Abstract

A simple method is presented for determining the waist position and the waist size of a Gaussian beam from measured spot sizes. The method does not require any least-squares process, and substitution of measured spot sizes directly into the formulas gives the waist parameters. Only few data are required to determine the parameters accurately as long as measured spot sizes contain small errors.

© 1986 Optical Society of America

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References

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  1. A. H. Firester, M. E. Heller, P. Sheng, “Knife-Edge Scanning Measurements of Subwavelength Focused Light Beams,” Appl. Opt. 16, 1971 (1977).
    [CrossRef] [PubMed]
  2. M. Mauck, “Knife-Edge Profiling of Q-Switched Nd:YAG Laser Beam and Waist,” Appl. Opt. 18, 599 (1979).
    [CrossRef] [PubMed]
  3. J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983).
    [CrossRef] [PubMed]
  4. D. K. Cohen, B. Little, F. S. Luecke, “Techniques for Measuring 1-μm Diam Gaussian Beams,” Appl. Opt. 23, 637 (1984).
    [CrossRef] [PubMed]
  5. H. Nagashima, H. Tokiwa, “A Way to Look Directly at the Power Distribution of the Laser Beam,” Trans. IECE Jpn. 57-C, 167 (1974), in Japanese.
  6. R. L. McCally, “Measurement of Gaussian Beam Parameters,” Appl. Opt. 23, 2227 (1984).
    [CrossRef] [PubMed]
  7. A. Yoshida, T. Asakura, “A Simple Technique for Quickly Measuring the Spot Size of Gaussian Laser Beams,” Opt. Laser Technol. 8, 273 (1976).
    [CrossRef]
  8. E. Stijns, “Measuring the Spot Size of a Gaussian Beam with an Oscillating Wire,” IEEE J. Quantum Electron. QE-16, 1298 (1980).
    [CrossRef]
  9. Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.
  10. Y. C. Kiang, R. W. Lang, “Measuring Focused Gaussian Beam Spot Sizes: A Practical Method,” Appl. Opt. 22, 1296 (1983).
    [CrossRef] [PubMed]
  11. Y. Suzaki, A. Tachibana, “Measurement of the Gaussian Laser Beam Divergence,” Appl. Opt. 16, 1481 (1977).
    [CrossRef] [PubMed]
  12. S. R. Mallinson, G. Warnes, “Optimization of Thick Lenses for Single-Mode Optical-Fiber Microcomponents,” Opt. Lett. 10, 238 (1985).
    [CrossRef] [PubMed]
  13. K. S. Lee, F. S. Barnes, “Microlenses on the End of Single-Mode Optical Fibers for Laser Applications,” Appl. Opt. 24, 3134 (1985).
    [CrossRef] [PubMed]
  14. W. H. Carter, “Focal Shift and Concept of Effective Fresnel Number for a Gaussian Laser Beam,” Appl. Opt. 21, 1989 (1982).
    [CrossRef] [PubMed]
  15. G. D. Sucha, W. H. Carter, “Focal Shift for a Gaussian Beam: An Experimental Study,” Appl. Opt. 23, 4345 (1984).
    [CrossRef] [PubMed]
  16. S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
    [CrossRef] [PubMed]
  17. J. T. Luxon, D. E. Parker, J. Karkheck, “Waist Location and Rayleigh Range for Higher-Order Mode Laser Beams,” Appl. Opt. 23, 2088 (1984).
    [CrossRef] [PubMed]
  18. S. Nemoto, T. Makimoto, “Generalized Spot Size for a Higher-Order Beam Mode,” J. Opt. Soc. Am. 69, 578 (1979).
    [CrossRef]
  19. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.
  20. R. M. Herman, J. Pardo, T. A. Wiggins, “Diffraction and Focusing of Gaussian Beams,” Appl. Opt. 24, 1346 (1985).
    [CrossRef] [PubMed]
  21. H. R. Bilger, T. Habib, “Knife-Edge Scanning of an Astigmatic Gaussian Beam,” Appl. Opt. 24, 686 (1985).
    [CrossRef] [PubMed]
  22. T. D. Baxter, T. T. Saito, G. L. Shaw, R. T. Evans, R. A. Motes, “Mode Matching for a Passive Resonant Ring Laser Gyroscope,” Appl. Opt. 22, 2487 (1983).
    [CrossRef] [PubMed]
  23. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]

