- Email: [email protected]
Research on speckle noise of laser Doppler velocimeter for the vehicle self-contained navigation
Optik 125 (2014) 5878–5883
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Optik journal homepage: www.elsevier.de/ijleo
Research on speckle noise of laser Doppler velocimeter for the vehicle self-contained navigation Jian Zhou ∗ , Xiaoming Nie, Xingwu Long College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China
a r t i c l e
i n f o
Article history: Received 22 October 2013 Accepted 31 May 2014 Keywords: Multipoint layer-type laser Doppler velocimeter Speckle broadening Vehicle-mounted self-contained navigation system
a b s t r a c t A novel multipoint layer-type laser Doppler velocimeter (MLLDV) is designed to measure the velocity of a vehicle for the self-contained navigation system. In order to investigate the speckle’s influence on the Doppler spectrum, formulas of time-lagged covariance and speckle broadening were derived for our MLLDV. Simulations and experiments are made for detailed analysis. The results show that the time-lagged covariance of photocurrent is directly proportional to the incident angle, and is inversely proportional to the elevation fluctuation of the ground together with the velocity of the vehicle. Speckle broadening is a function of the vehicle’s velocity, the 1/e2 Gaussian spot radius and the phase correlation length of the ground. For our MLLDV, Doppler frequency and Doppler broadening are both directly proportional to the velocity of the vehicle. Besides, the ratio between Doppler broadening and the corresponding Doppler frequency is about 0.72% when the speed of the vehicle varies from 0 to 9.6 m/s. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Since the laser Doppler velocimeter (LDV) was developed in the mid-1960s [1], it has a widespread ongoing application in many scientific fields for its advantages of good linearity, fast dynamic response, noncontact measurement and high resolution [2–4]. A typical application of LDV is to determine flow velocities. Recently, using an MLLDV to offer parameter of the velocity for vehiclemounted self-contained navigation system has been reported [5]. Compared with Global Position System (GPS), LDV is autonomous for navigation system. In addition, conventional speedometers derive the velocity information from the rotational speed of the wheels which has the disadvantage that any slip between wheel and road can not be detected; in addition, any distortion of the wheel will affect the measurement accuracy of the speedometer. But LDV shows excellent performance even under adverse environmental conditions. So MLLDV will be an important speed sensor in a self-contained navigation system. In this paper, the ground is chosen as a reference object and it is well known that reflection of coherent light from a rough surface produces speckle pattern which is a random interference pattern due to the random nature of the phase perturbations at different positions on the illuminated surface. If the surface is moving, the
speckle will also move in the same direction with a proportional velocity. That is to say, dynamic speckle becomes a disturbing signal, which will cause estimated error in velocity measurement. By now several studies have been made to discuss the speckle characteristics. R.F. Strean et al. pointed out that speckle from solid surface was an unnegligible noise in a scanning laser Doppler vibrometer [6]. Yura et al. analyzed the influence of partial coherence of the target using ABCD matrix [7]. Rough surface induced speckle effects on detection performance of a pulsed laser Doppler radar had been studied by Guanjun et al. [8]. However, few attempts have been made to investigate the speckle broadening and its influence on LDV in term of theory and experiment. The time-lagged (auto) covariance of the photocurrent from the detector is significant and useful, because the velocity information can be extracted according to the location of the peak of power spectrum. In the paper, the expressions of the time-lagged covariance and its autospectral density are derived based on light scattering theory of a random rough surface. In addition, according to the autospectral density function of covariance of the photocurrent, the formula of speckle broadening is given to analyze its influence on the measurement accuracy. Simulations and experiments are carried out to verify the correctness of the achieved results. 2. MLLDV for the vehicle self-contained navigation
∗ Corresponding author. Tel.: +86 13739091383; fax: +86 073184576314. E-mail address: [email protected] (J. Zhou). http://dx.doi.org/10.1016/j.ijleo.2014.07.048 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Now dual-beam LDV has become a choice for non-intrusive measurement in various areas. However, for a conventional
J. Zhou et al. / Optik 125 (2014) 5878–5883
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Fig. 3. A picture of the system.
Table 1 Parameters of the experiment. Surface type of the road
LDV
Fig. 1. Optical schematic of the conventional dual-beam LDV.
dual-beam LDV, two beams should be focused on the object to be measured (see Fig. 1), otherwise Doppler signal will be lost. In fact, the vehicle vibrates up and down ceaselessly on the road so that the ground leaves the intersection of the two beams sometimes. So the conventional dual-beam LDV is not suitable for the vehiclemounted self-contained navigation system. In order to enlarge the measurement range, a novel multipoint layer-type LDV is designed, which is shown in Fig. 2. The intersection of the two beams is often called probe volume in the dual-beam LDV. The height of probe volume of a single probe is given by [9] lm =
4f1 d sin(/2)
(1)
Wavelength Laser source Power Number of probes Measuring height of each probe Measuring range of Probe probe 1 Measuring range of probe 2 Diameter of the beam Focal length of the focusing lens The include angle Digitization rate
Cement 532 nm 50 mW 2 3.45 cm 0–3 cm (above ground level) 3–6 cm (above ground level) 0.2 mm 175 mm 2 degrees 100Hz
where is wavelength, f1 is the focal distance of the focusing lens, d is the diameter of the laser beam and is the include angle of the two beams. In general lm can reach several centimeters or even larger by adjusting d and . As shown in Fig. 2, several probes are distributed in vertical direction in different parts of the vehicle. Each probe has its own measurement range. For example the measurement range of Probe1 is 0 ∼ lm , the measurement range of Probe-2 is lm ∼ 2lm , etc., and the measurement range of Probe-n is (n − 1)lm ∼ nlm . No matter how the vehicle vibrates, the ground will be in the measurement range of one probe at least. So there is an effective output at least and which will be chosen out to extract the frequency for calculating the velocity of the vehicle. Fig. 3 is a picture of our system and the detail parameters are shown in Table 1. 3. Covariance and its autospectral density function For a single probe of MLLDV, the optical frame of the laser scattering intensity coherent detecting system is shown in Fig. 4, in which a is the receive-aperture of 1/e2 radius, is the limiting aperture of 1/e2 radius, s is the 1/e2 Gaussian spot radius and f2 is
Fig. 2. Schematic figure of MLLDV.
