## Laser Doppler distance sensor using phase evaluation

Optics Express, Vol. 17, Issue 4, pp. 2611-2622 (2009)

http://dx.doi.org/10.1364/OE.17.002611

Acrobat PDF (168 KB)

### Abstract

This paper presents a novel optical sensor which allows simultaneous measurements of axial position and tangential velocity of moving solid state objects. An extended laser Doppler velocimeter setup is used with two slightly tilted interference fringe systems. The distance to a solid state surface can be determined via a phase evaluation. The phase laser Doppler distance sensor offers a distance resolution of 150 nm and a total position uncertainty below 1 μm. Compared to conventional measurement techniques, such as triangulation, the distance resolution is independent of the lateral surface velocity. This advantage enables precise distance and shape measurements of fast rotating surfaces.

© 2009 Optical Society of America

## 1. Introduction

1. R. G. Dorsch, G. Häusler, and J. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. **33**, 1306-14 (1994). [CrossRef] [PubMed]

2. A. Kempe, S. Schlamp, and T. Rösgen, “Low-coherence interferometric tip-clearance probe,” Opt. Lett. **28**, 1323–1325 (2003). [CrossRef] [PubMed]

3. L. Sheng-Hua and L. Cheng-Chung, “Measuring large step heights by variable synthetic wavelength interferometry,” Meas. Sci. Technol. **13**, 1382–1387 (2002). [CrossRef]

4. G. Y. Sirat and D. Psaltis, “Conoscopic Holograms,” Opt. Commun. **65**, 243–249 (1988). [CrossRef]

5. J. Czarske, J. Môbius, K. Moldenhauer, and W. Ertmer, “External cavity laser sensor using synchronously-pumped laser diode for position measurements of rough surfaces,” Electron. Lett. **40**, 1584–1586 (2004). [CrossRef]

6. B. E. Truax, F. C. Demarest, and G. E. Sommargren, “Laser Doppler velocimeter for velocity and length measurements of moving surfaces,” Appl. Opt. **23**, 67–73 (1984). [CrossRef] [PubMed]

7. K. Matsubara, W. Stork, A. Wagner, J. Drescher, and K. D. Müller-Glaser, “Simultaneous measurement of the velocity and the displacement of the moving rough surface by a laser Doppler velocimeter,” Appl. Opt. **36**, 4516-20 (1997). [CrossRef] [PubMed]

8. T. Pfister, L. Büttner, and J. Czarske, “Laser Doppler profile sensor with sub-micrometre position resolution for velocity and absolute radius measurements of rotating objects,” Meas. Sci. Technol. **16**, 627–641 (2005). [CrossRef]

9. T. Pfister, L. Büttner, J. Czarske, H. Krain, and R. Schodl, “Turbo machine tip clearance and vibration measurements using a fibre optic laser Doppler position sensor,” Meas. Sci. Technol. **17**, 1693–1705 (2006). [CrossRef]

8. T. Pfister, L. Büttner, and J. Czarske, “Laser Doppler profile sensor with sub-micrometre position resolution for velocity and absolute radius measurements of rotating objects,” Meas. Sci. Technol. **16**, 627–641 (2005). [CrossRef]

9. T. Pfister, L. Büttner, J. Czarske, H. Krain, and R. Schodl, “Turbo machine tip clearance and vibration measurements using a fibre optic laser Doppler position sensor,” Meas. Sci. Technol. **17**, 1693–1705 (2006). [CrossRef]

*σ*is limited to about 1% of the measurement range

_{z,tot}*l*, currently.

_{z}*σ*/

_{z,tot}*l*can be considerably reduced compared to the previous sensor setup with frequency evaluation by using a steep gradient of the phase difference function.

_{z}## 2. Sensor principle

*f*. Thus, the measurement object velocity

*v*can be calculated by [10, 11

11. L. Büttner and J. Czarske, “Spatial resolving laser Doppler velocity profile sensor using slightly tilted fringe systems and phase evaluation,” Meas. Sci. Technol. **14**, 2111–2120 (2003). [CrossRef]

*d*is the mean fringe spacing due to the sensor setup. Conventionally, an average of the velocity over the measurement volume is obtained. In order to gain higher spatial resolution the two crossing laser beams have to be focused more strongly. However, this is accompanied with a higher velocity uncertainty due to increased wavefront curvature of the laser beams resulting in a stronger curvature of the fringe spacing function. Hence, with a conventional LDV, it is only possible to obtain either a high spatial resolution or a low velocity uncertainty [12

