Abstract

An instrument for angle measurement based on laser feedback has been designed. The measurement technique is based on the principle that when a wave plate placed into a feedback cavity rotates, its phase retardation varies. Phase retardation is a function of the rotating angle of the wave plate. Hence, the angle can be converted to phase retardation. The phase retardation is measured at certain characteristic points identified in the laser outputting curve that are then modulated by laser feedback. The angle of a rotating object can be measured if it is connected to the wave plate. The main advantages of this instrument are: high resolution, compact, flexible, low cost, effective power, and fast response.

© 2013 OSA

1. Introduction

High-precise small-angle measurement is very important in many application areas, such as robots, manufacturing automation, coordinate measuring machines, and control of adaptive optical system. Angle measurements are traditionally employed using autocollimators [1,2] and interferometers [3,4]. Although both traditional methods provide high resolution, devices based on these principles are usually large, making the devices hard to integrate with machines for online measurement in many cases. A new method of angle measurement, namely that based on the inter-reflection effect, is proposed to fill this gap [5]. The basic idea of this method is to use a differential detection scheme to largely reduce the inherent nonlinearity of the reflectance versus the angle of incidence so that the angular displacement of the laser beam can be accurately measured by the detection of the reflectance. For this method, in high measurement accuracy areas, elongated critical-angle prisms are unfortunately custom components and are not readily obtainable. These also require tight tolerances and, as a result, are expensive. Several other techniques have been proposed for angle measurement. These methods are based on single crystals of magnetic garnet [6], pattern projection techniques [7] double-exposure speckle photography [8], speckle shearing interferometry [9], photorefractive speckle correlation [10], Fourier transform [11], fringe [12] and circular Dammann gratings [13].

The methods listed above cannot simultaneously meet high angular resolution and low cost. In this paper, we propose an angle measurement method based on laser feedback. The advantages of this instrument are: high resolution, low cost, compact, flexible, effective power, and fast response.

Laser feedback was first observed in 1963 [14]. Since then, it has been used in the fields of velocity [15], displacement [16], absolute distance [17] and vibration measurement [18]. Our team began researching this phenomenon about 20 years ago. Some progress has been made experimentally and theoretically [1922]. We have designed various instruments to perform displacement [23], phase retardation [24] and internal stress measurements [25].

2. Setup and principle

2.1 Measurement setup

The angle measurement instrument is shown in Fig. 1 . A half-intracavity, single mode, linearly polarized He-Ne laser is used as light source. The ratio of gaseous pressure in the laser is He:Ne = 9:1 and Ne20:Ne22 = 1:1. The working wavelength is 632.8 nm. The laser cavity is made up of antireflection window film, mirror M1, and mirror M2 with reflectivities of 99.8 and 98.8%, respectively. The cavity length is 150 mm.

 

Fig. 1 Setup for angle measurement. D, photo detector; S, wave plate; θ: rotating angle; M1, M2, high reflectors; ME, feedback mirror; PZT, piezoelectric transducer; AMP, voltage amplification; DA, digital–to–analog conversion; AD, analog–to–digital conversion.

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The feedback cavity is used to reflect the laser back into the laser resonator and it is made up of M2 and feedback mirror ME, with the wave plate S between them. The feedback cavity length is 100 mm. ME has reflectivity of 15% and is used to reflect the laser beam back into the laser. A piezoelectric transducer (PZT) is used to tune, push and pull ME.

The laser outputting is detected by detector D and converted to voltage signal. Then, the voltage signal is observed by Oscilloscope and recorded in Computer by data acquisition card. AMP supplies triangular wave voltage and drives PZT. The maximum voltage applied to PZT is 100 V, which makes PZT a displacement of 0.5 μm.

When the length of the feedback cavity is scanned by PZT, laser-intensity transfer occurs, (see Fig. 2 ). There, the curve plots the laser intensity outputted from photo detector D. This is different from that of conventional laser feedback. There are dips at B and F point while the conventional laser feedback curves are similar to cosine form. The distance between points A and D or E and H is one period of λ/2. Through detecting the polarization states, we find the polarization states between AB and CD or EF and GH are mutually orthogonal. At points B and F, the polarization varies form one direction to its orthogonal direction. Thus, the points B and F are polarization flipping points.

 

Fig. 2 Phenomenon of laser intensity transfer.

