## Self-mixing interferometer: analysis of the output signals

Optics Express, Vol. 14, Issue 20, pp. 9188-9196 (2006)

http://dx.doi.org/10.1364/OE.14.009188

Acrobat PDF (186 KB)

### Abstract

This paper presents the calculation of amplitude and relative phase of signals at the three outputs available in a self-mixing interferometer, i.e. the front output on the target side, the back output on the rear of the chip, and, for diode laser, the junction voltage output. Front and rear outputs are observed to be in phase for the He-Ne laser while they are in phase opposition in the diode laser. This discrepancy can now be explained theoretically. It will also be shown how the junction voltage output is always in phase opposition with respect to the rear output. Experimental measurements were carried out on two sources: a laser diode and a He-Ne laser, validating the calculations.

© 2006 Optical Society of America

## 1. Introduction

1. S. Donati: “Laser interferometry by induced modulation of the cavity field,” J. Appl. Phys. **49**, 495–497 (1978). [CrossRef]

5. W. Wang, W. J. O. Boyle, K. T. W. Grattan, and A. W. Palmer: “Self-mixing Interference in a diode laser for optical sensing applications,” IEEE J. Lightwave Technol. **12**, 1577–1587 (1992). [CrossRef]

7. M. Norgia, S. Donati, and D. d’Alessandro, “Interferometric measurements of displacement on a diffusing target by a speckle-tracking technique,” IEEE J. Quantum Electron. **37**, 800–806 (2001). [CrossRef]

8. T. Bosch, N. Servagent, and S. Donati, “Optical feedback interferometry for sensing applications,” Opt. Eng. **40**, 20–27 (2001). [CrossRef]

*π*/λ is the wavenumber.

5. W. Wang, W. J. O. Boyle, K. T. W. Grattan, and A. W. Palmer: “Self-mixing Interference in a diode laser for optical sensing applications,” IEEE J. Lightwave Technol. **12**, 1577–1587 (1992). [CrossRef]

1. S. Donati: “Laser interferometry by induced modulation of the cavity field,” J. Appl. Phys. **49**, 495–497 (1978). [CrossRef]

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

5. W. Wang, W. J. O. Boyle, K. T. W. Grattan, and A. W. Palmer: “Self-mixing Interference in a diode laser for optical sensing applications,” IEEE J. Lightwave Technol. **12**, 1577–1587 (1992). [CrossRef]

_{1}and k

_{2}of fields at the two mirror outputs are oppositely directed, that is k

_{1}=-k

_{2}, their scalar products with the target displacement s has opposite sign. Now, the feedback in the Lang and Kobayashi equation [10

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

_{1}s and k

_{2}s interact with the

*same*(and single) cavity field, and they cannot generate two different signals modulated onto the same carrier. Both outputs derive from the same internal field, so only some extra field added to one output can bring the two outputs to become opposite in phase. This is just what is found in this paper: it is the reflection at the output mirror of returning field to give a contribution opposite to that coming out from the laser. The reflected field adds vectorially to the transmitted field exiting from the mirror. Then, depending on the relative amplitudes, the total field may or may not change sign with respect to the internal field. As it will be calculated in the next section, the reflected contribution increases respect to the transmitted contribution as the gain increases above the laser threshold. Only in semiconductor lasers the increase is large enough to produce the sign reversal, whereas in a He-Ne laser a gain large enough is never attained. This explains the different behavior of He-Ne lasers versus semiconductor lasers.

## 2. Analysis

_{0}, the electric field just before the exit of the output mirror and directed outward the laser, and the roundtrip in the cavity gain as well as the external back-and-forth path to the target (Fig. 2).

_{1}, t

_{2}(and r

_{1}, r

_{2}) are the field transmissions (and reflections) coefficients at mirrors 1 and 2. Later, we will use also the power coefficients, T

_{1}=

_{1}=

^{2iks}is the propagation term for the path to target and back;

^{γL}is the active medium gain (on the round trip 2L, for the field amplitude), equal to 1/r

_{1}r

_{2}because in the unperturbed regime of oscillation it shall be |r

_{1}r

_{2}e

^{γL+2ikL}|=1;

^{2ikL}is the phase of the roundtrip loop internal to the laser, with |e

^{2ikL}|=1, in the unperturbed regime of oscillation;

