Abstract

The feedback phenomenon of orthogonally polarized dual frequency laser has not been explained theoretically. This paper gives a model based on Lamb’s semi-classical gas-laser theory for the first time. The intensity reflectivity of the feedback mirror, the polarization characteristics of the dual frequency laser and external cavity length are considered besides the parameters studied before. The intensities of o-light and e-light are tuned by feedback mirror. The intensity alternation, leaning of curves and height difference of the two equal–intensity points etc. are discovered in the region of moderate optical feedback level. The experiments are done and the results are in good agreement with the theoretical model.

©2005 Optical Society of America

1. Introduction

Peek[1] first reported that the steady-state intensity of a laser could be modified by introducing coherent optical feedback from an external surface. The physical basis is the interference of the back-reflected field with the standing wave inside the laser resonant cavity. So the feedback effect is also called self-mixing interference. It has been studied hotly in the fields of displacement measurement[2, 3], imaging and vibration analysis[4], and microscope[5].

Theory models have been proposed to explain the feedback effect in a semi-conductor laser. The laser intensity modulated by feedback is sine-like[6] or sawtooth-like[7] with reference to different feedback level. For a gas laser, multiple reflection effect[8] and weak optical feedback[9] have been studied by neglecting many characteristics of a gas laser, such as gain saturation, polarization and mode competition. The simulation curves of these theory models are far different from the experimental curves. Especially, the feedback phenomenon of orthogonally polarized dual frequency laser has not been explained theoretically till now.

In this paper, a theory model based on semi-classical gas-laser theory of Lamb is described to explain the feedback effect of orthogonally polarized dual frequency laser for the first time. The effect of intensity reflectivity of the feedback mirror, the polarization of the dual frequency laser and external cavity length are considered besides the parameters studied before. The intensities of o-light and e-light is tuned by feedback mirror. The intensity alternation, leaning of curves and height difference of the two equal–intensity points etc. are discovered in the region of moderate optical feedback level. The experiments are done to prove the theory predicts and the results are in good agreement with the theoretical model. According to these results, the feedback effects of orthogonally polarized dual frequency laser are promising for application in precision measurement.

2. Theoretical model

A novel theory model is presented to explain the modified intensity due to the presence of optical feedback. The internal laser intensity of a two longitudinal mode laser is [10]

Io=1D(α1β2α2θ12)
Ie=1D(α2β1α1θ21),
D=β1β2θ12θ21

With

α1/2=α1/2ν1/22Q12,

These equations are suitable for dual frequency laser, too[11]. Where 1/2 means 1 for o-light and 2 for e-light, Io and Ie are the dimensionless intensities of o-light and e-light, α 1/2 is the unsaturated net gain, β 1/2 is the saturation parameter, θ 12 and θ 21 are cross-saturation coefficients, α'1/2 is the small signal gain, ν 1/2 is the frequency of the two lights, and Q 1/2 is the cavity quality factor. A schematic configuration of a He-Ne laser with the feedback mirror is shown in Fig. 1(a). L is the laser cavity length. The distance between the laser and the external mirror M3 is l.

 

Fig. 1. Schematic of (a) feedback effect in a He-Ne laser, (b) an equivalent system

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R 1 and R 2 representing the reflectivities of mirror M1 and M2, and neglecting all losses other than the transmissions of the laser-end mirrors, we may write[12]

Q0=(4πLλ)(1R1+1R2),

Here Q 0 is the quality factor of the cavity formed by M1 and M2. λ = c / ν is the wavelength. In the presence of an external mirror M3, the feedback beams reenters the laser and superposes with that of the internal laser field. Its phase is determined by the external cavity length l. The laser mirror M2 and the external mirror M3 form an external Fabry-Perot interferometer which now replaces the end mirror M2. The equivalent system is represented by Fig.1(b), in which the effective intensity reflectivity Rf of the end face of the laser is found to be[1]

Rf1/2=R2+(1R2){1(1R3)[1+R2R3+2(R2R3)12cosδ1/2]},

Where δ 1/2 = 4πl / λ 1/2 is the external phase difference between successive reflected beams, and R 3 is the intensity reflectivity of M3. The effective reflectivity of a laser mirror can thus be considerably changed by applying an external mirror with moderate reflectivity. That is how the intensity of the laser is modulated by feedback effect in our model.

