## Self-mixing in multi-transverse mode semiconductor lasers: model and potential application to multi-parametric sensing |

Optics Express, Vol. 20, Issue 6, pp. 6286-6305 (2012)

http://dx.doi.org/10.1364/OE.20.006286

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### Abstract

A general model is proposed for a Vertical Cavity Surface Emitting Laser (VCSEL) with medium aspect ratio whose field profile can be described by a limited set of Gauss-Laguerre modes. The model is adapted to self-mixing schemes by supposing that the output beam is reinjected into the laser cavity by an external target mirror. We show that the self-mixing interferometric signal exhibits features peculiar of the spatial distribution of the emitted field and the target-reflected field and we suggest an applicative scheme that could be exploited for experimental displacement measurements. In particular, regimes of transverse mode-locking are found, where we propose an operational scheme for a sensor that can be used to simultaneously measure independent components of the target displacement like target translations along the optical axis (longitudinal axis) and target rotations in a plane orthogonal to the optical axis (transverse plane).

© 2012 OSA

## 1. Introduction

1.. D. M. Kane and K. A. Shore, *Unlocking Dynamical Diversity. Optical Feedback Effects on Semiconductor Lasers* (John Wiley and Sons, 2005). [CrossRef]

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

3.. S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “Simultaneous measurement of linear and transverse displacements by laser self-mixing,” Appl. Opt. **48**, 1784–1789 (2009). [CrossRef] [PubMed]

4.. J. R. Tucker, J. L. Baque, Y. L. Lim, A. V. Zvyagin, and A. D. Rakic, “Parallel self-mixing imaging system based on an array of vertical-cavity surface-emitting lasers,” Appl. Opt. **46**, 6237–6246 (2007). [CrossRef] [PubMed]

6.. Y. L. Lim, R. Kliese, K. Bertling, K. Tanimizu, P. A. Jacobs, and A. D. Rakic, “Self-mixing flow sensor using a monolithic VCSEL array with parallel readout,” Opt. Express **18**, 11720–11727 (2010). [CrossRef] [PubMed]

*et al.*[7

7.. “Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A **104**, 127–131 (2003). [CrossRef]

8.. C-M. Wu and Y-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sens. Actuators, A **116**, 145–149 (2004). [CrossRef]

9.. W. S. Park and H. S. Cho, “Measurement of fine 6-degrees-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. **41**, 860–871 (2002). [CrossRef]

11.. S. Ottonelli, M. Dabbicco, F. De Lucia, M. di Vietro, and G. Scamarcio, “Laser-self-mixing interferometry for mechatronics applications,” Sensors **9**, 3527–3548 (2009). [CrossRef]

12.. F. P. Mezzapesa, L. Columbo, M. Brambilla, M. Dabbicco, A. Ancona, T. Sibillano, F. De Lucia, P. M. Lugará, and G. Scamarcio, “Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode,” Opt. Express **19**, 16160–16173 (2011). [CrossRef] [PubMed]

12.. F. P. Mezzapesa, L. Columbo, M. Brambilla, M. Dabbicco, A. Ancona, T. Sibillano, F. De Lucia, P. M. Lugará, and G. Scamarcio, “Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode,” Opt. Express **19**, 16160–16173 (2011). [CrossRef] [PubMed]

*μm*. On the one side they benefit of low laser threshold, large-scale integrability, low cost and commercial proliferation. On the others side, their emission profile is known to be interpretable in terms of a few families of frequency degenerate transverse modes [13

13.. C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett. **57**, 218–220 (1990). [CrossRef]

14.. H. Lia, T. L. Lucas, J. G. McInerney, and R. A. Morgan, “Transverse modes and patterns of electrically pumped vertical-cavity surface-emitting semiconductor lasers,” Chaos, Solitons Fractals **4**, 1619–1636 (1994). [CrossRef]

15.. J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E **62**, 8677–8699 (2000). [CrossRef]

17.. M. T. Cha and R. Gordon, “Spatially Filtered Feedback for Mode Control in Vertical-Cavity Surface-Emitting Lasers,” J. Lightwave Technol. **26**, 3893–3900 (2008). [CrossRef]

15.. J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E **62**, 8677–8699 (2000). [CrossRef]

17.. M. T. Cha and R. Gordon, “Spatially Filtered Feedback for Mode Control in Vertical-Cavity Surface-Emitting Lasers,” J. Lightwave Technol. **26**, 3893–3900 (2008). [CrossRef]

