Abstract

The actual focal plane for a laser beam under frequently encountered conditions is shifted from the plane given by geometrical optics. The concept of effective Fresnel number for an unapertured Gaussian laser beam is defined and used to examine focal shifts. It is found that the effective Fresnel number has a simple physical interpretation that makes clear its relationship to the focal shift phenomena. Expressions are obtained that relate the important Gaussian beam parameters to the effective Fresnel number, and a method of correcting for the focal shift in optical systems design is described.

© 1982 Optical Society of America

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References

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  1. G. Goubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, Ed. (Macmillan, New York, 1963), pp. 907–918, Eq. (24).
  2. A. G. van Nie, Philips Res. Rep. 19, 378 (1964), Eq. (38).
  3. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965), Eq. (65).
  4. D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  5. J. J. Stamnes, J. Opt. Soc. Am. 71, 15 (1981).
    [CrossRef]
  6. J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” submitted to Opt. Commun.
  7. E. Wolf, Y. Li, Opt. Commun. 39, 205 (1981).
    [CrossRef]
  8. Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
    [CrossRef]
  9. W. H. Carter, J. Opt. Soc. Am. 62, 1195 (1972).
    [CrossRef]
  10. R. J. Pressley, in CRC Handbook of Lasers, R. C. Weast, Ed., (CRC Press, Cleveland, 1971), p. 422.
  11. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 51.
  12. J. H. Erkkila, M. E. Rogers, J. Opt. Soc. Am. 71, 904 (1981).
    [CrossRef]
  13. D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), Secs. 4.2, 4.6, and 4.7.

1981 (4)

1972 (2)

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965), Eq. (65).

1964 (1)

A. G. van Nie, Philips Res. Rep. 19, 378 (1964), Eq. (38).

Avizonis, P. V.

Bell, W. E.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), Secs. 4.2, 4.6, and 4.7.

Carter, W. H.

Erkkila, J. H.

Goubau, G.

G. Goubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, Ed. (Macmillan, New York, 1963), pp. 907–918, Eq. (24).

Holmes, D. A.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 51.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965), Eq. (65).

Korka, J. E.

Li, Y.

E. Wolf, Y. Li, Opt. Commun. 39, 205 (1981).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Pressley, R. J.

R. J. Pressley, in CRC Handbook of Lasers, R. C. Weast, Ed., (CRC Press, Cleveland, 1971), p. 422.

Rogers, M. E.

Sinclair, D. C.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), Secs. 4.2, 4.6, and 4.7.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” submitted to Opt. Commun.

Stamnes, J. J.

J. J. Stamnes, J. Opt. Soc. Am. 71, 15 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” submitted to Opt. Commun.

van Nie, A. G.

A. G. van Nie, Philips Res. Rep. 19, 378 (1964), Eq. (38).

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 51.

Wolf, E.

E. Wolf, Y. Li, Opt. Commun. 39, 205 (1981).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965), Eq. (65).

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

E. Wolf, Y. Li, Opt. Commun. 39, 205 (1981).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Philips Res. Rep. (1)

A. G. van Nie, Philips Res. Rep. 19, 378 (1964), Eq. (38).

Other (5)

G. Goubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, Ed. (Macmillan, New York, 1963), pp. 907–918, Eq. (24).

J. J. Stamnes, B. Spjelkavik, “Focusing at Small Angular Apertures in the Debye and Kirchhoff Approximations,” submitted to Opt. Commun.

R. J. Pressley, in CRC Handbook of Lasers, R. C. Weast, Ed., (CRC Press, Cleveland, 1971), p. 422.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 51.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart, and Winston, New York, 1969), Secs. 4.2, 4.6, and 4.7.

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Figures (7)

Fig. 1
Fig. 1

Gaussian beam propagating in the +z direction and focused in the z = 0 plane.

Fig. 2
Fig. 2

Gaussian beam focused at the origin in object space which is transformed by a lens into a Gaussian beam focused at the origin in image space. In this figure k σ012 and k022 are the near-field distances from the origin in the object and image spaces, respectively.

