Abstract

A general expression is obtained for the exact computation of the fringe field in the intersection volume of two paraxial Gaussian beams for arbitrary beam waist positions and sizes. The expression is then simplified to allow easy fringe field computation while retaining good accuracy. By relating the simplified expression to the system parameters relevant to dual-beam laser Doppler velocimeters, simple design equations are obtained. These equations permit rapid evaluation of the fringe field variation along the major and minor axes of the intersection volume and clearly identify the system parameters controlling this variation.

© 1996 Optical Society of America

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References

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  1. S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,” J. Phys. D 6, 164–171 (1973).
    [CrossRef]
  2. S. Hanson, “Visualization of alignment errors and heterodyning constraints in laser Doppler velocimeters,” in The Accuracy of Flow Measurements by Laser Doppler Methods (Proceedings of the LDA-Symposium Copenhagen 1975, Skovlunde, Denmark, 1976), pp. 176–182.
  3. F. Durst, W. H. Stevenson, “Influence of Gaussian beam properties on laser Doppler signals,” Appl. Opt. 18, 516–524 (1979).
    [CrossRef] [PubMed]
  4. F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
    [CrossRef]
  5. P. C. Miles, P. O. Witze, “Fringe field quantification in an LDV probe volume by use of a magnified image,” Exp. Fluids 16, 330–335 (1994).
    [CrossRef]
  6. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]

1994 (1)

P. C. Miles, P. O. Witze, “Fringe field quantification in an LDV probe volume by use of a magnified image,” Exp. Fluids 16, 330–335 (1994).
[CrossRef]

1990 (1)

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

1979 (1)

1973 (1)

S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,” J. Phys. D 6, 164–171 (1973).
[CrossRef]

1966 (1)

Durst, F.

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

F. Durst, W. H. Stevenson, “Influence of Gaussian beam properties on laser Doppler signals,” Appl. Opt. 18, 516–524 (1979).
[CrossRef] [PubMed]

Hanson, S.

S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,” J. Phys. D 6, 164–171 (1973).
[CrossRef]

S. Hanson, “Visualization of alignment errors and heterodyning constraints in laser Doppler velocimeters,” in The Accuracy of Flow Measurements by Laser Doppler Methods (Proceedings of the LDA-Symposium Copenhagen 1975, Skovlunde, Denmark, 1976), pp. 176–182.

Kogelnik, H.

Li, T.

Miles, P. C.

P. C. Miles, P. O. Witze, “Fringe field quantification in an LDV probe volume by use of a magnified image,” Exp. Fluids 16, 330–335 (1994).
[CrossRef]

Müller, R.

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

Naqwi, A.

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

Stevenson, W. H.

Witze, P. O.

P. C. Miles, P. O. Witze, “Fringe field quantification in an LDV probe volume by use of a magnified image,” Exp. Fluids 16, 330–335 (1994).
[CrossRef]

Appl. Opt. (2)

Exp. Fluids (2)

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

P. C. Miles, P. O. Witze, “Fringe field quantification in an LDV probe volume by use of a magnified image,” Exp. Fluids 16, 330–335 (1994).
[CrossRef]

J. Phys. D (1)

S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,” J. Phys. D 6, 164–171 (1973).
[CrossRef]

Other (1)

S. Hanson, “Visualization of alignment errors and heterodyning constraints in laser Doppler velocimeters,” in The Accuracy of Flow Measurements by Laser Doppler Methods (Proceedings of the LDA-Symposium Copenhagen 1975, Skovlunde, Denmark, 1976), pp. 176–182.

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Figures (5)

Fig. 1
Fig. 1

Geometry and coordinate system definition for analysis of fringe spacing.

Fig. 2
Fig. 2

Dual-beam, path-compensated LDV. An equal path-length beam splitter is used, which ensures that the two beam waists lie in the same plane along the optical axis.

Fig. 3
Fig. 3

Geometry for evaluating the transformation of an off-axis Gaussian beam.

Fig. 4
Fig. 4

Geometric definitions used in the formulation of normalized coordinates.

Fig. 5
Fig. 5

Maximum percent error incurred when Eq. (A20) was used to estimate the fringe spacing for |(z02f)/zR2| ≤2.