The quality factor Q 1/2 of the laser cavity with optical feedback can be derived by replacing R 2 by R f1/2 in Eq. (3),

Q1/2=4πLλ1/21R1+1Rf1/2,

Replacing Q 0 by Q 1/2 in Eq. (2) and substituting Eq. (2) in Eq. (1), the modulated intensities of o-light and e-light with feedback effect are

Io=M1+c8DL(1R2)(1R3)(1+R2R3)N1+2R2R3(θ12cosδ2β2cosδ1)(1+R2R3)3+2(1+R2R3)(cosδ2+cosδ1)+4R2R3cosδ1cosδ2,
Ie=M2+c8DL(1R2)(1R3)(1+R2R3)N2+2R2R3(θ21cosδ1β1cosδ2)(1+R2R3)3+2(1+R2R3)(cosδ2+cosδ1)+4R2R3cosδ1cosδ2

with

N1=θ12β2
N2=θ21β1,
M1=Io0+α1β2α2θ12D+c8L(1R1)N1
M2=Ie0+α2β1α1θ21D+c8L(1R1)N2,

Where c is speed of light in vacuum. I e0 and I o0 are steady state intensities of e-light and o-light. For a certain laser, N 1, N 2 , M 1 and M 2 are constant. Consequently, we get the expression of intensities as function of δ 1/2(or l) for reasonable values of R 1, R 2 and R 3. Here

δ1=δ2+4πlcΔν,

With the frequency difference

Δν=ν1ν2,

Because the two frequencies (o-light and e-light) are orthogonally polarized, we use W.M.Doyle’s expressions[13] for α'1/2 , β 1/2 , θ 1/2 and θ 21 in our model which extend the semi-classical gas-laser theory of Lamb to describe the behavior of a gas laser having generalized polarization characteristics.

The computational intensities curve as a function of external cavity length l for gas lasers with ∆ν = 240MHz, R 1 = R 2 = 0.995, R 3 = 0.4 and L=170mm are shown in Fig. 2, in which the variation range of l is 2 μm and α 1 = 2.38×106 , α 2 = 2.41×106 , β 1 = 7.44×105 , β 2 = 7.5×105 , θ 12 = 5.35×105 , θ 2l = 5.39×105 . According to the Eq. (1), Eq. (7) and Eq. (8), the parameters’ values are D = 2.7×l011 , N 1 = -2.15×105 , N 2 = -2.06×105, M 1 = -2.38×1011, M 2 = -2.27×l011.

 

Fig. 2. Computer calculations of intensity variations versus external cavity length

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From Fig. 2 we can see that the output intensities of o-light and e-light change periodically as the external mirror moves along the laser’s axial direction. When one light reaches its peak, the other reaches its valley. There is obvious humpback on the opposite side of the two curves and both curves are leaning. The two equal–intensity points in one period have height difference. These phenomena are very different from the intensity tuning curve of weak optical feedback (sine-like or sawtooth-like) or normal dual frequency laser.

3. Experimental setup

In order to prove the theoretical model, an experimental setup is shown in Fig. 3. The operating wavelength of the half-intracavity He-Ne laser is 632.8nm. T is the discharge tube filled with Ne 20 : Ne 22 = 1:1 gas mixture to suppress the Lamb dip in the output intensity curve. The ration of gaseous pressure in laser is He : Ne = 7:1.

 

Fig. 3. Experimental setup. M1, M2, M3: mirrors; T: discharge tube; W: glass window coated with anti-reflective layer; Q: uniaxial quartz crystal; PZT: piezoelectric transducer; PBS: Wollaston prism; D1, D2: photoelectric detectors; C: signal processing circuit; F-P: Fabry-Perot scanning interferometer; OS: oscilloscope.

