The LSM interference (see Figure 2) takes place when a portion of the light emitted by a laser source is reflected back into the laser cavity by an external reflector, such as a mirror or a diffusive target, and interferes with the standing wave inside the laser cavity, causing an amplitude and frequency modulation of the output laser beam [11-12].

Several benefits derive from the LSM scheme. First, the optical alignment procedure is not so critical as in external interferometers, since there is only one measurement arm, the reference arm provided by the laser itself. Second, the reference arm is self-stabilized once the driving current and the heat sink temperature of the laser source are kept constant. Third, the output signal can be detected by a photodiode placed anywhere along the optical path. The geometry of the setup can be optimized if the signal is detected by means of the photodiode integrated into most commercial laser packages for power monitoring, which allows the laser to be used as source and detector at the same time. The basic setup, sketched in Figure 2(a), is thereby made of only a laser source, a collimating lens, a remote target, and a neutral attenuator if any adjustment of the feedback power is required. Actually, this setup can be considered as an evolution of the Michelson interferometer with the reference arm folded on itself toward the laser source, whose front facet serves as the beam splitter.

Fourth, the feedback regime, i.e. the relative amount of light coupled back into the laser, affects the characteristics of the output signal in a non-linear way, allowing for the identification of the sign of the displacement by means of a single quadrature. A useful classification of the feedback regimes for metrological purposes can be performed by adopting the C-value as selective parameter [13], where C is the feedback parameter [14] defined as follows: (1) C = ɛ R 3 R 2 ( 1 − R 2 ) 1 + α 2 x l ⋅ n eff

Expression (1) depends on a combination of laser dependent parameters (R₂ is power reflection coefficient of the front laser facet, l is the laser cavity length, n_eff is the effective refractive index of the active medium, α is the linewidth enhancement factor) and system adjustable parameters (R₃ is the power reflection coefficient of the target, also including the power attenuation possibly present in the external cavity, and ε < 1 is a constant referred to as the mode matching factor).

When C ≪ 1 (very weak feedback regime), the output waveform is represented by a sine function as in classical interferometry. In this regime the laser behavior is \\quite unperturbed with respect to isolated laser operations, and the symmetry of the waveform requires the duplication of the measurement channel in order to recover the direction of target displacement. With reference to the oscilloscope waveforms produced during a linear displacement of the target and reported in Figure 2(b), when 0.1 < C < 1 (weak feedback regime), the waveform becomes progressively distorted as C approaches unity, with an asymmetry related to the direction of motion of the external reflector, until for 1 < C < 4.6 (moderate feedback regime), the laser becomes bi-stable and the waveform is visibly sawtooth-like, with abrupt switches every time the phase changes by 2π. Lastly, for C > 4.6 (strong feedback regime, not represented in Figure 2(b), the system becomes multi-stable and the waveform collapses to a very noisy one [15].

On the basis of the above classification, operation in the moderate feedback regime represents the most convenient solution for displacement measurements along the longitudinal x-axis, since the module of the displacement can be simply recovered by the count of the total number of produced fringes, whereas the direction of displacement can be recovered via the sign of the sharp transitions of the sawtooth-like signal. In more detail, the standard analysis in the self-mixing configuration consists of the AC signal amplification by a trans-impedance amplifier, followed by the time-derivation of the output signal, which converts the sawtooth-like fringes in a series of positive and negative spikes, whose sign depends on the direction of the motion of the target. Finally, the net number of counts N = N₊ - N_-, given by the algebraic sum of the positive (N₊) /negative (N_-) counts, returns the linear displacement Δx as: (2) Δ x = N ⋅ λ / 2where N and λ are the net number of fringes and the wavelength of the laser.

A great variety of theoretical investigations and experimental improvements have been developed on this subject during the last twenty years. However, almost all of them take into account collimated laser beams [16-20], which, as we are going to demonstrate in the following, impose substantial limitations on applications of LSM to real world environments.