Equations (52)

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E ( x i , y i , z i ) = ( 2 π ) 1 / 2 1 w ( z i ) exp [ x i 2 + y i 2 w 2 ( z i ) ] × exp [ - j k z i + j φ ( z i ) - j k 2 x i 2 + y i 2 R ( z i ) ] ,
φ ( z i ) = tan - 1 ( z i z R i ) ,
R ( z i ) = z i + z R i 2 z i ,
w ( z i ) = w 0 i [ 1 + ( z i z R i ) 2 ] .
x 1 = x cos α + z sin α ,             y 1 = y , z 1 = - ( x - x w 1 ) sin α + ( z - z w 1 ) cos α ,
x 2 = x cos α - z sin α ,             y 2 = y , z 2 = ( x - x w 2 ) sin α + ( z - z w 2 ) cos α ,
θ i ( x , y , z ) = - k z i + φ ( z i ) - k 2 x i 2 + y i 2 R ( z i ) .
θ 1 ( x , y , z ) - θ 2 ( x , y , z ) = - k ( z 1 - z 2 ) + φ ( z 1 ) - φ ( z 2 ) - k 2 [ x 1 2 + y 1 2 R ( z 1 ) - x 2 2 + y 2 2 R ( z 2 ) ] = 2 n π .
L = ( d n d x ) - 1 = λ 2 sin α [ 1 - 1 2 tan α × ( x 1 z 1 z 1 2 + z R 1 2 - x 2 z 2 z 2 2 + z R 2 2 ) - ( G 1 + G 2 + X 1 + X 2 + Y 1 + Y 2 ) ] - 1 .
G i = z R i 2 k ( z R i 2 + z i 2 ) .
X i = 1 4 x i 2 ( z i 2 - z R i 2 ) ( z i 2 + z R i 2 ) 2 ,
Y i = 1 4 y i 2 ( z i 2 - z R i 2 ) ( z i 2 + z R i 2 ) 2 .
x i 2 max y i 2 max w 2 ( z i ) = w 0 i 2 ( 1 + z i 2 z R i 2 ) ,
X i max = Y i max = ( λ 2 π w 0 i ) 2 ( z i 2 - z R i 2 z i 2 + z R i 2 ) .
L = λ 2 sin α [ 1 + ( x 1 z 1 z 1 2 + z R 1 2 - x 2 z 2 z 2 2 + z R 2 2 ) 2 tan α - ( x 1 z 1 z 1 2 + z R 1 2 - x 2 z 2 z 2 2 + z R 2 2 ) ] .
L = λ 2 sin α [ 1 + x 1 z 1 ( z 1 2 + z R 1 2 ) tan α - x 1 z 1 ] .
1 L L z = - 1 f D f D z = z 1 cos α z 1 2 + z R 2 = cos α R 1
L = λ 2 sin α [ 1 + z cos 2 α ( z cos 2 α - z w ) z R 2 cos 2 α - z w ( z cos 2 α - z w ) ] .
L = λ 2 sin α [ 1 + ( z cos α z R ) 2 ] ,
L = λ 2 sin α { 1 - z ^ cos 2 α ( w 0 i d ) ( z 0 i - f z R i ) + z ^ 2 cos 4 α ( w 0 i d ) 2 [ 1 1 - z ^ cos 2 α ( w 0 i d ) ( z 0 i - f z R i ) ] } .
- 1 L L z = ( z 0 i - f ) cos 2 α f 2 ,
1 2 tan α ( x 1 z 1 z 1 2 + z R 1 2 - x 2 z 2 z 2 2 + z R 2 2 )
L = λ 2 sin α [ 1 - z ^ cos 2 α ( w 0 i d ) ( z 0 i - f z R i ) + z ^ 2 cos 4 α ( w 0 i d ) 2 ] .
L = λ 2 sin α [ 1 + z ^ 2 cos 4 α ( w 0 i d ) 2 ] ,
x 1 = x cos α ,             z 1 = - x sin α - z w 1 / cos α ,
L = λ 2 sin α [ 1 + x cos α 1 z 1 ( z 1 2 + z R 1 2 ) tan α - x cos α ] .
L = λ 2 R 1 R 1 sin α - x cos 2 α λ 2 R 1 R 1 sin α - x ,
L = λ 2 sin α [ 1 - x ^ cos 5 α ( w 0 i d ) ( z 0 i - f z R i ) ] ;