18.. F. Prati, A. Tesei, L. A. Lugiato, and R.J. Horowicz, “Stable states in surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1637–1654 (1994). [CrossRef]

19.. A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun. **115**, 297–302 (1995). [CrossRef]

20.. L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep. **219**, 293–310 (1992). [CrossRef]

21.. F. Prati, M. Travagnin, and L. A. Lugiato, “Logic gates and optical switching with vertical-cavity surface-emitting lasers,” Phys. Rev. A **55**, 690–700 (1997). [CrossRef]

22.. M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A **52**, 1728–1739 (1995). [CrossRef] [PubMed]

24.. J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt. **9**, 713–736 (1997). [CrossRef]

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

*Low Frequency Fluctuations*(see for example [1

1.. D. M. Kane and K. A. Shore, *Unlocking Dynamical Diversity. Optical Feedback Effects on Semiconductor Lasers* (John Wiley and Sons, 2005). [CrossRef]

1.. D. M. Kane and K. A. Shore, *Unlocking Dynamical Diversity. Optical Feedback Effects on Semiconductor Lasers* (John Wiley and Sons, 2005). [CrossRef]

*λ*/2.

18.. F. Prati, A. Tesei, L. A. Lugiato, and R.J. Horowicz, “Stable states in surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1637–1654 (1994). [CrossRef]

29.. G. Oppo and G. Dalessandro, “Gauss–Laguerre modes - a sensible basis for laser dynamics,” Opt. Commun. **88**, 130–136 (1992). [CrossRef]

## 2. Model

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

*Unlocking Dynamical Diversity. Optical Feedback Effects on Semiconductor Lasers* (John Wiley and Sons, 2005). [CrossRef]

18.. F. Prati, A. Tesei, L. A. Lugiato, and R.J. Horowicz, “Stable states in surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1637–1654 (1994). [CrossRef]

30.. L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A **37**, 3847–3866 (1988). [CrossRef] [PubMed]

33.. F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento **17**, 1–85 (1994). [CrossRef]

*z*with length

*l*and radius

*r*in the transverse (

_{c}*x, y*) plane. The target mirror has complex reflectivity

*R*and is placed at a distance

_{ext}*L – l*from the laser end facet whose mirror has in turn reflectivity

*R*and transmissivity

*T*= 1 –

*R*. The two mirrors constitute a cavity, external with respect to the laser one, and we assume that two lenses in the external optical path make up a self-imaging configuration, so that diffraction in this path can be neglected. This requires of course a suitable choice of the lens focal lengths

*f*

_{1,2}, of their separation

*d*and of the lens-laser

*d*and lens-target

_{d}*d*distances. A self-imaging configuration is realized for example when

_{t}*f*

_{1}=

*f*

_{2}=

*f*,

*d*=

_{l}*d*=

_{t}*f*,

*d*= 2

*f. E*,

_{F}*E*represent the forward and backward field envelopes respectively while

_{B}*Ȳ*

_{1}and

*Ȳ*

_{2}are the envelopes of the transmitted and injected fields that we suppose linearly polarized along a fixed direction in the transverse plane.

*h*>>

*r*.

_{c}*ω*

_{0}= −

*k*′

_{0}

*c*= −

*k*

_{0}

*c*/

*n*(0) = −

*k*

_{0}v is a reference frequency chosen arbitrarily, and v the light phase velocity in the medium.

*E*and

_{F}*E*satisfy the following paraxial Maxwell equations in the semiconductor medium [18

_{B}*g*is the phenomenological gain coefficient,

_{n}*α*is the Henry factor (≃ 1 ÷ 5),

*N*(

*z,t*) is the carrier density and

*N*

_{0}its transparency value (typical value: 1.4 × 10

^{24}

*m*

^{−3}).

*g*and the time derivatives, are given by [18

_{n}*φ*=

*tan*

^{−1}(

*y/x*),

*p*= 0,1,2,3...,

*m*= 0,±1,±2,±3,..:

*A*are also the eigenmodes of a resonator with spherical mirrors in the limit

_{p,m}*l*<<

*z*

_{0}, where

*z*

_{0}is the Rayleigh length [15

15.. J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E **62**, 8677–8699 (2000). [CrossRef]

30.. L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A **37**, 3847–3866 (1988). [CrossRef] [PubMed]

*p*and

*m*are the transverse mode indices and

*s*= 0,1,2,... is the longitudinal mode index. It is important to observe that the frequency of the GL modes depends on the transverse index