Fig. 4. Optical frame of the laser scattering intensity coherent detecting system.
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the focal distance of lens 2. The Green’s function for this detecting system is given by [7]
k2 2 k2 2 exp 4f1 f2 4f1 f2
G(r, p) =
f12 f22 p2 − 2r · p −
f2 2 r f1
(2)
where r is a vector in the object plane, p is a vector in the image plane, k = 2/ is the wavenumber. Assuming that the reflected optical field of the ground is U0 (r, t) = Ui (r) (r, t)
(3)
where Ui ( r) is the incident field, (r, t) = (r) exp[i(r)] is the
complex amplitude reflection coefficient, ( r) is the phase adjustment factor of the ground, ( r) = k(1 + cos ) h is the real reflection coefficient, is the incident angle and h is the elevation fluctuation of the ground. Based on the fringe pattern, the mean reflected interference fringe intensity in the r-plane is
k r 2 4P0 2r 2 x x 2 I0 (r) = Ui (r) = cos exp − 2 2 2
S
S
(4)
where P0 is the reflected power, kx = 2/, is the fringe period. Combining Eqs. (3) and (4), the reflected optical field can be written 2 U0 (r, t) = S
P0 cos
k r x x 2
exp
−
r2
S2
where rc = rh /[k(1 + cos ) h ] is a measure of the phase correlation length of the ground, rh is the correlation length of surface fluctuation of the ground, ω = f1 /(). The velocity information is usually not determined directly from the covariance but determined by the peak of the corresponding power spectrum. Therefore considering the autospectral density function of the covariance of photodetector current which is defined as
+∞
S(f ) = Ᏺ[Ci ( )] = (r, t)
(5)
2 I(p, t) = U0 (r, t)G(r, p)dr
(12)
−∞
(6)
4
S(f ) =
Then, the instantaneous photocurrent is given by
W (p)I(p, t)dp
Ci ( )e−2if d
where Ᏺ[] denotes the fast Fourier transformation, f is the frequency variation. Substuting Eq. (11) into Eq. (12), the autospectral density function can be written
So the intensity in the detector plane (p-plane) is
i(t) = ˛
Fig. 5. Normalized covariance of photocurrent (a) with relation to the elevation fluctuation of ground at different angles (b) with relation to the velocity of vehicle.
(7)
˛2 P02 (/f1 ) √ 2 2 2 f (1 + rc /s )(1 + s2 /ω2 )(1 + rc2 /s2 + rc2 /ω2 )
× e−((f −f0 )/ f )
2
(13)
where where ˛ is a conversion factor from power to current which depends on the quantum efficiency of the detector at the chosen wavelength, W(p) is the receive-aperture weighting function. It is easy to determine the time-lagged covariance of the resulting photocurrent from the detector which is centered at the corresponding geometrical-image point of the center of the target spot. The quantity is given by Ci ( ) = i(t)i(t + ) − i(t)i(t + )
(8)
where denotes the ensemble average of a stochastic variable. Substituting Eqs. (3), (6) and (7) into Eq. (8), the time-lagged covariance (detail derivation see Appendix A) can be obtained:
2
Ci ( ) = ˛
dp1 W (p1 )
⎧ vx ⎪ ⎨ f0 = v ⎪ ⎩ f = 2
(14)
rc + s2
where f0 is Doppler frequency, f is speckle broadening of the Doppler signal. It can be seen that the speckle broadening is a function of the velocity of the vehicle, the 1/e2 Gaussian spot radius and the phase correlation length of the ground. Obviously, in order to reduce the speckle broadening, the radius of the laser beam should be enlarged. 4. Simulations and experiments
dp2 W (p2 )K(p1 , p2 ; )
(9) 4.1. Simulations
where K(p1 , p2 ; )
2
∗ ∗ = dr 1 G(r 1 , p1 )Ui (r 1 ) dr 4 G (r 4 , p2 )Ui (r 4 )B (r 1 , r 4 − v ) (10)
The integration of time-lagged covariance is calculated based on the Mathematica program, with the final result that Ci ( ) =
4 ˛2 P02 (/f1 ) 2 2 2 2 (1 + rc /s )(1 + s /ω )(1 + rc2 /s2
×
1 + cos(vx /) −v /(r 2 + 2 ) c s e 2
+ rc2 /ω2 ) (11)
According to Eqs. (11) and (14), numerical calculation has been done to obtain the characteristic of time-lagged covariance and speckle broadening. In the simulation, assuming = 532 nm, rh = 1 m, f1 = 0.5 m and ˛ = 90%. The characteristic of time-lagged covariance is shown in Fig. 5, where Fig. 5(a) shows the relationship between normalized covariance N Ci ( ) and the elevation fluctuation of ground h at different angles and Fig. 5(b) shows the relationship between normalized covariance N Ci ( ) and the velocity of the vehicle v at = 45◦ and h = 0.2 m. Here, the timelagged covariance is normalized according to the peak value. It can be seen from Fig. 3(a) that the normalized time-lagged covariance decreases when the elevation fluctuation of ground h increases. In addition, the curve’s position of the normalized time-lagged covariance increases when the incident angle increases. The reason is the time-lagged covariance Ci ( ) is directly proportional to the
J. Zhou et al. / Optik 125 (2014) 5878–5883
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Fig. 6. Speckle broadening of Doppler frequency spectrum (a) with relation to the elevation fluctuation of ground (b) with relation to the 1/e2 Gaussian spot radius at different velocities.
