1. Introduction

Laser-Self-Mixing (LSM) is a well known phenomenon occurring when part of the radiation emitted by a laser source is coupled back into the laser cavity. Initially seen as a nuisance for the stability of laser emission, it was later identified as a simple and yet effective system to study nonlinear effects in laser dynamics [1]. In fact, depending on the relative amount of radiation coupled back into the laser, the system undergoes different dynamic regimes: from nearly linear (very weak feedback) to chaotic (strong feedback) behaviors [2]. In the ‘80s Lang and Kobayashi (LK) [3] formulated a field-delayed rate equation model based on the assumption that the radiation emitted by the laser diode re-enters the laser cavity after just one round trip forth to and back from a partially reflecting target and that the contribution of further reflections is negligible due to their large power attenuation. LSM can then be seen as the coherent superposition of the laser cavity modes (LCM) and the external cavity modes (ECM) established between the laser mirror and the target, whose relative phase is determined by the target position. However, since LSM interference takes place inside an active medium cavity, a too large feedback drives the system unstable, whereas a proper combination of laser-defined and external cavity parameters preserves the coherence of the superposition. The interference signal, resulting from a displacement of the target, thus modulates the optical power with a characteristic asymmetric fringe-like function, usefully exploitable for motion control applications in real-time with half-wavelength resolution [4]. This especially practical regime has become known as “the moderate feedback regime” and is identified by the dimensionless feedback coefficient C in the interval 1 < C < 4.6 [1].

The coefficient C can be defined as:

In the original LK scalar formalism a perfect translational symmetry in the transverse plane is assumed: the laser emission is a plane wave and the mirrors are plane mirrors orthogonal to the optic axis. The moderate coupling (or single ECM) operation is achieved by a proper choice of the parameters R ₃ and A. The experimental implementation of this regime in case of large variation of the external cavity length L, has usually involved collimated laser beams and intracavity dumping independent from the cavity length (either neutral density filters, polarization rotation optics or diffusive surfaces). In order to measure consistently the target displacement, the “moderate feedback regime” has to be preserved along the full continuous range between the maximum L_MAX and minimum L_min distance of the target from the lens. Since the feedback coefficient C is directly proportional to the cavity length L, the limiting values of the measurable displacement are bound to L_MAX/L_min ≤ 4.6 in order to embrace the moderate feedback regime. Accordingly, the reported longest continuous linear range achieved without any change in the experimental conditions (alignment, attenuation) is between L_min ~0.3 m and L_MAX ~1.4 m [6].

We have experimentally demonstrated recently the extension of the measurable displacement range up L_MAX/L_min = 18, without resorting to any intracavity optical component [7] and the measurement of off-axis target rotations up to ≈ ± 1°, by exploiting a divergent laser beam [8]. In this paper we present a simple model that quantitatively accounts for the previously reported experimental results. The model is based on considering the propagation of diverging Gaussian beams through finite aperture optical elements, thus allowing to calculate the effective feedback coefficient C as a function of the target displacement. In addition, by reconsidering the interaction of the Gaussian beam with a plane mirror, we demonstrate that wavefront curvature measurements can be performed by scanning LSM.

The paper is organized as follows: Section 2 reviews the matrix formalism for the propagation of a Gaussian beam and specializes the model to LSM interference. In this section we demonstrate that dynamic ranges of a few meters can be achieved with no need of optical attenuation or further adjustments of the initial alignment. Section 3 will provide evidence of the advantages of using divergent laser beams for measuring the yaw and pitch corrections to the longitudinal displacement of wobbling targets. Starting from the experimental results, in Section 4 we develop and validate a model for the interaction of a Gaussian beam with a plane mirror in a standing-wave cavity picture and present a new method for reconstructing the phase front of the Gaussian beam by LSM measurements. Finally, in Section 5 we draw conclusions and present potential applications of our findings.

2. Matrix formalism for the propagation of Gaussian beams along LSM interferometers

Detailed accounts of the ABCD matrix formalism applied to the propagation of Gaussian beams can be found in many excellent textbooks (see for example [9]), that is why in this Section only a brief summary of the main equations and of the adopted approximations is presented.