*p*and

*m*only via the combination

*q*= 2

*p*+ |

*m*|, thus introducing degeneracy. The degenerate family of order

*q*consists of

*q*+ 1 modes. In the following the transverse modes of the cavity will be denoted by the couple of indices (

*p,m*). In literature the mode (0,0) is usually denoted as

*TEM*

_{00}while the modes (0, ±

*m*) are indicated by

*TEM*

_{0m}and called doughnut modes because of their annular intensity distribution. From the limit

*h*>>

*l*, it directly follows that the frequency spacing between transverse modes Δ

*ω*= v

_{T}*/h*is much smaller than the longitudinal mode separation Δ

*ω*= v

_{L}*π*

*/l*. As it will be clear, this implies that even in the mean field limit that will be introduced soon, where a single longitudinal mode rules the system dynamics, several transverse modes can still compete to determine the dynamics of the transverse field profile.

*A*(

_{p,m}*ρ, φ*) form an orthonormal basis of the (

*ρ, φ*) plane, the fields

*E*can be expanded as: Inserting expressions (4) in Eqs. (1) and using the orthonormality of the Gauss-Laguerre functions we get: The boundary conditions at the two laser facets are: we set

_{F,B}*T*,

*g*and

_{n}l*δ*

_{0}and finite values for the ratios

*E*along

_{p,m}*z*. We refer the reader to the Appendix for a complete treatment of the model in absence of limiting approximations; we report here the final integro-differential equations equation for the latter quantity:

*τ*= 2

*L/c*is the external cavity round trip and the feedback coefficient

*k*(

*ρ, φ*) is given by: with the coefficient

*ɛ*representing coupling losses per external cavity round trip or the effect of a neutral attenuator in the external cavity, as it is often the case in experiments. The dependence of the complex target reflectivity

*R*from the spatial variables will be particularly useful for modeling the experimental configurations exploiting spatially structured optical elements (e.g. spatial filters, phase masks, transversely modulated mirrors etc...). We have also introduced in Eq. (6) the cavity roundtrip time

_{ext}*ps*), the photon life time

*ps*),

^{13}

*m*

^{3}

*s*

^{−1}). The coefficient

*α*accounts for distributed linear losses in the laser cavity. The reference frequency is set as that of the free running, single longitudinal mode laser frequency (

_{m}*ω*

_{0}=

*ω*

_{s̄,0}+

*α*/2

*τ*), where

_{p}*s̄*identifies a particular longitudinal resonance. We observe that although our model is suitable to describe the most general case of multiple reflections as well as non self-imaging configuration, in writing Eq. (6) we restrict here to the simpler case of single external reflection and self-imaging configuration. While the second hypothesis allows us to neglect diffraction in the external cavity and corresponds to a specific choice of the optical system, the first one leads us to consider only weak or moderate feedback regimes, which are the most interesting for self-mixing interferometry.

*I*= pump current,

*e*= electron charge and

*V*= volume of the active region.

*τ*is the carrier decay time (typical value: ∼ 1

_{e}*ns*). Note that Eqs. (6)–(7) reduce to the well known Lang-Kobayashi model [25

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

*m*takes only non-negative values. Obviously the eigenfrequencies are still given by Eq. (3) and the frequency degeneracy of each

*q*family is still

*q*+ 1. Since

*B*are real functions, this substitution allows for a straightforward separation of equations (8)–(9) in real and imaginary parts. Furthermore in terms of the new basis: The intensity distribution of the

_{p,m,o}*B*modes is reported for example in Fig. 3 of [33

_{p,m,o}33.. F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento **17**, 1–85 (1994). [CrossRef]

*p,m,o*} with the single progressive index

*i*.

## 3. Numerical simulations

*x,y*) field profiles, often leads to chaotic dynamics. Here we start by showing that there exist domains of transverse mode-locking, leading to a periodic regime, one above the free running laser (FRL) threshold in the case of

*weak*feedback and the other below the FRL threshold for

*strong*feedback. In the next section we will show how in these cases the information encoded in the field profile can be exploited to simultaneously and independently measure target displacements with more than one degree of freedom.

33.. F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento **17**, 1–85 (1994). [CrossRef]

*th*order Adams-Bashforth-Moulton predictor-corrector method. The integrals were computed by the Gaussian quadrature method.

### 3.1. Laser transverse dynamics in presence of uniform external feedback

*q*= 1) frequency degenerate modes for different values of the feedback strength

*k*and the pump

*I*at a fixed feedback delay time

_{p}*τ*given by the external cavity length

*L*(

*τ*= 2

*L/c*).

