Research on heterodyne efficiency of laser Doppler velocimeter

Optik 122 (2011) 2090–2093

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Research on heterodyne efficiency of laser Doppler velocimeter Guo Wei ∗ , Yu Wang, Jian Zhou, 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 26 August 2010 Accepted 15 January 2011

Keywords: Laser Doppler velocimeter Gaussian-Airy mode Heterodyne efficiency

a b s t r a c t Heterodyne efficiency is a very important factor in a laser Doppler velocimeter (LDV). Gaussian-Airy mode is put forward to analyze the heterodyne efficiency. And the calculate formulas and results of simulation of heterodyne efficiency are given. The results of numerical analysis show that heterodyne efficiency of LDV depends on the parameters—x0 , A,  and xa . Heterodyne efficiency can reach 81.45%, when x0 = 3.59, A = 5.04 and  = xa = 0. At last, the antenna theorem of LDV is derived, and a defocused telescope system is used to adjust the waist’s position and radius of the reference beam to match the signal beam. © 2011 Elsevier GmbH. All rights reserved.

1. Introduction Since the first application of a laser Doppler velocimeter (LDV) to fluid velocity measurement [1], utilizing an LDV to determine velocities of gas liquid and solid surface has become a widely used technique in industrial and research applications [2–4]. Compared with direct detection, heterodyne detection has the advantage of high accuracy, high transformation gain, good filtering performance and so on. In LDV, reference beam and signal beam mix on the photosensitive surface of avalanche photodiode. The beat frequency of the two beams is the Doppler frequency, which is in proportion to the velocity of the measured object. In order to make full use of the echo signal, an LDV is designed to maximize the heterodyne efficiency. So the paper analyzes the relationship between heterodyne efficiency and the optical parameters, and puts forward a method to maximize the heterodyne efficiency. 2. Laser Doppler velocimeter In this paper a reference-beam LDV has been designed to offer the vehicle’s velocity for a vehicle inertial navigation system. The optical schematic of this velocimeter is shown in Fig. 1. The light source is a 50 mW polarized solid-state laser operating in a single longitudinal mode and the TEM00 transverse mode. The output of the laser passes through the collimation and compression lens, which compresses the diameter of the laser beam and controls the divergence of the laser. The next element in the optical train is the beam splitter which divides the input beam into a transmitted

∗ Corresponding author. Tel.: +86 13875859029; fax: +86 073184576314. E-mail address: [email protected] (G. Wei). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.01.002

and reflected beam. The reflected beam passes through the attenuator on the mirror, then transmits along the negative direction and passes through the attenuator, beam splitter, Polaroid, optical filter and pinhole diaphragm onto the avalanche photodiode. We call the reflected beam “reference beam”. The transmitted beam passes through the diaphragm and the antenna on the ground. The echo signal collected by the antenna is partly reflected by the beam splitter. After that, it also passes through the Polaroid, optical filter and pinhole diaphragm onto the avalanche photodiode. We call the transmitted beam “signal beam”. As a result, reference beam and signal beam mix on the photosensitive surface of the avalanche photodiode. The Doppler frequency is given by fD =

2v cos  

(1)

where v is the velocity of the vehicle,  is the wavelength of the laser and  is the angle between incident light and the velocity vector of the vehicle. From Eq. (1), we know

v=

fD 2 cos 

(2)

3. Heterodyne efficiency of LDV The receiver of a reference-beam LDV is shown in Fig. 2. The signal beam from ground is far-field weak scattered light, which can be treated as a plane wave for approximation. And the reference beam is a Gaussian beam with TEM00 transverse mode. To assist in understanding the description, a coordinate system is established (Fig. 2). The direction of the optical axis is z axis, and the vertical direction is r axis. Based on the theory of Fraunhofer diffraction, we know that the signal beam (plane wave) becomes a light with Airy

