Abstract

A generalized method for determining the properties of a Gaussian laser beam focus including waist radius, position, and intensity distribution is presented. Of significance to in situ laser diagnostic methods is a unique analysis of illuminated area as a function of intensity in a plane of the laser beam. The results apply for nontruncating diffraction-limited optical systems and are important in determination of the optical sampling volume of laser instruments for particulate and gaseous species analysis.

© 1978 Optical Society of America

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References

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  1. B. Y. H. Liu, R. N. Berglund, J. K. Agarwal, Atmos. Environ. 8, 717 (1974).
    [CrossRef]
  2. E. D. Hirleman, S. L. K. Wittig, Sixteenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1977), p. 245.
    [CrossRef]
  3. E. D. Hirleman, S. L. K. Wittig, in CLEOS, 25–27 May 1976, Digest of Technical Papers (Optical Society of America, Washington, D.C., 1976), paper TUF4.
  4. W. M. Farmer, Appl. Opt. 15, 1984 (1976).
    [CrossRef] [PubMed]
  5. E. D. Hirleman, “Optical Technique for Particular Characterization in Combustion Environments: The Multiple Ratio Single Particle Counter,” Ph.D. Thesis, Purdue U., West Lafayette, Ind. (1977).
  6. L. D. Dickson, Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  7. D. J. Innes, A. L. Bloom, “Design of Optical Systems for Use With Laser Beams,” Spectra-Physics Laser Technical Bulletin No. 5 (Spectra-Physics, Inc., Mt. View, Calif., 1966).
  8. H. T. Hunt, P. A. Renard, R. G. Nelson, Appl. Opt. 15, 1458 (1976).
    [CrossRef] [PubMed]

1976

1974

B. Y. H. Liu, R. N. Berglund, J. K. Agarwal, Atmos. Environ. 8, 717 (1974).
[CrossRef]

1970

Agarwal, J. K.

B. Y. H. Liu, R. N. Berglund, J. K. Agarwal, Atmos. Environ. 8, 717 (1974).
[CrossRef]

Berglund, R. N.

B. Y. H. Liu, R. N. Berglund, J. K. Agarwal, Atmos. Environ. 8, 717 (1974).
[CrossRef]

Bloom, A. L.

D. J. Innes, A. L. Bloom, “Design of Optical Systems for Use With Laser Beams,” Spectra-Physics Laser Technical Bulletin No. 5 (Spectra-Physics, Inc., Mt. View, Calif., 1966).

Dickson, L. D.

Farmer, W. M.

Hirleman, E. D.

E. D. Hirleman, S. L. K. Wittig, in CLEOS, 25–27 May 1976, Digest of Technical Papers (Optical Society of America, Washington, D.C., 1976), paper TUF4.

E. D. Hirleman, “Optical Technique for Particular Characterization in Combustion Environments: The Multiple Ratio Single Particle Counter,” Ph.D. Thesis, Purdue U., West Lafayette, Ind. (1977).

E. D. Hirleman, S. L. K. Wittig, Sixteenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1977), p. 245.
[CrossRef]

Hunt, H. T.

Innes, D. J.

D. J. Innes, A. L. Bloom, “Design of Optical Systems for Use With Laser Beams,” Spectra-Physics Laser Technical Bulletin No. 5 (Spectra-Physics, Inc., Mt. View, Calif., 1966).

Liu, B. Y. H.

B. Y. H. Liu, R. N. Berglund, J. K. Agarwal, Atmos. Environ. 8, 717 (1974).
[CrossRef]

Nelson, R. G.

Renard, P. A.

Wittig, S. L. K.

E. D. Hirleman, S. L. K. Wittig, Sixteenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1977), p. 245.
[CrossRef]

E. D. Hirleman, S. L. K. Wittig, in CLEOS, 25–27 May 1976, Digest of Technical Papers (Optical Society of America, Washington, D.C., 1976), paper TUF4.

Appl. Opt.

Atmos. Environ.

B. Y. H. Liu, R. N. Berglund, J. K. Agarwal, Atmos. Environ. 8, 717 (1974).
[CrossRef]

Other

E. D. Hirleman, S. L. K. Wittig, Sixteenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1977), p. 245.
[CrossRef]

E. D. Hirleman, S. L. K. Wittig, in CLEOS, 25–27 May 1976, Digest of Technical Papers (Optical Society of America, Washington, D.C., 1976), paper TUF4.

E. D. Hirleman, “Optical Technique for Particular Characterization in Combustion Environments: The Multiple Ratio Single Particle Counter,” Ph.D. Thesis, Purdue U., West Lafayette, Ind. (1977).

D. J. Innes, A. L. Bloom, “Design of Optical Systems for Use With Laser Beams,” Spectra-Physics Laser Technical Bulletin No. 5 (Spectra-Physics, Inc., Mt. View, Calif., 1966).

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Figures (5)

Fig. 1
Fig. 1

Gaussian beam properties for a single-lens system indicating coordinate systems and nomenclature.

Fig. 2
Fig. 2

Nondimensional output waist position νo.

Fig. 3
Fig. 3

Nondimensional output waist radius μo.

Fig. 4
Fig. 4

Intensity contours near the output waist of a nontruncated diffraction-limited Gaussian beam focus, relative to peak intensity at focal center.

Fig. 5
Fig. 5

Cross-sectional area enclosed by laser focus intensity contours of fraction peak intensity I/Io normalized to area within the 1/e2 intensity contour.

Equations (18)

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I ( r , z ) I ( o , z ) = exp [ 2 r 2 / w ( z ) 2 ] ,
z o = f [ z i ( z i f ) + f F 2 ] ( z i f ) 2 + f F 2 [ Eq . ( 46 ) 6 ] ,
w o = w i f [ ( z i f ) 2 + f F 2 ] 1 / 2 [ Eq . ( 47 ) 6 ] ,
η ( z i ) / f ,
γ ( π w i 2 ) / ( z i λ ) ;
ν z / ( z i ) ,
μ [ w ( z ) ] / w i .
ν o = η ( 1 + γ 2 ) 1 η 2 ( 1 + γ 2 ) 2 η + 1 ,
μ o = [ ( η 1 ) 2 + η 2 γ 2 ] 1 / 2 ,
ξ 1 γ μ o 2 ( z z 0 z i ) ,
ρ r / ( w o ) ,
I ( ρ , ξ ) I ( o , o ) = 1 ( 1 + ξ 2 ) exp [ 2 ρ 2 / ( 1 + ξ 2 ) ] .
A ( I / I o ) = 4 o ξ max o ρ max d ρ d ξ
A ( 1 / e 2 ) = 10.2278 [ ( π w o 3 ) / λ ] ,
ln A ( I / I o ) A ( 1 / e 2 ) = 2.0992 + ln I / I o .
0 ln I / I o 2 , A ( I / I o ) A ( 1 / e 2 ) = + 0.2205 [ ln ( I / I o ) ] + 0.0464 [ ln ( I / I o ) ] 2 + 0.0455 [ ln ( I / I o ) ] 3 ,
2 < ln I / I o 6 , A ( I / I o ) A ( 1 / e 2 ) = 11.38 16.73 [ ln ( I / I o ) ] + 9.433 [ ln ( I / I o ) ] 2 2.280 [ ln ( I / I o ) ] 3 + 0.2246 [ ln ( I / I o ) ] 4 ,
6 < ln ( I / I o ) < , A ( I / I o ) A ( 1 / e 2 ) = 0.1226 ( I o / I ) .

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