*τ*= 0.16

*ns*which corresponds to an external cavity of

*L*= 2.4

*cm*. In this figure, as in the following ones, we adopt the physical units for the time

*t*and longitudinal coordinate

*z*. The red crosses represent the values of

*k*and

*I*considered in the simulations. We were able to distinguish five different dynamical domains denoted by the letters

_{p}*A*,

*B*,

*C*,

*D*,

*E*and depicted in Fig. 2(a).

*A*and

*D*. The former corresponds to weak or very weak feedback (

*k*< 0.01) and pump

*I*above the FRL threshold (

_{p}*I*> 1), the latter corresponds to strong feedback (

_{p}*k*> 0.1) and

*I*slightly below (less than ≃ 15%) the FRL threshold. In the regions

_{p}*B*and

*C*the mode competition always leads to an irregular dynamics. In the region

*E*the feedback is insufficient to trigger lasing emission.Although we did not perform a systematic study, we noted that the extension of the different dynamical domains in the (

*k,I*) plane varies with

_{p}*τ*but maintains the same qualitative appearance.

#### 3.1.1. Dynamics of the (*q* = 1) frequency degenerate family. Above the FRL threshold

*k*= 0 (no feedback) the two modes oscillate regularly in anti-phase with average value, amplitude and Fourier spectrum which depend on

*I*. We interpret this in terms of spatial hole burning, causing the carrier density and the gain to increase where the field intensity is smaller so that the system tries to change the lasing regions as proposed in [31

_{p}31.. M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A **49**, 1427–1451 (1994). [CrossRef] [PubMed]

35.. C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, “Restless optical vortex,” Phys. Rev. A **47**, R1616–R1619 (1993). [CrossRef] [PubMed]

*γ*. The fact that the anti-phase dynamics has been also reported in different models like those in [26, 27

27.. M. S. Torre, C. Masoller, and P. Mandel, “Transverse mode dynamics in vertical-cavity surface-emitting lasers with optical feedback,” Phys. Rev. A **66**, 053817 (2002). [CrossRef]

*B*a continuum of new frequencies is present in the power spectrum. For increasing values of

*I*the width and height of the Fourier spectrum maximum increases and blue-shifts towards the closest external cavity resonance

_{p}*ω*

_{1}= 2

*π/τ*.

*C*we observe that the number of peaks in the FFT increases by increasing

*k*showing again an irregular system dynamics (see for example Fig. 3(a)). The central frequencies of their envelopes gradually shift towards the closest external cavity resonances while their linewidths decrease accordingly.

*A*, i.e. when the feedback strength

*k*is smaller than a critical value. This is confirmed by the results in Fig. 2(b) which also show that the critical

*k*depends on

*τ*.

*q*= 1) modes,

*TEM*

_{10}(

*B*

_{2}) and

*TEM*

_{01}(

*B*

_{3}), at steady state. In the inset we can see the averaged (50

*ns*) intensity profile of each mode and of the total intensity (sum of the modal intensities). The two modes evolve with a frequency close to the relaxation oscillation frequency (see right panel of Fig. 3(b)). Note that they are not exactly in anti-phase since the total intensity, is not constant in time. As expected the departure from the perfect anti-phase dynamics reduces by increasing the photon-to-carrier lifetime ratio

*γ*, i.e. by reducing the spatial hole burning characteristic time scale. Moreover, as it happens in absence of feedback (

*k*= 0), the average value and the amplitude of these periodic oscillations increase with the pump

*I*.

_{p}#### 3.1.2. Dynamics of the (*q* = 1) frequency degenerate family. Below the FRL threshold

*D*where the system exhibits again regular anti-phase oscillations with a frequency that approaches the cavity resonance 2

*π/τ*= 39.27

*GHz*(see Fig. 3(c)). This evidence might be explained if we consider an effective photon-to-carrier lifetime ratio, and consequently an effective relaxation oscillation frequency, calculated assuming the cavity length equal to the external cavity’s. Being the new relaxation frequency much larger, we can thus expect the mode competition to be ruled by the new, slower time scale that becomes the effective cavity round trip time

*τ*. Moreover we note that the amplitude and average value of the periodic oscillations increase with the pump

*I*and at regime the mode with higher intensity is determined by the initial noise distribution. We plan to further investigate the process of generation of a periodic modal intensity oscillation at a frequency much higher than the typical semiconductor laser relaxation’s (few

_{p}*GHz*) because of its potential applications into the field of optical communication (see for example [36

