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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 6286–6305
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Self-mixing in multi-transverse mode semiconductor lasers: model and potential application to multi-parametric sensing

L. Columbo, M. Brambilla, M. Dabbicco, and G. Scamarcio  »View Author Affiliations


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

The self-mixing interferometry in semiconductor lasers has been largely exploited in the field of high precision metrology to realize compact, contactless, sensors for real time measurements of absolute distance, vibrations, rotations, flow rate [1

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

, 2

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]

].

In order to simultaneously measure target displacements with more than one degree of freedom (e.g. target displacements along the optical axis (longitudinal displacements), target displacements in the plane orthogonal to the optical axis (transverse plane), target rotations around the optical axis, etc..) one can either combine in suitable geometry two or more conventional diode lasers [3

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]

] or implement an array of mutually independent single transverse mode VCSEL [4

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

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]

]. In all these cases the sensing system is based on customized multi-sources assemblies.

A typical challenge to optical interferometry is the measurement of the purely transverse degrees of freedom of motion of the target, usually identified as straightness, flatness and roll. Several optical schemes have been proposed for measuring small roll rotations, possibly in combination with other degrees of freedom. They can be roughly divided into two categories, those relying on the polarization state of the laser radiation and those relying on tilting the wave vector with respect to the surface normal. Detection of roll angle by measurement of the phase shift between two orthogonal laser modes was demonstrated by Liu 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]

] and the principle was further developed by Wu and Chuang [8

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

]. However, additional phase sensitive optical elements must be inserted in the cavity and the heterodyne detection scheme requires two acousto-optic modulators for frequency shifting the laser mode. To the latter category belong all those detection schemes based on multiple laser sources aimed at pyramidal or prismatic reflective target [9

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

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]

]. These schemes also allow for multiple degrees of freedom measurements, however they suffer of limited resolution and/or limited angular range because of the geometrical boundaries.

Hence, the realization of a similar multi-tasking would be very interesting in terms of cost, compactness and riconfigurability, if attained with a single semiconductor laser emitting a more structured field, so that the multiplicity of measurements can be ascribed to the field/feedback properties, rather than on the number of emitters. The simultaneous measurement of multiple target longitudinal displacements by self-mixing interferometry in a single laser diode has been recently reported by us [12

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]

]. However in [12

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]

] the laser was single mode (both longitudinal and transverse) and the target included independently moving parts.

Here, we introduce a model for laser whose emitted field is characterized by a number of transverse modes and we numerically investigate how the mode dynamics can respond in a viable fashion to the target feedback, so to allow for concurrent measurements of independent displacements.

On the one hand, there is a vast literature on the theoretical analysis of free running low- and mid- Fresnel number VCSEL dynamics in absence of external feedback (see for example [18

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

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]

]). In all these cases the complex electric field profile is suitably described as the superposition of few linearly polarized Gauss-Laguerre or Bessel modes which may nonlinearly compete to give a regular spatio-temporal behavior (stationary patterns, mode-locking, self-pulsing, etc..). These phenomena of spatio-temporal self-organization can have in principle novel applications in the field of optical information processing. In fact, in the earliest stages of the field of ”Transverse Nonlinear Optics” (see [20

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

] and papers cited therein), a number of in-principle applications ranging from optical transistoring to associative memories were proposed, although in those days the technology was unripe for transfer to the VCSEL devices (see for example [21

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]

]). Other studies on transverse dynamics were proposed in the case of VCSEL modeling, including peculiarities such as field polarizations, residual anisotropies or spin flipping [22

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

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]

]. On the other hand, theoretical works based on the 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]

] studied the semiconductor laser dynamics subjected to optical feedback in the plane wave approximation (i.e. assuming a transversely uniform field profile) in different feedback regimes. The effect of feedback on the laser characteristics depends on both the amount of the re-injected power and the round trip time of the field in the external cavity. The optical feedback may lead to stable single frequency emission or complex behaviors like chaotic intensity fluctuations or 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]

] and papers cited therein). The most interesting results for sensing applications have been obtained for weak or moderate feedback regimes [1

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

] where the laser emits on a single frequency at steady state. In this case, the interference fringes in the field intensity generated by the target motion along the optical axis are separated by λ/2.