1985 (4)

1984 (4)

1983 (4)

1982 (1)

1980 (1)

E. Stijns, “Measuring the Spot Size of a Gaussian Beam with an Oscillating Wire,” IEEE J. Quantum Electron. QE-16, 1298 (1980).
[CrossRef]

1979 (2)

1977 (2)

1976 (1)

A. Yoshida, T. Asakura, “A Simple Technique for Quickly Measuring the Spot Size of Gaussian Laser Beams,” Opt. Laser Technol. 8, 273 (1976).
[CrossRef]

1974 (1)

H. Nagashima, H. Tokiwa, “A Way to Look Directly at the Power Distribution of the Laser Beam,” Trans. IECE Jpn. 57-C, 167 (1974), in Japanese.

1973 (1)

Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.

1966 (1)

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.

Asakura, T.

A. Yoshida, T. Asakura, “A Simple Technique for Quickly Measuring the Spot Size of Gaussian Laser Beams,” Opt. Laser Technol. 8, 273 (1976).
[CrossRef]

Barnes, F. S.

Baxter, T. D.

Bilger, H. R.

Carter, W. H.

Cohen, D. K.

Evans, R. T.

Firester, A. H.

Fujii, Y.

Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.

Garetz, B. A.

Habib, T.

Heller, M. E.

Herman, R. M.

Higashino, K.

Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.

Karkheck, J.

Khosrofian, J. M.

Kiang, Y. C.

Kogelnik, H.

Lang, R. W.

Lee, K. S.

Li, T.

Little, B.

Luecke, F. S.

Luxon, J. T.

Makimoto, T.

Mallinson, S. R.

Matsui, H.

Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.

Mauck, M.

McCally, R. L.

Motes, R. A.

Nagashima, H.

H. Nagashima, H. Tokiwa, “A Way to Look Directly at the Power Distribution of the Laser Beam,” Trans. IECE Jpn. 57-C, 167 (1974), in Japanese.

Nemoto, S.

Obayashi, S.

Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.

Pardo, J.

Parker, D. E.

Saito, T. T.

Self, S. A.

Shaw, G. L.

Sheng, P.

Stijns, E.

E. Stijns, “Measuring the Spot Size of a Gaussian Beam with an Oscillating Wire,” IEEE J. Quantum Electron. QE-16, 1298 (1980).
[CrossRef]

Sucha, G. D.

Suzaki, Y.

Tachibana, A.

Tokiwa, H.

H. Nagashima, H. Tokiwa, “A Way to Look Directly at the Power Distribution of the Laser Beam,” Trans. IECE Jpn. 57-C, 167 (1974), in Japanese.

Warnes, G.

Wiggins, T. A.

Yoshida, A.

A. Yoshida, T. Asakura, “A Simple Technique for Quickly Measuring the Spot Size of Gaussian Laser Beams,” Opt. Laser Technol. 8, 273 (1976).
[CrossRef]

Appl. Opt. (16)

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

A. H. Firester, M. E. Heller, P. Sheng, “Knife-Edge Scanning Measurements of Subwavelength Focused Light Beams,” Appl. Opt. 16, 1971 (1977).
[CrossRef] [PubMed]

W. H. Carter, “Focal Shift and Concept of Effective Fresnel Number for a Gaussian Laser Beam,” Appl. Opt. 21, 1989 (1982).
[CrossRef] [PubMed]

S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
[CrossRef] [PubMed]