Given the optic axis x, the fundamental Gaussian beam at wavelength λ is completely defined by any one of the following parameters: the beam waist w ₀, the Rayleigh range x ₀ or the beam divergence θ ₀, related by $w_0^2=(\lambda x_0/\pi )={(\pi {\theta }_0/\lambda )}^{-2}$. Since a semiconductor diode laser usually emits on an elliptic mode with different divergences in the planes parallel and orthogonal to the junction, two of the above parameters, one for each plane, will be necessary to specify the fundamental Gaussian beam. By taking the origin of the coordinate system along the beam axis at the beam waist, that we assume to coincide with the center of the laser cavity, and aligning the junction plane with the xy plane, the intensity distribution of the Gaussian beam along the beam propagation is given by:

The propagation of the Gaussian beam through a linear optical system can be conveniently described by propagating its complex parameter q(x) = x + i x₀ according to the ABCD law, namely q ₂ = (Aq ₁ + B) / (Cq ₁ + D), where q ₁ and q ₂ define the complex Gaussian parameter before and after, respectively, an optical element characterized by the unitary 2 × 2 matrix [(A, B),(C, D)]. For the elliptical Gaussian beam, the parameter propagation can be calculated either via a tensor formalism [10] or considering the propagation of two complex parameters, q_y(x) and q_z(x). With reference to Fig. 1 , where the schematics of our experimental apparatus is drawn, the only optical element crossed by the beam is the aspherical lens, whose matrix we take as the simple thin-lens matrix [(1, 0),(−f ⁻¹, 1)]. The mirror reflection is accounted for by the identity matrix and its small tilt angles: θ _y (pitch) around the y axis and θ _z (yaw) around the z axis, contribute to displace the backward beam centre off-axis at the lens position according to y ₀ ~Lθ _z and z ₀ ~Lθ _y . The complex Gaussian beam parameters of the backward beam at the lens (q_L) and at the original waist position (q_D) are then given by:

 

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

Download Full Size | PDF

A general feature of the Gaussian beam is that the power loss through any on-axis circular aperture of radius a is the exp[−2a ²/(w _y(x)w _z(x))] fraction of the incident power. Since a divergent Gaussian beam will grow in size propagating forth and back along the cavity, the longer the cavity the higher the power loss at any of the finite aperture optical elements, the most important of which are in our case the collimation lens and the diode facet itself (the losses at the mirror are in fact mostly due to the non unitary target reflectance R ₃). According to this simple model we can calculate the contribution to the feedback power ratio due to the Gaussian beam profile as:

In Fig. 2(a) we plot the function κ′(L) for three values of the diode-lens distance d = f + Δd (which also correspond to different beam angular widths after the lens). The plots show immediately that κ' is constant only for an optimally collimated laser beam (d = f). For a diverging beam (Δd < 0) κ' will decrease monotonically with the external cavity length. This behavior has been confirmed experimentally by measuring the amplitude of the LSM signal, that is given by the so called modulation coefficient m = |P_F − P ₀|/(FP₀) = 2κ′(L)τ_p/τ_c [4], where τ_p,c are the photon cavity lifetime and round-trip time, respectively, both typically in the 1-10 ps range for a Fabry-Perot diode laser and F is a form function of the order unit dependent on C. In Fig. 2(b) the normalized value of m/m_MAX = κ′(L)/κ′_MAX is reported as a function of the external cavity length for different diode-lens distances. It appears to remain nearly constant (within 10%) for d ~f, whereas its value is reduced more than 60% along one meter for d ~ 0.99f, in good qualitative agreement with the model. Two main difficulties prevent from a quantitative comparison near the collimation condition: the uncertainty in the measurement of the actual diode-lens distance d and the deflection of the backward beam. Our alignment procedure guarantees the matching of the initial collimation condition (d = f) to within ± 3 μm whereas the lens displacement Δd could be measured with an experimental error of ± 0.5 μm. Even such a small uncertainty would result into a 10% variation of κ′ along a one meter displacement of the target when d ~f. More important, the well collimated laser beam would suffer relatively larger backward coupling losses at the diode facet for the smallest deviation from the beam axis, with respect to a diverging beam (see also Section 3).

 

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

Download Full Size | PDF

The observed decrease of the diode-to-diode coupled amplitude is sufficient to compensate for the linear increase of C with L. Figure 3(a) shows the most relevant result of our approach, namely that the value of the feedback coefficient C can be tailored by adjusting the beam angular width after the lens without resorting to any variable filter. For a perfectly collimated beam, the value of C increases monotonically with the target distance as expected from Eq. (1). However, even for a slightly diverging beam the value of C increases to a maximum and then decreases because of the exponential loss experienced by the ECM.