12. J. W. Czarske, “Laser Doppler velocimetry using powerful solid-state light sources, Review Paper,” Meas. Sci. Technol. **17**, R71–R91 (2006). [CrossRef]

11. L. Büttner and J. Czarske, “Spatial resolving laser Doppler velocity profile sensor using slightly tilted fringe systems and phase evaluation,” Meas. Sci. Technol. **14**, 2111–2120 (2003). [CrossRef]

*z*of the scattering object. Assuming plane wavefronts, the phase difference

*φ*can be described as:

*s*is the slope of the phase difference function

*φ*(

*z*) and

*φ*

_{0}the phase offset in the center of the measurement volume (

*z*= 0). By evaluating this phase difference, the position

*z*inside the measurement volume can be determined using the inverse function of Eq. (2). With the known working distance

*D*

_{0}between sensor front face and measurement volume, also the distance

*D*=

*D*

_{0}+

*z*of the measurement object with respect to the sensor can be determined. Consequently, only the position

*z*is considered in the following.

*z*inside the measurement volume, the local fringe spacing can be taken into account allowing a more precise velocity determination compared to a conventional LDV. Thus, Eq. (1) can be transformed to:

*φ*(

*z*) should have a high slope

*s*(cp. section 4). However, for an unambiguous determination of the position the calibration curve has to be bijective. Hence, the range of the phase difference inside the measurement volume usually has to be restricted to 2

*π*. Therefore, an optimum slope

*s*exists, which is given by [11

_{opt}11. L. Büttner and J. Czarske, “Spatial resolving laser Doppler velocity profile sensor using slightly tilted fringe systems and phase evaluation,” Meas. Sci. Technol. **14**, 2111–2120 (2003). [CrossRef]

*l*is the length of the measurement volume in z-direction.

_{z}*φ*(

*z*) by increasing the tilting angle

*ψ*between the two interference fringe systems, see Fig. 1. With a higher slope of the calibration curve, a lower measurement uncertainty can be achieved if

*σ*remains constant, see Eq. (5). This possibility of reducing the measurement uncertainty without decreasing the size of the measurement volume significantly is a key advantage of the phase sensor compared to the laser Doppler distance sensor with frequency evaluation [8

_{φ}8. T. Pfister, L. Büttner, and J. Czarske, “Laser Doppler profile sensor with sub-micrometre position resolution for velocity and absolute radius measurements of rotating objects,” Meas. Sci. Technol. **16**, 627–641 (2005). [CrossRef]

*s*, the determination of the position is no longer unambiguous within the whole measurement volume. Hence, one additional information is needed, which can be obtained by a further interference fringe system employing a third discriminable laser wavelength. The three interference fringe systems have to be adjusted in such a way, that in addition to the steeper phase function

_{opt}*φ*

_{1}(

*z*) one bijective phase function

*φ*

_{2}(

*z*) is obtained, see Fig. 2. Thus, it is possible to determine the object position first roughly via the calibration function

*φ*

_{2}(

*z*) and secondly very accurate via the steeper calibration function

*φ*

_{1}(

*z*). Consequently, a more precise position measurement is possible. Nevertheless, the requirement of a third laser wavelength demands a higher technical effort in practice.

## 3. Sensor setup

*φ*of the two scattered light signals and the position

*z*inside the measurement volume. Tilting the dichroic mirror results in a change of the angle between both interference fringe systems. Consequently, the slope

*s*of the calibration function

*φ*(

*z*) can be tuned appropriately. The phase difference offset

*φ*

_{0}can be changed by tilting the glass plates inside the Keplerian telescope [11

**14**, 2111–2120 (2003). [CrossRef]

13. L. Büttner and J. Czarske, “Passive directional discrimination in laser-Doppler anemometry by the two-wavelength quadrature homodyne technique,” Appl. Opt. **42**, 3843–3852 (2003). [CrossRef] [PubMed]

*f*

_{1,2}where calculated with a least square fit of the fast Fourier transformed photo detector signals. For phase estimation the cross-correlation function of the two photo detector signals was calculated. Via a cosine least square fit, the time shift of the maximum of the cross-correlation function was determined, which is proportional to the phase difference

*φ*.