Y. C. Kiang, R. W. Lang, “Measuring Focused Gaussian Beam Spot Sizes: A Practical Method,” Appl. Opt. 22, 1296 (1983).
[CrossRef] [PubMed]

T. D. Baxter, T. T. Saito, G. L. Shaw, R. T. Evans, R. A. Motes, “Mode Matching for a Passive Resonant Ring Laser Gyroscope,” Appl. Opt. 22, 2487 (1983).
[CrossRef] [PubMed]

J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983).
[CrossRef] [PubMed]

D. K. Cohen, B. Little, F. S. Luecke, “Techniques for Measuring 1-μm Diam Gaussian Beams,” Appl. Opt. 23, 637 (1984).
[CrossRef] [PubMed]

J. T. Luxon, D. E. Parker, J. Karkheck, “Waist Location and Rayleigh Range for Higher-Order Mode Laser Beams,” Appl. Opt. 23, 2088 (1984).
[CrossRef] [PubMed]

R. L. McCally, “Measurement of Gaussian Beam Parameters,” Appl. Opt. 23, 2227 (1984).
[CrossRef] [PubMed]

G. D. Sucha, W. H. Carter, “Focal Shift for a Gaussian Beam: An Experimental Study,” Appl. Opt. 23, 4345 (1984).
[CrossRef] [PubMed]

H. R. Bilger, T. Habib, “Knife-Edge Scanning of an Astigmatic Gaussian Beam,” Appl. Opt. 24, 686 (1985).
[CrossRef] [PubMed]

R. M. Herman, J. Pardo, T. A. Wiggins, “Diffraction and Focusing of Gaussian Beams,” Appl. Opt. 24, 1346 (1985).
[CrossRef] [PubMed]

K. S. Lee, F. S. Barnes, “Microlenses on the End of Single-Mode Optical Fibers for Laser Applications,” Appl. Opt. 24, 3134 (1985).
[CrossRef] [PubMed]

Y. Suzaki, A. Tachibana, “Measurement of the Gaussian Laser Beam Divergence,” Appl. Opt. 16, 1481 (1977).
[CrossRef] [PubMed]

M. Mauck, “Knife-Edge Profiling of Q-Switched Nd:YAG Laser Beam and Waist,” Appl. Opt. 18, 599 (1979).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

E. Stijns, “Measuring the Spot Size of a Gaussian Beam with an Oscillating Wire,” IEEE J. Quantum Electron. QE-16, 1298 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Laser Technol. (1)

A. Yoshida, T. Asakura, “A Simple Technique for Quickly Measuring the Spot Size of Gaussian Laser Beams,” Opt. Laser Technol. 8, 273 (1976).
[CrossRef]

Opt. Lett. (1)

Trans. IECE Jpn. (2)

Y. Fujii, S. Obayashi, K. Higashino, H. Matsui, “CO2 Laser Beam Guide by Ge Lenses and Measurement of Spotsize,” Trans. IECE Jpn. 56-C, 563 (1973), in Japanese.

H. Nagashima, H. Tokiwa, “A Way to Look Directly at the Power Distribution of the Laser Beam,” Trans. IECE Jpn. 57-C, 167 (1974), in Japanese.

Other (1)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.

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Figures (7)

Fig. 1
Fig. 1

Variations of s±(z) for (a) s1 > s2 and (b) s1 < s2.

Fig. 2
Fig. 2

Data points in the z-s plane. Solid curve represents s(z) calculated by the proposed method.

Fig. 3
Fig. 3

Data points in the z-s plane. Solid curve represents s(z) calculated by the proposed method. Position z = 0 coincides with the laser head.

Fig. 4
Fig. 4

Fictitious data points for a given spot size (solid curve). Broken curve represents a spot size calculated by the proposed method.

Fig. 5
Fig. 5

Δs0 vs z0 curve for determining the waist parameters by an iterative least-squares method applied to data points (zi,si), i = 1–6 in Fig. 2. R is the correlation coefficient between (ziz0)2 and s i 2.