In this paper, we cast a model suitable to provide a general description of the two dimensional spatio-temporal field dynamics in a transversally extended laser with moderate aperture with cylindrical symmetry, such as a graded index-guided VCSEL, subject to optical feedback. The cross size of the device is assumed to be such that the emission, at least not very far from threshold, can be described in terms of a few Gauss-Laguerre (GL) modes grouped in frequency degenerate families [18

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

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

]. The basic model is developed as to encompass multiple reflections from the external cavity and to account for laser-target settings exceeding a self-imaging configuration. We also allow for a spatially distributed feedback that can describe either a multi-sections external mirror or a mask in the external cavity. While the number of modes is in principle not limited by the approach, in the following analysis of the steady state regimes and laser dynamics, we prefer to restrict ourselves to a small number (1–3) of linearly polarized modes, single reflection in the external cavity and self-imaging configurations, in order to maintain an intuitive picture of the modeling and identify new metrologically interesting properties of the reinjected device.

Focusing on the first frequency degenerate family of GL modes, we find parametric regimes where transverse mode-locking is found and we can in principle simultaneously and independently measure target longitudinal displacements and rotations in the transverse plane.

2. Model

The semiconductor laser dynamics, in presence of a spatially uniform feedback, has been widely described by the Lang-Kobayashi model, consisting of two coupled delayed rate equations for the amplitude of the electric field and the carriers density [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]

]. It holds in the single longitudinal mode and plane wave approximations and in the hypothesis of negligible multiple reflections in the external cavity between the laser exit facet and the target mirror. Its predictions for weak and moderate feedback are in very good agreement with a number of experimental evidences in the field of sensing and metrology [1

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

].

Here we extend this model, to describe semiconductor laser multi-transverse-mode dynamics by making use of previously approaches developed in absence of optical feedback [18

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

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

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

], and to account for the presence of optical elements in the external cavity, as well as for multiple reflections therein [26

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. 3, 353-3-58 (1997).

]. We also assume that the target mirror re-injecting to the laser may exhibit a deterministically designed spatial modulation of the reflectivity.

We consider a mid-area surface emitting semiconductor laser subjected to optical re-injection from an external reflecting target as sketched in Fig. 1. The resonator is supposed to exhibit cylindrical symmetry with respect to the propagation axis z with length l and radius rc in the transverse (x, y) plane. The target mirror has complex reflectivity Rext and is placed at a distance 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 f1,2, of their separation d and of the lens-laser dd and lens-target dt distances. A self-imaging configuration is realized for example when f1 = f2 = f, dl = dt = f, d = 2f. EF, EB represent the forward and backward field envelopes respectively while Ȳ1 and Ȳ2 are the envelopes of the transmitted and injected fields that we suppose linearly polarized along a fixed direction in the transverse plane.

Fig. 1. Sketch of a semiconductor laser subject to optical feedback provided by an external target in a self-imaged configuration.

In presence of weakly index guiding in the transverse plane we assume a background refractive index that varies according to the law [18

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]

]:
n(r)=n(0)1r2/h2
where n(0)=cμ0ε0, r=x2+y2 represents the radial coordinate and h >> rc.

Expressed in terms of the forward and backward propagating fields introduced above, the field inside the laser cavity and the external cavity realized by the laser facet and the target mirror, are:
E(x,y,t)=EF(x,y,z,t)exp(i(k0z+ω0t))+EB(x,y,z,t)exp(i(k0z+ω0t))+c.c.Y(x,yz,t)=Y¯1(x,y,z,t)exp(i(k0z+ω0t))+Y¯2(x,y,z,t)exp(i(k0z+ω0t))+c.c.
where ω0 = −k0c = −k0c/n(0) = −k0v is a reference frequency chosen arbitrarily, and v the light phase velocity in the medium.

In the slowly varying envelope approximations EF and EB satisfy the following paraxial Maxwell equations in the semiconductor medium [18

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]

]:
12ik0(2k0r2h2)EF,B±EF,Bz+1vEF,Bt=12gn(1+iα)(NN0)EF,B
(1)

The parameter gn is the phenomenological gain coefficient, α is the Henry factor (≃ 1 ÷ 5), N(z,t) is the carrier density and N0 its transparency value (typical value: 1.4 × 1024m−3).