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The relationship between polarization flipping point and phase retardation magnitude is analyzed as Fig. 3 . The top curves in Fig. 3 are o light and e light intensity curves without laser feedback. The second curve is o light and e light intensity curves with laser feedback. The third curve is polarization states of laser outputting. The lowest curve is the voltage applized to PZT. In the feedback cavity, laser passes through the wave plate twice, so, the retardation of o light and e light in Fig. 3 is 2δ, whereδ is phase retardation of the wave plate. According to Fig. 3, the relationship between flipping point and phase retardation is calculated:

δ=(tBCtAD+tFGtEH)×90o
where δ is phase retardation of the wave plate. Point C has the same intensity of point B; similarly, G is that point corresponding to F. tBC is the time interval between points C and B; similar definitions hold for tAD, tFG, and tEH as for tBC.

 

Fig. 3 Relationship between phase retardation and polarization flipping point.

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The measurement accuracy of this setup for phase retardation is 0.5° and the repeatability is 0.05°. The long term stability is 0.19°.

2.2 Principle

When the wave plate S placed in the feedback cavity is rotated, the phase retardation of the wave plate is changed. Thus, the variation of phase retardation reflects the rotation angle of the wave plate. The variation of phase retardation is the comprehensive effects of the thickness and refractive variation, and interference influence.

Firstly, the changes of thickness and refractive index are considered. The refractive index of ordinary (o) light and extraordinary (e) light varies when the wave plate is rotating. The phase retardation is given by:

δ1=2πdλ[(neecosθenoocosθo)+(tanθotanθe)sinθ(neno)]+δ0,

where δ1 represents the phase retardation when the wave plate rotates without interference effect, δ0 represents the phase retardation when the rotating angle is zero, d is the thickness of the wave plate, λ is the laser wavelength, θ is the rotating angle, θo and θe are the angles of refraction of o light and e light, respectively, when the rotating angle is θ, no and ne refractive indices of o light and e light, respectively, when the rotating angle is zero, and noo and nee refractive indices of o light and e light, respectively, when the rotating angle is θ. If the tilt axis is perpendicular to the ray axis, noo and nee is given by:

noo=no,nee=n2on2e+(n2on2e)sin2θn2o,

The experimental results of phase retardation depending thickness and refractive index are shown in Fig. 4 .

 

Fig. 4 Phase retardation depending thickness and refractive index.

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Actually, multi-beam interference will appear in the transmitted light. Here, we use the secondary refraction from the inner surface of the wave plate to analyze the influence of interference on phase retardation.

The upper and lower surfaces of the wave plate are highly parallel. The refracted light of the same polarization directions will interfere with each other. From the refractive index ellipsoid, we can analyze the effects of secondary refraction. The secondary refraction is as shown in Fig. 5 . From the refractive index formula, the relations among θ, θo, and θe are given:

 

Fig. 5 Schematic of optical interference of wave plate.

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noosin(θo)=sinθ,neesin(θe)=sinθ,

Based on the refractive index ellipsoid, noo and nee is given by:

noo=no,nee=[cos2(θe)ne2+sin2(θe)no2]1/2,

whereas, based on the Fresnel formula, we obtain the various reflectivities and transmittances of o light and e light denoted by ro, re, to, and te:

roab=sin(θθo)sin(θ+θo),reab=tg(θθe)tg(θ+θe),roba=sin(θθo)sin(θ+θo),reba=tg(θθe)tg(θ+θe),toab=2sin(θo)cosθsin(θ+θo),teab=2sin(θe)cosθsin(θ+θe)cos(θθe),toba=2sinθcosθosin(θ+θo),teba=2sinθcosθesin(θ+θe)cos(θθe),
where the added subscripts a-b and b-a refer to the two propagation directions for light, i.e., from air to the wave plate, and from the wave plate to air at the surface.

The light rays passing through the wave plate will interfere with each other. The light field is given by:

Eoo=E'ooe(jωtφo)=Eotoabtobaexp(iknoodcos(θo))[1+robaexp(2iknoodcos(θo))],Eee=E'eee(jωtφe)=Eeteabtebaexp(ikneedcos(θe))[1+rebaexp(2ikneedcos(θe))],δ2=φeφo,
where δ2 represents the phase retardation when the wave plate rotates with interference effect, Eoo and Eee are the electric vectors of the o light and e light, respectively, through the wave plate, E′oo and E′ee amplitudes of the o light and e light, respectively, and φo and φe are the phases for o light and e light, respectively. Eo and Ee are the respective initial electric vectors, k = 2π/λ, of the o incident light and e incident light.