phase correlation length rc . The smaller of the incident angle and the more heavily fluctuations of the ground surface, the shorter of the phase correlation length and the smaller of the time-lagged covariance. It can be found in Fig. 5(b) that the normalized timelagged covariance is inversely proportional to the velocity of the vehicle. That is because the correlation of two adjacent Doppler signals becomes bad at a certain value of time-lag when the velocity of the vehicle increases. The result of the numerical analysis of speckle broadening f is shown as a function of the elevation fluctuation of ground h , the vehicle’s velocity v and the 1/e2 Gaussian spot radius s in Fig. 6. It can be found from Fig. 6(a) that although speckle broadening increases when the elevation fluctuation of ground increases, the variation of speckle broadening is very small. Fig. 6(b) shows that speckle broadening decreases when the 1/e2 Gaussian spot radius increases. In addition, the curve’s position of speckle broadening increases when the vehicle’s velocity increases. In fact, the speckle broadening can be diminished by enlarging the 1/e2 Gaussian spot radius. Besides speckle, there are several factors contributing to the Doppler broadening, such as transit time, instrument, signal processing [10] etc. (detail description see Appendix B). For our MLLDV, comparison of the contributions to the frequency spectrum broadening of these factors is shown in Fig. 7 which includes speckle. It can be seen that the instrument broadening is a major factor in Doppler broadening. The contribution of speckle is larger than that of signal processing, while the contribution of transit time is much less than that of the other factors and therefore it can be neglected. The results indicate that speckle broadening is an unnegligible noise, which affects the measurement accuracy of MLLDV.
Fig. 7. Speckle, transit time, instrument and signal processing algorithm broadening as a function of the velocity of vehicle for an MLLDV.
Fig. 8. The measured result of our MLLDV (a) Doppler signal (b) frequency spectrum.
4.2. Experiments A typical measurement of the velocity for a vehicle by our MLLDV system is shown in Fig. 8, where Fig. 8(a) shows the measured Doppler burst and Fig. 8(b) shows the corresponding frequency spectrum. We can see from Fig. 8 that Doppler burst is obvious and Doppler broadening fD is 21 kHz when Doppler frequency fD is 2.227 MHz. A series of Doppler signals at different velocities of the vehicle have been obtained. Doppler frequencies and Doppler broadenings measured with the signal processing algorithm are shown in Fig. 9. Fig. 9(a) shows the relationship between the Doppler frequency and the velocity of the vehicle, while Fig. 9(b) shows the relationship between the Doppler broadening and the velocity. The measured Doppler frequency and the velocity of the vehicle keep a good linear relationship, which is good agreement with the theoretical formula [10]. It can be seen from Fig. 9(b) that the Doppler broadening is directly proportional to the velocity of the vehicle. The physical reason is obvious: when the velocity of the vehicle increases, all of the Doppler broadenings including speckle broadening, transit time broadening, instrument broadening and signal processing broadening increase. The ratio between Doppler frequency and the corresponding Doppler broadening is about 0.72%. In fact, because the rotating speed of the motor cannot exceed 3200 r/min and the radius of the wheel is 18 cm, the maximum speed of our electric vehicle (shown in Fig. 3) is 9.6 m/s. More experiments will be carried out when the vehicle runs at a high speed.
Fig. 9. The acquired results by experiments (a) relationship between Doppler frequency and the velocity (b) relationship between Doppler broadening and the velocity.
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J. Zhou et al. / Optik 125 (2014) 5878–5883
5. Conclusion
For the vehicle moving with a constant velocity, assuming that the time evolution of the reflected phase is given by
The paper demonstrates that speckle can cause the broadening of frequency spectrum in MLLDV by theory and experiment. Simulations and experiments show that the time-lagged covariance of the photocurrent is directly proportional to the incident angle, and is inversely proportional to the elevation fluctuation of the ground and the velocity of the vehicle. Speckle broadening is a function of the velocity of the vehicle, the 11/e2 Gaussian spot radius and the phase correlation length of the ground. The speckle broadening can be diminished by enlarging the 1/e2 Gaussian spot radius when speckle broadening is a major factor among the several broadenings. For our MLLDV, Doppler frequency and Doppler broadening are both directly proportional to the velocity of the vehicle. Besides, the ratio between the Doppler broadening and the corresponding Doppler frequency is about 0.72% when the speed of the electric vehicle varies from 0 to 9.6 m/s. In addition, more research on speckle noise under the condition of angular motion and experiments when the vehicle runs at a higher speed should be carried out.
(r, t + ) =
(r − v , t)
(A5)
And the correlation function is defined as ∗
B (r 1 , r 2 ) = (r 1 , t)
(r 2 , t) =
8
e−(2(r 1 −r 2 )
k2 rc2
2
)/rc2
(A6)
Combining Eqs. (A5) and (A6), Eq. (A4) reduces to
2
Ci ( ) = ˛
dp1 W (p1 )
dp2 W (p2 )
dr 1
dr 2
dr 3
dr 4 G(r 1 , p1 )G∗ (r 2 , p1 )G(r 3 , p2 )G∗ (r 4 , p2 ) · Ui (r 1 )Ui∗ (r 4 )
×
× Ui∗ (r 2 )Ui (r 3 ) (r 1 , t)
= ˛2
∗
dp1 W (p1 )
(r 4 − v , t)
dp2 W (p2 )
∗
dr 1
(r 2 , t) (r 3 − v , t)
dr 2
dr 3
× dr 4 G(r 1 , p1 )G∗ (r 2 , p1 )G(r 3 , p2 )G∗ (r 4 , p2 )
Acknowledgment ∗
2
·B (r 1 , r 4 − v )B (r 2 , r 3 − v ) = ˛ The authors gratefully acknowledge the support from the Fund of National Natural Science Foundation of China under Grant no. 61308060.
