## Continual mechanical vibration trajectory tracking based on electro-optical heterodyne interferometry |

Optics Express, Vol. 22, Issue 7, pp. 7799-7810 (2014)

http://dx.doi.org/10.1364/OE.22.007799

Acrobat PDF (1714 KB)

### Abstract

Vibration is one of the confused problems in many fields. To give a comprehensive analysis of vibration, an electro-optical heterodyne interferometry with temporal intensity analysis method that can track the trajectory of the vibration dynamically has been built in this paper. The carrier frequency is introduced by the electrically controlled electro-optical frequency shifter. The trajectory is obtained by using temporal evolution of the light intensity in heterodyne interferometry. The instantaneous displacement of the vibration is extracted with spectral analysis technique. No target mirror and moving parts are required in our self-developed system. The principle and system configuration are described. The simulations and the preliminary experiments have been performed and the results show that this trajectory tracking system is high-efficiency, low-cost, jamproof, robust, precise and simple.

© 2014 Optical Society of America

## 1. Introduction

1. S. I. Stepanov, I. A. Sokolov, G. S. Trofimov, V. I. Vlad, D. Popa, and I. Apostol, “Measuring vibration amplitudes in the picometer range using moving light gratings in photoconductive GaAs:Cr,” Opt. Lett. **15**(21), 1239–1241 (1990). [CrossRef] [PubMed]

3. W. C. Wang, C. H. Hwang, and S. Y. Lin, “Vibration measurement by the time-averaged electronic speckle pattern interferometry methods,” Appl. Opt. **35**(22), 4502–4509 (1996). [CrossRef] [PubMed]

5. C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. **76**(5), 055110 (2005). [CrossRef]

6. K. A. Stetson and W. R. Brohinsky, “Fringe-shifting technique for numerical analysis of time-average holograms of vibrating objects,” J. Opt. Soc. Am. A **5**(9), 1472–1476 (1988). [CrossRef]

7. K. Kyuma, S. Tai, K. Hamanaka, and M. Nunoshita, “Laser Doppler velocimeter with a novel optical fiber probe,” Appl. Opt. **20**(14), 2424–2427 (1981). [CrossRef] [PubMed]

8. K. A. Stetson, “Method of vibration measurements in heterodyne interferometry,” Opt. Lett. **7**(5), 233–234 (1982). [CrossRef] [PubMed]

9. G. E. Sommargren, “Up/down frequency shifter for optical heterodyne interferometry,” J. Opt. Soc. Am. **65**(8), 960–961 (1975). [CrossRef]

10. O. B. Wright, “Stabilized dual-wavelength fiber-optic interferometer for vibration measurement,” Opt. Lett. **16**(1), 56–58 (1991). [CrossRef] [PubMed]

11. Y. Park and K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett. **36**(3), 331–333 (2011). [CrossRef] [PubMed]

**EOHI**) with temporal intensity analysis method is established. The carrier frequency is introduced by using a lithium niobate crystal (LiNbO

_{3}) which is located in the center of two transversely electric fields [12

12. C. F. Buhrer, L. R. Bloom, and D. H. Baird, “Electro-optic light modulation with cubic crystals,” Appl. Opt. **2**(8), 839–846 (1963). [CrossRef]

## 2. Description of EOHI system

### 2.1 The theoretical basis of heterodyne interferometry

_{3}crystal is placed in the center of two transversely electric fields which are alternating in sinusoid with phase delay π/2. When a beam of monochromatic linearly polarized light passes through the crystal, the emergent light contains left- and right-circularly polarized light components with different frequency shift. The amount of frequency shift is determined by the rotating frequency of the electric fields which are electrically controlled. By turning the rotating frequency of the electric fields up or down, the carrier frequency can be changed in a very large range. The phase which is converted from the vibration trajectory is retrieved by inverse Fourier transform of a filtered spectrum obtained by Fourier transformation of the time series intensity signal.

### 2.2 The properties of LiNbO_{3} crystal in two transversely electric fields

_{3}belongs to trigonal system. In the absence of electric field, the indicatrix of LiNbO

_{3}is a rotating ellipsoid with the three-fold axis as its rotation axis. As shown in Fig. 1, Cartesian coordinate system is built along the three-fold axis (z-axis). The x- and the y-axis are the coordinate axes in the crystal. The profile, perpendicular to z-axis, is a circle at z = 0, which means n

_{x}= n

_{y}= n

_{0}. In this case, there is no intrinsic birefringence for a beam of light passing through the crystal along the three-fold axis which is called degeneracy state.