 

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

Download Full Size | PDF

Notably, whatever the choice of d < f − x₀ and irrespective of the diode or lens parameters, the ECM total attenuation R ₃/A that maximizes the ratio L_MAX/L_min is that for which C_MAX = 4.6 and the corresponding useful range is reported in Fig. 3(b). To stay in the moderate feedback regime, an ideally collimated laser beam requires large attenuation and projects the achievable range far beyond the coherence length of the diode laser (typically 10⁴ mm implying L_MAX < 5 × 10³ mm). As the diode is put closer to the lens, the useful range gets shorter and closer to the laser source, but L_MAX/L_min ≈10² with an impressive twenty-fold improvement compared to the perfect collimation case. It has been our choice to experimentally optimize d in order to work without filters (A = 1) along the optical path, at the same time minimizing L_min.

At the end of this section it has to be noted that having adopted the thin-lens approximation and having neglected the diffraction effects accompanying the power losses experienced by the backward propagating beam at the finite aperture optics, resulted in a manageable analytical model without hampering its predictive power. We checked the analytical results presented in this section by numerically solving the Kirchoff-Huygens integral [11], which properly accounts for the diffraction effects, and also considering the actual physical parameters of the lens used in the experiments. The numerical calculation of κ′ is only about a factor two constantly higher than the analytical results in the range − 300 μm < Δd < − 5 μm, but confirmed the same dependence on both the diode-lens distance and the external cavity length.

3. Linear and angular measurements with a divergent Gaussian beam: experimental results

The experimental confirmation of the above model was realized with the set-up sketched in Fig. 1. The laser diode is located in a collimation tube (see Fig. 1) which is mounted onto a 5-axis kinematic mount to finely adjust its position and orientation. The linear stage (parallel to the x axis) allows to vary the external cavity length L. The target is a round mirror mounted onto a gimbal mount fixed at a translation stage (parallel to the y axis) to change the distance of the gimbal point from the laser beam.

3.1 Improved linear displacements

To validate the model calculation based on the Gaussian beam approach, we verified the extended linear displacement range predicted in the previous section. The length of our linear stage allowed to prove a continuous range up to about 2 m. Minimum and maximum tested limits are L_min = 0.1 m and L_MAX = 1.8 m at Δd = − 320 μm and R₃ ~0.04, however reaching the notable value L_MAX/L_min ≈18 [7], much larger than the previously reported ranges. In this experimental condition, the feedback coefficient is expected to reach its maximum value around L ~200 mm diminishing for both longer and shorter distance. The experimental estimation of the feedback coefficient can either be carried out by numerical fitting of the SLM signal as in ref [12], or more simply by following the experimental procedure suggested in par. 7.4.2 of ref [1]. In Fig. 4 we plot the estimated value of C obtained by analyzing the LSM traces of an oscillating target at different distances from the lens, together with those calculated from our model. In spite of the relatively low accuracy of the experimental fitting (about 10%), our results validate the Gaussian beam model and forecast the extension of the measuring range up to the laser coherence length provided the implementation of a variable gain amplifier to compensate for the reduction of the signal amplitude expected in case of a diverging beam (see Fig. 2).

 

Fig. 4 Red empty circles and purple full triangles mark the values of the extracted C, obtained by fitting the LSM traces recorded for two different values of the diode-lens distance d. Δd is measured with an experimental error of ± 0.5 μm. The black straight line and the dotted curve show the theoretical trends calculated for the two different values of d.

Download Full Size | PDF

The experimental data in Fig. 4, demonstrate that also for our diode laser, as well as for most semiconductor lasers, the “moderate feedback regime” extends also beyond the theoretical critical value of C = 4.6, as already recognized in [6]. Such occurrence would of course proportionally extend the achievable ratio L_MAX/L_min to values larger than 4.6 also for collimated laser beams. However, the same ratio calculated for a diverging Gaussian beam happen scale with a power law of C, thus further improving the benefit already shown in Fig. 3b.

3.2 Improved angular measurement

One important consequence of resorting to a larger beam angular width is that a diverging beam preserves the integrity of the LSM fringes for pitch and yaw target rotations, up to angle amplitudes comparable to the beam angular width. A direct measurement of the maximum target rotation around the z-axis (yaw) is reported in Fig. 5(a) for different beam divergences θ_L, calculated from the estimated diode-lens distance using the imaging lens equation as ${\theta }_L=\left({\theta }_0/f\right)\sqrt{{(\Delta d)}^2+x_0^2}$.