## 4. Theoretical measurement uncertainty

*σ*of repeated measurements at a fixed axial position of the measurement object. Using Eq. (2) the position uncertainty can be described by:

_{z}*SNR*is the signal-to-noise ratio and

*N*the number of statistically independent sampling points. The minimum position uncertainty can be written now as:

*φ*, the two Doppler frequencies

*f*

_{1,2}have to be equal. Therefore, both interference fringe spacing functions

*d*

_{1,2}have to be identical (

*d*

_{1}(

*z*) =

*d*

_{2}(

*z*)). Assuming ideal Gaussian laser beams with their beam waists located in the center of the measurement volume, the interference fringe spacing functions

*d*

_{1,2}(

*z*) are given by [14]:

*θ*denotes the half crossing angle of the partial laser beams and

_{i}*w*

_{0i}the beam waist radius for the respective laser wavelength

*i*= 1,2. Since a diffraction grating is used for beam splitting with sin

*α*

_{1}/

*λ*

_{1}= sin

*α*

_{2}/

*λ*

_{2}, the minimum fringe spacings at the center of the measurement volume, i.e.

*z*= 0, are equal [13

13. L. Büttner and J. Czarske, “Passive directional discrimination in laser-Doppler anemometry by the two-wavelength quadrature homodyne technique,” Appl. Opt. **42**, 3843–3852 (2003). [CrossRef] [PubMed]

*f*

_{1,2}are not equal for

*z*≠ 0 resulting in a phase drift Δ

*φ*during the measurement time causing a systematic phase deviation [11

**14**, 2111–2120 (2003). [CrossRef]

13. L. Büttner and J. Czarske, “Passive directional discrimination in laser-Doppler anemometry by the two-wavelength quadrature homodyne technique,” Appl. Opt. **42**, 3843–3852 (2003). [CrossRef] [PubMed]

*φ*= 0 the two modulation frequencies

*f*

_{1,2}and thus the interference fringe spacings

*d*

_{1,2}have to be equal. Therefore, the terms in the square brackets in Eq. (8) have to be equal for any position

*z*. In approximation, this requirement is fulfilled when the beam waists of the two wavelengths

*w*

_{01,2}match with [11

**14**, 2111–2120 (2003). [CrossRef]

**42**, 3843–3852 (2003). [CrossRef] [PubMed]

^{-4}… 10

^{-3}has been achieved by adjusting the sensor setup according to Eq. (10). Hence, it can bes assumed that

*d*

_{1}(

*z*) ≈

*d*

_{2}(

*z*) ≈

*d*(

*z*) and therefore the systematic phase drift described by Eq. (9) can be neglected.

*z*, also the velocity of the measurement object can be determined. Based on Eq. (1) the relative measurement uncertainty for the velocity

*σ*/

_{v}*v*can be calculated by the Gaussian uncertainty propagation assuming the statistical independence of fringe spacing

*d*and Doppler frequency

*f*

*σ*= (

_{d}*∂d*/

*∂z*)

*σ*and Eq. (5) the relative measurement uncertainty for the velocity can be written as:

_{z}*∂d*/

*∂z*in the center of the measurement range are approximately zero, the second term inside the square root of Eq. (12) can be neglected and the relative measurement uncertainty for the velocity can be approximated by:

15. J. Czarske, “Statistical frequency measuring error of the quadrature demodulation technique for noisy single-tone pulse signals,” Meas. Sci. Technol. **12**, 597–614 (2001). [CrossRef]

## 5. Experimental results

*R*-values (arithmetical mean deviation of the roughness profile) between 0.1 μm and 3.6 μm. The metal objects were mounted on a motor with stabilized rotation frequency, which was moved through the measurement volume in z-direction by a motorized translation stage. Thus, measurements at defined axial positions and with well known object velocities can be accomplished. At each position 25 individual measurements were carried out in order to obtain the statistical uncertainties. For each individual measurement, the Doppler frequencies

_{a}*f*

_{1,2}and the phase difference

*φ*were calculated from the whole time domain signals corresponding to an averaging over the 12 mm broad tip of the metal objects.

*l*= 800 μm corresponding to a slope

_{z}*s*= 0.45°/μm. Secondly, a sensor setup with a slope of

_{I}*s*= 5.5°/μm was arranged corresponding to a bijective range of only 65 μm. The phase difference functions for the two setups are shown in Fig. 4. For the latter setup, the measurements were carried out only for the bijective range around

_{II}*z*= 0 in order limit the technical effort. In future, also measurements within the whole measurement range of 800 μm will be investigated by using a third interference fringe system as explained in section 2.