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A plane mirror M1 and a concave mirror M2 which has a radius of 1m form the laser cavity. Their reflectivities are R 1 = R 2 = 0.995 and L=170mm M3 is external mirror with a reflectivity of R 3 = 0.4 that reflects light back into the internal cavity. M2 and M3 together form the feedback external cavity, whose length is l=95mm. The plate Q is a birefringence component made of quartz crystal by which a frequency is split into two orthogonally polarized frequencies (o-light and e-light). The frequency difference of the two frequencies can be changed from 40MHz to one longitudinal mode spacing by changing the angle θ between the crystalline axis of Q and the laser axis. D1, D2 are photoelectric detectors that is used to detect the output intensities of two orthogonally polarized lights, respectively. The computer’s function is to get the signal from A/D and control the input of D/A. The laser modes are observed by F-P and OS. A voltage given by a D/A card and amplified by a PI amplifier is used to drive the PZT. The voltage of the D/A card changes about 1V and the PZT changes a distance of λ / 2.

4. Experimental results

The experimental curves for ∆ν = 240MHz are shown in Fig. 4. Compared Fig. 4 with Fig. 2, we can see that the experimental waveforms have the same characteristics as the theoretical simulation curves. The modulation intensities of o-light and e-light change periodically and inversely. The humpback exists and the curves are leaning which are in good agreement with the simulated curves. The equal-intensity point appears when o-light and e-light have the same gain in the equivalent system shown in Fig. 1(b). The two equal–intensity points in one period have height difference. Consequently, the theory predict is observed in the experiment.

 

Fig. 4. Experimental waveforms of intensity variations versus external cavity length

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5. Conclusions

Our theoretical model based on Lamb semi-classical gas-laser theory gives a new idea to explain many phenomena of a dual frequency laser with moderate optical feedback regime. The theoretical analysis and comparison with the experimental results on the feedback effect of orthogonally polarized dual frequency laser are presented in this paper and they are in good agreement with each other. The output intensities of the two orthogonally polarized frequencies change periodically as the external mirror moves along the laser’s axial direction. There is obvious humpback on the opposite side of the two curves and both curves are leaning. This is very different from the intensity tuning curve of weak optical feedback or common dual frequency laser. The equal-intensity point appears when o-light and e-light have the same gain in the equivalent system. There is height difference between the two equal-intensity points in one period. When one light reaches its peak, the other reaches its valley. In other words, the external cavity length l changes every λ / 2, the laser intensity produce a fringe for o-light and e-light, respectively. This result can be used to measure displacement of external mirror M3 and our results presented in this paper will also advance the research of dual frequency self-mixing interferometer.

Acknowledgments

This work was supported by the project 60438010 Nature Science Foundation of China.

References and links

1. Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967). [CrossRef]  

2. Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003). [CrossRef]  

3. J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995). [CrossRef]  

4. A. Bearden, MP. O’Neill, LC. Osborne, and TL. Wong, “Imaging and vibrational analysis with laser-feedback interferometry,” Opt. Lett. 18, 238–240 (1993). [CrossRef]   [PubMed]  

5. T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995). [CrossRef]  

6. W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994). [CrossRef]  

7. T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999). [CrossRef]  

8. G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003). [CrossRef]  

9. L. Fei and S. Zhang, “Self-mixing interference effects of orthogonally polarized dual frequency laser,” Opt. Express 12, 6100–6105 (2004). URL: http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6100 [CrossRef]   [PubMed]  

10. Willis E. Lamb, “Theory of an optical Maser,” Phys. Rev. 134, A1429–A1440 (1964). [CrossRef]  

11. Y. JiangRing Laser Gyroscopes. (Tsinghua University Press, Beijing,1985), Chap.3.

12. L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001). [CrossRef]  

13. W. M. Doyle and M. B. White, “Effects of atomic degeneracy and cavity anisotropy on the behavior of a gas laser,” Phys. Rev. 147, 359–367(1966). [CrossRef]  