^{γL/2+iks}=1/√(r

_{1}r

_{2}) in the unperturbed regime of oscillation;

1. S. Donati: “Laser interferometry by induced modulation of the cavity field,” J. Appl. Phys. **49**, 495–497 (1978). [CrossRef]

4. S. Shinoara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi: “Compact and versatile self-mixing type semiconductor laser doppler velocimemeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. **38**, 574–577 (1989). [CrossRef]

^{2}is the effective gain and is the usually accepted approximation of the more rigorous α/(1+βE

^{2}), Γ is the field loss per unit time and is equal to - (ln r

_{1}r

_{2}) (c/2L) when mirrors dominate, L is the laser cavity optical length (n

_{eff}L when the effective index is considered) and, Re{ΔE} (equal to - A [E

_{0}

_{1}] cos 2ks, as from Eq. (1)) is the self-mixing or induced modulation term.

_{0}as: E

_{0}=√(α - Γ)/β. Then, we allow for the returning field ΔE to be different from zero (but small) and solve Eq. (3) with E=E

_{0}+ΔE

_{sm}, obtaining the perturbed field solution ΔE

_{sm}, in the self-mixing condition as:

_{1}r

_{2}) (c/2L) and rearranging terms, we can rewrite this equation as:

_{1}=

_{1}=

_{1}and E

_{2}in the self-mixing condition, their expressions are still given by Eq. (2) with E

_{0}replaced by E

_{0}+ΔE

_{sm}:

_{01}and E

_{02}, as well their impressed self-mix modulations ΔE

_{sm1}and ΔE

_{sm2}are in phase, and differ in amplitude just by a factor (t

_{2}/t

_{1})√(r

_{1}/r

_{2}). Thus, for these contributions, the associated modulation indexes, ΔE

_{sm1}/E

_{01}and ΔE

_{sm2}/E

_{02}are just the same.

_{ref}that the returning field experiences (point 5, bottom path in Fig. 2) when impinging on mirror M1. This field ΔE

_{ref}is easily written as:

_{ref}to the output E

_{1}gives for the total field exiting from the front mirror:

_{tot}/E

_{0}. Using Eq. (6), Eq. (7), Eq. (9),

_{1}and,

_{1}these quantities are written as:

_{1,2}=0.3 for the cleaved facets). Then, the quantity subtracted to unity at the right-hand side of Eq. (12) is easily larger than one above threshold and front and rear outputs are in phase opposition. For a high-gain medium we may also expect the following behavior at increasing pump current. At threshold, where gain and losses are equal, i.e. 2γL≈- ln R

_{1}R

_{2}, Eq. (12) shows that outputs signals are equal (m1/m2≈1) and in phase. If we increase the gain we reach a value 2γL≈- ln R

_{1}R

_{2}+R

_{1}/T

_{1}, at which m1=0, i.e. the front signal disappears. Increasing the gain further above threshold, i.e. 2γL≫- ln R

_{1}R

_{2}, the front output recovers in amplitude but becomes in phase opposition respect to the rear output signal (m

_{1}/m

_{2}<0), see Eq. (12).

_{r}is the recombination time, d is the active region thickness, G is the modal (power) gain (with G(N - N

_{0})γ=c), and N

_{0}is the carrier density at transparency. As the electric field E=E

_{0}is perturbed by the self-mixing phenomenon to the new value E=E

_{0}+ΔE, where ΔE is given by Eq. (4) or Eq. (5), the small signal solution for ΔN is found from Eq. (13) as:

11. R. Juskaitis, N. P. Rea, and T. Wilson: “Semiconductor laser confocal microscopy,” Appl. Opt. **33**, 578–584 (1994). [CrossRef] [PubMed]

12. R. H. Webb and F. J. Rogomentich: “Microlaser microscope using self-mixing detection for confocality,” Opt. Lett. **20**, 533–535 (1995). [CrossRef] [PubMed]

_{ph}=σ

_{diff}=dV

_{ak}/dI=2kT/eI

_{dc}found across the laser diode carries Johnson noise, which is given, i

_{n}rms current, by in=[4kTB/r

_{diff}]

^{1/2}, or by i

_{n}=[2eI

_{dc}B]

^{1/2}after substituting the value of r

_{diff}. Thus, the current noise across the junction is the shot noise of the bias current, a quantity much larger than current I

_{ph}detected by the photodiode and accompanied by the shot noise i

_{n}=[2eI

_{ph}B]

^{1/2}.