*R*= 0.2 μm and

_{a}*R*= 3.6 μm, respectively. It can be seen that the measured position uncertainties

_{a}*σ*(

_{z}*z*) of the two different test objects are similar, see Fig. 5(a). In the center of the measurement range (

*z*= 0) a minimum position uncertainty of

*σ*= 0.98 μm was obtained. This value matches the theoretical uncertainty estimation of

_{z}*σ*=0.78 μm well, when using Eq. (7) with a

_{ztheor}*SNR*= 5 dB, a number of data points

*N*= 18000 and a slope |

*s*

^{-1}| =0.45°/μm, which are typical values occurring at these experiments. At the outer regions the statistical uncertainties increase due to the lower

*SNR*and the lower numbers of fringes caused by the decreasing intersection area of the two partial laser beams. The mean value of the statistical position uncertainty over the whole measurement range is ⟨

*σ*ߩ ≥ 2.2 μm for both test objects.

_{z}*z*(

*z*) are completely different from each other, which is due to the speckle pattern caused by coherent scattering at the random structures of the solid state surfaces. The maximum absolute values of the systematic deviations within the measurement range are Δ

*z*= 8 μm for

_{max}*R*= 0.2 μm and Δ

_{a}*z*= 14 μm for

_{max}*R*= 3.6 μm.

_{a}*Guide to the Expression of Uncertainty in Measurement*(GUM), a total measurement uncertainty

*σ*can be calculated. Assuming that the probability density of the systematic deviations is uniformly distributed in the interval [-Δ

_{z,tot}*z*, Δ

_{max}*z*], the Gaussian uncertainty propagation result in:

_{max}*σ*= 5.2 μm and

_{z,tot}*σ*= 8.7 μm, respectively.

_{z,tot}*R*= 0.2 μm are presented in Fig. 5(c) and 5(d). A minimum position uncertainty of

_{a}*σ*= 140 nm and a corresponding mean value of ⟨

_{z}*σ*⟩ = 280 nm were achieved over a measurement range of 65 μm. Due to the speckle effect at random surface structures the maximum systematic position deviation Δ

_{z}*z*= 1.5 μm is again significantly higher than the statistical uncertainties. Using Eq. (14), a total measurement uncertainty of

_{max}*σ*= 0.91 μm could be obtained with the steep phase function setup of the phase sensor.

_{z,tot}*s*by a factor

*k*= 12.2 from

*s*= 0.45°/μm to

_{I}*s*= 5.5 °/μm a reduction of the mean statistical position uncertainty by a factor

_{II}*k*= 7.9 was obtained. Also the maximum systematic position deviation and thus the total measurement uncertainty are reduced by a factor

*k*= 9.3. This is in good agreement with theory, see Eq. (2), since

*σ*is indirectly proportional to the slope

_{z}*s*of the phase difference function assuming that

*s*remains constant. In a subsequent experiment the influence of the slope

_{φ}*s*on the position uncertainty was investigated in more detail. Therefore, measurements with different slopes

*s*were carried out. Figure 6(a) shows the measured statistical uncertainties in comparison with a 1/

*s*- regression proving that theory and experimental results agree well. Also the systematic deviations behave in the same manner resulting in a 1/

*s*- behavior also for the total measurement uncertainty, see Fig. 6(b). These results confirm that the measurement uncertainty can be reduced directly by increasing the slope

*s*of the phase difference function as predicted. However, to expand this reduced uncertainty to the whole measurement range of 800 μm, a third interference fringe system has to be applied, see Fig. 2. Respective experiments with an extended setup are planned for the future.

*s*= 0.45°/μm. Figure 7 shows the obtained total measurement uncertainties

_{I}*σ*within the whole measurement range of 800 μm according to Eq. (14), which are between 5.2 μm and 8.2 μm. No significant dependence on the surface roughness could be observed. In addition, for several test objects, measurements with different object velocities were accomplished. According to theory the object velocity should have no influence on the position uncertainty, see Eq. (7). The measurement results confirm this feature of the phase sensor, see Fig. 7.