*x*and

*y*are the same directions as they are in the crystal (see Fig. 1),

*E*is the amplitude of the applied electric fields, and

_{m}*ω*is the rotating frequency of the electric fields. The linear electro-optic or Pockels effect is taken into account and according to the symmetry of LiNbO

_{m}_{3}(trigonal system), the properties of the electro-optic tensors are [13

13. I. P. Kaminow and E. H. Turner, “Electrooptic light modulators,” Appl. Opt. **5**(10), 1612–1628 (1966). [CrossRef] [PubMed]

*n*is the refractive index of the x- and the y-axis in the absence of the electric fields,

_{0}**represents the synthetic electric field with the components expressed in Eq. (1). From Eq. (3), it is clear that the effect of two transversely electric fields is rotating the three-fold axis with an angle**

*E**θ*(Fig. 2) given byIt can be seen from Fig. 2 that the LiNbO

_{3}crystal in the center of two transversely electric fields can be regarded as a rolling wave-plate with phase delay

*Г*. The phase delay between the fast and the slow directions of polarization is [14]where

*d*is the length of the crystal along the three-fold axis,

*E*is the amplitude of the electric fields and

_{m}*λ*is the wavelength of the incident light.

_{3}along its three-fold axis, under the action of two transversely electric fields, the crystal can be regarded as a rolling wave-plate whose angular frequency is

*ω*and phase delay is

_{m}/2*Г*.

### 2.3 Method to introduce carrier frequency

_{3}crystal with two transversely electric fields and a stationary quarter-wave plate (QWP). The two electric fields are alternating in sinusoid with phase delay π/2.

*ω*is the frequency of the incident light. Based on the analysis in Section 2.2, the LiNbO

_{3}crystal in the center of two transversely electric fields can be regarded as a rolling wave plate. The effect of this crystal to the incident light can be denoted as Jones matrixwhere

*ω*is the rotating frequency of the electric fields. Hence, when the incident linearly polarized light passes through this crystal along its three-fold axis, the matrix of the emergent light isFrom Eq. (8), it can been seen that the first item represents a linearly polarized light with the same polarization direction and frequency as the incident light, the second item is a right-circularly polarized light with frequency shift

_{m}*ω*and the third item is a left-circularly polarized light with frequency shift

_{m}*-ω*(see Fig. 3). Then, the light is incident on the QWP after passing through the crystal. The QWP is placed with an angle 45° between the directions of its optical axis and y-axis. The Jones matrix of this QWP isThen, the light represented by Eq. (8) passes through the QWP, and the emergent light isIt can be seen from Eq. (10), the first item is a left-circularly polarized light which has the same frequency as the incident light, the second and the third items are a pair of linearly polarized lights whose polarization directions are perpendicular and the frequency difference of this pair of lights is

_{m}*2ω*as shown in Fig. 3.

_{m}*2ω*is obtained. The first item of Eq. (10) is considered as background light. By setting the voltage of the two electric fields to the crystal’s half-wave voltage, the first item can be zero. However, this is not a necessary procedure, because the carrier frequency is far away from the frequency of the incident light (

_{m}*ω>>2ω*). By means of spectral analysis, the first item can be easily removed.

_{m}### 2.4 System configuration and signal processing procedure

*I*and

_{1}*I*are the intensities of the two linearly polarized lights respectively, and

_{2}*I*is the intensity of the left-circularly polarized light,

_{3}*φ*is the initial phase difference between each two lights according to the subscripts,

*2ω*is the carrier frequency which is generated by the frequency difference between the two perpendicular linearly polarized lights. The two perpendicular linearly polarized lights are reflected from the reference arm and the measuring arm respectively. The last two items in Eq. (11), which are generated by the frequency shifted light and the original frequency light, are considered as the influence of the beam with frequency of

_{m}*ω*. It is clear that the carrier frequency item in Eq. (11) and the other items are in different frequency band, thus, the carrier frequency item can be easily obtained from the frequency spectrum by band-pass filtering. Hence, to make it easy to understand, Eq. (11) is simplified aswhere

*I*is the sum of the intensities which are irrelevant to the carrier frequency,

_{0}*I*is the amplitude of the carrier frequency and

_{c}*Ф*is the initial phase which usually can be ignored.