36.. E. K. Lau, X. Zhao, H.-K. Sung, D. Parekh, C. Chang-Hasnain, and M. C. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100 – *GHz* resonance frequencies and 80 – *GHz* intrinsic bandwidths,” Opt. Express **16**, 6609–6618 (2008). [CrossRef] [PubMed]

#### 3.1.3. Dynamics of *q* = 2 frequency degenerate family

*q*= 2 family are active. In general, here the dynamics is irregular but we were able to identify special conditions where a transverse mode-locking can be found provided that the mode with central maximum (

*B*

_{4}mode) is suppressed. In this case the remaining

*B*

_{5}and

*B*

_{6}modes, which breaks the azimutal symmetry, evolve in anti-phase and we may in principle extend to this case all the results reported in the next section about simultaneous measurements of target displacements valid for the

*TEM*

_{10}(

*B*

_{2}) and

*TEM*

_{01}(

*B*

_{3}) in the phase-locking regime.

*q*= 0), (

*q*= 1) and (

*q*= 2) families compete to rule the system dynamics we may get a phase-locking regime provided large losses are added to the modes that gain in the central region (

*B*

_{4}and

*B*

_{0}modes). This suggests that for a VCSEL emitting several mode families (as it is often the case with off-the-shelf commercial multimode VCSELs), a locking regime might be found among modes with

*m*≠ 0, if one can suppress the modes with axial maxima.

## 4. Applications to displacements measurements

*TEM*

_{01}and

*TEM*

_{10}modes in the regimes discussed in the previous section.

### 4.1. Simultaneous measure of target rotations in the transverse plane and target longitudinal displacements using spatially modulated feedback

*A*and

*D*in Fig. 2(a). We note that physically the latter regime requires a high

*k*and this implies a laser with a low internal mirror reflectivity (

*R*< 0.5) as can be achieved in optically pumped devices [37

37.. G Slekys, I Ganne, I Sagnes, and R Kuszelewicz, “Optical pattern formation in passive semiconductor microresonators,” J. Opt. B: Quantum Semiclassical Opt. **2**, 443–446 (2000). [CrossRef]

38.. A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D: Appl. Phys. **37**, R75–R85 (2004). [CrossRef]

*R*> 0.95) leads to much lower values of

*k*. In this case the available operational regime is that corresponding to region

*A*. At the end of this section we show that the same measurement technique remains unchanged and can be performed in a similar way with analogous results, although a reduced sensitivity is predicted.

### 4.2. Measure of target rotations in the transverse plane

*k*= 0 (i.e. there is no reflection), while in the gray parts

*k*= 0.4. The mirror is then rotated by steps of Θ = 0.17

*rad*each 32

*ns*around the optical axis. As Fig. 4(a) shows, for each target rotation step we observe a clear jump in the intensities of the

*TEM*

_{10}and the

*TEM*

_{01}modes that corresponds to the rotation of the field profile shown in the bottom panel of Fig. 4(a). The steep slopes following each rotation define the fastest response time of the system in the

*ns*range (∼

*τ*), corresponding to a maximum detectable rotation frequency of about 50 MHz (see Fig 4b). Although such upper limit is not achievable in a mechanical rotator, we note that the steady state reached afterwards by the system (after some ringing) ensures that much slower rotation frequencies will be equally detected.

_{e}### 4.3. Measure of target displacement in the longitudinal direction

*dL*. We start with an external cavity length

*L*= 2.4

*cm*, which corresponds to

*dL*= 0, where the modal intensities oscillate at a frequency close to the first cavity resonance of ∼ 39

*GHz*. By gradually increasing

*dL*we observe at

*GHz*. This change is periodic with the target shift and allows to measure translations with the same

*L*of the external cavity is different from an odd integer multiple of ≃

*λ*/4 the presence of the dominant frequency peak at around the inverse of the first cavity resonance might be justified as in paragraph 3.1.2: the system is assimilated to a laser with effective cavity equal to the external one. On the contrary, when

*L*is an odd integer multiple of ≃

*λ*/4 the interference is almost destructive and we may expect that the oscillations of the modal intensities to drop towards the relaxation oscillation frequency, thus showing a dominant low frequency peak.

*dL*. It is easy to identify the

*λ*/2 periodicity typical of feedback interferometry when measuring displacements. As in standard self-mixing (single transverse mode lasers) the intensity shift allows to discriminate between the directions of the translation.

*quantifiers*can be potentially associated with a measure of the target longitudinal displacements with a submicrometric resolution: the average total intensity (as in the single mode laser diode) and the frequency of the modal intensity oscillations.