## 3. Experiment

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

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

^{-3}to ≈0.2. Because of the negative-resistance V-I characteristics, it was not possible to vary the supply current (and hence γL) appreciably.

_{1}=0.970±0.005, whereas the rear mirror reflectivity was calculated from the ratio P

_{1}/P

_{2}=(1 - R

_{1})/(1 - R

_{2}) of front-to-rear dc powers. Using the measured value P

_{1}/P

_{2}=18 (±0.5), we found R2=0.9983±0.0001. This provides us with the value of loss ln R

_{1}R

_{2}=- 0.032. The measured value of modulation index ratio was m

_{1}/m

_{2}=0.67±0.01 (the data of Fig. 3). If we go on the diagram of Fig. 5, we can see that we need 2γL=0.05±0.001 to fit theoretical data. A roundtrip gain (in power) of ≈5% per pass is just what typically expected [13] from a He-Ne laser in normal working conditions, so we may conclude that an adequate fitting is achieved for the He-Ne.

_{eff}=3.32 for the effective refractive index [14

14. M. Kumar, J. T. Boyd, H. E. Jackson, and B. L. Weiss: “Birefringent properties of GaAlAs multiple quantum well planar optical waveguides,” IEEE J. Quantum Electron. **28**, 1678–1689 (1992). [CrossRef]

_{1}=R

_{2}=0.29 for the two cleaved untreated facets and the total loss is lnR

_{1}R

_{2}=- 2.48.

_{dc}≈70mA (the threshold being at 40mA). At a target distance of s=40cm, on a loudspeaker driven at 30Hz for a swing of ≈18µm peak-to-peak, the value of modulation index ratio was m

_{1}/m

_{2}=-0.57. We found that front and rear outputs were in phase opposition independent from the feedback level (for C=10

^{-3}to >1), and distance (from 20 to 150cm). A sample of the several measurements is shown in Fig. 4, where the front PD1 and rear PD2 output signals are compared together with the signal taken at the junction voltage V

_{ak}. Also for V

_{ak}the phase relationship respect to PD2 were observed not to change at different target distance and strength of injection.

_{dc}=61mA. At this current, from Eq. (10) the corresponding gain value is evaluated as (2γL + lnR

_{1}R

_{2})-1=R

_{1}/T

_{1}whence 2γL=T

_{1}/R

_{1}- ln R

_{1}R

_{2}=4.94.

_{dc}=40.5mA to the maximum permissible current I

_{dc}=90mA, the modulation-index ratio m1/m2 varied with continuity from +1, the value expected for 2γL=- ln R

_{1}R

_{2}=2.48, to about -3 (see Fig. 4), in accordance with the qualitative trend of Eq. (10) and Eq. (11), in which the sign of the quantity (2γL + ln R

_{1}R

_{2})-1 - R

_{1}/T

_{1}determines the phase concordance or opposition as pointed out in the discussion following Eq. (12). Now, recalling Eq. (13), we see that the gain per unit length γ=G(N-N

_{0})/c can be expressed at the equilibrium (dN/dt=0) as γ=A I

_{dc}+B, where A and B are two constants. Fitting them to the curve of Fig. 4 we can plot the experimental results for the laser diode as in Fig. 6 (left). By inspection, the agreement is satisfactory and thus confirms the theoretical explanation presented above.

## 4. Conclusions

## Acknowledgments

## References and links

1. | S. Donati: “Laser interferometry by induced modulation of the cavity field,” J. Appl. Phys. |

2. | S. Donati: Electrooptical iInstrumentation, (Prentice Hall, 2004), Chap. 2.2. |

3. | S. Donati, G. Giuliani, and S. Merlo: “Laser diode feedback interferometer for measurement of displacement without ambiguity,” IEEE J. Quantum Electron. |

4. | S. Shinoara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi: “Compact and versatile self-mixing type semiconductor laser doppler velocimemeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. |

5. | W. Wang, W. J. O. Boyle, K. T. W. Grattan, and A. W. Palmer: “Self-mixing Interference in a diode laser for optical sensing applications,” IEEE J. Lightwave Technol. |