G. Wei et al. / Optik 122 (2011) 2090–2093

v

Vehicle

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where h is heterodyne efficiency and S is the photosensitive surface of the avalanche photodiode. If kr2 /2f = kr2 /2R(zf ), that is to say

avalanche photodiode Solid-state laser

R(zf ) = f optical filter

the phase difference between signal beam and reference beam will be a constant. In order to simplify the calculation, define two parameters as follows:

diaphragm collimating and compacting lens

polaroid semitransparent mirror diaphragm

attenuator

(11)

mirror

x= antenna

ground

A=

kdr 2f

(12)

k2 d2 ω2 (zf )

(13)

4f 2

Using Eqs. (5)–(9) and (11)–(13) √ Af/d ω0 =  2 1 + (Af/d2 )

Fig. 1. Optical schematic diagram of LDV.

(Af/d2 )

zf = f

(14)

2

1 + (Af/d2 )

(15)

2

Generally, 0 ≤ A ≤ 40 and f/d2  1, then Eq. (15) reduces to zf = 0

From Eqs. (9) and (16), we know f = zl , R(zf ) = ∞ and ω(zf ) = ω0 . It means that the maximal heterodyne efficiency can be achieved when the waist’s position of the reference beam coincides with the photosensitive surface of the detector. Using Eqs. (3) and (4), Eq. (10) becomes

Fig. 2. Receiver of LDV.

distribution after passing the diaphragm. So the model of this kind of LDV is Gaussian-Airy model. So the electric field of signal beam in complex notation is given by Us (r) = 2

J1 (kdr/2f ) exp kdr/2f



j

kr 2  + jkf − 2 2f

(16)



(3)

 r 2 0

h =

0

0

2J1 (kdr/2f ) kdr/2f

∞ 2  2J (kdr/2f ) 2 1

0

0

kdr/2f



rdrdϕ

r 2 0

0



2 2 2

exp − 4fk2 ωd2 r(z 0

)

f

2 rdrdϕ



2 2 2

exp − 2fk2 ωd2 r(z

f

)

(17) rdrdϕ

Combining Eqs. (12) and (13), Eq. (17) reduces to

 8 A

x0 0



2

J1 (x) exp − xA



2 dx

where k is wave number, d is the diameter of diaphragm, f is the focal distance of lens and J1 is the first order Bessel function. And the electric field of reference beam is given by

h =

r2 Ur (r) = exp − 2 exp ω (zf )

where x0 = kdr0 /(2f) and r0 is the radius of the photosensitive surface of the avalanche photodiode. But when the optical axis of signal beam and that of reference beam are not coincident strictly, the heterodyne efficiency becomes







 

−j k

r2 + zf 2R(zf )





−˚

(4)

where zo =

1 2 kω 2 0



ω(z) = ω0 R(z) = z0

1+

z z0

˚ = arctan

+

h =

z 2

(6)

z0

z0

(7)

z

z

8 A

x0 0



2x2 0 A





(18)



2

J1 (x)J0 2 df  exp − xA



1 − exp −

2x2 0 A





2 dx (19)

where  is the mismatch angle of the two beams. 4. Numerical analysis

(8)

z0

zf = f − zl

(9)

z0 is the confocal parameter, ω0 is radius of waist, ω(z) is radius of equiphase wave surface at z point and ˚ is the initial phase. David Fink has given the design formulas of heterodyne efficiency when the phase difference between signal beam and reference beam is a constant [5]



 

Us (r) · Ur (r) dS 2 D h =

∞ ∞

Us (r) 2 dS |Ur (r)|2 dS · D



(5)



1 − exp −

0

0

(10)

Based on the calculation formula of heterodyne efficiency, we know heterodyne efficiency depends on the optical parameter of detector x0 , the optical parameter of Gaussian beam A and the mismatch angle . Figs. 3–5 show the numerical curves of GaussianAiry heterodyne efficiency. Figs. 3 and 4 show the change curves of heterodyne efficiency h along with x0 and A, respectively, when the mismatch angle  is different at each time. And Fig. 5 shows the graph of relation between heterodyne efficiency h and the mismatch angle . From the results of numerical analysis, we know that the mismatch angle has a great influence on the heterodyne efficiency,

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G. Wei et al. / Optik 122 (2011) 2090–2093

Fig. 3. Numerical curves that Gaussian-Airy heterodyne efficiency with parameter x0 different.