We observe at this point that the ”cold” eigenmodes of the empty laser cavity, which satisfy the equations obtained from Eq. (1) by setting equal to zero the coefficient gn and the time derivatives, are given by [18

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]

]:
𝒜p,m(ρ,ϕ,z)=Ap,m(ρ,ϕ)exp(i(2p+|m|+1)z/h)
where φ = tan−1(y/x), ρ=r/W0=rk0/2h is a normalized radial coordinate and we introduced the Gauss-Laguerre modes of index p = 0,1,2,3...,m = 0,±1,±2,±3,..:
Ap,m(ρ,ϕ)=2π(2ρ2)|m|/2[p!(p+|m|)!]1/2Lp|m|(2ρ2)eρ2eimϕ
(2)

The laser cavity eigenfrequencies are then:
ωs,q=ωs+ωq=vlsπ+vh(2p+|m|+1)
(3)
where 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 = 2p + |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 TEM00 while the modes (0, ±m) are indicated by TEM0m 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 ΔωT = v/h is much smaller than the longitudinal mode separation ΔωL = vπ/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.

In this approximation, which is very well verified in the case of VCSEL, the forward and backward field amplitudes are equal to their common average Ep,m along 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:
dEp,mdt=(iqΔωT12τp(1+iα))Ep,m+12Gn(1+iα)02πdϕ0ρdρAp,m*(NN0)E+1τc02πdϕ0ρdρAp,m*k(ρ,ϕ)E(ρ,ϕ,tτ)exp(iω0τ)
(6)
where τ = 2L/c is the external cavity round trip and the feedback coefficient k(ρ, φ) is given by:
k(ρ,ϕ)=ε(1R)Rext(ρ,ϕ)R
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 Rext 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 τc=2lv (typical value: 8ps), the photon life time τp=1vT+αm (typical value: 1.6ps), Gn=2gnlτc (typical value: 8 × 1013m3s−1). The coefficient αm 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 (ω0 = ωs̄,0 + α/2τp), where 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.

The model is self-consistent when we add the rate equation for the carrier density:
dN(ρ,ϕ,t)dt=IeVN(ρ,ϕ,t)τen(0)2ε02ω0Gn(N(ρ,ϕ,t)N0)|E(ρ,ϕ,t)|2
(7)
with I = pump current, e = electron charge and V = volume of the active region. τe is the carrier decay time (typical value: ∼ 1ns). 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]

] in the plane wave approximation and spatially uniform feedback.

Fig. 3. Parametric regime: α = 5, α0 = 3.0, Ψ = 50.0, γ = 0.001, τ = 0.16ns. (a) Region C (Ip = 2.4, k = 0.4). Left: Temporal evolution of the modal intensities of the TEM10 (B2) and TEM01 (B3) modes. Right: Fast Fourier trasform (FFT) of the TEM10 modal intensity. We plot the FFT of a single transverse mode, the FFT of the other mode is analogous. The dashed lines denote the external cavities resonances ωi. (b) Region A (Ip = 1.05, k = 0.0008) and (c) Region D (Ip = 0.9, k = 0.4). Left: Temporal evolution of the modal intensities of the TEM10 and TEM01 modes. The blue trace represents the sum of the modal intensities. The intensity profile of the two modes is reported in the inset together with the total intensity distribution averaged over ∼ 50ns. The dimension of the transverse plane is 5W0 × 5W0 in the simulation. Right: FFT of the TEM10 modal intensity. The dashed line in part (b) denotes the relaxation oscillation frequency ωr while in part (c) it denotes the first external cavity resonance.

3. Numerical simulations

We preliminary performed a wealth of checkpoints on the laser without feedback, in order to reproduce results obtained in literature for GL-multimode lasers [33

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

].

3.1. Laser transverse dynamics in presence of uniform external feedback

In the case of interest we mainly investigated the dynamics of the two (q = 1) frequency degenerate modes for different values of the feedback strength k and the pump Ip at a fixed feedback delay time τ given by the external cavity length L (τ = 2L/c).

In Fig. 2(a) we summarize our most significant results for τ = 0.16ns which corresponds to an external cavity of L = 2.4cm. 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 Ip considered in the simulations. We were able to distinguish five different dynamical domains denoted by the letters A, B, C, D, E and depicted in Fig. 2(a).