Since a collimated laser beam will bounce back and forth along the external cavity without significant power loss, a reduction of the feedback power in order to satisfy the moderate feedback condition is often needed. Different approaches have already been used, e.g. the insertion of an optical attenuator in the external cavity or the adoption of a weakly diffusive target. However, the solutions adopted so far always introduce a fixed attenuation irrespective of the target distance x. Being C directly proportional to x, the system will necessarily cross the upper threshold C = 4.6 for long enough cavities, thereby becoming unstable and limiting the useful measurement range to the condition x_max/x_min = 4.6. The longest reported linear measurement range, achieved without any change of experimental conditions (optics alignment, feedback attenuation, etc.), is Δx = 1.2 m [21].

At variance to this approach, we exploit an alternative one [22], which consists of the reduction of the feedback power by inserting power losses that are dependent on the external cavity length, thus balancing the linear increase of the feedback parameter C. The easiest way to achieve length-dependent losses consists of using a slight divergent laser beam. In this context, the plane wave approximation is firmly unable to correctly model the operation of LSM displacement sensors when the laser beam is not perfectly collimated. To counter this problem, we exploited the ABCD matrix formalism to develop a simulation model of the laser/mirror interaction in the Gaussian beam approximation.

The aim of this Section is to summarize the main equations of the model. 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 w₀, that we assume to coincide with the center of laser cavity, the Rayleigh range x₀ or the beam divergence θ₀, related by w 0 2 = ( π θ 0 / λ ) − 2 = ( λ x 0 / π ). Since a semiconductor diode laser usually emits an asymmetric beam 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 elliptical Gaussian mode.

The propagation of the Gaussian beam through a linear optical system can be conveniently described by propagating its complex parameter q(x) = x + j x₀ according to the ABCD law, namely q₂ = (Aq₁ +B)/ (Cq₁ +D), where q₁ and q₂ defines the complex Gaussian parameter before and after, respectively, an optical system characterized by the linear 2×2 matrix [(A, B),(C, D)]. For the elliptical Gaussian beam, the parameter propagation can be defined by the separate consideration of the propagation of two complex parameters, q_y(x) and q_z(x). With reference to Figure 2(a), where the schematics of the LSM interferometer are presented, 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)] with f the nominal focal length of the lens. 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 according to y₀ ≈ xθ_z and z₀ ≈ xθ_y. The complex Gaussian beam parameters of the backward beam at the lens (q_x) and at the original waist position (q_D) are then given by: (3) q Ly , z = q 0 y , z ( f − 2 x ) + df + 2 x ( f − d ) f − d − q 0 y , zand (4) q Dy , z = q 0 y , z [ 2 x ( d − x ) − f ( 2 d − x ) ] − 2 ( d − f ) [ d ( f − x ) + fx ] 2 x ( d − f ) − f ( 2 d − f ) − 2 ( f − x ) q 0 y , zrespectively, where d is the distance of the lens from the laser. The beam waists can be calculated accordingly, provided that w y , z 2 ( x ) = − λ / π ℑ [ q y , z − 1 ( x ) ].

The important element to consider is that the power lost by a Gaussian beam through any on-axis circular aperture of radius a is the exp[−2a²/(w_y(x) w_z(x))] fraction of the incident power. Since a divergent Gaussian beam will grow in size propagating back and forth along the cavity, the longer the cavity, the higher the fractional 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 reflection coefficient R₃). By taking the origin of the coordinate system at the beam waist position and aligning the junction plane of the diode laser with one of the Cartesian planes, the contribution to the feedback power ratio due to the Gaussian beam profile can be calculated as: (5) P F P 0 = exp [ − 2 a 2 w y ( d ) w z ( d ) − 2 a 2 w y ( d + 2 x ) w z ( d + 2 x ) − 2 w 0 y w 0 z w y ( 2 d + 2 x ) w z ( 2 d + 2 x ) ]and the feedback parameter C in eq. (1) can be re-written as: (6) C = P F P 0 R 3 R 2 ( 1 − R 2 ) 1 + α 2 x l ⋅ n eff

In this way C becomes dependent on the laser parameters w_0y,z and λ, and can be corrected by a proper choice of the lens parameters (f, a) and of the lens-diode distance d. Figure 3 shows the most relevant result of our approach, namely that a careful choice of the beam angular width after the lens allows a compensation of the linear increase of the C-parameter with the target distance x and of the feedback power without any optical filter. C is directly proportional to x as expected from Eq. (1) only for a well - collimated laser beam having d / f ≈ 1, e.g. the case referred in literature as constant mode-matching factor ε.