_{z,tot}*v*, the tooth tip position was measured simultaneously by the phase sensor with a steep phase function (

*s*= 5.5°/μm) and two commercial triangulation sensors manufactured by

_{II}*Micro-Epsilon*, one with a measurement rate of 2.5 kHz (TS

_{2.5k}) and the other one with 20 kHz (TS

_{20k}). The latter one comprises an elliptical laser spot in order to reduce the influence of the speckle effect on the measurement result. Both triangulation sensors have a measurement range of 2 mm. Regarding Fig. 8, the position uncertainty of both triangulation sensors worsens with increasing velocity

*v*of the brass tooth, which is due to the decreasing averaging time. The position uncertainty of the triangulation sensor TS

_{2.5k}increases from

*σ*= 2.1μm at

_{z}*v*= 0.5 m/s to

*σ*= 6.6μm at

_{z}*v*= 7.5 m/s. Due to the higher measurement rate and the special shape of the laser spot, the measurement uncertainty of the triangulation sensor TS

_{20k}is significantly lower and increases only from

*σ*= 0.1 μm at

_{z}*v*= 0.5 m/s to

*σ*= 1.2 μm at

_{z}*v*= 12.5 m/s. In contrast to the triangulation sensors, the position uncertainty

*σ*of the phase sensor remains nearly constant at

_{z}*σ*= 500 nm over the whole velocity range proving that its position uncertainty is indeed independent of the object velocity, see Fig. 8. Hence, the phase sensor is predestined for precise position measurements on fast moving objects. This feature opens up new applications such as process control of turbo machines, vacuum pumps and at grinding and turning.

_{z}*d*

_{1,2}resulting in a mean relative measurement uncertainty of

*σ*/

_{v}*v*= 3.5×10

^{-4}. Due to the simultaneous measurement of axial position and tangential velocity, the diameter and thus the two-dimensional shape of rotating test objects can be calculated as described in [8

**16**, 627–641 (2005). [CrossRef]

## 6. Conclusions

*σ*< 1 μm within a measurement range of 65 μm was achieved. In the future, this measurement range can be extended without increasing the measurement uncertainty significantly by using a third interference fringe system. Since also the surface velocity is determined, the absolute radius and thus the two-dimensional shape of rotating objects can be calculated additionally. It was shown that the velocity can be measured with a relative uncertainty

_{z,tot}*σ*/

_{v}*v*= 3.5 × 10

^{-4}within the whole measurement volume. Furthermore, it was demonstrated that the position uncertainty is independent of the object velocity, which is an important advantage compared to many other optical sensors, such as triangulation. Hence, interesting applications of the phase sensor are distance and shape measurements of fast rotating objects such as turbo machines and vacuum pumps or at grinding and turning.

## Acknowledgments

## References and links

1. | R. G. Dorsch, G. Häusler, and J. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. |

2. | A. Kempe, S. Schlamp, and T. Rösgen, “Low-coherence interferometric tip-clearance probe,” Opt. Lett. |

3. | L. Sheng-Hua and L. Cheng-Chung, “Measuring large step heights by variable synthetic wavelength interferometry,” Meas. Sci. Technol. |

4. | G. Y. Sirat and D. Psaltis, “Conoscopic Holograms,” Opt. Commun. |

5. | J. Czarske, J. Môbius, K. Moldenhauer, and W. Ertmer, “External cavity laser sensor using synchronously-pumped laser diode for position measurements of rough surfaces,” Electron. Lett. |

6. | B. E. Truax, F. C. Demarest, and G. E. Sommargren, “Laser Doppler velocimeter for velocity and length measurements of moving surfaces,” Appl. Opt. |

7. | K. Matsubara, W. Stork, A. Wagner, J. Drescher, and K. D. Müller-Glaser, “Simultaneous measurement of the velocity and the displacement of the moving rough surface by a laser Doppler velocimeter,” Appl. Opt. |

8. | T. Pfister, L. Büttner, and J. Czarske, “Laser Doppler profile sensor with sub-micrometre position resolution for velocity and absolute radius measurements of rotating objects,” Meas. Sci. Technol. |

9. | T. Pfister, L. Büttner, J. Czarske, H. Krain, and R. Schodl, “Turbo machine tip clearance and vibration measurements using a fibre optic laser Doppler position sensor,” Meas. Sci. Technol. |

10. | H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, |

11. | L. Büttner and J. Czarske, “Spatial resolving laser Doppler velocity profile sensor using slightly tilted fringe systems and phase evaluation,” Meas. Sci. Technol. |

12. | J. W. Czarske, “Laser Doppler velocimetry using powerful solid-state light sources, Review Paper,” Meas. Sci. Technol. |

13. | L. Büttner and J. Czarske, “Passive directional discrimination in laser-Doppler anemometry by the two-wavelength quadrature homodyne technique,” Appl. Opt. |

