Abstract

In most laser Doppler velocimetry (LDV) systems, the frequency of one of the two laser beams that intersect to create the probe volume is shifted with an acousto-optic element. It is shown here that Bragg shifting can impose a problematic fluctuation in intensity on the frequency-shifted beam, producing spurious velocity measurements. This fluctuation occurs at twice the Bragg cell frequency, and its relative amplitude to the time average intensity is a function of the ratio of the laser beam diameter to the Bragg cell acoustic wavelength. A physical model and a configuration procedure to minimize adverse effects of the intensity modulations are presented.

© 2009 Optical Society of America

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References

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  1. F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, 1976).
  2. P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).
  3. B. E. A. Saleh and M. C. Teich, “Acousto-Optics,” in Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 804-833.
  4. S. J. Lascos and D. T. Cassidy, “Optical phase and intensity modulation from a rotating optical flat: effect on noise in degree of polarization measurements,” Appl. Opt. 48, 1697-1704(2009).
    [CrossRef] [PubMed]

2009 (1)

2004 (1)

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).

Abramski, K. M.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).

Antonczak, A. J.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).

Cassidy, D. T.

Durst, F.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, 1976).

Kaczmarek, P. R.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).

Lascos, S. J.

Melling, A.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, 1976).

Rogowski, T.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, “Acousto-Optics,” in Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 804-833.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, “Acousto-Optics,” in Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 804-833.

Whitelaw, J. H.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, 1976).

Appl. Opt. (1)

Opt. Appl. (1)

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, “Laser Doppler vibrometry with acoustooptic frequency shift,” Opt. Appl. 34, 373-384 (2004).

Other (2)

B. E. A. Saleh and M. C. Teich, “Acousto-Optics,” in Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 804-833.

F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, 1976).

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Figures (12)

Fig. 1
Fig. 1

Bragg cell schematic. An incident laser beam enters the acousto-optic element at a shallow angle θ. A portion of the beam is reflected and frequency shifted.

Fig. 2
Fig. 2

Amplitudes of the bracketed intensity-reflectance terms in Eq. (10). Note the primary steady term (dashed line) is constant and the time-dependent term (solid line) decreases as the ratio of laser beam diameter to Bragg cell acoustic wavelength increases.

Fig. 3
Fig. 3

Amplitude ratio of the time varying term to the sum of the steady terms and its envelope in Eq. (10). Note that the relative amplitude of the time-dependent term decreases as Λ / ( π L ) .

Fig. 4
Fig. 4

Intensity-reflectance steady term ratio for a nonoptimized beam ( λ = 514.5 nm , λ B = 488 nm ) to that of a Bragg-optimized beam ( λ = λ B = 488 nm ) using Eq. (11). Note the decrease in relative intensity reflectance for the nonoptimized beam as the ratio of laser beam diameter to Bragg cell acoustic wavelength increases.

Fig. 5
Fig. 5

Intensity reflectance time-dependent term coefficients for a nonoptimized beam ( λ = 514.5 nm , λ B = 488 nm ) and a Bragg- optimized beam ( λ = λ B = 488 nm ) using Eq. (11). Note that the amplitude of the nonoptimized beam does not diminish compared to that of the Bragg-optimized beam, though they become out of phase as the ratio of laser beam diameter to Bragg cell acoustic wavelength increases.

Fig. 6
Fig. 6

Amplitude ratios of the time varying term to the sum of the steady terms for a nonoptimized beam ( λ = 488 nm , λ B = 514.5 nm ), a Bragg-optimized beam ( λ = λ B = 514.5 nm ), and a system-optimized beam ( λ = 488 nm , λ B = 501 nm ) using Eq. (11). Note that both the Bragg-optimized beam and the system-optimized beam decrease as the ratio of laser beam diameter to Bragg cell acoustic wavelength increases, while the nonoptimized beam experiences a local minimum.

Fig. 7
Fig. 7

Envelopes of the time-dependent-to-steady-term amplitude ratios shown in Fig. 6. Note that both the Bragg-optimized beam ( λ = λ B = 488 nm ) and the system-optimized beam ( λ = 514.5 nm , λ B = 501 nm ) decrease as the ratio of laser beam diameter to Bragg cell acoustic wavelength increases, while the nonoptimized beam ( λ = 488 nm , λ B = 514.5 nm ) experiences a local minimum. Experimental results for the Bragg-optimized and non-Bragg-optimized wavelength conditions are plotted as + and ×, respectively, and are discussed in the Section 4.