 

Fig. 5 (a) Experimental maximum target rotation around the z-axis (yaw) at which the LSM signal preserved the saw-tooth line shape (moderate feedback regime) for different values of the laser beam angular width. The vertical axis has the origin ρ = 0 when the mirror is oriented parallel to the yz plane. The cavity length was L = 300 mm and the set-up was adjusted for each θ_L to have the laser beam pointing perpendicular to the gimbal center point of the target mount. (b) Calculated 3D plot of C as a function of the target rotation for a cavity length of 300 mm. The upper dark surface corresponds to 1 < C < 4.6. All parameters have the same values as for Fig. 2.

Download Full Size | PDF

The dependence of κ' and C on the target yaw and pitch rotations can be calculated by considering the power loss of a decentered Gaussian beam whose intensity distribution is given by Eq. (2). In the small angle and small aperture approximation, such losses contribute the following multiplicative factor:

This special feature of the divergent Gaussian beam, to tolerate angular misalignments of the target, explains the experimental method we used for measuring small-angle rotations around any axis in the yz plane. The measurement of such a rotation can be obtained by performing linear displacement measurements of two distinct points in the plane of the reflective target by means of two laser sources L_i (i = 1,2). If the two beams of wavelength λ ₁ ~λ ₂ are aligned parallel to the x axis and are separated by a distance w in the yz plane, the target rotation is given by the difference of the counted LSM fringes N_i as θ ~(N₁λ ₁ – N₂λ ₂)/(2w). Yaw and pitch rotation measurement in the range ± 0.45° were demonstrated by this method with an angular resolution < 1 × 10⁻³ ° and have been previously reported in ref [8].

Like all interferometric fringe-counting methods, each LSM channel is characterized by an intrinsic sensitivity of λ/2 and by an accuracy σ = N_i σ_λ /2, where N_i is the number of LSM fringes and σ_λ is the uncertainty in the knowledge of the laser wavelength, mostly dependent on the effective diode temperature. Both passive and active diode temperature stabilization techniques have been investigated achieving a best value of σ ~7 μm along a 1 meter displacement [8]. Concerning the angular measurement, the sensitivity of our method depends on the distance between the two LSM channels and for a reasonable compact device it is worse than in commercially available sensors based for example on detecting the beam displacement on a CCD. On the other hand, our method for measuring the angular deviation of the target is fully interferometric and covers a range one order of magnitude larger, only limited by the angular divergence of the beams.

4. Model for the interaction of a Gaussian beam with a plane mirror

The Gaussian beam model presented and validated in the previous sections provides a simple and yet quantitative interpretation of the intuitive geometric picture that only the portion of the beam striking the mirror in close proximity to its normal direction will be reflected back inside the laser and can contribute to the LSM effect.

Figure 6(a,b) provides perhaps the most direct evidence of this understanding. The divergent Gaussian beam was carefully aligned to point the center of the gimbal mirror mount and the mirror was given a continuous clockwise rotation around its normal orientation as sketched in Fig. 6(c). The two LSM traces, recorded for the same rotation angle but with a different beam angular width, put in evidence two distinct features: i) the sign of the derivative changes halfway the rotation, at the crossing of the ρ = 0° orientation and ii) the frequency of the LSM fringes is positively (negatively) chirped for increasing (decreasing) absolute values of the rotation angle, in spite of the constant rotation speed of the mirror.

 

Fig. 6 Derivative of the LSM signal for a gimbal rotations from + 0.22° to – 0.22° for two different laser beam divergences: (a) θ_L ~0.7°; (b) θ_L ~0.4°. (c) . Schematics of the alignment of a divergent Gaussian beam with the center of the gimbal mirror mount: D is the distance of the plane mirror surface from the beam waist, x is the phase front curvature in the approximation x >> x ₀ (spherical phase front) and ρ is the angle of which the target is rotated, the red curves are the phase fronts of the Gaussian beam.

Download Full Size | PDF

The Gaussian beam model accounts for both the observed features when considering the phase front curvature of the beam given by R(x) = x[1 + (x ₀/x)²], x being measured from the beam waist. For x >> x ₀, R(x) ≈x (spherical wave) and the distance of the plane mirror surface from the waist can be approximated by D ≈x cosρ .