Fig. 2. Parametric regime: α = 5, α0 = 3.0, Ψ = 50.0, γ = 0.001. (a) τ = 0.16ns. Summary of the numerical results in the (k,Ip) plane for the family (q = 1). The red crosses represent the values of k and Ip considered in the simulations. The vertical and horizontal gray lines denote the k = 0 and Ip = 0 axis. The different dynamical domains A, B, C, D and E separated by the black continuous lines are described in the text. The red crosses corresponding to region A are not shown in the picture for sake of clarity. (b) Ip = 1.05. Each symbol represents a numerical simulation. The red circles correspond to regular oscillations of the field intensity while the black squares are associated to an irregular dynamical behavior.

As described in the following, we observed a regular system dynamics only in region A and D. The former corresponds to weak or very weak feedback (k < 0.01) and pump Ip above the FRL threshold (Ip > 1), the latter corresponds to strong feedback (k > 0.1) and Ip slightly below (less than ≃ 15%) the FRL threshold. In the regions 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,Ip) plane varies with τ but maintains the same qualitative appearance.

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

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

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

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

]. It follows that the frequency of this periodic dynamics must be close to the relaxation oscillation frequency which is ωr=γ(Ip1) at the lowest oder in γ. The fact that the anti-phase dynamics has been also reported in different models like those in [26

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. 3, 353-3-58 (1997).

, 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]

] indicates that this is a robust dynamical feature of the system.

In region B a continuum of new frequencies is present in the power spectrum. For increasing values of Ip the width and height of the Fourier spectrum maximum increases and blue-shifts towards the closest external cavity resonance ω1 = 2π/τ.

In region 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.

Above the FRL threshold we observed a phase-locking dynamics only in region 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.

3.1.2. Dynamics of the (q = 1) frequency degenerate family. Below the FRL threshold

3.1.3. Dynamics of q = 2 frequency degenerate family

We finally observe that when the six transverse modes of the (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 (B4 and B0 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

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

4.2. Measure of target rotations in the transverse plane

In order to make our device sensitive to target rotations in the transverse plane, we now assume that the target mirror is not uniform but has a spatial modulation of its reflectivity such as that sketched in Fig. 4(a). The two white quarters of the circle correspond to 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 = 0.17rad each 32ns around the optical axis. As Fig. 4(a) shows, for each target rotation step we observe a clear jump in the intensities of the TEM10 and the TEM01 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 (∼ τe), 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.

Fig. 4. Parametric regime as in Fig. 3(c). (a) Top: Time plot of the modal intensities during a feedback mask rotation from Θ = 0 (left inset mask) to Θ = π (right inset mask) (rotation steps of Θ = 0.17rad each 32ns). Bottom: Single frame extracted from the numerical simulation of the field intensity variation ( Media 1. (b) Time plot of the modal intensities during a feedback mask rotation from Θ = 0 to Θ = 3π (rotation steps of Θ = 0.05rad each 0.32ns).

We observe at this point that the proposed system is not able to detect the direction of the target rotation. Of course one could think to realize more complex spatial profiles for the feedback masks, e.g. by breaking the clockwise-anticlockwise symmetry, which would change the spatial hole burning process for either TEM01 and TEM10 mode in a different way for clockwise and anticlockwise target rotations, thus allowing to discriminate between the two.

4.3. Measure of target displacement in the longitudinal direction

Fig. 5 Parametric regime: α = 5, α0 = 3.0, Ψ = 50.0, γ = 0.001, Ip = 0.9, k = 0.4. (a) Primary (low frequency) peak variation in the FFT of the TEM10 modal intensity (that of TEM01 mode is very similar) with the displacement dL. (b) Variation of the average total intensity with the displacement dL. The circles highlight the frequency discontinuity and the corresponding values of the average total intensity.

In Fig. 5(b) we plot the average total intensity versus target longitudinal displacement 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.

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

As anticipated in the beginning of this section we get similar results also above the FRL threshold where transverse mode-locking occurs. The results are reported in Fig. 7.

An inspection of Fig. 7(a) and Fig. 7(b) shows that the quantity Idiff 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 Idiff 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 Idiff. Moreover, as observed below the FRL threshold, the quantity Isum remains unchanged by the rotation.

Slightly different is the situation regarding the measurement of target translation (see Fig. 7(d)). The half wavelength periodicity in the Isum signal is still evident, but the contrast is reduced by roughly a factor 2. The quantity Idiff 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.

A comment is in order to address the actual experimental realization and benefit of the proposed scheme.