14. | P. Miles and P. O. Witze, “Evaluation of the Gaussian beam model for prediction of LDV fringe fields,” in |

15. | J. Czarske, “Statistical frequency measuring error of the quadrature demodulation technique for noisy single-tone pulse signals,” Meas. Sci. Technol. |

16. | T. Pfister, P. Günther, L. Büttner, and J. Czarske, “Shape and vibration measurement of fast rotating objects employing novel laser Doppler techniques,” in |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(280.3340) Remote sensing and sensors : Laser Doppler velocimetry

(150.5495) Machine vision : Process monitoring and control

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 2, 2008

Revised Manuscript: January 12, 2009

Manuscript Accepted: January 12, 2009

Published: February 6, 2009

**Citation**

P. Günther, T. Pfister, L. Büttner, and J. Czarske, "Laser Doppler distance sensor using phase evaluation," Opt. Express **17**, 2611-2622 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2611

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### References

- R. G. Dorsch, G. Häusler, and J. Herrmann, "Laser triangulation: fundamental uncertainty in distance measurement," Appl. Opt. 33, 1306-14 (1994). [CrossRef] [PubMed]
- A. Kempe, S. Schlamp, and T. Rösgen, "Low-coherence interferometric tip-clearance probe," Opt. Lett. 28, 1323-1325 (2003). [CrossRef] [PubMed]
- L. Sheng-Hua and L. Cheng-Chung, "Measuring large step heights by variable synthetic wavelength interferometry," Meas. Sci. Technol. 13, 1382-1387 (2002). [CrossRef]
- G. Y. Sirat and D. Psaltis, "Conoscopic Holograms," Opt. Commun. 65, 243-249 (1988). [CrossRef]
- J. Czarske, J. Möbius, K. Moldenhauer, and W. Ertmer, "External cavity laser sensor using synchronouslypumped laser diode for position measurements of rough surfaces," Electron. Lett. 40, 1584-1586 (2004). [CrossRef]
- B. E. Truax, F. C. Demarest, and G. E. Sommargren, "Laser Doppler velocimeter for velocity and length measurements of moving surfaces," Appl. Opt. 23, 67-73 (1984). [CrossRef] [PubMed]
- K. Matsubara, W. Stork, A. Wagner, J. Drescher, and K. D. Müller-Glaser, "Simultaneous measurement of the velocity and the displacement of the moving rough surface by a laser Doppler velocimeter," Appl. Opt. 36, 4516-20 (1997). [CrossRef] [PubMed]
- T. Pfister, L. Büttner, and J. Czarske, "Laser Doppler profile sensor with sub-micrometre position resolution for velocity and absolute radius measurements of rotating objects," Meas. Sci. Technol. 16, 627-641 (2005). [CrossRef]
- T. Pfister, L. Büttner, J. Czarske, H. Krain, and R. Schodl, "Turbo machine tip clearance and vibration measurements using a fibre optic laser Doppler position sensor," Meas. Sci. Technol. 17, 1693-1705 (2006). [CrossRef]
- H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer Verlag, Berlin, 2003).
- L. Büttner and J. Czarske, "Spatial resolving laser Doppler velocity profile sensor using slightly tilted fringe systems and phase evaluation," Meas. Sci. Technol. 14, 2111-2120 (2003). [CrossRef]
- J. W. Czarske, "Laser Doppler velocimetry using powerful solid-state light sources, Review Paper," Meas. Sci. Technol. 17, R71-R91 (2006). [CrossRef]
- L. Büttner and J. Czarske, "Passive directional discrimination in laser-Doppler anemometry by the twowavelength quadrature homodyne technique," Appl. Opt. 42, 3843-3852 (2003). [CrossRef] [PubMed]
- P. Miles and P. O. Witze, "Evaluation of the Gaussian beam model for prediction of LDV fringe fields," in Proc. 8th Int. Symposium an Application of Laser Techniques to Fluid Mechanics Lisabon/Portugal, vol. 40.1, pp. 1-8 (1996).
- J. Czarske, "Statistical frequency measuring error of the quadrature demodulation technique for noisy single-tone pulse signals," Meas. Sci. Technol. 12, 597-614 (2001). [CrossRef]
- T. Pfister, P. Günther, L. Büttner, and J. Czarske, "Shape and vibration measurement of fast rotating objects employing novel laser Doppler techniques," Proc. SPIE 6616, 66,163S1-12 (2007).

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