_{12}*Δz(t)*denote the instantaneous vibration displacement, then the phase change which is caused by the displacement

*Δz(t)*can be expressed aswhere

*λ*is the wavelength of the incident light and the time series intensity function which contains the vibration can be written aswhere

*f*is the carrier frequency whose value is twice the rotating frequency of the electric field (

_{0}*f*). By using of Euler’s formula, Eq. (14) can be expanded aswhere

_{0}= ω_{m}/π***represents the complex conjugate and

*c*can be expressed aswhere

*Ф(t)*is the total phase in the cosine function in Eq. (14) as expressed in Eq. (17)To calculate the phase

*Ф(t)*, Fourier transform is acted on Eq. (15) and the result iswhere capital letters represent the frequency spectrum of each item in Eq. (15) correspondingly. In frequency domain, since the three items in Eq. (18) are in different frequency band, a proper filter, whose center frequency is the carrier frequency, is adopted to screen out

*C*. Simultaneously, the influence of the beam with frequency of

*ω*and any other undesired frequency can be removed together. Then inverse Fourier transform is conducted on the filtered spectrum. Thus, in time domain,

*c(t)*is obtained and the total phase

*Ф(t)*in Eq. (17) can be denoted as [15

15. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

*Ф(t)*which is expressed in Eq. (17). Thus, the truncated phase of Eq. (17) is obtained. Phase unwrapping technique is used to obtain the total phase. Then the carrier frequency (

*2πf*) is removed, the remained part is the phase converted from vibration displacement. Finally, according to Eq. (13) and the incident wavelength, the instantaneous vibration displacement

_{0}t*Δz*can be measured. The signal processing flowchart is shown in Fig. 5.

## 3. Simulations and experiments

### 3.1 Simulations

### 3.2 Experiments

_{3}crystal and a QWP. The size of the crystal is 5 mm × 5 mm × 30 mm and the length of the three-fold axis is 30 mm. The sinusoidal alternating electric fields with a frequency of 5,903 Hz are generated between every two opposite 5 mm × 30 mm surfaces. The voltage between one pair of electrodes is 171 Volt. The phase difference between two electric fields is π/2 which is realized by the drive power supply. Thus, the frequency of the carrier is 11,806 Hz according to Eq. (10). The angles of QWP and polarizers are adjusted as the directions shown in Fig. 3 and Fig. 4. A Si-biased detector DET36A (U.S. THORLABS Company) is adopted at a sampling frequency of 50,000 Hz to collect the intensity in time sequence. The sampling time is 20.9715 s.

## 4. Discussion

_{m}(whose value is half of the carrier frequency) exist in the simulation spectrum, whereas the electric frequency signals (ω

_{m}) are remarkable in the measured spectrum. These differences are generated due to the voltage settings. In the simulation, the voltage is assumed to be the half-wave voltage. In this case, the first item of Eq. (10) vanishes. As a result of the interference between the second (ω + ω

_{m}) and the third (ω-ω

_{m}) items in Eq. (10), there is only the carrier frequency (2ω

_{m}) in the spectrum. In terms of the intensity, the third, the fifth, and the sixth items in Eq. (11) vanish. On the contrast, when the voltage is not the half-wave voltage, the first item of Eq. (10) cannot be ignored. And the electric frequency signals ω

_{m}has the influence on the intensity captured by the detector, as described in Eq. (11). As shown in the experimental results, besides the carrier frequency (2ω

_{m}), the electric frequency signals (ω

_{m}) is appeared due to the existing of the frequency shifted lights (ω + ω

_{m}or ω-ω

_{m}) and the original frequency light (ω). In our experiment, the high modulation degree of the carrier frequency signal (2ωm) can be obtained due to the amplitude of the frequency shifted light (ω + ω

_{m}and ω-ω

_{m}) can be controlled to be equal. As a comparison, the frequency signals ω

_{m}is not suitable to be used as the carrier frequency, because the amplitudes of the frequency shifted lights (ω + ω

_{m}or ω-ω

_{m}) and the original frequency light (ω) are not controllable.

*2πf*) is subtracted from the total phase (

_{0}t*Ф(t)*), the linear fitting may not figure out the drift of the carrier frequency. In the low speed vibration measurement, linear fitting can provide an acceptable result. But in the high speed vibration measurement, the carrier frequency need to be pretty high, so frequency stabilization of the power supply and real-time feedback on the carrier frequency are two proper methods to be taken in our self-developed system and these will be studied in our next work.