dp1 W (p1 )
dp2 W (p2 )
2
∗ ∗ × dr 1 G(r 1 , p1 )Ui (r 1 ) dr 4 G (r 4 , p2 )Ui (r 4 )B (r 1 , r 4 − v )
Appendix A. The time-lagged covariance of the photocurrent
= ˛2
dp2 W (p2 )K(p1 , p2 ; )
dp1 W (p1 )
(A7)
The time-lagged covariance of the resulting photocurrent from the detector is given by Ci ( ) = i(t)i(t + ) − i(t)i(t + )
(A1)
Substituting Eqs. (3), (6) and (7) into Eq. (8) can obtain
Ci ( ) = i(t)i(t + )−i(t)i(t + ) = ˛
= ˛
W (p1 )
× dp1 ˛
2
U0 (r, t)G(r, p1 )dr dp1 ˛
W (p2 )
×
W (p2 )
2
W (p2 )I(p2 , t + )dp2 − ˛
W (p1 )I(p1 , t)dp1 ˛
W (p1 )I(p1 , t)dp1 ˛
2
U0 (r, t + )G(r, p2 )dr dp2 − ˛
U0 (r, t + )G(r, p2 )dr dp2 = ˛2
dp1 W (p1 )
W (p1 )
2
U0 (r, t)G(r, p1 )dr
dr 2 G∗ (r 2 , p1 )
dr 1 G(r 1 , p1 )
W (p2 )I(p2 , t + )dp2
dp2 W (p2 )
dr 4 G∗ (r 4 , p2 ) · [U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + ) − U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + )]
dr 3 G(r 3 , p2 )
(A2)
Based on the theory of complex Gaussian square, U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + ) = U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + ) + U0 (r 1 , t)U0∗ (r 4 , t + )U0∗ (r 2 , t)U0 (r 3 , t + ) (A3)
Then, Eq. (A2) reduces to
Ci ( ) = ˛2
dp1 W (p1 )
dr 1 G(r 1 , p1 )
dr 2 G∗ (r 2 , p1 )
dp2 W (p2 )
·U0 (r 1 , t)U0∗ (r 4 , t + )U0∗ (r 2 , t)U0 (r 3 , t + ) = ˛2 p2 )G∗ (r 4 , p2 ) · U1 (r 1 )Ui∗ (r 4 )Ui∗ (r 2 )Ui (r 3 ) (r 1 , t)
∗
dp1 W (p1 )
(r 4 , t + )
∗
dr 4 G∗ (r 4 , p2 )
dr 3 G(r 3 , p2 )
dp2 W (p2 )
dr 1
(r 2 , t) (r 3 , t + )
dr 2
dr 3
dr 4 G(r 1 , p1 )G∗ (r 2 , p1 )G(r 3 , (A4)
J. Zhou et al. / Optik 125 (2014) 5878–5883
where
Appendix B. Several factors contributing to Doppler broadening
K(p1 , p2 ; )
2
∗ ∗ = dr 1 G(r 1 , p1 )Ui (r 1 ) dr 4 G (r 4 , p2 )Ui (r 4 )B (r 1 , r 4 − v ) (A8)
The integration of K( p1 , p2 ; ) is calculated based on the Mathematica program with the result K(p1 , p2 ; ) = ×e
642 rc8 2 (1 − 4A rc4 )(4U 2
− W 2)
(T 2 U+T 2 U+T 2 U+T 2 U+4WGU 2 −T1 T3 W −T2 T4 W −WGW 2 )/(4U 2 −W 2 ) 1
2
3
4
1
(A9)
1 2 + 2 ω2 rc
(A10)
2f1 F AF 2 rc2 − Q f2
(A11)
s2
U=
+
W=
(A12)
4A2 rc4 − 1
(A13)
8rc2 F2
(A14)
Q
T1 =
FHx − 2Arc2 FJx Q
(A15)
T2 =
FHy − 2Arc2 FJy Q
(A16)
T3 =
FJx − 2Arc2 FHx Q
(A17)
T4 =
FJy − 2Arc2 FHy Q
(A18)
Hx = vx
Hy = vy Jx = Jy =
1 2rc2 1 2rc2
+
+
1 s2 1
(A19)
(A20)
s2
vx
(A21)
vy
(A22)
2rc2
In practice, it is good choice to assume that the detector aperture is larger than the imaged spot, and set W( p) = 1 for calculating the time-lagged covariance. Substituting Eqs. (A9)–(A22) into Eq. (A7), the final result of the time-lagged covariance with the Mathematica program ˛2 P02 (/f1 )
4
(1 + rc2 /s2 )(1 + s2 /ω2 )(1 + rc2 /s2 + rc2 /ω2 )
×
In the model of laser Doppler, Doppler frequency is a function of the included angle between laser source and detector. And the detector receives the scattering in a small solid angle, so the aperture size of the detector causes the broadening of the frequency spectrum. This is the instrument broadening.
Fast Fourier transform (FFT) is utilized to extract the Doppler frequency. Its frequency resolution is given by f =
fs N
(B1)
where fs is the sampling frequency and N is the number of the sampling data. It seems that the frequency resolution can be improved by decreasing the sampling frequency fs or increasing the number of the sampling data N. However, fs cannot be decreased because of the wide range of the measured velocity, and N cannot be increased because of the huge amount of the computation. The limited frequency resolution of this method broadened the frequency spectrum. This is the signal processing broadening. References
2rc2
Ci ( ) =
The signal produced by a particle only lasts a limited period of time, for the time of the single particle going through the spot is limited. The time is called transit time. According to the uncertainty principle, the limited transit time causes the broadening of the frequency spectrum. This is the transit time broadening.