The phase retardation including the interference, thickness variation and refractive index variation can be rewritten as:

δ=δ2+δ1=φeφo+2dλ[(neecosθenoocosθo)+(tanθotanθe)sinθ(neno)],
which describes the relationship between phase retardation and rotating angle of wave plate.

3. Results

In our experiment, simultaneous measurements were taken of the phase retardation using the laser feedback instrument and rotating angles using the high-precise goniometer. The experimental data for a six-degree angular displacement are shown in Fig. 6 . The phase retardation oscillates as the wave plate rotates; its variation is similar to a cosine waveform with attenuating peaks.

 

Fig. 6 Experimental results of angle dependent phase retardation.

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In practical applications of angle measurements, the wave plate is first placed in the feedback cavity, as shown in Fig. 1, with the wave plate surface aligned perpendicular to the ray direction. Next, the rotating object is connected to the wave plate. The phase retardation of wave plate is measured in real time as the object is rotating. The angular resolution of this laser feedback instrument is 0.00003° over the range ± 15°.

4. Summary

In this paper, an instrument for angle measurement is introduced. For this technique, the angle is converted to a phase retardation, which is measured using laser feedback. The measurement instrument is compact, flexible, low cost, and has effective power and a fast response. It is a useful tool for precision measurement and can also be used for establishing calibration and metrological standards.

Acknowledgments

This work is supported by the Key Program of the National Natural Science Foundation of China (NSFC) (No. 61036016) and Scientific and Technological Achievements Transformation and Industrialization Project by the Beijing Municipal Education Commission.

References and links

1. F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54(12), 1648–1652 (1983). [CrossRef]  

2. G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55(5), 747–750 (1984). [CrossRef]  

3. G. Hussain and M. Ikram, “Optical measurements of angle and axis of rotation,” Opt. Lett. 33(21), 2419–2421 (2008). [CrossRef]   [PubMed]  

4. P. Paolino and L. Bellon, “Single beam interferometric angle measurement,” Opt. Commun. 280(1), 1–9 (2007). [CrossRef]  

5. P. S. Huang, S. Kiyono, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31(28), 6047–6055 (1992). [CrossRef]   [PubMed]  

6. S. Li, C. Yang, E. Zhang, and G. Jin, “Compact optical roll-angle sensor with large measurement range and high sensitivity,” Opt. Lett. 30(3), 242–244 (2005). [CrossRef]   [PubMed]  

7. T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng. 40(3), 426–432 (2001). [CrossRef]  

8. K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng. 36(4), 331–344 (2001). [CrossRef]  

9. R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol. 24(5), 257–261 (1992). [CrossRef]  

10. R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng. 37(11), 2988–2997 (1998). [CrossRef]  

11. Z. Ge and M. Takeda, “High-resolution two-dimensional angle measurement technique based on fringe analysis,” Appl. Opt. 42(34), 6859–6868 (2003). [CrossRef]   [PubMed]  

12. Y. Nakano and K. Murata, “Talbot interferometry for measuring the small tilt angle variation of an object surface,” Appl. Opt. 25(15), 2475–2477 (1986). [CrossRef]   [PubMed]  

13. F. J. Wen and P. S. Chung, “Use of the circular Dammann grating in angle measurement,” Appl. Opt. 47(28), 5197–5200 (2008). [CrossRef]   [PubMed]  

14. P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Sci. 17, 180–182 (1963).

15. S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38(2), 574–577 (1989). [CrossRef]  

16. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]  

17. Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18(4), 582–595 (1982).

18. P. A. Roos, M. Stephens, and C. E. Wieman, “Laser Vibrometer based on optical-feedback-induced frequency modulation of a single-mode laser diode,” Appl. Opt. 35(34), 6754–6761 (1996). [CrossRef]   [PubMed]  

19. Y. D. Tan, S. L. Zhang, and Y. N. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009). [CrossRef]   [PubMed]  

20. L. Cui and S. Zhang, “Semi-Classical theory model for feedback effect of orthogonally polarized dual frequency He-Ne laser,” Opt. Express 13(17), 6558–6563 (2005). [CrossRef]   [PubMed]  

21. Z. G. Xu, S. L. Zhang, Y. Li, and W. Du, “Adjustment-free cat’s eye cavity He-Ne laser and its outstanding stability,” Opt. Express 13(14), 5565–5573 (2005). [CrossRef]   [PubMed]  