### 4.4. Concurrent measurements of multiple target displacements: target rotations in the transverse plane and target longitudinal displacements

*I*=|

_{sum}*g*

_{1}|

^{2}+ |

*g*

_{2}|

^{2}, and the normalized difference,

*I*=(|

_{diff}*g*

_{1}|

^{2}– |

*g*

_{2}|

^{2})/

*I*, during a target rotation of

_{sum}*π*/2 in the transverse plane for a fixed external cavity length (Fig. 6(a)) and a target longitudinal displacement of 1

*μm*for a fixed angular position of the mask (Fig. 6(b)). We see that

*I*is not affected by the rotation, while

_{sum}*I*allows to measure the rotation steps, while on the contrary, during the translation

_{diff}*I*quantifies the motion while the

_{sum}*I*remains unaffected. This clearly shows that the two quantities represent a couple of suitable quantifiers of multiple target displacements.

_{diff}*I*oscillates at a frequency close to the relaxation oscillation frequency, contrary to what happens below the FRL threshold (see Fig. 6). Nonetheless, by low-pass filtering the

_{diff}*I*temporal trace, as shown in Fig. 7(c), we still get a proper measure of the rotational steps of the target. From an experimental point of view this corresponds to use a ”slow” detector to measure

_{diff}*I*. Moreover, as observed below the FRL threshold, the quantity

_{diff}*I*remains unchanged by the rotation.

_{sum}*I*signal is still evident, but the contrast is reduced by roughly a factor 2. The quantity

_{sum}*I*is not constant, but exhibits a slight modulation (< 1.5%) which is anyway much smaller than the mean value jumps due to rotation. Thus, such a side effect cannot shade out the independence of the two quantifiers with respect to the two target motions.

_{diff}3.. S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “Simultaneous measurement of linear and transverse displacements by laser self-mixing,” Appl. Opt. **48**, 1784–1789 (2009). [CrossRef] [PubMed]

39.. D. Guo, M. Wang, and S. Tan, “Self-mixing interferometer based on sinusoidal phase modulating technique,” Opt. Express **13**, 1537–1543 (2005). [CrossRef] [PubMed]

41.. G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng. **40**, 95–99 (2001). [CrossRef]

*TEM*01 and

*TEM*10 modes are simultaneously lasing with the same average power (see for example [24

24.. J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt. **9**, 713–736 (1997). [CrossRef]

*I*and

_{sum}*I*. The response time of the system averaged over 1

_{diff}*μs*would allow rotation speed measurement up about 10

^{5}

*rpm*by counting the oscillation frequency of any of the two modes (as in Fig. 4(b)). At the same time, any possible longitudinal displacement of the target could be detected with sub-wavelength resolution by measuring the total power modulation (as in Fig. 6(b) and Fig. 7(d)). High precision drilling and milling applications could possibly benefit from resorting to a single device to control both rotation speed and penetration depth. Second, while the conventional self-mixing system, based on a laser diode or on a single longitudinal mode VCSEL, only requires the reading of the internal photodiode current, or even just terminal voltage variation, the proposed scheme, based on a multimode VCSEL, introduces a mechanical complication given by the presence of a polarizing beam-splitter and two external detectors. However, such a technical drawback is rewarded by having simultaneous access to two different degrees-of-freedom of motion with a single device, the revolution speed being also entirely in the transverse plane, a plane typically unaccessible by interferometric measurements.

17.. M. T. Cha and R. Gordon, “Spatially Filtered Feedback for Mode Control in Vertical-Cavity Surface-Emitting Lasers,” J. Lightwave Technol. **26**, 3893–3900 (2008). [CrossRef]

42.. S. Wolff and H. Fouckhardt, “Intracavity stabilization of broad area lasers by structured delayed optical feedback,” Opt. Express **7**, 222–227 (2000). [CrossRef] [PubMed]

## 5. Conclusions

## Appendix

*s̄*(

*ω*

_{0}=

*ω*

_{s̄,}_{0}) and we additionally suppose valid the mean field limit, i.e. small laser facets transmission, small gain per cavity round trip and small cavity detuning. Under this approximation by summing up side by side Eqs. (12)–(13), taking the average along

*z*and using the boundary conditions (10)–(11): where we supposed that

*N*is almost constant along

*z*and we set

*Ē*(

*ρ,φ*) = ∑

*(*

_{p,m}E_{p,m}A_{p,m}*ρ, ϕ*). These latter two hypothesis are very well verified in a semiconductor laser cavity with a longitudinal dimension of few microns like a VCSEL.