B.3. Signal processing broadening
f1 F= f2 ω2 Q =
B.1. Transit time broadening
B.2. Instrument broadening
where A=
5883
1 + cos(vx /) −(v )/r 2 + 2 c s e 2
(A23)
[1] Y. Yeh, H.Z. Cummins, Localized fluid flow measurements with an He–Ne laser spectrometer, Appl. Phys. Lett. 176 (4) (1964) 176–178. [2] R.S. Hansen, C. Pedersen, All semiconductor laser Doppler anemometer at 1.55 m, Opt. Express 16 (22) (2008) 18288–18295. [3] M.A. Gondala, J. Mastromarino, U.K.A. Klein, Laser Doppler velocimeter for remote measurement of polluted water and aerosols discharges, Opt. Lasers Eng. 38 (2002) 589–600. [4] B.E. Truax, F.C. Demarest, G.E. Sommargren, Laser Doppler velocimeter for velocity and length measurements of moving surfaces, Appl. Opt. 23 (1) (1984) 67–73. [5] J. Zhou, W. Yang, X. Long, Research on multipoint layer-type laser Doppler selfvelocimeter, Opt. Eng. 49 (4) (2010), 044301(1)–(8). [6] R.F. Streana, L.D. Mitchell, A.J. Barker, Global noise characteristics of a laser Doppler vibrometer-I, Theory Opt. Laser Eng. 30 (1998) 127–139. [7] H.T. Yura, S.G. Hanso, L. Lading, Laser Doppler velocimeter: analytical solution to the optical system including the effects of partial coherence of the target, J. Opt. Soc. Am. A 12 (1995) 2040–2047. [8] G. Guanjun, S. Yun, Rough surfaces induced speckle effects on the detection performance of pulsed laser radar, Acta Phys. Sin. 53 (7) (2004) 2089–2093 (in Chinese). [9] J. Zhou, Q. Feng, S. Ma, et al., Error analysis of reference-beam laser Doppler velocimeter, High Power Laser Part. Beams 22 (11) (2010) 2581–2587 (in Chinese). [10] Y. Zhong, G. Zhang, C. Leng, et al., A differential laser Doppler system for onedimensional in-plane motion measurement of MEMS, Measurement 40 (2007) 623–627.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Research on speckle noise of laser Doppler velocimeter for the vehicle self-contained navigation Jian Zhou ∗ , Xiaoming Nie, Xingwu Long College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China
a r t i c l e
i n f o
Article history: Received 22 October 2013 Accepted 31 May 2014 Keywords: Multipoint layer-type laser Doppler velocimeter Speckle broadening Vehicle-mounted self-contained navigation system
a b s t r a c t A novel multipoint layer-type laser Doppler velocimeter (MLLDV) is designed to measure the velocity of a vehicle for the self-contained navigation system. In order to investigate the speckle’s influence on the Doppler spectrum, formulas of time-lagged covariance and speckle broadening were derived for our MLLDV. Simulations and experiments are made for detailed analysis. The results show that the time-lagged covariance of photocurrent is directly proportional to the incident angle, and is inversely proportional to the elevation fluctuation of the ground together with the velocity of the vehicle. Speckle broadening is a function of the vehicle’s velocity, the 1/e2 Gaussian spot radius and the phase correlation length of the ground. For our MLLDV, Doppler frequency and Doppler broadening are both directly proportional to the velocity of the vehicle. Besides, the ratio between Doppler broadening and the corresponding Doppler frequency is about 0.72% when the speed of the vehicle varies from 0 to 9.6 m/s. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Since the laser Doppler velocimeter (LDV) was developed in the mid-1960s [1], it has a widespread ongoing application in many scientific fields for its advantages of good linearity, fast dynamic response, noncontact measurement and high resolution [2–4]. A typical application of LDV is to determine flow velocities. Recently, using an MLLDV to offer parameter of the velocity for vehiclemounted self-contained navigation system has been reported [5]. Compared with Global Position System (GPS), LDV is autonomous for navigation system. In addition, conventional speedometers derive the velocity information from the rotational speed of the wheels which has the disadvantage that any slip between wheel and road can not be detected; in addition, any distortion of the wheel will affect the measurement accuracy of the speedometer. But LDV shows excellent performance even under adverse environmental conditions. So MLLDV will be an important speed sensor in a self-contained navigation system. In this paper, the ground is chosen as a reference object and it is well known that reflection of coherent light from a rough surface produces speckle pattern which is a random interference pattern due to the random nature of the phase perturbations at different positions on the illuminated surface. If the surface is moving, the
speckle will also move in the same direction with a proportional velocity. That is to say, dynamic speckle becomes a disturbing signal, which will cause estimated error in velocity measurement. By now several studies have been made to discuss the speckle characteristics. R.F. Strean et al. pointed out that speckle from solid surface was an unnegligible noise in a scanning laser Doppler vibrometer [6]. Yura et al. analyzed the influence of partial coherence of the target using ABCD matrix [7]. Rough surface induced speckle effects on detection performance of a pulsed laser Doppler radar had been studied by Guanjun et al. [8]. However, few attempts have been made to investigate the speckle broadening and its influence on LDV in term of theory and experiment. The time-lagged (auto) covariance of the photocurrent from the detector is significant and useful, because the velocity information can be extracted according to the location of the peak of power spectrum. In the paper, the expressions of the time-lagged covariance and its autospectral density are derived based on light scattering theory of a random rough surface. In addition, according to the autospectral density function of covariance of the photocurrent, the formula of speckle broadening is given to analyze its influence on the measurement accuracy. Simulations and experiments are carried out to verify the correctness of the achieved results. 2. MLLDV for the vehicle self-contained navigation
∗ Corresponding author. Tel.: +86 13739091383; fax: +86 073184576314. E-mail address: [email protected] (J. Zhou). http://dx.doi.org/10.1016/j.ijleo.2014.07.048 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Now dual-beam LDV has become a choice for non-intrusive measurement in various areas. However, for a conventional
J. Zhou et al. / Optik 125 (2014) 5878–5883
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Fig. 3. A picture of the system.
Table 1 Parameters of the experiment. Surface type of the road
LDV
Fig. 1. Optical schematic of the conventional dual-beam LDV.