Fig. 8
Fig. 8

Intensity reflectance steady term ratio for two system-optimized beams λ = 514.5 nm λ B = 501 nm and λ = 488 nm , λ B = 501 nm using Eq. (11). Note the relative intensity reflectance for both beams are identical and much stronger than that for the nonoptimized beam in Fig. 4 as the ratio of laser beam diameter to Bragg cell acoustic wavelength increases.

Fig. 9
Fig. 9

Temporal intensity spectrums of the Bragg-shifted 488 nm beam with (a)  2 L / Λ 15 and (b)  2 L / Λ 30 . The Bragg angle was optimized for 488 nm .

Fig. 10
Fig. 10

Temporal intensity spectrums of the Bragg-shifted 514.5 nm beam with (a)  2 L / Λ 15 and (b)  2 L / Λ 30 . The Bragg angle was optimized for 488 nm .

Fig. 11
Fig. 11

Temporal intensity spectrums of the 488 nm LDV signal beam with (a)  2 L / Λ 15 and (b)  2 L / Λ 30 . The Bragg angle was optimized for 488 nm .

Fig. 12
Fig. 12

Temporal intensity spectrums of the 514.5 nm LDV signal with (a)  2 L / Λ 15 and (b)  2 L / Λ 30 . The Bragg angle was optimized for 488 nm . The frequency peaks are (1) frequency downmixed peak at 33 MHz , (2) LDV peak at 47 MHz , (3) Bragg intensity modulation peak at 80 MHz , (4) frequency upmixed peak at 127 MHz

Equations (14)

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δ f = λ 2 sin ( φ 2 ) .
v x = ( f f B ) δ f .
n ( x ) = n o Δ n o cos ( q x Ω t ) ,
r B = L / 2 L / 2 exp ( j 2 k x sin θ ) d r d x d x ,
r B = j 1 2 d r d n q Δ n o L { exp ( j Ω t ) sinc [ ( 2 k sin θ q ) L 2 π ] exp ( j Ω t ) sinc [ ( 2 k sin θ + q ) L 2 π ] } ,
| r B | 2 = ( 1 2 d r d n q Δ n o L ) 2 { sinc 2 [ ( 2 k sin θ q ) L 2 π ] + sinc 2 [ ( 2 k sin θ + q ) L 2 π ] 2 sinc [ ( 2 k sin θ q ) L 2 π ] sinc [ ( 2 k sin θ + q ) L 2 π ] cos ( 2 Ω t ) } .
sin θ B = λ 2 Λ ,
d r d n = 1 2 n o sin 2 θ .
| r B | 2 = ( 2 π Δ n o n o L Λ λ 2 ) 2 { 1 + sinc 2 ( 2 L Λ ) 2 sinc ( 2 L Λ ) cos ( 2 Ω t ) } .
R B = sin 2 ( 2 π Δ n o n o L Λ λ 2 ) { 1 + sinc 2 ( 2 L Λ ) - 2 sinc ( 2 L Λ ) cos ( 2 Ω t ) } .
R B ( λ ) = sin 2 ( 1 2 d r d n q Δ n o L ) { sinc 2 [ 1 2 ( λ B λ - 1 ) 2 L Λ ] + sinc 2 [ 1 2 ( λ B λ + 1 ) 2 L Λ ] 2 sinc [ 1 2 ( λ B λ 1 ) 2 L Λ ] sinc [ 1 2 ( λ B λ + 1 ) 2 L Λ ] cos ( 2 Ω t ) } .
2 ( λ B λ 1 λ B λ + 1 ) sin [ 1 2 ( λ B λ 1 ) π 2 L Λ ] sin 2 [ 1 2 ( λ B λ - 1 ) π 2 L Λ ] + ( λ B λ 1 λ B λ + 1 ) 2 ,
λ B λ 1 = λ B λ + 1 ,
sin θ B = [ Λ ( 1 λ 1 + 1 λ 2 ) ] 1 .

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