( 4 M + 1 ) g ( ) / 2 R c ,
P ( R ) = 1 - J 0 2 ( R c ) - J 1 2 ( R c ) ,
α = g ( ) f λ / 2 π r 1 ,

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