6. | S. Donati and S. Merlo, “A PC-interfaced, compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. |

7. | M. Norgia, S. Donati, and D. d’Alessandro, “Interferometric measurements of displacement on a diffusing target by a speckle-tracking technique,” IEEE J. Quantum Electron. |

8. | T. Bosch, N. Servagent, and S. Donati, “Optical feedback interferometry for sensing applications,” Opt. Eng. |

9. | S. Donati and C. Mirasso, eds., “Optical Chaotic Cryptography,” IEEE J. Quantum Electron. |

10. | R. Lang and K. Kobayashi: “External optical feedback effects on semiconductor laser properties,” IEEE J. Quantum Electron. |

11. | R. Juskaitis, N. P. Rea, and T. Wilson: “Semiconductor laser confocal microscopy,” Appl. Opt. |

12. | R. H. Webb and F. J. Rogomentich: “Microlaser microscope using self-mixing detection for confocality,” Opt. Lett. |

13. | see Ref.2, Sect.A1.1, A1.2. |

14. | M. Kumar, J. T. Boyd, H. E. Jackson, and B. L. Weiss: “Birefringent properties of GaAlAs multiple quantum well planar optical waveguides,” IEEE J. Quantum Electron. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(250.0250) Optoelectronics : Optoelectronics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 12, 2006

Revised Manuscript: July 31, 2006

Manuscript Accepted: July 31, 2006

Published: October 2, 2006

**Citation**

Enrico M. Randone and Silvano Donati, "Self-mixing interferometer: analysis of the output signals," Opt. Express **14**, 9188-9196 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9188

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### References

- S. Donati: "Laser interferometry by induced modulation of the cavity field," J. Appl. Phys. 49, 495-497 (1978). [CrossRef]
- S. Donati: Electrooptical iInstrumentation, (Prentice Hall, 2004), Chap. 2.2.
- S. Donati, G. Giuliani, and S. Merlo: "Laser diode feedback interferometer for measurement of displacement without ambiguity," IEEE J. Quantum Electron. 31,113-119 (1995). [CrossRef]
- S. Shinoara, H. Naito, H. Yoshida, H. Ikeda, and M. Sumi: "Compact and versatile self-mixing type semiconductor laser doppler velocimemeter with direction discrimination circuit," IEEE Trans. Instrum. Meas. 38, 574-577 (1989). [CrossRef]
- W. Wang, W. J. O. Boyle, K. T. W. Grattan, and A. W. Palmer: "Self-mixing Interference in a diode laser for optical sensing applications," IEEE J. Lightwave Technol. 12, 1577-1587 (1992). [CrossRef]
- S. Donati, and S. Merlo, "A PC-interfaced, compact laser-diode feedback interferometer for displacement measurements," IEEE Trans. Instrum. Meas. 45, 942-947 (1996). [CrossRef]
- M. Norgia, S. Donati, and D. d'Alessandro, "Interferometric measurements of displacement on a diffusing target by a speckle-tracking technique," IEEE J. Quantum Electron. 37, 800-806 (2001). [CrossRef]
- T. Bosch, N. Servagent, and S. Donati, "Optical feedback interferometry for sensing applications," Opt. Eng. 40, 20-27 (2001). [CrossRef]
- S. Donati, and C. Mirasso, eds., "Optical Chaotic Cryptography," IEEE J. Quantum Electron. 38, 1338 (2002).
- R. Lang, and K. Kobayashi: "External optical feedback effects on semiconductor laser properties," IEEE J. Quantum Electron. 16, 347-355 (1980). [CrossRef]
- R. Juskaitis, N. P. Rea, and T. Wilson: "Semiconductor laser confocal microscopy," Appl. Opt. 33, 578-584 (1994). [CrossRef] [PubMed]
- R. H. Webb, and F. J. Rogomentich: "Microlaser microscope using self-mixing detection for confocality," Opt. Lett. 20, 533-535 (1995). [CrossRef] [PubMed]
- see Ref.2, Sect.A1.1, A1.2.
- M. Kumar, J. T. Boyd, H. E. Jackson, and B. L. Weiss: "Birefringent properties of GaAlAs multiple quantum well planar optical waveguides," IEEE J. Quantum Electron. 28, 1678-1689 (1992). [CrossRef]

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