*E*the field

*Ē*we finally get the following system of coupled, nonlinear, integro-differential equations for the field modal amplitudes: where we reset the reference frequency as the free running, single longitudinal mode laser frequency (

*ω*

_{0}=

*ω*

_{s̄,0}+

*α*/2

*τ*) and we exploit the relation

_{p}*ω*–

_{s̄,q}*ω*

_{s̄,}_{0}–

*α*/2

*τ*≃

_{p}*ω*–

_{s̄,q}*ω*

_{s̄,}_{0}=

*q*Δ

*ω*.

_{T}*A*in free space [43]. Clearly we get the self-imaging configuration when 𝒟 is the identity operator. This for example allowed us to estimate the effect on the laser characteristics of a non perfect self-imaging, which is likely to occur experimentally.

_{p,m}*Y*

_{2,p,m}(

*l,t – τ*) =

*Y*(

_{p,m}*t – τ*). Finally, we may write in an alternative form Eq. (16):

*k*are given by:

_{s}## Acknowledgments

*RBFR*08

*E*7

*VA*_001.

## References and links

1.. | D. M. Kane and K. A. Shore, |

2.. | S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback inteferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. |

3.. | S. Ottonelli, M. Dabbicco, F. De Lucia, and G. Scamarcio, “Simultaneous measurement of linear and transverse displacements by laser self-mixing,” Appl. Opt. |

4.. | J. R. Tucker, J. L. Baque, Y. L. Lim, A. V. Zvyagin, and A. D. Rakic, “Parallel self-mixing imaging system based on an array of vertical-cavity surface-emitting lasers,” Appl. Opt. |

5.. | Y. L. Lim, M. Nikolic, K. Bertling, R. Kliese, and A. D. Rakic, “Self-mixing imaging sensor using a monolithic VCSEL array with parallel readout,” Opt. Express |

6.. | Y. L. Lim, R. Kliese, K. Bertling, K. Tanimizu, P. A. Jacobs, and A. D. Rakic, “Self-mixing flow sensor using a monolithic VCSEL array with parallel readout,” Opt. Express |

7.. | “Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A |

8.. | C-M. Wu and Y-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sens. Actuators, A |

9.. | W. S. Park and H. S. Cho, “Measurement of fine 6-degrees-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. |

10.. | C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express |

11.. | S. Ottonelli, M. Dabbicco, F. De Lucia, M. di Vietro, and G. Scamarcio, “Laser-self-mixing interferometry for mechatronics applications,” Sensors |

12.. | F. P. Mezzapesa, L. Columbo, M. Brambilla, M. Dabbicco, A. Ancona, T. Sibillano, F. De Lucia, P. M. Lugará, and G. Scamarcio, “Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode,” Opt. Express |

13.. | C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, l. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett. |

14.. | H. Lia, T. L. Lucas, J. G. McInerney, and R. A. Morgan, “Transverse modes and patterns of electrically pumped vertical-cavity surface-emitting semiconductor lasers,” Chaos, Solitons Fractals |

15.. | J. U. Nöckel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E |

16.. | S.-H. Park, Y. Park, H. Kim, H. Jeon, S. M. Hwang, J. K. Lee, S. H. Nam, B. C. Koh, J. Y. Sohn, and D. S. Kim, “Microlensed vertical-cavity surface-emitting laser for stable single fundamental mode operation,” Appl. Phys. Lett. |

17.. | M. T. Cha and R. Gordon, “Spatially Filtered Feedback for Mode Control in Vertical-Cavity Surface-Emitting Lasers,” J. Lightwave Technol. |

18.. | F. Prati, A. Tesei, L. A. Lugiato, and R.J. Horowicz, “Stable states in surface-emitting semiconductor lasers,” Chaos, Solitons Fractals4, 1637–1654 (1994). [CrossRef] |

19.. | A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun. |

20.. | L. A. Lugiato, “Spatio-temporal structures. Part I,” Phys. Rep. |

21.. | F. Prati, M. Travagnin, and L. A. Lugiato, “Logic gates and optical switching with vertical-cavity surface-emitting lasers,” Phys. Rev. A |

22.. | M. San Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A |

23.. | F. Prati, G. Tissoni, M. San Miguel, and N. B Abraham, “Vector vortices and polarization state of low-order transverse modes in a VCSEL,” Opt. Commun. |

24.. | J Martń-Regalado, S. Balle, M. San Miguel, A. Valle, and L. Pesquera, “Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain-guided devices,” Quantum Semi-classic. Opt. |