dual-beam LDV, two beams should be focused on the object to be measured (see Fig. 1), otherwise Doppler signal will be lost. In fact, the vehicle vibrates up and down ceaselessly on the road so that the ground leaves the intersection of the two beams sometimes. So the conventional dual-beam LDV is not suitable for the vehiclemounted self-contained navigation system. In order to enlarge the measurement range, a novel multipoint layer-type LDV is designed, which is shown in Fig. 2. The intersection of the two beams is often called probe volume in the dual-beam LDV. The height of probe volume of a single probe is given by [9] lm =
4f1 d sin(/2)
(1)
Wavelength Laser source Power Number of probes Measuring height of each probe Measuring range of Probe probe 1 Measuring range of probe 2 Diameter of the beam Focal length of the focusing lens The include angle Digitization rate
Cement 532 nm 50 mW 2 3.45 cm 0–3 cm (above ground level) 3–6 cm (above ground level) 0.2 mm 175 mm 2 degrees 100Hz
where is wavelength, f1 is the focal distance of the focusing lens, d is the diameter of the laser beam and is the include angle of the two beams. In general lm can reach several centimeters or even larger by adjusting d and . As shown in Fig. 2, several probes are distributed in vertical direction in different parts of the vehicle. Each probe has its own measurement range. For example the measurement range of Probe1 is 0 ∼ lm , the measurement range of Probe-2 is lm ∼ 2lm , etc., and the measurement range of Probe-n is (n − 1)lm ∼ nlm . No matter how the vehicle vibrates, the ground will be in the measurement range of one probe at least. So there is an effective output at least and which will be chosen out to extract the frequency for calculating the velocity of the vehicle. Fig. 3 is a picture of our system and the detail parameters are shown in Table 1. 3. Covariance and its autospectral density function For a single probe of MLLDV, the optical frame of the laser scattering intensity coherent detecting system is shown in Fig. 4, in which a is the receive-aperture of 1/e2 radius, is the limiting aperture of 1/e2 radius, s is the 1/e2 Gaussian spot radius and f2 is
Fig. 2. Schematic figure of MLLDV.
Fig. 4. Optical frame of the laser scattering intensity coherent detecting system.
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the focal distance of lens 2. The Green’s function for this detecting system is given by [7]
k2 2 k2 2 exp 4f1 f2 4f1 f2
G(r, p) =
f12 f22 p2 − 2r · p −
f2 2 r f1
(2)
where r is a vector in the object plane, p is a vector in the image plane, k = 2/ is the wavenumber. Assuming that the reflected optical field of the ground is U0 (r, t) = Ui (r) (r, t)
(3)
where Ui ( r) is the incident field, (r, t) = (r) exp[i(r)] is the
complex amplitude reflection coefficient, ( r) is the phase adjustment factor of the ground, ( r) = k(1 + cos ) h is the real reflection coefficient, is the incident angle and h is the elevation fluctuation of the ground. Based on the fringe pattern, the mean reflected interference fringe intensity in the r-plane is
k r 2 4P0 2r 2 x x 2 I0 (r) = Ui (r) = cos exp − 2 2 2
S
S
(4)
where P0 is the reflected power, kx = 2/, is the fringe period. Combining Eqs. (3) and (4), the reflected optical field can be written 2 U0 (r, t) = S
P0 cos
k r x x 2
exp
−
r2
S2
where rc = rh /[k(1 + cos ) h ] is a measure of the phase correlation length of the ground, rh is the correlation length of surface fluctuation of the ground, ω = f1 /(). The velocity information is usually not determined directly from the covariance but determined by the peak of the corresponding power spectrum. Therefore considering the autospectral density function of the covariance of photodetector current which is defined as
+∞
S(f ) = Ᏺ[Ci ( )] = (r, t)
(5)
2 I(p, t) = U0 (r, t)G(r, p)dr
(12)
−∞
(6)
4
S(f ) =
Then, the instantaneous photocurrent is given by
W (p)I(p, t)dp
Ci ( )e−2if d
where Ᏺ[] denotes the fast Fourier transformation, f is the frequency variation. Substuting Eq. (11) into Eq. (12), the autospectral density function can be written
So the intensity in the detector plane (p-plane) is
i(t) = ˛
Fig. 5. Normalized covariance of photocurrent (a) with relation to the elevation fluctuation of ground at different angles (b) with relation to the velocity of vehicle.
(7)
˛2 P02 (/f1 ) √ 2 2 2 f (1 + rc /s )(1 + s2 /ω2 )(1 + rc2 /s2 + rc2 /ω2 )
× e−((f −f0 )/ f )
2
(13)
where where ˛ is a conversion factor from power to current which depends on the quantum efficiency of the detector at the chosen wavelength, W(p) is the receive-aperture weighting function. It is easy to determine the time-lagged covariance of the resulting photocurrent from the detector which is centered at the corresponding geometrical-image point of the center of the target spot. The quantity is given by Ci ( ) = i(t)i(t + ) − i(t)i(t + )
(8)
where denotes the ensemble average of a stochastic variable. Substituting Eqs. (3), (6) and (7) into Eq. (8), the time-lagged covariance (detail derivation see Appendix A) can be obtained:
2
Ci ( ) = ˛
dp1 W (p1 )
⎧ vx ⎪ ⎨ f0 = v ⎪ ⎩ f = 2
(14)
rc + s2
where f0 is Doppler frequency, f is speckle broadening of the Doppler signal. It can be seen that the speckle broadening is a function of the velocity of the vehicle, the 1/e2 Gaussian spot radius and the phase correlation length of the ground. Obviously, in order to reduce the speckle broadening, the radius of the laser beam should be enlarged. 4. Simulations and experiments
dp2 W (p2 )K(p1 , p2 ; )
(9) 4.1. Simulations
where K(p1 , p2 ; )
2
∗ ∗ = dr 1 G(r 1 , p1 )Ui (r 1 ) dr 4 G (r 4 , p2 )Ui (r 4 )B (r 1 , r 4 − v ) (10)
The integration of time-lagged covariance is calculated based on the Mathematica program, with the final result that Ci ( ) =
4 ˛2 P02 (/f1 ) 2 2 2 2 (1 + rc /s )(1 + s /ω )(1 + rc2 /s2
×
1 + cos(vx /) −v /(r 2 + 2 ) c s e 2
+ rc2 /ω2 ) (11)
According to Eqs. (11) and (14), numerical calculation has been done to obtain the characteristic of time-lagged covariance and speckle broadening. In the simulation, assuming = 532 nm, rh = 1 m, f1 = 0.5 m and ˛ = 90%. The characteristic of time-lagged covariance is shown in Fig. 5, where Fig. 5(a) shows the relationship between normalized covariance N Ci ( ) and the elevation fluctuation of ground h at different angles and Fig. 5(b) shows the relationship between normalized covariance N Ci ( ) and the velocity of the vehicle v at = 45◦ and h = 0.2 m. Here, the timelagged covariance is normalized according to the peak value. It can be seen from Fig. 3(a) that the normalized time-lagged covariance decreases when the elevation fluctuation of ground h increases. In addition, the curve’s position of the normalized time-lagged covariance increases when the incident angle increases. The reason is the time-lagged covariance Ci ( ) is directly proportional to the
J. Zhou et al. / Optik 125 (2014) 5878–5883
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Fig. 6. Speckle broadening of Doppler frequency spectrum (a) with relation to the elevation fluctuation of ground (b) with relation to the 1/e2 Gaussian spot radius at different velocities.