If d / f < 1, the output beam can be made to diverge and the C-parameter decreases sharply with the mirror distance x until no more filters are required to keep its value within the moderate feedback regime. In this case, only a small fraction of the backward beam re-enters the laser cavity: the greater the laser divergence, the smaller the feedback radiation coupled with the laser [22-23]. Only the fraction of the laser beam orthogonal to the plane target is reflected back following the same optical path toward the collimation lens and contributes to the LSM signal. The remaining part of the laser beam, truncated by the finite aperture of the collimating lens does not contribute to the self-mixing interaction.

The measurement technique for the assessment of longitudinal displacements (see Section 2.2) can be easily replicated in order to also evaluate yaw and pitch rotations of the moving target. Each rotation angle of the target can be obtained by simultaneously measuring the linear displacement of three distinct points in the plane of the target. This requires the presence of three identical Self-Mixing Interferometers SMI_i (i = 1,2,3), each made of a laser diode L_i with integrated monitor photodiode, a lens and a single reflective or retroreflective target M.

With reference to Figure 4, SMI₁ and SMI₃, aligned with the x - axis and placed at a distance of d 13 = L 1 L 3 ¯ along the z-axis, will measure ϑ_y according to Δϑ_y = arctan[ (Δx₃ − Δx₁)/d₁₃]. Analogously, SMI₁ and SMI₂, aligned with the x - axis and placed at a distance of d 12 = L 1 L 2 ¯ along the y-axis, will measure ϑ_z according to Δϑ_z = arctan[ (Δx₂ − Δx₁)/d₁₂]. Since the small angle approximation holds throughout the angular range of measurement, the three DOFs are then derived from the following relations: (7) { Δ X = N 1 ⋅ λ 1 / 2 Δ ϑ z ≅ [ N 2 ⋅ λ 2 − N 1 ⋅ λ 1 ] / [ 2 d 12 ] Δ ϑ y ≅ [ N 3 ⋅ λ 3 − N 1 ⋅ λ 1 ] / [ 2 d 13 ]

The working principle is demonstrated by inspection of the oscilloscope waveforms reported in Figure 5, where the combined pitch and yaw rotation [Figure 5(b)] results in a different number of interference fringes counted by the three detectors with respect to the pure translational motion [Figure 5(a)].

The prototype of the sensor is schematically illustrated in Figure 6. It is composed of a laser head, which consists of three parallel laser diodes mounted side-by-side in a “L”-like configuration, and a target made of a plane squared mirror of 70 mm side. The sources are Distributed-Feedback (DFB) laser diodes, with nominal wavelength of 1310 nm, biased at a current I = (23.00 ± 0.01) mA, and equipped with a monitor photodiode integrated into the laser package and an AR-coated aspheric lens.

The photocurrent produced by the detector is firstly AC-coupled to the trans-impedance amplifier of gain = 10⁵ V/A, and then fed into a signal processing board interfaced to a computer for the fringe count. In order to avoid the presence of a variable attenuator, the moderate feedback condition has been achieved by properly defocusing the laser source, following the principle introduced in Section 2.3.

The target is fixed onto a rotation stage, mounted itself onto a 1-meter long linear stage. The minimum distance between the target and the laser head is 20 cm. A commercial 6-axis measurement system (API 6D Laser) has been used as the reference meter. Its nominal resolution is 0.02 μm for longitudinal displacement and 3 × 10^-5° for angular rotations.

For validating the proposed system as linear displacement sensor, the linear stage was moved along the x-axis in the range -10³ – 10³ mm at a speed of 10 mm/s. Then, the system was tested as a rotation sensor by keeping the target at constant positions (120 cm from the laser head) and moving the rotation platform by increasing the yaw or pitch angles in both clockwise (-) or anti-clockwise (+) directions in the range ± 0.45°.

Figure 7 shows the measured linear displacement Δx [Figure 7(a)] and rotations [Figure 7(b)] as compared with the reference values simultaneously measured by the API system. The system response is linear over a longitudinal dynamic range of six orders of magnitude and an angular dynamic range of three orders of magnitude. The maximum continuous linear and angular displacements were only limited by the length of the linear stage and the reference standards.