[ r 2 r 2 ] = [ 1 0 - 1 f 1 ] [ r 1 r 1 ] .
R 2 = f R 1 cos α R 1 cos α - f ,
( r 2 , top - r 2 , center ) = w 1 cos α ( 1 1 - sin α cos α w 1 R 1 ) w 1 cos α ( 1 + tan α w 1 R 1 ) ,
( r 2 , center - r 2 , bottom ) = w 1 cos α ( 1 1 + sin α cos α w 1 R 1 ) w 1 cos α ( 1 - tan α w 1 R 1 ) .
w 2 = ( r 2 , top - r 2 , center ) + ( r 2 , center - r 2 , bottom ) 2 w 1 cos α .
z 0 2 = R 2 [ 1 + ( λ R 2 π w 2 2 ) 2 ] ,
w 0 2 2 = w 2 2 [ 1 + ( π w 2 2 λ R 2 ) 2 ] ,
z 0 2 - f = f 2 [ z 0 1 - f - f ( z R 1 2 cos 2 α z 0 1 2 + z R 1 2 cos 2 α ) ( cos 6 α - 1 ) ] z R 1 2 cos 2 α + ( z 0 1 - f ) 2 + f 2 ( z R 1 2 cos 2 α z 0 1 2 + z R 1 2 cos 2 α ) ( cos 6 α - 1 ) ,
w 0 2 2 = f 2 w 0 1 2 cos 4 α z R 1 2 cos 2 α + ( z 0 1 - f ) 2 + f 2 ( z R 1 2 cos 2 α z 0 1 2 + z R 1 2 cos 2 α ) ( cos 6 α - 1 ) .
ɛ = ( z R 1 2 cos 2 α z 0 1 2 + z R 1 2 cos 2 α ) ( cos 6 α - 1 ) ,
z 0 2 - f = f 2 ( z 0 1 - f ) z R 1 2 cos 2 α + ( z 0 1 - f ) 2 [ h ( ɛ ) g ( ɛ ) ] ,
w 0 2 2 = f 2 w 0 1 2 cos 4 α z R 1 2 cos 2 α + ( z 0 1 - f ) 2 [ 1 g ( ɛ ) ] .
g ( ɛ ) = 1 + f 2 ɛ z R 1 2 cos 2 α + ( z 0 1 - f ) 2 ,
h ( ɛ ) = 1 - f ɛ ( z 0 1 - f ) .
l = 2 w ( z i ) sin α | x = z = 0
L = λ 2 sin α { 1 - z ^ cos 2 α ( w 0 2 d ) ( z 0 2 - f z R 2 ) [ g ( ɛ ) 1 / 2 h ( ɛ ) ] + z ^ 2 cos 2 α ( w 0 2 d ) 2 [ g ( ɛ ) ] × ( 1 1 - z ^ cos 2 α ( w 0 2 d ) ( z 0 2 - f z R 2 ) [ g ( ɛ ) 1 / 2 h ( ɛ ) ] ) } .
ɛ ( z R 1 2 cos 2 α f 2 ) ( cos 6 α - 1 ) .
g ( ɛ ) cos 6 α { 1 + [ ( z 0 2 - f ) z R 2 ] 2 } 1 + cos 6 α ( ( z 0 2 - f ) z R 2 ) 2 ,
h ( ɛ ) 1 - f ( cos 6 α - 1 ) ( z 0 2 - f ) { 1 + cos 6 α [ ( z 0 2 - f ) z R 2 ] 2 } .
L = λ 2 sin α { 1 - z ^ cos 2 α ( w 0 2 d ) ( z 0 2 - f z R 2 ) + z ^ 2 cos 4 α ( w 0 2 d ) 2 [ 1 1 - z ^ cos 2 α ( w 0 2 d ) ( z 0 2 - f z R 2 ) ] } .
L = λ 2 sin α { 1 + [ z 1 x cos α ( z 1 2 + z R 1 2 ) tan α ] + [ z 1 x cos α ( z 1 2 + z R 1 2 ) tan α ] 2 + } .
x ^ x x max = x cos w 0 1 z R 1 ( z 1 2 + z R 1 2 ) 1 / 2 ,
L = λ 2 sin α { 1 - x ^ cos 2 α ( w 0 2 d ) ( z 0 2 - f z R 2 ) [ g ( ɛ ) 1 / 2 h ( ɛ ) ] + x ^ 2 cos 4 α ( w 0 2 d ) 2 ( z 0 2 - f z R 2 ) 2 [ g ( ɛ ) 1 / 2 h ( ɛ ) ] 2 - } .
L = λ 2 sin α [ 1 - x ^ cos 5 α ( w 0 2 d ) ( z 0 2 - f z R 2 ) ] .

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