Abstract

The unique properties of Gaussian beams must be considered when designing and using laser Doppler anemometer systems. This paper presents an analysis of Gaussian beam effects in LDA systems in order to quantify their influence on the Doppler signal. The analytical results, which are verified by experiment, show that both axial and lateral frequency gradients can exist in the probe volume of improperly aligned LDA systems. The effects of lens aberrations and of optical path changes due to the insertion of planar optical elements in the beam are also considered.

© 1979 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. 44, 455 (1965).
  2. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  3. J. A. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
    [CrossRef] [PubMed]
  4. L. D. Dickson, Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  5. S. Hanson, J. Phys. D— 6, 164 (1973).
    [CrossRef]
  6. B. Eliasson, R. Däindliker, Opt. Acta 21, 119 (1974).
    [CrossRef]
  7. W. Kleen, R. Müller, “Optische Resonatoren und Aubreitungsgesetze für Laserstrahlen,” in Laser, G. Grau, Ed. (Springer, Berlin, 1969).
    [CrossRef]
  8. A. Boivin, E. Wolf, Phys. Rev. B 138, 1561 (1965) (see also Spectra-Physics Technical Bulletin 5, Spectra-Physics, Inc., 1966).
    [CrossRef]
  9. F. Durst, W. H. Stevenson, Appl. Opt. 15, 137 (1976).
    [CrossRef] [PubMed]

1976 (1)

1974 (1)

B. Eliasson, R. Däindliker, Opt. Acta 21, 119 (1974).
[CrossRef]

1973 (1)

S. Hanson, J. Phys. D— 6, 164 (1973).
[CrossRef]

1970 (1)

1969 (1)

1966 (1)

1965 (2)

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

A. Boivin, E. Wolf, Phys. Rev. B 138, 1561 (1965) (see also Spectra-Physics Technical Bulletin 5, Spectra-Physics, Inc., 1966).
[CrossRef]

Arnaud, J. A.

Boivin, A.

A. Boivin, E. Wolf, Phys. Rev. B 138, 1561 (1965) (see also Spectra-Physics Technical Bulletin 5, Spectra-Physics, Inc., 1966).
[CrossRef]

Däindliker, R.

B. Eliasson, R. Däindliker, Opt. Acta 21, 119 (1974).
[CrossRef]

Dickson, L. D.

Durst, F.

Eliasson, B.

B. Eliasson, R. Däindliker, Opt. Acta 21, 119 (1974).
[CrossRef]

Hanson, S.

S. Hanson, J. Phys. D— 6, 164 (1973).
[CrossRef]

Kleen, W.

W. Kleen, R. Müller, “Optische Resonatoren und Aubreitungsgesetze für Laserstrahlen,” in Laser, G. Grau, Ed. (Springer, Berlin, 1969).
[CrossRef]

Kogelnik, H.

Li, T.

Müller, R.

W. Kleen, R. Müller, “Optische Resonatoren und Aubreitungsgesetze für Laserstrahlen,” in Laser, G. Grau, Ed. (Springer, Berlin, 1969).
[CrossRef]

Stevenson, W. H.

Wolf, E.

A. Boivin, E. Wolf, Phys. Rev. B 138, 1561 (1965) (see also Spectra-Physics Technical Bulletin 5, Spectra-Physics, Inc., 1966).
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. (1)

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

J. Phys. D (1)

S. Hanson, J. Phys. D— 6, 164 (1973).
[CrossRef]

Opt. Acta (1)

B. Eliasson, R. Däindliker, Opt. Acta 21, 119 (1974).
[CrossRef]

Phys. Rev. B (1)

A. Boivin, E. Wolf, Phys. Rev. B 138, 1561 (1965) (see also Spectra-Physics Technical Bulletin 5, Spectra-Physics, Inc., 1966).
[CrossRef]

Other (1)

W. Kleen, R. Müller, “Optische Resonatoren und Aubreitungsgesetze für Laserstrahlen,” in Laser, G. Grau, Ed. (Springer, Berlin, 1969).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Main components of differential Doppler anemometer.

Fig. 2
Fig. 2

Parameters describing Gaussian beam characteristics.

Fig. 3
Fig. 3

Gaussian laser beam focused by a single lens.

Fig. 4
Fig. 4

General LDA optical configuration

Fig. 5
Fig. 5

Fringe model indicating variations in Doppler frequency: beam waists on same side of focal plane.

Fig. 6
Fig. 6

Diagram to explain error in mean Doppler frequency.