Improvements in the angular measurement range can be achieved by using three solid retroreflectors (CC), one for each laser, in place of the plane mirror (PT), since in this case the backward optical path is not affected by the target orientation [24].

The only reported measurement of the tilt angle of a remote target via LSM technology was realized by Giuliani et al. [25], who exploited the parabolic dependence of the DC power from the tilt angle when the laser diode was driven in the coherence-collapse (high feedback) regime. The technique shows a good sensitivity (0.1 arcsec) but a small angular range (0.06°), and requires the target sitting at a fixed distance from the laser. At the contrary, the proposed three DOFs sensor presents several benefits which can be summarized as follows:

Operations in moderate feedback regime allow avoiding the signal instabilities typical of the coherence collapse and, above all, make possible a digital treatment of the signal in place of the analogical signal processing exploited in [25].

Additionally, a comparison with commercially available instruments employed for industrial metrological applications shows that the demonstrated sensor would fill the existing gap between the sophisticated and costly multi DOFs instruments capable of nanometric resolution and the off-the-shelf instruments capable of sub-millimetric resolution. The intermediate micrometric resolution range is today covered by one DOF linear optical encoders which are unable to handle target rotations. Besides, encoders make use of an “active” reading head with cables and electro-optical components, whose installation on machine tools must satisfy several mechanical constraints. At the contrary, the target of the LSM sensor is “passive”, thus providing a more robust system and a greater flexibility in the sensor integration on machine tool without significant design constraints. These features make the LSM sensor suitable for industrial applications as machine tool calibration and diagnostic.

Every interferometer (both classic or LSM - based) is basically insensitive to any transverse displacement, since the spatial phase k·Δr - where Δr is the displacement vector and k the wave vector – is null and thus no interference fringe is produced. Two possible solutions can be adopted to overcome this drawback:

The projection of a component of the transverse motion along the direction of the laser beam, via additional optical elements along the light path or specific geometrical configurations.

The first solution, commonly explored in literature, is usually achieved by inserting a Wollaston prism on the movable target [26]. When the Wollaston prism is moved along with the stage, a difference in the optical path occurs and the number of interferometric fringes detected by an external receiver is used to measure the straightness of the stage. The measurement resolution is typically some micrometers, and can be improved via the insertion of additional optical elements [27-28]. However, this approach requires expensive setups and cannot be easily extended in a multi – DOF measuring system because of the presence of a fixed element beyond the movable stage, which strictly limits the spatial optimization of the apparatus.

At the opposite, the second approach allows an easy integration with the LSM scheme and only requires a single reference plane - that defined by the laser sources – since the signal detection via the monitor photodiode allows the integration of source and detection elements into a single housing.

With reference to the two SMIs illustrated in Figure 8(a), SMI₁ is able to detect purely linear displacement Δx along the x-axis whereas it is totally blind to any transverse motion. A second interferometer SMI₄, rigidly tilted by a small angle α with respect to the x-axis in the xy plane, identifies a new optical axis x'. This expedient makes the interferometer sensitive to displacements along both the linear (x) and transverse (y) axis, since both cause a change Δl₄ = ± N₄ × λ₄ /2 of the optical path length along x'.

Only SMI₄ is thus needed for measuring a pure Δy displacement, whereas a combined linear / transverse displacement requires the comparison of the two SMIs readings in order to be properly measured according to: (8) { Δ x = Δ l 1 Δ y = Δ l 4 / sin α − Δ l 1 / tan ( α )

The same considerations apply for the measurement of flatness, via a third interferometer SMI₅, tilted by an angle β with respect to the x – axis in the xz plane [see Figure 8(b]. In terms of number of interference fringes, the measurement displacements can be written as: (9) { Δ x = N 1 ⋅ λ 1 / 2   Δ y = N 4 ⋅ λ 4 / [ 2 sin ( α ) ] − N 1 ⋅ λ 1 / [ 2 tan ( α ) ] Δ z = N 5 ⋅ λ 5 / [ 2 sin ( β ) ] − N 1 ⋅ λ 1 / [ 2 tan ( β ) ]

Similarities with the measurement technique for yaw and pitch rotations are remarkable: straightness and flatness displacements can be obtained by three identical self-mixing interferometers SMI_i (i = 1,4,5), via the simultaneous measurement of the linear displacement of three distinct points on a specially designed target.