22. W. Mao, S. L. Zhang, L. Cui, and Y. D. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006). [CrossRef]   [PubMed]  

23. X. J. Wan, D. Li, and S. L. Zhang, “Quasi-common-path laser feedback interferometry based on frequency shifting and multiplexing,” Opt. Lett. 32(4), 367–369 (2007). [CrossRef]   [PubMed]  

24. W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum. 83(1), 013101 (2012). [CrossRef]   [PubMed]  

25. W. X. Chen, S. L. Zhang, and X. W. Long, “Internal stress measurement by laser feedback method,” Opt. Lett. 37(13), 2433–2435 (2012). [CrossRef]   [PubMed]  

References

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  1. F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum.54(12), 1648–1652 (1983).
    [CrossRef]
  2. G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum.55(5), 747–750 (1984).
    [CrossRef]
  3. G. Hussain and M. Ikram, “Optical measurements of angle and axis of rotation,” Opt. Lett.33(21), 2419–2421 (2008).
    [CrossRef] [PubMed]
  4. P. Paolino and L. Bellon, “Single beam interferometric angle measurement,” Opt. Commun.280(1), 1–9 (2007).
    [CrossRef]
  5. P. S. Huang, S. Kiyono, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt.31(28), 6047–6055 (1992).
    [CrossRef] [PubMed]
  6. S. Li, C. Yang, E. Zhang, and G. Jin, “Compact optical roll-angle sensor with large measurement range and high sensitivity,” Opt. Lett.30(3), 242–244 (2005).
    [CrossRef] [PubMed]
  7. T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
    [CrossRef]
  8. K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng.36(4), 331–344 (2001).
    [CrossRef]
  9. R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol.24(5), 257–261 (1992).
    [CrossRef]
  10. R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
    [CrossRef]
  11. Z. Ge and M. Takeda, “High-resolution two-dimensional angle measurement technique based on fringe analysis,” Appl. Opt.42(34), 6859–6868 (2003).
    [CrossRef] [PubMed]
  12. Y. Nakano and K. Murata, “Talbot interferometry for measuring the small tilt angle variation of an object surface,” Appl. Opt.25(15), 2475–2477 (1986).
    [CrossRef] [PubMed]
  13. F. J. Wen and P. S. Chung, “Use of the circular Dammann grating in angle measurement,” Appl. Opt.47(28), 5197–5200 (2008).
    [CrossRef] [PubMed]
  14. P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Sci.17, 180–182 (1963).
  15. S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
    [CrossRef]
  16. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31(1), 113–119 (1995).
    [CrossRef]
  17. Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron.18(4), 582–595 (1982).
  18. P. A. Roos, M. Stephens, and C. E. Wieman, “Laser Vibrometer based on optical-feedback-induced frequency modulation of a single-mode laser diode,” Appl. Opt.35(34), 6754–6761 (1996).
    [CrossRef] [PubMed]
  19. Y. D. Tan, S. L. Zhang, and Y. N. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express17(16), 13939–13945 (2009).
    [CrossRef] [PubMed]
  20. L. Cui and S. Zhang, “Semi-Classical theory model for feedback effect of orthogonally polarized dual frequency He-Ne laser,” Opt. Express13(17), 6558–6563 (2005).
    [CrossRef] [PubMed]
  21. Z. G. Xu, S. L. Zhang, Y. Li, and W. Du, “Adjustment-free cat’s eye cavity He-Ne laser and its outstanding stability,” Opt. Express13(14), 5565–5573 (2005).
    [CrossRef] [PubMed]
  22. W. Mao, S. L. Zhang, L. Cui, and Y. D. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express14(1), 182–189 (2006).
    [CrossRef] [PubMed]
  23. X. J. Wan, D. Li, and S. L. Zhang, “Quasi-common-path laser feedback interferometry based on frequency shifting and multiplexing,” Opt. Lett.32(4), 367–369 (2007).
    [CrossRef] [PubMed]
  24. W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum.83(1), 013101 (2012).
    [CrossRef] [PubMed]
  25. W. X. Chen, S. L. Zhang, and X. W. Long, “Internal stress measurement by laser feedback method,” Opt. Lett.37(13), 2433–2435 (2012).
    [CrossRef] [PubMed]

2012

W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum.83(1), 013101 (2012).
[CrossRef] [PubMed]

W. X. Chen, S. L. Zhang, and X. W. Long, “Internal stress measurement by laser feedback method,” Opt. Lett.37(13), 2433–2435 (2012).
[CrossRef] [PubMed]

