Abstract

We analyze the laser-self-mixing process in the Gaussian beam approximation and reformulate the expression of the feedback coefficient C in terms of the effective feedback power coupled back into the laser diode. Our model predicts a twenty-fold increase of the ratio between the maximum and the minimum measurable displacements judged against the current plane-wave model. By comparing the interaction of collimated or diverging Gaussian laser beams with a plane mirror target, we demonstrate that diverging beams tolerate larger wobbling during the target displacement and allow for measurement of off-axis target rotations up to the beam angular width. A novel method for reconstructing the phase front of the Gaussian beam by self-mixing scanning measurements is also presented.

© 2010 OSA

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References

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  1. D. M. Kane and K. A. Shore eds., Unlocking dynamical diversity – Optical feedback effects on semiconductor diode lasers (J. Wiley and Sons, 2005).
  2. R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
    [CrossRef]
  3. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [CrossRef]
  4. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
    [CrossRef]
  5. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” J. Quantum Electron. 18(2), 259–264 (1982).
    [CrossRef]
  6. S. Donati, G. Giuliani, and S. Merlo, “Laser Diode Feedback Interferometer Measurement of Displacement without Ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
    [CrossRef]
  7. S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
    [CrossRef]
  8. S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
    [CrossRef]
  9. A. E. Siegman, Lasers, (University Science Books, 1986).
  10. Y. Cai and Q. Lin, “Decentered elliptical Gaussian beam,” J. Appl. Opt. 41(21), 4336–4340 (2002).
    [CrossRef]
  11. K. Tanaka, N. Saga, and K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 24(8), 1098–1101 (1985).
    [CrossRef] [PubMed]
  12. G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41(9), 1157–1167 (2005).
    [CrossRef]
  13. N. R. Barbeau, “Power deposited by a Gaussian beam on a decentered circular aperture,” Appl. Opt. 34(28), 6443–6445 (1995).
    [CrossRef] [PubMed]

2009

S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
[CrossRef]

2008

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

2005

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41(9), 1157–1167 (2005).
[CrossRef]

2002

Y. Cai and Q. Lin, “Decentered elliptical Gaussian beam,” J. Appl. Opt. 41(21), 4336–4340 (2002).
[CrossRef]

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
[CrossRef]

1995

N. R. Barbeau, “Power deposited by a Gaussian beam on a decentered circular aperture,” Appl. Opt. 34(28), 6443–6445 (1995).
[CrossRef] [PubMed]

S. Donati, G. Giuliani, and S. Merlo, “Laser Diode Feedback Interferometer Measurement of Displacement without Ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[CrossRef]

1986

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

1985

1982

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” J. Quantum Electron. 18(2), 259–264 (1982).
[CrossRef]

1980

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

Barbeau, N. R.

Bes, C.

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41(9), 1157–1167 (2005).
[CrossRef]

Bosch, T.

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41(9), 1157–1167 (2005).
[CrossRef]

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
[CrossRef]

Cai, Y.

Y. Cai and Q. Lin, “Decentered elliptical Gaussian beam,” J. Appl. Opt. 41(21), 4336–4340 (2002).
[CrossRef]

Chraplyvy, A. R.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

Dabbicco, M.

S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
[CrossRef]

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

De Lucia, F.

S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
[CrossRef]

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

di Vietro, M.

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

Donati, S.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
[CrossRef]

S. Donati, G. Giuliani, and S. Merlo, “Laser Diode Feedback Interferometer Measurement of Displacement without Ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[CrossRef]

Giuliani, G.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
[CrossRef]

S. Donati, G. Giuliani, and S. Merlo, “Laser Diode Feedback Interferometer Measurement of Displacement without Ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[CrossRef]

Hauchi, K.

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” J. Quantum Electron. 18(2), 259–264 (1982).
[CrossRef]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, “Decentered elliptical Gaussian beam,” J. Appl. Opt. 41(21), 4336–4340 (2002).
[CrossRef]

Merlo, S.

S. Donati, G. Giuliani, and S. Merlo, “Laser Diode Feedback Interferometer Measurement of Displacement without Ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[CrossRef]

Mezzapesa, F. P.

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

Norgia, M.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
[CrossRef]

Ottonelli, S.

S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
[CrossRef]

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

Plantier, G.

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41(9), 1157–1167 (2005).
[CrossRef]

Saga, N.

Scamarcio, G.

S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
[CrossRef]

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

Tanaka, K.

Tkach, R. W.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

G. Plantier, C. Bes, and T. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41(9), 1157–1167 (2005).
[CrossRef]

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

S. Donati, G. Giuliani, and S. Merlo, “Laser Diode Feedback Interferometer Measurement of Displacement without Ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995).
[CrossRef]

IEEE Photon. Technol. Lett.

S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact Three Degrees-of-Freedom Motion Sensor Based on the Laser-Self-Mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008).
[CrossRef]

J. Appl. Opt.

Y. Cai and Q. Lin, “Decentered elliptical Gaussian beam,” J. Appl. Opt. 41(21), 4336–4340 (2002).
[CrossRef]

J. Lightwave Technol.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002).
[CrossRef]

J. Quantum Electron.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” J. Quantum Electron. 18(2), 259–264 (1982).
[CrossRef]

SPIE

S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “All-interferometric 6-Degree-of-freedom sensor based on the laser-self-mixing,” SPIE 7389 (2009).
[CrossRef]

Other

A. E. Siegman, Lasers, (University Science Books, 1986).

D. M. Kane and K. A. Shore eds., Unlocking dynamical diversity – Optical feedback effects on semiconductor diode lasers (J. Wiley and Sons, 2005).