Fig. 3
Fig. 3

Lens focusing a Gaussian beam at the origin.

Fig. 4
Fig. 4

Table of numerical data calculated from Eqs. (18), (21), (22), and (23) for given N(−d). Sketches corresponding to those data appear in Fig. 5.

Fig. 5
Fig. 5

Sketches showing the range of the focal spot along the z axis relative to the position of the lens plane, focal plane, and geometrical focal plane for three values of effective Fresnel number N(−d) in the lens plane.

Fig. 6
Fig. 6

|N′(−d)| as a function of |N(−d)| from Eq. (25).

Fig. 7
Fig. 7

Illustrating the two solutions for focusing a Gaussian beam onto a given plane while maintaining a specified geometrical Fresnel number in the lens aperture.

Equations (42)

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G ( x ) = k 2 2 π γ ( z ) exp { ik [ z + r 2 2 R ( z ) ] } exp [ r 2 2 σ 2 ( z ) ] ,
R ( z ) = z + k 2 σ 0 4 / z ,
σ ( z ) = σ 0 2 + z 2 k 2 σ 0 2 ,
γ ( z ) = k 2 σ 0 2 + ikz
| z | k σ 0 2
| z | k σ 0 2 .
θ = tan 1 1 k σ 0
R ( z ) ~ z ,
R ( z ) ~ k 2 σ 0 4 / z
N = a 2 λ R < 1 ,
N ( z ) = σ 2 ( z ) λ R ( z ) .
N ( z ) = z / ( 2 π k σ 0 2 ) .
| N ( z ) | = σ 2 ( z ) λ | R ( z ) | < 1 2 π ,
R ( z ) / z = 1 + 1 ( 2 π ) 2 N 2 ( z ) ,
σ ( z ) σ 0 = 1 + 4 π 2 N 2 ( z ) ,
N ( z ) = z z N ( z ) .
δ = | R ( d ) | d .
δ d = 1 ( 2 π ) 2 N 2 ( d ) 0 .
N ( d ) = σ 2 ( d ) λ R ( d ) .
N ( δ ) = 1 / [ ( 2 π ) 2 N ( d ) ] .
R ( δ ) d = 1 + 1 4 π 2 N 2 ( d ) ,
σ ( δ ) σ 0 = 1 + 1 4 π 2 N 2 ( d ) .
| z | d < h d = 1 2 π | N ( d ) | .
N ( z ) = σ 2 ( z ) λ z
N ( z ) = N ( z ) + 1 4 π 2 N ( z ) .
σ ( d ) = λ d π
R 2 ( d ) + ( 2 π ) 2 d N 2 ( d ) R ( d ) + ( 2 π ) 2 d 2 N 2 ( d ) = 0 ,
R ( d ) = 2 π 2 d N 2 ( d ) [ 1 1 1 π 2 N 2 ( d ) ] ,
N ( d ) = 1 2 π .
N ( d ) = ( 1 ) / π ,
R ( d ) = 2 k σ 0 2 .
σ 0 = σ 2 ( d ) / 2 .
θ = tan 1 1 / ( kd ) .
1 1 1 π 2 N 2 ( d ) = 1 { 1 1 2 π 2 N 2 ( d ) 1 8 π 2 N 4 ( d ) O [ 1 N 6 ( d ) ] }
R ( d ) ~ d { 1 + 1 4 π 2 N 2 ( d ) + O [ 1 N 4 ( d ) ] } , far field ,
R ( d ) ~ d { 4 π 2 N 2 ( d ) 1 1 4 π 2 N 2 ( d ) O [ 1 N 4 ( d ) ] } , near field ,
N ( d ) N ( d ) , far field .
2 π N ( d ) 1 2 π N ( d ) , near field .
σ 0 = λ d | N ( d ) | , near field ,
θ = tan 1 1 kd 2 π | N ( d ) | , near field .
σ 0 = λ d ( 2 π ) 2 | N ( d ) | , far field ,
θ = tan 1 2 π | N ( d ) | kd , far field .

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