## 5. Conclusion

## Acknowledgment

## References and links

1. | S. I. Stepanov, I. A. Sokolov, G. S. Trofimov, V. I. Vlad, D. Popa, and I. Apostol, “Measuring vibration amplitudes in the picometer range using moving light gratings in photoconductive GaAs:Cr,” Opt. Lett. |

2. | R. L. Powell and K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. |

3. | W. C. Wang, C. H. Hwang, and S. Y. Lin, “Vibration measurement by the time-averaged electronic speckle pattern interferometry methods,” Appl. Opt. |

4. | R. Sato and K. Nagaoka, “Motion trajectory measurement of NC machine tools using accelerometers,” Int. J. Automation Technol. |

5. | C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. |

6. | K. A. Stetson and W. R. Brohinsky, “Fringe-shifting technique for numerical analysis of time-average holograms of vibrating objects,” J. Opt. Soc. Am. A |

7. | K. Kyuma, S. Tai, K. Hamanaka, and M. Nunoshita, “Laser Doppler velocimeter with a novel optical fiber probe,” Appl. Opt. |

8. | K. A. Stetson, “Method of vibration measurements in heterodyne interferometry,” Opt. Lett. |

9. | G. E. Sommargren, “Up/down frequency shifter for optical heterodyne interferometry,” J. Opt. Soc. Am. |

10. | O. B. Wright, “Stabilized dual-wavelength fiber-optic interferometer for vibration measurement,” Opt. Lett. |

11. | Y. Park and K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett. |

12. | C. F. Buhrer, L. R. Bloom, and D. H. Baird, “Electro-optic light modulation with cubic crystals,” Appl. Opt. |

13. | I. P. Kaminow and E. H. Turner, “Electrooptic light modulators,” Appl. Opt. |

14. | C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optics effect,” Appl. Phys. Lett. |

15. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

**OCIS Codes**

(040.2840) Detectors : Heterodyne

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

(120.7280) Instrumentation, measurement, and metrology : Vibration analysis

(230.2090) Optical devices : Electro-optical devices

(250.4110) Optoelectronics : Modulators

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 13, 2014

Revised Manuscript: March 16, 2014

Manuscript Accepted: March 20, 2014

Published: March 27, 2014

**Citation**

Shengjia Wang, Zhan Gao, Guangyu Li, Ziang Feng, and Qibo Feng, "Continual mechanical vibration trajectory tracking based on electro-optical heterodyne interferometry," Opt. Express **22**, 7799-7810 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7799

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### References

- S. I. Stepanov, I. A. Sokolov, G. S. Trofimov, V. I. Vlad, D. Popa, I. Apostol, “Measuring vibration amplitudes in the picometer range using moving light gratings in photoconductive GaAs:Cr,” Opt. Lett. 15(21), 1239–1241 (1990). [CrossRef] [PubMed]
- R. L. Powell, K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55(12), 1593–1597 (1965). [CrossRef]
- W. C. Wang, C. H. Hwang, S. Y. Lin, “Vibration measurement by the time-averaged electronic speckle pattern interferometry methods,” Appl. Opt. 35(22), 4502–4509 (1996). [CrossRef] [PubMed]
- R. Sato, K. Nagaoka, “Motion trajectory measurement of NC machine tools using accelerometers,” Int. J. Automation Technol. 5(3), 387–394 (2011).
- C. H. Liu, W. Y. Jywe, C. C. Hsu, T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005). [CrossRef]
- K. A. Stetson, W. R. Brohinsky, “Fringe-shifting technique for numerical analysis of time-average holograms of vibrating objects,” J. Opt. Soc. Am. A 5(9), 1472–1476 (1988). [CrossRef]
- K. Kyuma, S. Tai, K. Hamanaka, M. Nunoshita, “Laser Doppler velocimeter with a novel optical fiber probe,” Appl. Opt. 20(14), 2424–2427 (1981). [CrossRef] [PubMed]
- K. A. Stetson, “Method of vibration measurements in heterodyne interferometry,” Opt. Lett. 7(5), 233–234 (1982). [CrossRef] [PubMed]
- G. E. Sommargren, “Up/down frequency shifter for optical heterodyne interferometry,” J. Opt. Soc. Am. 65(8), 960–961 (1975). [CrossRef]
- O. B. Wright, “Stabilized dual-wavelength fiber-optic interferometer for vibration measurement,” Opt. Lett. 16(1), 56–58 (1991). [CrossRef] [PubMed]
- Y. Park, K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett. 36(3), 331–333 (2011). [CrossRef] [PubMed]
- C. F. Buhrer, L. R. Bloom, D. H. Baird, “Electro-optic light modulation with cubic crystals,” Appl. Opt. 2(8), 839–846 (1963). [CrossRef]
- I. P. Kaminow, E. H. Turner, “Electrooptic light modulators,” Appl. Opt. 5(10), 1612–1628 (1966). [CrossRef] [PubMed]
- C. F. Buhrer, D. Baird, E. M. Conwell, “Optical frequency shifting by electro-optics effect,” Appl. Phys. Lett. 1(2), 46–49 (1962).
- M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

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