2009

2008

2007

2006

2005

2003

2001

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
[CrossRef]

K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng.36(4), 331–344 (2001).
[CrossRef]

1998

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
[CrossRef]

1996

1995

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31(1), 113–119 (1995).
[CrossRef]

1992

P. S. Huang, S. Kiyono, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt.31(28), 6047–6055 (1992).
[CrossRef] [PubMed]

R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol.24(5), 257–261 (1992).
[CrossRef]

1989

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

1986

1984

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum.55(5), 747–750 (1984).
[CrossRef]

1983

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum.54(12), 1648–1652 (1983).
[CrossRef]

1982

Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron.18(4), 582–595 (1982).

1963

P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Sci.17, 180–182 (1963).

Bellon, L.

P. Paolino and L. Bellon, “Single beam interferometric angle measurement,” Opt. Commun.280(1), 1–9 (2007).
[CrossRef]

Chen, W. X.

W. X. Chen, S. L. Zhang, and X. W. Long, “Internal stress measurement by laser feedback method,” Opt. Lett.37(13), 2433–2435 (2012).
[CrossRef] [PubMed]

W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum.83(1), 013101 (2012).
[CrossRef] [PubMed]

Chung, P. S.

Cui, L.

Deslattes, R. D.

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum.55(5), 747–750 (1984).
[CrossRef]

Dharmsaktu, K. S.

K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng.36(4), 331–344 (2001).
[CrossRef]

Donati, S.

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31(1), 113–119 (1995).
[CrossRef]

Du, W.

Ganesan, A. R.

R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol.24(5), 257–261 (1992).
[CrossRef]

Ge, Z.

Giuliani, G.

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31(1), 113–119 (1995).
[CrossRef]

Greivenkamp, J. E.

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
[CrossRef]

Huang, P. S.

Hussain, G.

Ikeda, H.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

Ikram, M.

Ito, M.

Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron.18(4), 582–595 (1982).

Jin, G.

Kamada, O.

King, P. G. R.

P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Sci.17, 180–182 (1963).

Kiyono, S.

Koboyashi, Y.

Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron.18(4), 582–595 (1982).

Kumar, A.

K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng.36(4), 331–344 (2001).
[CrossRef]

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
[CrossRef]

Li, D.

Li, H. H.

W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum.83(1), 013101 (2012).
[CrossRef] [PubMed]

Li, S.

Li, Y.

Long, X. W.

W. X. Chen, S. L. Zhang, and X. W. Long, “Internal stress measurement by laser feedback method,” Opt. Lett.37(13), 2433–2435 (2012).
[CrossRef] [PubMed]

W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum.83(1), 013101 (2012).
[CrossRef] [PubMed]

Luther, G. G.

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum.55(5), 747–750 (1984).
[CrossRef]

Mao, W.

Merlo, S.

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31(1), 113–119 (1995).
[CrossRef]

Murata, K.

Naito, H.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

Nakamura, H.

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
[CrossRef]

Nakano, Y.

Paolino, P.

P. Paolino and L. Bellon, “Single beam interferometric angle measurement,” Opt. Commun.280(1), 1–9 (2007).
[CrossRef]

Pati, G. S.

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
[CrossRef]

Roos, P. A.

Sasaki, O.

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
[CrossRef]

Schuda, F. J.

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum.54(12), 1648–1652 (1983).
[CrossRef]

Shinohara, S.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

Singh, K.

K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng.36(4), 331–344 (2001).
[CrossRef]

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
[CrossRef]

Sirohi, R. S.

R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol.24(5), 257–261 (1992).
[CrossRef]

Stephens, M.

Steward, G. J.

P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Sci.17, 180–182 (1963).

Sumi, M.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

Suzuki, T.

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
[CrossRef]

Takeda, M.

Tan, B. C.

R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol.24(5), 257–261 (1992).
[CrossRef]

Tan, Y. D.

Towler, W. R.

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum.55(5), 747–750 (1984).
[CrossRef]

Tripathi, R.

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
[CrossRef]

Wan, X. J.

Wen, F. J.

Wieman, C. E.

Xu, Z. G.

Yamamoto,

Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron.18(4), 582–595 (1982).

Yang, C.

Yoshida, H.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

Zhang, E.

Zhang, S.

Zhang, S. L.

Zhang, Y. N.

Appl. Opt.

IEEE J. Quantum Electron.