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Figures (8)

Fig. 1
Fig. 1

(a). Schematics of the set-up. The laser source is a Fabry-Perot (FP) laser diode with nominal wavelength λ = 833 nm equipped with a collimating lens and monitor photodiode. The target is a plane mirror fixed at a gimbal motorized rotation stage, mounted onto a translation stage (y axis) perpendicular to the optical axis. The whole system is placed onto a 1-meter long linear stage (x axis). L measures the distance between the collimating lens and the target. The zoomed area shows the collimation tube holding the laser diode. The collimating lens can be finely translated along the beam axis and its effective distance from the laser beam waist is d.

Fig. 2
Fig. 2

(a) Calculated dependence of the diode-to-diode coupled field amplitude κ ′ for the case of an elliptic Gaussian beam. The parameters used in the calculation are q 0 y = i 9.17 μm, q 0 z = i 1.74 μm, α = 4, R 2 = 0.31, τl = 2.4 ps, f = 8 mm, a = 4 mm, R 3/A = 10−3. Different curves are calculated for different values of the diode-lens distance d as indicated in the plots. Δd = d - f is the extent of the translation of the collimating lens from the position of ideal collimation d = f; Δd <(>) 0 identifies a diverging (converging) laser beam after the lens. (b) Experimental measurement of the LSM signal amplitude, normalized to its short-cavity value (see text). The nominal focal length of the collimating lens is f = 8 mm, whereas Δd was measured with an experimental error of ± 0.5 μm.

Fig. 3
Fig. 3

(a) Calculated dependence of the feedback coefficient C for the case of an elliptic Gaussian beam. The parameters used in the calculation are the same of the Fig. 2. Different lines are calculated for different values of the diode-lens distance d as indicated in the plots. Δd is the extent of the translation of the collimating lens from the initial position d = f. (b). Calculated plots of LMAX and Lmin vs. diode-lens distance d. The attenuation parameter R 3/A for each point has been adapted to preserve the moderate feedback regime along the full range.

Fig. 4
Fig. 4

Red empty circles and purple full triangles mark the values of the extracted C, obtained by fitting the LSM traces recorded for two different values of the diode-lens distance d. Δd is measured with an experimental error of ± 0.5 μm. The black straight line and the dotted curve show the theoretical trends calculated for the two different values of d.

Fig. 5
Fig. 5

(a) Experimental maximum target rotation around the z-axis (yaw) at which the LSM signal preserved the saw-tooth line shape (moderate feedback regime) for different values of the laser beam angular width. The vertical axis has the origin ρ = 0 when the mirror is oriented parallel to the yz plane. The cavity length was L = 300 mm and the set-up was adjusted for each θL to have the laser beam pointing perpendicular to the gimbal center point of the target mount. (b) Calculated 3D plot of C as a function of the target rotation for a cavity length of 300 mm. The upper dark surface corresponds to 1 < C < 4.6. All parameters have the same values as for Fig. 2.

Fig. 6
Fig. 6

Derivative of the LSM signal for a gimbal rotations from + 0.22° to – 0.22° for two different laser beam divergences: (a) θL ~0.7°; (b) θL ~0.4°. (c) . Schematics of the alignment of a divergent Gaussian beam with the center of the gimbal mirror mount: D is the distance of the plane mirror surface from the beam waist, x is the phase front curvature in the approximation x >> x 0 (spherical phase front) and ρ is the angle of which the target is rotated, the red curves are the phase fronts of the Gaussian beam.

Fig. 7
Fig. 7

(a) Calculated π-shifted wavefronts of a laser beam for a diode-lens distance d = 7.6 mm and for an external cavity length L = 300 mm. Continuous lines are tangent to the first three phase-fronts and indicate the position of the mirror surface upon rotation around the gimbal axis (the origin of the coordinate system). Positive angles correspond to the upward LSM fringes in Fig. 6(a,b) and negative angles correspond to the downward LSM fringes. (b) Number of LSM fringes counted for a given mirror rotation (from 0° to + 0.22°) for two different values of the diode-laser distance (red hollow square Δd = −100 μm, blue full circles Δd = −150 μm). The continuous lines are the best fit of the experimental data with the equation N = 2R(1 − cosρ)/λ. = 2(L - df/Δd)(1 − cosρ)/λ. Red curve, R = 938.5 mm corresponding to a value of Δd = − 99 μm; blue curve, R = 710.3 mm, corresponding to Δd = −153 μm.

Fig. 8
Fig. 8

Experimentally measured (symbols) and calculated (lines) paraboloidal phase fronts of our laser beam for two different divergences: θL = 0.4° (red line and open circles) continuous phase front, and θL = 0.7° (blue dashed line and square symbols).

Equations (7)

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C=ε(1R2)1+α2τlR2R3AτL
I(x,y,z)=2P0πwy(x)wz(x)exp[2(yy0)2wy2(x)2(zz0)2wz2(x)]
qLy,z=q0y,z(f2L)+df+2L(fd)fdq0y,z
qDy,z=q0y,z[2L(df)f(2df)]2(df)[d(fL)+fL]2L(df)f(2df)2(fL)q0y,z
PFP0=[1exp(2a2wy(d)wz(d))][1exp(2a2wy(d+2L)wz(d+2L))]       [1exp(2w0yw0zwy(2d+2L)wz(2d+2L))]
C=PFP0(1R2)1+α2τlR2R3AτL=κ'τL1+α2τl
2πwy(d+2L)wz(d+2L)0adz0adyexp[2(yLθy)2wy(d+2L)22(zLθz)2wz(2d+2L)2]2πwy(2d+2L)wz(2d+2L)0w0zdz0w0ydyexp[2(ydθy)2wy(2d+2L)22(zdθz)2wz(2d+2L)2]

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