phase correlation length rc . The smaller of the incident angle and the more heavily fluctuations of the ground surface, the shorter of the phase correlation length and the smaller of the time-lagged covariance. It can be found in Fig. 5(b) that the normalized timelagged covariance is inversely proportional to the velocity of the vehicle. That is because the correlation of two adjacent Doppler signals becomes bad at a certain value of time-lag when the velocity of the vehicle increases. The result of the numerical analysis of speckle broadening f is shown as a function of the elevation fluctuation of ground h , the vehicle’s velocity v and the 1/e2 Gaussian spot radius s in Fig. 6. It can be found from Fig. 6(a) that although speckle broadening increases when the elevation fluctuation of ground increases, the variation of speckle broadening is very small. Fig. 6(b) shows that speckle broadening decreases when the 1/e2 Gaussian spot radius increases. In addition, the curve’s position of speckle broadening increases when the vehicle’s velocity increases. In fact, the speckle broadening can be diminished by enlarging the 1/e2 Gaussian spot radius. Besides speckle, there are several factors contributing to the Doppler broadening, such as transit time, instrument, signal processing [10] etc. (detail description see Appendix B). For our MLLDV, comparison of the contributions to the frequency spectrum broadening of these factors is shown in Fig. 7 which includes speckle. It can be seen that the instrument broadening is a major factor in Doppler broadening. The contribution of speckle is larger than that of signal processing, while the contribution of transit time is much less than that of the other factors and therefore it can be neglected. The results indicate that speckle broadening is an unnegligible noise, which affects the measurement accuracy of MLLDV.
Fig. 7. Speckle, transit time, instrument and signal processing algorithm broadening as a function of the velocity of vehicle for an MLLDV.
Fig. 8. The measured result of our MLLDV (a) Doppler signal (b) frequency spectrum.
4.2. Experiments A typical measurement of the velocity for a vehicle by our MLLDV system is shown in Fig. 8, where Fig. 8(a) shows the measured Doppler burst and Fig. 8(b) shows the corresponding frequency spectrum. We can see from Fig. 8 that Doppler burst is obvious and Doppler broadening fD is 21 kHz when Doppler frequency fD is 2.227 MHz. A series of Doppler signals at different velocities of the vehicle have been obtained. Doppler frequencies and Doppler broadenings measured with the signal processing algorithm are shown in Fig. 9. Fig. 9(a) shows the relationship between the Doppler frequency and the velocity of the vehicle, while Fig. 9(b) shows the relationship between the Doppler broadening and the velocity. The measured Doppler frequency and the velocity of the vehicle keep a good linear relationship, which is good agreement with the theoretical formula [10]. It can be seen from Fig. 9(b) that the Doppler broadening is directly proportional to the velocity of the vehicle. The physical reason is obvious: when the velocity of the vehicle increases, all of the Doppler broadenings including speckle broadening, transit time broadening, instrument broadening and signal processing broadening increase. The ratio between Doppler frequency and the corresponding Doppler broadening is about 0.72%. In fact, because the rotating speed of the motor cannot exceed 3200 r/min and the radius of the wheel is 18 cm, the maximum speed of our electric vehicle (shown in Fig. 3) is 9.6 m/s. More experiments will be carried out when the vehicle runs at a high speed.
Fig. 9. The acquired results by experiments (a) relationship between Doppler frequency and the velocity (b) relationship between Doppler broadening and the velocity.
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5. Conclusion
For the vehicle moving with a constant velocity, assuming that the time evolution of the reflected phase is given by
The paper demonstrates that speckle can cause the broadening of frequency spectrum in MLLDV by theory and experiment. Simulations and experiments show that the time-lagged covariance of the photocurrent is directly proportional to the incident angle, and is inversely proportional to the elevation fluctuation of the ground and the velocity of the vehicle. Speckle broadening is a function of the velocity of the vehicle, the 11/e2 Gaussian spot radius and the phase correlation length of the ground. The speckle broadening can be diminished by enlarging the 1/e2 Gaussian spot radius when speckle broadening is a major factor among the several broadenings. For our MLLDV, Doppler frequency and Doppler broadening are both directly proportional to the velocity of the vehicle. Besides, the ratio between the Doppler broadening and the corresponding Doppler frequency is about 0.72% when the speed of the electric vehicle varies from 0 to 9.6 m/s. In addition, more research on speckle noise under the condition of angular motion and experiments when the vehicle runs at a higher speed should be carried out.
(r, t + ) =
(r − v , t)
(A5)
And the correlation function is defined as ∗
B (r 1 , r 2 ) = (r 1 , t)
(r 2 , t) =
8
e−(2(r 1 −r 2 )
k2 rc2
2
)/rc2
(A6)
Combining Eqs. (A5) and (A6), Eq. (A4) reduces to
2
Ci ( ) = ˛
dp1 W (p1 )
dp2 W (p2 )
dr 1
dr 2
dr 3
dr 4 G(r 1 , p1 )G∗ (r 2 , p1 )G(r 3 , p2 )G∗ (r 4 , p2 ) · Ui (r 1 )Ui∗ (r 4 )
×
× Ui∗ (r 2 )Ui (r 3 ) (r 1 , t)
= ˛2
∗
dp1 W (p1 )
(r 4 − v , t)
dp2 W (p2 )
∗
dr 1
(r 2 , t) (r 3 − v , t)
dr 2
dr 3
× dr 4 G(r 1 , p1 )G∗ (r 2 , p1 )G(r 3 , p2 )G∗ (r 4 , p2 )
Acknowledgment ∗
2
·B (r 1 , r 4 − v )B (r 2 , r 3 − v ) = ˛ The authors gratefully acknowledge the support from the Fund of National Natural Science Foundation of China under Grant no. 61308060.