References

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  1. Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
    [Crossref]
  2. Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003).
    [Crossref]
  3. J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
    [Crossref]
  4. A. Bearden, MP. O’Neill, LC. Osborne, and TL. Wong, “Imaging and vibrational analysis with laser-feedback interferometry,” Opt. Lett. 18, 238–240 (1993).
    [Crossref] [PubMed]
  5. T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995).
    [Crossref]
  6. W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
    [Crossref]
  7. T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
    [Crossref]
  8. G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003).
    [Crossref]
  9. L. Fei and S. Zhang, “Self-mixing interference effects of orthogonally polarized dual frequency laser,” Opt. Express 12, 6100–6105 (2004). URL: http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6100
    [Crossref] [PubMed]
  10. Willis E. Lamb, “Theory of an optical Maser,” Phys. Rev. 134, A1429–A1440 (1964).
    [Crossref]
  11. Y. JiangRing Laser Gyroscopes. (Tsinghua University Press, Beijing,1985), Chap.3.
  12. L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001).
    [Crossref]
  13. W. M. Doyle and M. B. White, “Effects of atomic degeneracy and cavity anisotropy on the behavior of a gas laser,” Phys. Rev. 147, 359–367(1966).
    [Crossref]

2004 (1)

2003 (2)

Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003).
[Crossref]

G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003).
[Crossref]

2001 (1)

L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001).
[Crossref]

1999 (1)

T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
[Crossref]

1995 (2)

J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
[Crossref]

T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995).
[Crossref]

1994 (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
[Crossref]

1993 (1)

1967 (1)

Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[Crossref]

1966 (1)

W. M. Doyle and M. B. White, “Effects of atomic degeneracy and cavity anisotropy on the behavior of a gas laser,” Phys. Rev. 147, 359–367(1966).
[Crossref]

1964 (1)

Willis E. Lamb, “Theory of an optical Maser,” Phys. Rev. 134, A1429–A1440 (1964).
[Crossref]

Alkemade, C. Th. J.

Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[Crossref]

Bearden, A.

Bolwjin, P.T.

Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[Crossref]

Boyle, W. J. O.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
[Crossref]

Brarden, A.

T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995).
[Crossref]

Ding, Y.

Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003).
[Crossref]

Doyle, W. M.

W. M. Doyle and M. B. White, “Effects of atomic degeneracy and cavity anisotropy on the behavior of a gas laser,” Phys. Rev. 147, 359–367(1966).
[Crossref]

Fei, L.

Grattan, K. T. V.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
[Crossref]

Hirabayashi, S.

T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
[Crossref]

Jiang, Y.

Y. JiangRing Laser Gyroscopes. (Tsinghua University Press, Beijing,1985), Chap.3.

Kao, J.

J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
[Crossref]

Kikuchi, M.

J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
[Crossref]

Lamb, Willis E.

Willis E. Lamb, “Theory of an optical Maser,” Phys. Rev. 134, A1429–A1440 (1964).
[Crossref]

Li, L.

L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001).
[Crossref]

Li, S.

L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001).
[Crossref]

Li, Y.

Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003).
[Crossref]

Liu, G.

G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003).
[Crossref]

Maruyama, T.

T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
[Crossref]

O’Neill, MP.

Osborne, LC.

Ozono, S.

J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
[Crossref]

Palmer, A. W.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
[Crossref]

Peek, Th. H.

Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[Crossref]

Sabato, S.L.

T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995).
[Crossref]

Sasaki, O.

T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
[Crossref]

Suzuki, T.

T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
[Crossref]

Wang, W. M.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
[Crossref]

White, M. B.

W. M. Doyle and M. B. White, “Effects of atomic degeneracy and cavity anisotropy on the behavior of a gas laser,” Phys. Rev. 147, 359–367(1966).
[Crossref]

Wong, T. L.

T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995).
[Crossref]

Wong, TL.

Yamaguchi, I.

J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
[Crossref]

Zhang, S.

L. Fei and S. Zhang, “Self-mixing interference effects of orthogonally polarized dual frequency laser,” Opt. Express 12, 6100–6105 (2004). URL: http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6100
[Crossref] [PubMed]

G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003).
[Crossref]

Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003).
[Crossref]

L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001).
[Crossref]

Zhu, J.