With reference to Fig. 7(a) , where π-separated gaussian phase-fronts are drawn for a given diode-lens distance, during a mirror rotation around the gimbal axis, one new self-mixing fringe will develop every time the mirror surface swept a 2π phase shift, thus passing from being tangent to one phase-front (the normal incidence portion of the beam) to the closest next. The total number of recorded fringes from the neutral (ρ = 0°) mirror orientation will then be N = Δϕ/(2π) = 2x(1 − cosρ)/λ = 2(L − df/Δd)(1 - cosρ)/λ and are plotted in Fig. 7(b) together with the experimental results taken at the corresponding values of d.

 

Fig. 7 (a) Calculated π-shifted wavefronts of a laser beam for a diode-lens distance d = 7.6 mm and for an external cavity length L = 300 mm. Continuous lines are tangent to the first three phase-fronts and indicate the position of the mirror surface upon rotation around the gimbal axis (the origin of the coordinate system). Positive angles correspond to the upward LSM fringes in Fig. 6(a,b) and negative angles correspond to the downward LSM fringes. (b) Number of LSM fringes counted for a given mirror rotation (from 0° to + 0.22°) for two different values of the diode-laser distance (red hollow square Δd = −100 μm, blue full circles Δd = −150 μm). The continuous lines are the best fit of the experimental data with the equation N = 2R(1 − cosρ)/λ. = 2(L - df/Δd)(1 − cosρ)/λ. Red curve, R = 938.5 mm corresponding to a value of Δd = − 99 μm; blue curve, R = 710.3 mm, corresponding to Δd = −153 μm.

Download Full Size | PDF

The good agreement of our interaction model with the experimental data suggested a way to reconstruct the beam phase-front by scanning LSM interference. At any given position of the mirror rotation axis translated along the y-axis, fixed angle rotations ρ are performed at a constant angular speed. Individual LSM fringes on each trace occur when the mirror plane surface is tangent to laser phase fronts separated by Δϕ = π. By measuring the tangent angles for several positions of the rotation axis, the tangent envelope of the wave front can be reconstructed and the beam phase curvature can be calculated from the parametric data set.

Whereas a detailed account of the experimental procedure and of the extraction algorithm will be published elsewhere, the proof of principle of this sensor for a one-dimensional scan is presented in Fig. 8 . The lines represent the paraboloidal phase-front of the Gaussian beam, solution of the equation x + y ²/2R(x) = nλ (n is an integer), for two different values of Δd. The symbols are the experimental data taken by scanning the rotation axis of the mirror along y for 1 mm with a step of about 0.1 mm. It is worth noting that the scanning procedure allows the reconstruction of the wavefronts with a sensitivity depending primarily on the scanning step size, in principle overtaking the λ/2 intrinsic resolution of the self-mixing itself.

 

Fig. 8 Experimentally measured (symbols) and calculated (lines) paraboloidal phase fronts of our laser beam for two different divergences: θ_L = 0.4° (red line and open circles) continuous phase front, and θ_L = 0.7° (blue dashed line and square symbols).

Download Full Size | PDF

5. Conclusion

In this paper we revised the common assumption of a constant mode-matching factor when evaluating the feedback coefficient C in the laser-self-mixing signal analysis. We demonstrated that the relevant feedback power ratio can be tailored by changing the distance between the diode laser and the collimation lens. In particular, by reducing the distance below the focal length, the resulting divergent laser beam allows for compensate the linear increase of C with the target distance, by increasing the power diffraction losses of the feedback beam at the lens and the diode pupils. We experimentally demonstrated a four-fold increment of the ratio L _MAX/L _min obtained without resorting to any intracavity element and theoretically predicted its possible increment up to a value of one hundred.

The Gaussian beam approximation accounts for all the experimental observations and, based on this model, we demonstrated the principle of a novel scanning self-mixing laser phase-front sensor capable of sub-wavelength resolution.

The observed increment of the accessible displacement range projects the field of applications of LSM sensors up to the coherence length of semiconductor lasers, thus making them appealing also for ranges nowadays presided over by gas-lasers. Moreover, a diverging laser beam tolerates misalignments of the moving target without deterioration of the signal-to-noise ratio, that compares favourably with the critically alignment required by a conventional interferometer when dealing with fast moving objects.