25.. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser proprieties,” IEEE J. Quantum Electron. |

26.. | J. Y. Law and G. P. Agrawal, “Effects of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. |

27.. | M. S. Torre, C. Masoller, and P. Mandel, “Transverse mode dynamics in vertical-cavity surface-emitting lasers with optical feedback,” Phys. Rev. A |

28.. | K. Green, B. Krauskopf, and D. Lenstra, “External cavity mode structure of a two-mode VCSEL subject to optical feedback,” Opt. Commun. |

29.. | G. Oppo and G. Dalessandro, “Gauss–Laguerre modes - a sensible basis for laser dynamics,” Opt. Commun. |

30.. | L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, “Role of transverse effects in laser instabilities,” Phys. Rev. A |

31.. | M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A |

32.. | A. B. Coates, C. O. Weiss, C. Green, E. J. DAngelo, J. R. Tredicce, M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Prati, A. J. Kent, and G.-L. Oppo, “Dynamical transverse laser patterns. II. Experiments,” Phys. Rev. A |

33.. | F. Prati, M. Brambilla, and L. A. Lugiato, “Pattern formation in lasers,” Riv. Nuovo Cimento |

34.. | S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Kndl, M. Miller, and R. Jger, “Cavity solitons as pixels in semiconductor microcavities,” Nature |

35.. | C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, “Restless optical vortex,” Phys. Rev. A |

36.. | E. K. Lau, X. Zhao, H.-K. Sung, D. Parekh, C. Chang-Hasnain, and M. C. Wu, “Strong optical injection-locked semiconductor lasers demonstrating > 100 – |

37.. | G Slekys, I Ganne, I Sagnes, and R Kuszelewicz, “Optical pattern formation in passive semiconductor microresonators,” J. Opt. B: Quantum Semiclassical Opt. |

38.. | A. C. Tropper, H. D. Foreman, A. Garnache, K. G. Wilcox, and S. H. Hoogland, “Vertical-external-cavity semiconductor lasers,” J. Phys. D: Appl. Phys. |

39.. | D. Guo, M. Wang, and S. Tan, “Self-mixing interferometer based on sinusoidal phase modulating technique,” Opt. Express |

40.. | F. A. Chollet, G. M. Hegde, A. K. Asundi, and A. Q. Liu, “Simple extra-short external cavity laser self-mixing interferometer for acceleration sensing,” Proc. SPIE |

41.. | G. Giuliani, S. Donati, M. Passerini, and T. Bosch, “Angle measurement by injection detection in a laser diode,” Opt. Eng. |

42.. | S. Wolff and H. Fouckhardt, “Intracavity stabilization of broad area lasers by structured delayed optical feedback,” Opt. Express |

43.. | A. E. Siegman, |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(280.3420) Remote sensing and sensors : Laser sensors

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: October 11, 2011

Revised Manuscript: December 9, 2011

Manuscript Accepted: January 2, 2012

Published: March 5, 2012

**Citation**

L. Columbo, M. Brambilla, M. Dabbicco, and G. Scamarcio, "Self-mixing in multi-transverse mode semiconductor lasers: model and potential application to multi-parametric sensing," Opt. Express **20**, 6286-6305 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6286

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### References

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- J. R. Tucker, J. L. Baque, Y. L. Lim, A. V. Zvyagin, and A. D. Rakic, “Parallel self-mixing imaging system based on an array of vertical-cavity surface-emitting lasers,” Appl. Opt.46, 6237–6246 (2007). [CrossRef] [PubMed]
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- Y. L. Lim, R. Kliese, K. Bertling, K. Tanimizu, P. A. Jacobs, and A. D. Rakic, “Self-mixing flow sensor using a monolithic VCSEL array with parallel readout,” Opt. Express18, 11720–11727 (2010). [CrossRef] [PubMed]
- “Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A104, 127–131 (2003). [CrossRef]
- C-M. Wu and Y-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sens. Actuators, A116, 145–149 (2004). [CrossRef]
- W. S. Park and H. S. Cho, “Measurement of fine 6-degrees-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng.41, 860–871 (2002). [CrossRef]
- C. J. Chen, P. D. Lin, and W. Y. Jywe, “An optoelectronic measurement system for measuring 6-degree-of-freedom motion error of rotary parts,” Opt. Express15, 14601–14617 (2007). [CrossRef] [PubMed]
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- S. Wolff and H. Fouckhardt, “Intracavity stabilization of broad area lasers by structured delayed optical feedback,” Opt. Express7, 222–227 (2000). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986).

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