First of all, the requirement of a cooperative target, like a structured mirror, is necessary to avoid speckle interference, but will restrict the range of possible applications. Speckles arising from using a diffusive target would most probably disrupt the mode dynamics especially in the case of frequency degenerate modes. Although most laser self-mixing applications operate with diffusive surfaces, a number of relevant applications have been demonstrated for mirror like target (see for example [3

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

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

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]

]).

As far as the laser source is concerned, small-size VCSEL with few GL modes competing just above threshold are readily commercially available. However, the residual birefringence of the device structure usually breaks the intrinsic cylindrical symmetry of the system in such a way that the GL modes belonging to the same family are no longer frequency degenerate and have orthogonal polarizations. Nevertheless there are critical points in the injection current/temperature parameter space (the polarization switching points) where TEM01 and TEM10 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]

]). Their different polarization would thus make experimentally easier the independent measurement of their individual power required to calculate the sum and difference Isum and Idiff. The response time of the system averaged over 1μs would allow rotation speed measurement up about 105rpm 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.

Fig. 6. Parametric regime: α = 5, γ = 0.001, Ip = 0.9, k = 0.4. Modal intensities sum (Isum) and normalized difference (Idiff) during: (a) a target rotation of π/2 in the transverse plane for a fixed external cavity length L = 2.4cm (We rotate the feedback mask of Θ = 0.17rad each 48ns); (b) a target longitudinal displacement of 1μm (dL = 0 corresponds to L = 2.4cm, λ = 830nm) for a fixed angle Θ = 0rad.
Fig. 7. Parametric regime: α = 5, γ = 0.001, Ip = 1.05, k = 0.002. Plot of Isum and Idiff during: (a) a target rotation of π/2 in the transverse plane for a fixed external cavity length L = 2.4cm (We rotate the feedback mask of Θ = 0.17rad each 160ns); (d) a target longitudinal displacement of 1μm (dL = 0 corresponds to L = 2.4cm, λ = 830nm) for a fixed angle Θ = π/4. In part (a) for sake of clarity not all the data points in the Idiff trace are plotted. The plot in part (b) represents a zoom of the traces in part (a). Figure 7(c) has been obtained by low-pass filtering the oscillations shown in part (a) with a cut-off at 1MHz, corresponding to a detector integration time of about 1μs.

Finally, since spatially filtered optical feedback is already a widely used technique to stabilize the mode emission in multimode VCSELs [17

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]

] as well as in broad-area diode lasers [42

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]

], we believe that the proposed detection method could be experimentally implemented using commercial devices, attaining mode control as well as two degrees of freedom measurement capability at the same time.

5. Conclusions

In conclusion we developed a general model to describe the effect of spatially structured feedback on mid-area semiconductor laser which quite generally describes the dynamics of an arbitrary number of transverse and longitudinal modes. For simplicity and in order to demonstrate applicative potentials of such systems, we reported results in the case of single reflection, self-imaging external cavity, mean field approximation (single longitudinal mode) and we limit the number of transverse modes to two (first family of frequency degenerate modes). Our simulations reveal the existence of dynamical regimes of transverse mode-locking where informations on target longitudinal displacements and target rotations in the transverse plane are easily and independently encoded in the phase and/or amplitude field profile by introducing a spatially structured feedback mirror. This indication is thus very promising for applications in metrology since would allow to conceive a single-emitter device to simultaneous measure target motion with different degrees of freedom.

Appendix

In this Appendix we report the detailed derivation of the equation (6) that describes the dynamics of a mid-area semiconductor laser, e.g. a VCSEL, subjected to optical feedback in the mean field limit.

Introduced the auxiliary field amplitudes:
εF;p,m=exp(iδ0+ln(R)(zl)/l)εF;p,m+zlTRY2,p,mεB;p,m=exp(ln(R)z/l)εB;p,m
which obey the periodic boundary conditions:
εF;p,m(0,t)=εB;p,m(0,t)
(10)
εF;p,m(l,t)=εB;p,m(l,t)
(11)
we can recast Eqs. (5) as:
εF;p,mz+1vεF;p,mtzlTRY2,p,mzzvlTRY2,p,mt=Tl((εF;p,mzlTRY2,p,m)(iδ0T+ln(R)T)+(l2Tgn(1+iα)p,m(εF;p,mzlTRY2,p,m)exp(2i(qq)zh)02πdϕ0ρdρ𝒜p,m*(z)𝒜p,m(z)(NN0))+Y2,p,mR)
(12)
εB;p,mz+1vεB;p,mt=Tl(εB;p,mln(R)T+(l2Tgn(1+iα)p,mεB;p,m02πdϕ0ρdρ𝒜p,m*(z)𝒜p,m(z)(NN0)))
(13)