Fig. 6
Fig. 6

Result of an iterative least-squares method applied separately to the left-hand and right-hand data in Fig. 2. Curves marked left and right are obtained from the left-hand and right-hand data, respectively.

Fig. 7
Fig. 7

Δs0 vs z0 curve for determining the waist parameters by an iterative least-squares method applied to four data points (zi,si), i = 1,4,7,10 in Fig. 3. Correlation coefficient R between (ziz0)2 and s i 2 is also shown.

Tables (2)

Tables Icon

Table I Experimental Data of a Spot Size

Tables Icon

Table II Data Combinations and Calculated Values of z 0 ± , s 0 ±

Equations (26)

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s ( z ) = s 0 [ 1 + ( z - z 0 k s 0 2 ) 2 ] 1 / 2 ,
( z - z 0 ) 2 = ( k s 0 ) 2 [ s 2 ( z ) - s 0 2 ] .
( z 1 - z 0 ) 2 = ( k s 0 ) 2 ( s 1 2 - s 0 2 ) , ( z 2 - z 0 ) 2 = ( k s 0 ) 2 ( s 2 2 - s 0 2 ) ,
z 0 ± = 2 u r - q [ t ± v ( 1 - r ) 1 / 2 ] q 2 + 4 r ,
s 0 ± = s ¯ { r [ p ± 2 ( 1 - r ) 1 / 2 ] q 2 + 4 r } 1 / 2 ,
s ¯ = ( s 1 s 2 ) 1 / 2 ,
p = s 1 s 2 + s 2 s 1 ,             q = s 1 s 2 - s 2 s 1 ,
r = ( z 1 - z 2 k s 1 s 2 ) 2 ,             t = z 1 s 2 s 1 - z 2 s 1 s 2 ,
u = z 1 + z 2 ,             v = z 1 - z 2 .
z 0 + > z 0 - when s 1 > s 2 , z 0 + < z 0 - when s 1 < s 2 , s 0 + > s 0 - .
s 2 ( z ) = s 0 2 + ( z - z 0 ) 2 / ( k s 0 ) 2 ,
w ( z ) = w 0 { 1 + [ 2 ( z - z 0 ) k w 0 2 ] 2 } 1 / 2 ,
z 0 ± = 2 u r - q [ t ± v ( 1 - r ) 1 / 2 ] q 2 + 4 r ,
w 0 ± = w ¯ { r [ p ± 2 ( 1 - r ) 1 / 2 ] q 2 + 4 r } 1 / 2 ,
w ¯ = ( w 1 w 2 ) 1 / 2 ,
p = w 1 w 2 + w 2 w 1 ,             q = w 1 w 2 - w 2 w 1 ,
r = [ 2 ( z 1 - z 2 ) k w 1 w 2 ] 2 ,             t = z 1 w 2 w 1 - z 2 w 1 w 2 ,
u = z 1 + z 2 ,             v = z 1 - z 2 .
z 0 = z 1 + ( k s 1 s 2 ) s 1 2 / ( s 1 2 + s 2 2 ) = z 2 - ( k s 1 s 2 ) s 2 2 / ( s 1 2 + s 2 2 ) ,
s 0 = s 1 s 2 / ( s 1 2 + s 2 2 ) 1 / 2 .
z 0 = z 1 + k s 1 2 / 2 = z 2 - k s 2 2 / 2 ,
s 0 = s 1 / 2 = s 2 / 2 ,
k s 0 2 = z 0 - z 1 = z 2 - z 0 .
s 1 = s 0 [ 1 + ( z 1 - z 0 k s 0 2 ) 2 ] 1 / 2 ,
s 2 = s 0 [ 1 + ( z 2 - z 0 k s 0 2 ) 2 ] 1 / 2 ,
s 1 / s 0 = s 2 / s 0 = 0.

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