Fig. 7
Fig. 7

Experimental system.

Fig. 8
Fig. 8

Measured frequency in probe volume.

Fig. 9
Fig. 9

Comparison between theory and experiments.

Fig. 10
Fig. 10

Influence of improperly aligned LDA-systems on mean Doppler frequency in the center of measuring volume.

Fig. 11
Fig. 11

Frequency variation in probe volume for large z0.

Fig. 12
Fig. 12

Normalized data for large z0.

Fig. 13
Fig. 13

Effect of abberations on frequency variation in probe volume.

Fig. 14
Fig. 14

Effect of inserting water basin on frequency variation in probe volume.

Fig. 15
Fig. 15

Change of beam waist position due to planar refractive element.

Fig. 16
Fig. 16

Fringe model for beam waists on opposite sides of focal plane.

Fig. 17
Fig. 17

Geometry for theoretical analysis on path uncompensated systems.

Fig. 18
Fig. 18

Generalized ellipse.

Fig. 19
Fig. 19

Central plane of probe volume: beam waists on opposite sides of focal plane.

Equations (35)

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θ = ( 2 λ ) / ( π w 0 ) .
w 2 ( z ) = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ] .
R ( z ) = z [ 1 + ( π w 0 2 λ z ) 2 ] .
z 1 = f + ( z 0 - f ) f 2 ( z 0 - f ) 2 + ( π w 0 2 λ ) 2 ,
w 1 = w 0 f [ ( z 0 - f ) 2 + ( π w 0 2 λ ) 2 ] 1 / 2 .
1 ν D d ν D d z = - 1 / R ,
π w 1 2 λ f = π w 0 2 / λ f ( z 0 - f f ) 2 + ( π w 0 2 λ f ) 2 ,
( z 1 - f f ) = ( z 0 - f f ) ( z 0 - f f ) 2 + ( π w 0 2 λ f ) 2 .
R f = - ( z 1 - f ) f { 1 + [ π w 1 2 λ ( f - z 1 ) ] 2 } 1 / 2 .
R / f = ( - f / ( z 0 - f ) ,
f ν D d ν D d z = z 0 - f f .
Δ x = λ / ( 2 sin α )
V x ( ν D λ ) / ( 2 sin α ) .
z 0 = f ;             z 0 = f - 100 mm ;             z 0 = f + 100 mm .
Δ z 0 = d ( 1 - tan α tan β ) d ( 1 - m 1 m 2 ) ;             Δ z 0 0.33 d .
( Δ ν D Δ z ) Δ z 0 = ν D f [ ( z 0 - f ) f ] - ν D Δ z 0 f 2 .
( Δ ν D Δ z ) - ( Δ ν D Δ z ) Δ z 0 = ν D · Δ z 0 f 2 ν D · d · { 1 - [ ( m 1 ) / ( m 2 ) ] } f 2 .
| ( Δ ν D Δ z ) - ( Δ ν D Δ z ) Δ z 0 | 0.39 × 10 - 2 kHz mm ,
ϕ P 1 = ϕ 1 + k R 1 ,
ϕ P 2 = ϕ 2 - k R 2 .
Δ ϕ = ϕ P 1 - ϕ P 2 = k ( R 1 + R 2 ) + ( ϕ 1 - ϕ 2 ) .
z 2 a 2 + x 2 b 2 = 1.
x = ± ( R 2 - c 2 ) 1 / 2 .
x n = ± [ ( n λ 2 ) 2 - c 2 ] 1 / 2 ,
( d x n ) / ( d n ) = ( n λ 2 / 4 ) / x .
d x n d n | x = x 0 = Δ x = λ 2 sin α ,
ν D = ( 2 V x sin α ) / λ ,
d d x ( d x n d n ) = - n λ 2 4 x 2 + λ 2 4 y d n d x .
d d x ( d x n d n ) = λ 2 x [ - ( x 2 + c 2 ) 1 / 2 x + x ( x 2 + c 2 ) 1 / 2 ] .
d d x ( d x n d n ) - 1 x d x d n | x = x 0 .
d ν D d x = d d x ( V x L ) = - V x L 2 d L d x = - ν D L ( - L x ) = ν D x .
1 ν D d ν D d x = 1 x = 1 R sin α ,
( d ν D ) / ν D = ( 2 w ) / ( R sin α cos α ) ,
( d ν D ) / ( ν D ) = ( 2 w ) / ( R α ) ,
( d ν D ) / ν D = ( 2 λ ) / ( π w 0 α ) .

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