We chose as reference frequency that of the fundamental Gaussian mode of given longitudinal index (ω0 = ωs̄,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):
dEp,mdt=vT2l(Ep,m(iθT)+lTgn(1+iα)02πdϕ0ρdρAp,m*(NN0)E¯+Y2,p,mR)
where we supposed that Ep,m=1l0lεF;p,mdz1l0lεB;p,mdz, that the carrier density N is almost constant along z and we set Ē(ρ,φ) = ∑p,m Ep,mAp,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.

In order to account for multiple reflections in the external cavity one may write:
Y2(l,t)=exp(i(k0k0)l)n(0)Y¯2(l+L,tτ/2)exp(2ik0L)=Rext1Rexp(2ik0l)exp(iω0τ)EF(l,tτ)RextRexp(iω0τ)Y2(l,tτ)
According to the mean field limit we can set:
Y2(l,t)=εRext1Rexp(iω0τ)E(tτ)εRextRexp(iω0τ)Y2(l,tτ)
(15)

In deriving Eq. (15) we neglected diffraction in the external cavity by implicitly assuming a self-imaging configuration. As anticipated in the introduction, although all the results reported in this paper refer to this simpler case, our model accounts for non self-imaging cavity if Eq. (15) is replaced with:
Y2(l,t)=εRext1Rexp(iω0τ)p,mEp,m(tτ)𝒟Ap,mεRextRexp(iω0τ)p,mY2,p,m(l,tτ)𝒟Ap,m
where 𝒟 denotes the propagation operator of the Gauss-Laguerre functions Ap,m in free space [43

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

]. 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.

Introducing relation (15) in Eq. (14) we obtain:
dEp,mdt=(iqΔωT12τp(1+iα))Ep,m+12Gn(1+iα)02πdϕ0ρdρAp,m*(NN0)E+1τc(εRext(1R)Rexp(iω0τ)Ep,m(tτ)εRext1Rexp(iω0τ)Yp,m(l,tτ))
(16)
where we put Y2,p,m(l,t – τ) = Yp,m(t – τ). Finally, we may write in an alternative form Eq. (16):
dEp,m(t)dt=(iqΔωT12τp(1+iα))Ep,m(t)+12Gn(1+iα)02πdϕ0ρdρAp,m*(N(ρ,ϕ,t)N0)E(ρ,ϕ,t)+1τc(s=1ksEp,m(tsτ)exp(isω0τ))
(17)
where the feedback coefficients ks are given by:
ks=εs(1R)RextR(RRext)s1,s𝒩

From Eq. (16) we derive Eq. (6) by further assuming a spatially modulated target and a single reflection in the external cavity.

Acknowledgments

Lorenzo Columbo and Massimo Brambilla also acknowledge the Progetto FIRB – PROGRAMMA ”FUTURO IN RICERCA” Anno 2008 – Protocollo: RBFR08E7VA_001.

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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

  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]
  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. Express17, 5517–5525 (2009). [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. Express18, 11720–11727 (2010). [CrossRef] [PubMed]
  7. “Z. Liu, D. Lin, H. Jiang, and C. Yin, “Roll angle interferometer by means of wave plates,” Sens. Actuators, A104, 127–131 (2003). [CrossRef]
  8. C-M. Wu and Y-T. Chuang, “Roll angular displacement measurement system with microradian accuracy,” Sens. Actuators, A116, 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]
  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. Express15, 14601–14617 (2007). [CrossRef] [PubMed]
  11. S. Ottonelli, M. Dabbicco, F. De Lucia, M. di Vietro, and G. Scamarcio, “Laser-self-mixing interferometry for mechatronics applications,” Sensors9, 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. Express19, 16160–16173 (2011). [CrossRef] [PubMed]
  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 Fractals4, 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. E62, 8677–8699 (2000). [CrossRef]
  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.80, 183–185 (2002). [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]
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