G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003).
[Crossref]

Am. J. Phys. (1)

Th. H. Peek, P.T. Bolwjin, and C. Th. J. Alkemade, “ Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[Crossref]

J. Lightwave Technol. (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, “Self-mixing interference inside a single-mode diode-laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587(1994).
[Crossref]

J. Microscopy. (1)

T. L. Wong, S.L. Sabato, and A. Brarden, “PHOEBE, a prototype scanning laser-feedback microscope for imaging biological cells in aqueous media,” J. Microscopy. 177, 162–170 (1995).
[Crossref]

Meas. Sci, Technol. (1)

J. Kao, M. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci, Technol. 6, 45–52 (1995).
[Crossref]

Opt. Commun. (2)

G. Liu, S. Zhang, and J. Zhu, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221, 387–393 (2003).
[Crossref]

L. Li, S. Zhang, and S. Li, “The new phenomena of orthogonally polarized lights in laser feedback,” Opt. Commun. 200, 303–307 (2001).
[Crossref]

Opt. Eng. (2)

T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Opt. Eng. 38, 543–548 (1999).
[Crossref]

Y. Ding, S. Zhang, and Y. Li, “Displacement sensors based on feedback effect of orthogonally polarized lights of frequency-split HeNe lasers,” Opt. Eng. 42, 2225–2228 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (2)

Willis E. Lamb, “Theory of an optical Maser,” Phys. Rev. 134, A1429–A1440 (1964).
[Crossref]

W. M. Doyle and M. B. White, “Effects of atomic degeneracy and cavity anisotropy on the behavior of a gas laser,” Phys. Rev. 147, 359–367(1966).
[Crossref]

Other (1)

Y. JiangRing Laser Gyroscopes. (Tsinghua University Press, Beijing,1985), Chap.3.

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Figures (4)

Fig. 1.
Fig. 1. Schematic of (a) feedback effect in a He-Ne laser, (b) an equivalent system
Fig. 2.
Fig. 2. Computer calculations of intensity variations versus external cavity length
Fig. 3.
Fig. 3. Experimental setup. M1, M2, M3: mirrors; T: discharge tube; W: glass window coated with anti-reflective layer; Q: uniaxial quartz crystal; PZT: piezoelectric transducer; PBS: Wollaston prism; D1, D2: photoelectric detectors; C: signal processing circuit; F-P: Fabry-Perot scanning interferometer; OS: oscilloscope.
Fig. 4.
Fig. 4. Experimental waveforms of intensity variations versus external cavity length

Equations (15)

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I o = 1 D ( α 1 β 2 α 2 θ 12 )
I e = 1 D ( α 2 β 1 α 1 θ 21 ) ,
D = β 1 β 2 θ 12 θ 21
α 1 / 2 = α 1 / 2 ν 1 / 2 2 Q 12 ,
Q 0 = ( 4 πL λ ) ( 1 R 1 + 1 R 2 ) ,
R f 1 / 2 = R 2 + ( 1 R 2 ) { 1 ( 1 R 3 ) [ 1 + R 2 R 3 + 2 ( R 2 R 3 ) 1 2 cos δ 1 / 2 ] } ,
Q 1 / 2 = 4 πL λ 1 / 2 1 R 1 + 1 R f 1 / 2 ,
I o = M 1 + c 8 DL ( 1 R 2 ) ( 1 R 3 ) ( 1 + R 2 R 3 ) N 1 + 2 R 2 R 3 ( θ 12 cos δ 2 β 2 cos δ 1 ) ( 1 + R 2 R 3 ) 3 + 2 ( 1 + R 2 R 3 ) ( cos δ 2 + cos δ 1 ) + 4 R 2 R 3 cos δ 1 cos δ 2 ,
I e = M 2 + c 8 DL ( 1 R 2 ) ( 1 R 3 ) ( 1 + R 2 R 3 ) N 2 + 2 R 2 R 3 ( θ 21 cos δ 1 β 1 cos δ 2 ) ( 1 + R 2 R 3 ) 3 + 2 ( 1 + R 2 R 3 ) ( cos δ 2 + cos δ 1 ) + 4 R 2 R 3 cos δ 1 cos δ 2
N 1 = θ 12 β 2
N 2 = θ 21 β 1 ,
M 1 = I o 0 + α 1 β 2 α 2 θ 12 D + c 8 L ( 1 R 1 ) N 1
M 2 = I e 0 + α 2 β 1 α 1 θ 21 D + c 8 L ( 1 R 1 ) N 2 ,
δ 1 = δ 2 + 4 π l c Δ ν ,
Δ ν = ν 1 ν 2 ,

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