Two preliminary considerations may help the introduction of our solution for the DOF most elusive to interference. First, roll ϑ_x, that is the rotation along the x - axis, can be basically measured similarly to the yaw or pitch rotation (see Paragraph 3.1), i.e. by means of a differential technique by two parallel interferometers SMI_i and SMI_j. This means Δϑ_x &prop; (Δl_i − Δl_j)/d_ij. Second, roll ϑ_x is a transverse angular displacement, thus requiring the tilt of the two laser beams with respect to the x-axis as for the straightness and flatness (see Paragraph 4.2). This means Δϑ_x ≈ [Δl_i − Δl_j]/ [d_ij·sin(ξ)], with ξ the tilt angle of the mirror employed in the roll measurement, i.e. ξ = α or β in case of M₂ or M₃, respectively.

One can exploit, at this aim, the laser source L₅ already used for flatness measurement, and a new laser diode L₆, both pointed against the mirror M₃ and separated along the y-axis by a distance d₅₆. A pure roll rotation can thus be derived from the differential readings of the two SMI_i (i = 5,6) by means of the following expression Δϑ_x ≈ [Δl₆ − Δl₅]/ [d₅₆·sin(β)] or, in terms of number of fringes, as: (10) Δ ϑ x ≅ [ N 6 ⋅ λ 6 − N 4 ⋅ λ 4 ] / [ 2 ⋅ d 46 ⋅ sin ( α ) ]

The prototype of the sensor, illustrated in Figure 9 (a), is similar to that previously described in Figure 6. The only difference consists in the target configuration, which is given by three reciprocally tilted mirrors [Figure 9(b)]: M₁ is a squared mirror of side 20 mm lying in the plane yz, M₂ is a rectangular mirror of dimension 50 mm × 20 mm, tilted at angle α = (2.1 ± 0.1)° around the z – axis, M₃ is a rectangular mirror (70 mm × 50 mm) tilted at angle β = (1.8 ± 0.1)° around the y - axis.

To validate the proposed system as a transverse displacement sensor, the interferometers SMI_i (i = 4-6) were tested for displacements Δy and Δz in the range ± 1 mm [Figure 10(a)] and for rotation Δϑ_x in the range ± 0.4° [Figure 10(b)], at a fixed distance laser/ target L = 120 cm. The linearity of the response is guaranteed over the full measurement range, that is mainly limited by the available stages and reference meter ranges.

Most practical applications require a real-time control of small five DOFs deviations of a moving target while it is moving along the main x-axis. Interestingly, the proposed technique allows the simultaneous measurement of more DOFs, as demonstrated in Figure 12 where a linear increasing translations Δx is imposed to the target together with a fixed yaw rotation Δϑ_z of about 0.12° (Figure 12 a) or a fixed straightness displacement of Δy = - 0.5 mm [Figure 12(b)]. The results shows that the combination of linear and angular motion can be handled consistently by SMIs without a significant loss of accuracy.

In the previous sections, we reported on the profiling of the LSM technique for the measurement of the six DOFs by using an intuitive geometrical approach for each DOF. Nonetheless, identical results can be achieved in an analytical frame by means of a unique mathematical algorithm able to convert the six counts of fringes N_i (i=1…6) into the direct measurement of linear and angular displacements of the target, and vice-versa. This kind of analysis becomes necessary in presence of random displacement of the target, which can present simultaneously up to six DOFs and thus requires a re-definition of the measurement expressions for taking into account the mutual interactions between the different DOFs.

Figure 13 reports a possible configuration of a unique target for a 6-DOFs self-mixing system, chosen in order to optimize the geometry and the compactness of the system. The target is composed by a squared mirror M₁ lying in the yz plane for the measurement of linear displacement, yaw and pitch, a rectangular mirror M₂ tilted by a yaw α for the measurement of straightness, and a second rectangular mirror M₃ tilted by a pitch β for the measurement of flatness and roll.