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron.31(1), 113–119 (1995).
[CrossRef]

Y. Koboyashi, Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron.18(4), 582–595 (1982).

IEEE Trans. Instrum. Meas.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas.38(2), 574–577 (1989).
[CrossRef]

New Sci.

P. G. R. King and G. J. Steward, “Metrology with an optical maser,” New Sci.17, 180–182 (1963).

Opt. Commun.

P. Paolino and L. Bellon, “Single beam interferometric angle measurement,” Opt. Commun.280(1), 1–9 (2007).
[CrossRef]

Opt. Eng.

T. Suzuki, H. Nakamura, O. Sasaki, and J. E. Greivenkamp, “Small-rotating-angle measurement using an imaging method,” Opt. Eng.40(3), 426–432 (2001).
[CrossRef]

R. Tripathi, G. S. Pati, A. Kumar, and K. Singh, “Object tilt measurement using a photo-refractive speckle correlator: theoretical and experimental analysis,” Opt. Eng.37(11), 2988–2997 (1998).
[CrossRef]

Opt. Express

Opt. Laser Technol.

R. S. Sirohi, A. R. Ganesan, and B. C. Tan, “Tilt measurement using digital speckle shear interferometry,” Opt. Laser Technol.24(5), 257–261 (1992).
[CrossRef]

Opt. Lasers Eng.

K. S. Dharmsaktu, A. Kumar, and K. Singh, “Measurement of tilt of a diffuse object by double-exposure speckle photography using speckle fanning in a photo refractive BaTio3 crystal,” Opt. Lasers Eng.36(4), 331–344 (2001).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

W. X. Chen, H. H. Li, S. L. Zhang, and X. W. Long, “Measurement of phase retardation of waveplate online based on laser feedback,” Rev. Sci. Instrum.83(1), 013101 (2012).
[CrossRef] [PubMed]

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum.54(12), 1648–1652 (1983).
[CrossRef]

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum.55(5), 747–750 (1984).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Setup for angle measurement. D, photo detector; S, wave plate; θ: rotating angle; M1, M2, high reflectors; ME, feedback mirror; PZT, piezoelectric transducer; AMP, voltage amplification; DA, digital–to–analog conversion; AD, analog–to–digital conversion.

Fig. 2
Fig. 2

Phenomenon of laser intensity transfer.

Fig. 3
Fig. 3

Relationship between phase retardation and polarization flipping point.

Fig. 4
Fig. 4

Phase retardation depending thickness and refractive index.

Fig. 5
Fig. 5

Schematic of optical interference of wave plate.

Fig. 6
Fig. 6

Experimental results of angle dependent phase retardation.

Equations (8)

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δ=( t BC t AD + t FG t EH )× 90 o
δ 1 = 2πd λ [ ( n e e cos θ e n o o cos θ o )+( tan θ o tan θ e )sinθ( n e n o ) ]+ δ 0 ,
n o o = n o , n e e = n 2 o n 2 e +( n 2 o n 2 e ) sin 2 θ n 2 o ,
n o o sin( θ o )=sinθ, n e e sin( θ e )=sinθ,
n oo = n o , n ee = [ cos 2 ( θ e ) n e 2 + sin 2 ( θ e ) n o 2 ] 1/2 ,
r oab = sin(θ θ o ) sin(θ+ θ o ) , r eab = tg(θ θ e ) tg(θ+ θ e ) , r oba = sin(θ θ o ) sin(θ+ θ o ) , r eba = tg(θ θ e ) tg(θ+ θ e ) , t oab = 2sin( θ o )cosθ sin(θ+ θ o ) , t eab = 2sin( θ e )cosθ sin(θ+ θ e )cos(θ θ e ) , t oba = 2sinθcos θ o sin(θ+ θ o ) , t eba = 2sinθcos θ e sin(θ+ θ e )cos(θ θ e ) ,
E oo = E ' oo e (jωt φ o ) = E o t oab t oba exp( ik n oo d cos( θ o ) )[ 1+ r oba exp( 2ik n oo dcos( θ o ) ) ], E ee = E ' ee e (jωt φ e ) = E e t eab t eba exp( ik n ee d cos( θ e ) )[ 1+ r eba exp( 2ik n ee dcos( θ e ) ) ], δ 2 = φ e φ o ,
δ= δ 2 + δ 1 = φ e φ o + 2d λ [ ( n ee cos θ e n oo cos θ o )+( tan θ o tan θ e )sinθ( n e n o ) ],

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