Fig. 6. Numerical curves that Gaussian-Airy heterodyne efficiency with parameter xa different.

So we can put Eqs. (20) and (21) as the antenna theorem of the reference-beam LDV. When the optical axes of the two beams are parallel (xa is the distance between the optical axis of the signal beam and that of the reference beam), the change curve of heterodyne efficiency with different xa is shown in Fig. 6. From Fig. 6, we know that the heterodyne efficiency of LDV decreases when the distance of the two optical axes becomes large. So we should do our effort to make the optical axes of the two beams coincident. 5. Technological approach A defocused telescope system (Fig. 7) is usually used to adjust the position and radius of waist of Gaussian beam. The position parameter si and radius of waist ω02 are given by [6]

Fig. 4. Numerical curves that Gaussian-Airy heterodyne efficiency with parameter A different.

Si =

f12 f22 [S0 + f1 (f1 − )/] f2 ( − f2 ) + 2 2 } 2   {[S0 + f1 (f1 − )/] + Z01

ω02 = ω01





MT +

 s − i f1 f1 f2

(22)

2

s0 (MT + /f1 ) + si (1/MT + /f2 ) − s0 si /f1 f2 + l + Z01

2 1/2 (23)

where  = f1 + f2 − l Z01 = MT =

2 ω01

 −f2 f1

Fig. 5. Numerical curves that Gaussian-Airy heterodyne efficiency with parameter  different.

and when  = 0, x0 = 3.59, A = 5.04, LDV achieves the maximal heterodyne efficiency. According to Eq. (14) and x0 = kdr0 /(2f), ω0 and r0 are given by ω0 = 0.714 r0 = 1.143

f d

f d

(20) (21)

Fig. 7. Transformation of Gaussian beam through telescope system.

(24) (25) (26)

G. Wei et al. / Optik 122 (2011) 2090–2093

 is defocus length, Z01 is the Rayleigh length in object space and MT is the magnification of the telescope system. Based on Eqs. (22) and (23), we can change the defocus length and the position of the telescope system to adjust the waist’s position and radius of the reference beam for matching the signal beam. 6. Conclusion The paper presents an LDV for vehicle inertial navigation system. And the heterodyne efficiency of the reference-beam LDV is analyzed. The maximal heterodyne efficiency can be achieved when the waist’s position of the reference beam coincides with the photosensitive surface of the detector. The results of numerical analysis show that the heterodyne efficiency depends on four parameters—x0 , A, , and xa . Based on the antenna theorem, ω0 and r0 are chosen to

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achieve maximal heterodyne efficiency. At last, we also demonstrated that a defocused telescope system can be used to adjust the waist’s position and radius of the reference beam for matching the signal beam. References [1] Y. Yeh, H.Z. Cummings, Localized flow measurements with a He–Ne laser spectrometer, Appl. Phys. Lett. 176 (4) (1964) 176–178. [2] René Skov Hansen, Christian Pedersen, All semiconductor laser Doppler anemometer at 1.55 ␮m, Opt. Express 16 (22) (2008) 18288–18295. [3] I. Stec, S. Hertegård, J.E. Juto, Laser Doppler measurements of the vocal fold blood micro-circulation, J. Voice 21 (3) (2007) 345–354. [4] Béatrice Augere, Jean-Pierre Cariou, All-fiber 1.5 ␮m CW coherent laser anemometer for in-flight measurements, in: Laser Radar Technology and Applications VIII, Proceedings of SPIE, vol. 5086, 2003, pp. 121–128. [5] Fink David, Coherent detection signal-to-noise, Appl. Opt. 14 (3) (1975) 689–690. [6] Zongfeng Ma, Chunxi Zhang, Shuang Gao, et al., Mixing efficiency in coherent Lidar receiver, Opt. Tech. 32 (2006) 377–379 (in Chinese).