The main steps in the development of the algorithm are reported in the following. First, the relative change of the optical path Δδ_ij = δ_ij − δ_ij,0 between the laser source L_i and the target M_j is provided by Δδ_ij = N_i · λ_i, where δ_ij and δ_ij,0 are the optical path at the end and at the start of the motion, respectively, and the net count of interference fringes N_i produced during the motion only depends by the initial and final position assumed by the target.

Second, the optimal target for 6-DOFs measurement is assumed to be composed of three planes with no reciprocal symmetry. The lack of symmetry is validated when there is a single intersection point between the three planes M_j or, analytically, when their normal versors r̂_j r̂_j are linearly independent. Thereinafter, M₁ will be used for linear displacement (i = 1,2,3, j = 1), M₂ for straightness (i = 4, j = 2) and M₃ for flatness and roll (i = 5,6, j = 3), so that the couple of indexes (i,j) identify the six interferometric channels SMI_ij, each composed by a laser head and a plane target.

Third, in the LSM scheme it will be assumed that δ_ij agrees with twice (considering the forward and backward path) the geometrical distance between the point-like laser source and the plane of the target, since the relevant optical path in the LSM scheme is given by the ray orthogonal to the mirror, reflected backward following the same parallel direction. In a fixed right-handed reference frame, we have: (14) Δ δ ij = 2 ( t → i − d → i ) ⋅ r ^ jwhere the vector d i → (i = 1..6; six laser sources) defines the position of the laser source L_i pointed against the target M_j, the vector t j → (j = 1..3; three reflective targets) defines the position of a generic point T_j of the mirror M_j whose normal versor is r̂_j. The change in the optical path can then be obtained as: (15) N i ⋅ λ i / 2 = ( t → j − d → i ) ⋅ r ^ j − ( t j 0 → − d i → ) ⋅ r ^ j0A simplification of equation (15) can be introduced replacing the three vectors t j → by a single vector t⃗ which defines the position of the intersection point T between the three planes of the mirrors. The existence and the uniqueness of the intersection point is derived from the linear independency of the versors r̂_j. Thereby, equation (15) can be re-written as: (16) N i ⋅ λ i / 2 = ( t → − d → i ) ⋅ r ^ j − ( t 0 → − d i → ) ⋅ r ^ j0where the wavelengths λ_i, the initial linear position t 0 → and the initial angular position r̂_j0 of the target, and the laser positions d i → are exactly known.

Equation (16) is the core of the LSM analytical approach, and can be exploited in two complementary ways:

to recover the 6 counts N_i once known the final target position t⃗_\and orientation r̂_j. In this case equation (16) represents a parametric system of 6 equations (i = 1..6) in 6 variables, which can be exactly solved;

For example, in presence of pure longitudinal and transverse displacements, the versors r̂_j describing the angular orientation of the target can be assumed constant during the motion, so that r̂_j = r̂_j0. As a result, equation (16) simplifies to: (17) N i ⋅ λ i / 2 = ( t → − t → 0 ) ⋅ r ^ j 0

By making explicit the index i = 1,4,5 of the lasers involved in the measurement of translations, and by writing the normal versors r̂_j0 [see Figure 9(b)] as: (18) r ^ 10 = ( 1 , 0 , 0 ) ; r ^ 20 = ( cos α , sin α , 0 ) ; r ^ 30 = ( cos β , 0 , − sin β )the indexed equation (17) gives rise to the following system: (19) { N 1 ⋅ λ 1 / 2 = Δ x N 4 ⋅ λ 4 / 2 = Δ x ⋅ cos ( α ) + Δ y ⋅ sin ( α ) N 5 ⋅ λ 5 / 2 = Δ x ⋅ cos ( β ) − Δ z ⋅ sin ( β )

Equation (19) allows retrieving the translational DOFs by means of a system analogous to equation (7), except for a minus sign in the flatness Δz coming from a correct consideration of the angle sign in the right-handed reference frame.

A similar procedure provides the estimation of rotational DOFs which, as an intermediate step, requires the knowledge of the rotated versors r̂_j from the global rotation matrix, given as the right- product of the yaw/pitch/roll rotations matrices. Details can be found in [23].