dp1 W (p1 )
dp2 W (p2 )
2
∗ ∗ × dr 1 G(r 1 , p1 )Ui (r 1 ) dr 4 G (r 4 , p2 )Ui (r 4 )B (r 1 , r 4 − v )
Appendix A. The time-lagged covariance of the photocurrent
= ˛2
dp2 W (p2 )K(p1 , p2 ; )
dp1 W (p1 )
(A7)
The time-lagged covariance of the resulting photocurrent from the detector is given by Ci ( ) = i(t)i(t + ) − i(t)i(t + )
(A1)
Substituting Eqs. (3), (6) and (7) into Eq. (8) can obtain
Ci ( ) = i(t)i(t + )−i(t)i(t + ) = ˛
= ˛
W (p1 )
× dp1 ˛
2
U0 (r, t)G(r, p1 )dr dp1 ˛
W (p2 )
×
W (p2 )
2
W (p2 )I(p2 , t + )dp2 − ˛
W (p1 )I(p1 , t)dp1 ˛
W (p1 )I(p1 , t)dp1 ˛
2
U0 (r, t + )G(r, p2 )dr dp2 − ˛
U0 (r, t + )G(r, p2 )dr dp2 = ˛2
dp1 W (p1 )
W (p1 )
2
U0 (r, t)G(r, p1 )dr
dr 2 G∗ (r 2 , p1 )
dr 1 G(r 1 , p1 )
W (p2 )I(p2 , t + )dp2
dp2 W (p2 )
dr 4 G∗ (r 4 , p2 ) · [U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + ) − U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + )]
dr 3 G(r 3 , p2 )
(A2)
Based on the theory of complex Gaussian square, U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + ) = U0 (r 1 , t)U0∗ (r 2 , t)U0 (r 3 , t + )U0∗ (r 4 , t + ) + U0 (r 1 , t)U0∗ (r 4 , t + )U0∗ (r 2 , t)U0 (r 3 , t + ) (A3)
Then, Eq. (A2) reduces to
Ci ( ) = ˛2
dp1 W (p1 )
dr 1 G(r 1 , p1 )
dr 2 G∗ (r 2 , p1 )
dp2 W (p2 )
·U0 (r 1 , t)U0∗ (r 4 , t + )U0∗ (r 2 , t)U0 (r 3 , t + ) = ˛2 p2 )G∗ (r 4 , p2 ) · U1 (r 1 )Ui∗ (r 4 )Ui∗ (r 2 )Ui (r 3 ) (r 1 , t)
∗
dp1 W (p1 )
(r 4 , t + )
∗
dr 4 G∗ (r 4 , p2 )
dr 3 G(r 3 , p2 )
dp2 W (p2 )
dr 1
(r 2 , t) (r 3 , t + )
dr 2
dr 3
dr 4 G(r 1 , p1 )G∗ (r 2 , p1 )G(r 3 , (A4)
J. Zhou et al. / Optik 125 (2014) 5878–5883
where
Appendix B. Several factors contributing to Doppler broadening
K(p1 , p2 ; )
2
∗ ∗ = dr 1 G(r 1 , p1 )Ui (r 1 ) dr 4 G (r 4 , p2 )Ui (r 4 )B (r 1 , r 4 − v ) (A8)
The integration of K( p1 , p2 ; ) is calculated based on the Mathematica program with the result K(p1 , p2 ; ) = ×e
642 rc8 2 (1 − 4A rc4 )(4U 2
− W 2)
(T 2 U+T 2 U+T 2 U+T 2 U+4WGU 2 −T1 T3 W −T2 T4 W −WGW 2 )/(4U 2 −W 2 ) 1
2
3
4
1
(A9)
1 2 + 2 ω2 rc
(A10)
2f1 F AF 2 rc2 − Q f2
(A11)
s2
U=
+
W=
(A12)
4A2 rc4 − 1
(A13)
8rc2 F2
(A14)
Q
T1 =
FHx − 2Arc2 FJx Q
(A15)
T2 =
FHy − 2Arc2 FJy Q
(A16)
T3 =
FJx − 2Arc2 FHx Q
(A17)
T4 =
FJy − 2Arc2 FHy Q
(A18)
Hx = vx
Hy = vy Jx = Jy =
1 2rc2 1 2rc2
+
+
1 s2 1
(A19)
(A20)
s2
vx
(A21)
vy
(A22)
2rc2
In practice, it is good choice to assume that the detector aperture is larger than the imaged spot, and set W( p) = 1 for calculating the time-lagged covariance. Substituting Eqs. (A9)–(A22) into Eq. (A7), the final result of the time-lagged covariance with the Mathematica program ˛2 P02 (/f1 )
4
(1 + rc2 /s2 )(1 + s2 /ω2 )(1 + rc2 /s2 + rc2 /ω2 )
×
In the model of laser Doppler, Doppler frequency is a function of the included angle between laser source and detector. And the detector receives the scattering in a small solid angle, so the aperture size of the detector causes the broadening of the frequency spectrum. This is the instrument broadening.
Fast Fourier transform (FFT) is utilized to extract the Doppler frequency. Its frequency resolution is given by f =
fs N
(B1)
where fs is the sampling frequency and N is the number of the sampling data. It seems that the frequency resolution can be improved by decreasing the sampling frequency fs or increasing the number of the sampling data N. However, fs cannot be decreased because of the wide range of the measured velocity, and N cannot be increased because of the huge amount of the computation. The limited frequency resolution of this method broadened the frequency spectrum. This is the signal processing broadening. References
2rc2
Ci ( ) =
The signal produced by a particle only lasts a limited period of time, for the time of the single particle going through the spot is limited. The time is called transit time. According to the uncertainty principle, the limited transit time causes the broadening of the frequency spectrum. This is the transit time broadening.
B.3. Signal processing broadening
f1 F= f2 ω2 Q =
B.1. Transit time broadening
B.2. Instrument broadening
where A=
5883
1 + cos(vx /) −(v )/r 2 + 2 c s e 2
(A23)
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